Chapter 7: Informal Logical Fallacies This chapter is based on Fundamental Methods of Logic, by Matthew Knachel. I. Logical Fallacies: Formal and Informal Generally and crudely speaking, a logical fallacy is just a bad argument. Bad, that is, in the logical sense of being incorrect—not bad in sense of being ineffective or unpersuasive. Alas, many fallacies are quite effective in persuading people; that is why they’re so common. Often, they’re not used mistakenly, but intentionally—to fool people, to get them to believe things that maybe they shouldn’t. The goal of this chapter is to develop the ability to recognize these bad arguments for what they are so as not to be persuaded by them. There are formal and informal logical fallacies. The formal fallacies are simple: they’re just invalid deductive arguments. Consider the following: If the Democrats retake Congress, then taxes will go up. But the Democrats won’t retake Congress. Therefore, taxes won’t go up. This argument is invalid. It’s got an invalid form: If A then B; not A; therefore, not B. Any argument of this form is fallacious, an instance of “Denying the Antecedent.” We can leave it as an exercise for the reader to fill in propositions for A and B to get true premises and a false conclusion. Intuitively, it’s possible for that to happen: maybe a Republican Congress will raise taxes. Our concern in this chapter is not with formal fallacies—deductive arguments that are bad because they have a bad form—but with informal fallacies. These arguments are bad, roughly, because of their content, their context, and/or their mode of delivery. Since we can’t judge inductive arguments based on their form, we need to learn a variety of ways in which inductive arguments fail. In other words, any time an inductive argument is weak instead of strong, that argument has committed a fallacy; fallacies are names given to common types of problematic reasoning. There are a lot of different ways of defining and characterizing informal fallacies, and lot of different ways of organizing them into groups. Since Aristotle first did it in his Sophistical Refutations, authors of logic books have been defining and classifying the informal fallacies in various ways. These remarks are offered as a kind of disclaimer: the reader is warned that the particular presentation of the fallacies in this chapter will be unique and will disagree in various ways with other presentations, reflecting as it must the author’s own idiosyncratic interests, understanding, and estimation of what is important. This is as it should be and always is. The interested reader is encouraged to consult alternative sources for further edification. II. Fallacies of Relevance We will discuss six informal fallacies under this heading. What they all have in common is that they involve arguing in such a way that issue that’s supposed to be under discussion is somehow sidestepped, avoided, or ignored. These fallacies are called “Fallacies of Relevance” because they involve arguments that are bad insofar as the reasons given are logically irrelevant to the issue at hand. People who use these techniques with malicious intent are attempting to distract their audience from the central questions they’re supposed to be addressing, allowing them to appear to win an argument that they haven’t really engaged in. Appeal to Emotion Fallacies There are as many different appeal to emotion fallacies as there are human emotions. Fear is perhaps the most commonly exploited emotion for politicians. Political ads inevitably try to suggest to voters that one’s opponent will take away medical care or leave us vulnerable to terrorists, or some other scary outcome—usually without a whole lot in the way of substantive proof that these fears are at all reasonable. Commercial advertisers do it, too. Think of all the ads with sexy models schilling for cars or beers or whatever. What does sexiness have to do with how good a beer tastes? Nothing. The ads are trying to engage your emotions to get you thinking positively about their product. Any time an arguer is trying to get you to accept their conclusion (buy their product, vote for their candidate) based on an emotion rather than on logic, an appeal to emotion fallacy has been committed. We usually name them according to which emotion the arguer is trying to stir up in the audience. We’ll cover three appeals to emotion here. Appeal to Pity Suppose you’re one of those sleazy personal injury lawyers—an “ambulance chaser.” You’ve got a client who was grocery shopping at Wal-Mart, and in the produce aisle she slipped on a grape that had fallen on the floor and injured herself. On the day of the trial, what do you do? How do you coach your client? Tell her to wear her nicest outfit, to look her best? Of course not! You wheel her into the courtroom in a wheelchair (whether she needs it or not); you put one of those foam neck braces on her, maybe give her an eye patch for good measure. You tell her to periodically emit moans of pain. When you’re summing up your case before the jury, you spend most of your time talking about the horrible suffering your client has undergone since the incident in the produce aisle: the hospital stays, the grueling physical therapy, the addiction to pain medications, etc., etc. All of this is a classic fallacious appeal to emotion—specifically, in this case, pity. The people you’re trying to convince are the jurors. The conclusion you have to convince them of, presumably, is that Wal-Mart was negligent and hence legally liable in the matter of the grape on the floor. The details don’t matter, but there are specific conditions that have to be met—proved beyond a reasonable doubt—in order for the jury to find Wal-Mart guilty. But you’re not addressing those (probably because you can’t). Instead, you’re trying to distract the jury from the real issue by playing to their emotions. You’re trying to get them feeling sorry for your client, in the hopes that those emotions will cause them to bring in the verdict you want. All appeal to pity fallacies work in the same way. Instead of presenting logical reasons to accept the conclusion, appeals to pity try to stir up your emotions of sympathy, feeling sorry for someone. Arguers using this technique are hoping you’ll make a decision with your heart, rather than your head. Appeal to Force Perhaps the least subtle of the fallacies is the appeal to force, in which you attempt to convince your interlocutor to believe something by threatening him. Threats pretty clearly distract one from the business of dispassionately appraising premises’ support for conclusions. It’s an appeal to emotion fallacy, as the arguer is trying to generate a sense of fear in the audience, with the hope that they’ll act out of fear. Like the appeal to pity, the appeal to force fallacy is often used when there is no good logical evidence for the conclusion, so the arguer tries to force the audience to accept the conclusion anyway. There are many examples of this technique throughout history. In totalitarian regimes, there are often severe consequences for those who don’t toe the party line (see George Orwell’s 1984 for a vivid, though fictional, depiction of the phenomenon). The Catholic Church used this technique during the infamous Spanish Inquisition: the goal was to get non-believers to accept Christianity; the method was to torture them until they did. An example from more recent history: in 2001, after 9/11, President George W. Bush gave a speech to congress that was broadcast. In it he addresses both the American people, and nations around the world—in part to convince them to join the war against terror. He said, “Every nation, in every region, now has a decision to make. Either you are with us, or you are with the terrorists. From this day forward, any nation that continues to harbor or support terrorism will be regarded by the United States as a hostile regime.” Either you help us, or you will become our enemy. The appeal to force is not usually subtle. But there is a very common, very effective debating technique that belongs under this heading, one that is a bit less overt than explicitly threatening someone who fails to share your opinions. It involves the sub-conscious, rather than conscious, perception of a threat. Here’s what you do: during the course of a debate, make yourself physically imposing; sit up in your chair, move closer to your opponent, use hand gestures, like pointing right in their face; cut them off in the middle of a sentence, shout them down, be angry and combative. If you do these things, you’re likely to make your opponent very uncomfortable—physically and emotionally. They might start sweating a bit; their heart may beat a little faster. They’ll get flustered and maybe trip over their words. They may lose their train of thought; winning points they may have made in the debate will come out wrong or not at all. You’ll look like the more effective debater, and the audience’s perception will be that you made the better argument. But you didn’t. You came off better because your opponent was uncomfortable. The discomfort was not caused by an actual threat of violence; on a conscious level, they never believed you were going to attack them physically. But you behaved in a way that triggered, at the sub-conscious level, the types of physical/emotional reactions that occur in the presence of an actual physical threat. This is the more subtle version of the appeal to force. It’s very effective and quite common (watch cable news talk shows and you’ll see it). Argumentum ad Populum Many of the fallacies have Latin names, as identifying fallacies has been an occupation of logicians since ancient times, and because ancient and medieval European work comes down to us in Latin, which was the language of European scholarship for centuries. The Latin name of this fallacy literally means “argument to the people.” The emotion ad populum fallacies stir up is the human desire to be liked, loved, respected, admired—in short, to be popular. There is a group out there, and you’re not in it, you’re being left out. Sometimes this group is “everyone,” and variety of ad populum is called the bandwagon appeal. Bandwagon arguments are an extremely common technique, especially for advertisers. They appeal to people’s underlying desire to fit in, to do what everybody else is doing, not to miss out. The advertisement assures us that a certain television show is #1 in the ratings—with the tacit conclusion being that we should be watching, too. But this is a fallacy. We’ve all known it’s a fallacy since we were little kids, the first time we did something wrong because all of our friends were doing it, too, and our moms asked us, “If all of your friends jumped off a bridge, would you do that too?” Bandwagon appeals want you to join the crowd; but sometimes, an ad populum instead wants you to join a special, “elite” group of people. These arguments commit the appeal to snobbery variety of ad populum. It’s not that everyone is doing it, whatever “it” is, but it’s only done by people who are somehow better than everyone else. It’s appealing to your vanity, by convincing you that if you buy this product, or watch this television show, you too can be better than the crowd. For example, there was a commercial for Grey Poupon mustard You can watch this commercial for yourself at the following link: Grey Poupon commercial. back in the 1980s that shows a rich man in the back of a car eating an elaborate meal while his chauffeur drives him around. The chauffeur reaches into the glove box, and pulls out a jar of this mustard. Another fancy car pulls up next to them, and the man being driven around in that car asks “Pardon me, would you have any Grey Poupon?” “But of course” is the answer. Not everyone is using this mustard, only the elite. Counterargument fallacies There are three types of fallacies in this chapter that we’ll call “counterargument fallacies.” A counterargument is an attempt to respond to or refute someone else’s argument. There are perfectly logical ways to approach someone’s argument when you disagree, and of course, there are perfectly illogical ways as well. It is important to note that any fallacy can appear in a counterargument. These two fallacies, however, only appear in counterarguments. It is thus often relevant to examine the original argument before dissecting the counterargument to see why it fails. Our three counterargument fallacies are called the “straw man,” “ad hominem,” and “red herring.” Straw Man This fallacy involves the misrepresentation of an opponent’s viewpoint—an exaggeration or distortion of it that renders it indefensible, something nobody thinking logically would agree with. You make it look as though your opponent is arguing for something absurd, then declare that you don’t agree with his position—except it isn’t really his position, it’s one that you made up. You create your own version of their view, and then defeat the new creation instead of what your opponent actually said. Thus, you merely appear to defeat your opponent: your real opponent doesn’t hold the view you imputed to him; instead, you’ve defeated a distorted version of it, one of your own making, one that is easily dispatched. Instead of taking on the real man, you construct one out of straw, thrash it, and pretend to have achieved victory. It works if your audience doesn’t realize what you’ve done, if they believe that your opponent really holds the crazy view. Politicians are frequently victims (and practitioners) of this tactic. After his 2005 State of the Union Address, President George W. Bush’s proposals were characterized thus: George W. Bush's State of the Union Address, masked in talk of “freedom" and “democracy,” was an outline of a brutal agenda of endless war, global empire, and the destruction of what remains of basic social services. International Action Center, Feb. 4 2005, http://iacenter.org/folder06/stateoftheunion.htm Well, who’s not against “endless war” and “destruction of basic social services”? But of course this characterization is a gross exaggeration of what was actually said in the speech, in which Bush declared that we must “confront regimes that continue to harbor terrorists and pursue weapons of mass murder” and rolled out his proposal for privatization of Social Security accounts. Whatever you think of those actual policies, you need to do more to undermine them than to mis-characterize them as “endless war” and “destruction of social services.” That’s distracting your audience from the real substance of the issues. In 2009, during the (interminable) debate over President Obama’s healthcare reform bill—the Patient Protection and Affordable Care Act—former vice-presidential candidate Sarah Palin took to Facebook to denounce the bill thus: The America I know and love is not one in which my parents or my baby with Down Syndrome will have to stand in front of Obama's “death panel” so his bureaucrats can decide, based on a subjective judgment of their “level of productivity in society,” whether they are worthy of health care. Such a system is downright evil. Yikes! That sounds like the evilest bill in the history of evil! Bureaucrats euthanizing Down Syndrome babies and their grandparents? Holy Cow. ‘Death panel’ and ‘level of productivity in society’ are even in quotes. Did she pull those phrases from the text of the bill? Of course she didn’t. This is a complete distortion of what’s actually in the bill (the kernel of truth behind the “death panels” thing seems to be a provision in the Act calling for Medicare to fund doctor-patient conversations about end-of-life care); the non-partisan fact-checking outfit Politifact named it their “Lie of the Year” in 2009. Palin is not taking on the bill or the president themselves; she’s confronting a made-up version, defeating it (which is easy, because the made-up bill is evil as heck), and pretending to have won the debate. But this distraction only works if her audience believes her straw man is the real thing. Alas, many did, and the provision funding end-of-life care was taken out of the bill. This is why these fallacies are used so frequently: they often work. Argumentum ad Hominem Like ad populum fallacies, everybody always uses the Latin for this one—usually shortened to just ‘ad hominem’, which means ‘at the person.’ You commit this fallacy when, instead of attacking your opponent’s views, you attack your opponent himself. This fallacy comes in a lot of different forms; there are a lot of different ways to attack a person while ignoring (or downplaying) their actual arguments. To organize things a bit, we’ll divide the various ad hominem attacks into two groups: Abusive and Circumstantial. 1. Abusive Abusive ad hominem is the more straightforward of the two. The simplest version is simply calling your opponent names instead of debating him. During the 2016 Republican presidential primary, Donald Trump came up with catchy little nicknames for his opponents, which he used just about every time he referred to them. If you pepper your descriptions of your opponent with tendentious, unflattering, politically charged language, you can get a rhetorical leg-up. Mind you, simply insulting someone is not a logical fallacy. It becomes a fallacy when you try to dismiss their argument altogether, or get other people to dismiss their arguments, based on the insults you’re hurling. Trump used the nicknames in order to try to discredit his opponents so their arguments would have less of an effect. Another abusive ad hominem attack is “guilt by association.” Here, instead of directly insulting your opponent, you tarnish your opponent by associating them or their views with someone or something that your audience despises. Consider the following: Former Vice President Dick Cheney was an advocate of a strong version of the so-called Unitary Executive interpretation of the Constitution, according to which the president’s control over the executive branch of government is quite firm and far-reaching. The effect of this is to concentrate a tremendous amount of power in the Chief Executive, such that those powers arguably eclipse those of the supposedly co-equal Legislative and Judicial branches of government. You know who else was in favor of a very strong, powerful Chief Executive? That’s right, Hitler. We just compared Dick Cheney to Hitler. Ouch. Nobody likes Hitler, so. Not every comparison like this is fallacious, of course. But in this case, where the connection is particularly flimsy, we’re clearly pulling a fast one. Comparing your opponent to Hitler—or the Nazis—is quite common. Some clever folks came up with a fake-Latin term for the tactic: Argumentum ad Nazium (cf. the real Latin phrase, ad nauseum—to the point of nausea). Such comparisons are so common that author Mike Godwin formulated “Godwin's Law of Nazi Analogies: As an online discussion grows longer, the probability of a comparison involving Nazis or Hitler approaches one.” (“Meme, Counter-meme,” Wired, 10/1/94) 2. Circumstantial The circumstantial ad hominem fallacy is not as blunt an instrument as its abusive counterpart. It also involves attacking one’s opponent, focusing on some aspect of his person—their circumstances—as the core of the criticism. Specifically, you’re trying to dismiss their argument by saying they have something to gain. You’re questioning their motives in making the argument in the first place To see what we’re talking about, consider this argument: A recent study from scientists at the University of Minnesota claims to show that glyphosate—the main active ingredient in the widely used herbicide Roundup—is safe for humans to use. But guess whose business school just got a huge donation from Monsanto, the company that produces Roundup? That’s right, the University of Minnesota. Ever hear of conflict of interest? This study is junk, just like the product it’s defending. This is a fallacy. It doesn’t follow from the fact that the University received a grant from Monsanto that scientists working at that school faked the results of a study. The fact of the grant does raise a red flag; there may be some conflict of interest at play. But raising the possibility of a conflict is not enough, on its own, to show that the study in question can be dismissed out of hand. It may be appropriate to subject it to heightened scrutiny, but we cannot shirk our duty to assess its arguments on their merits. Most people argue for conclusions they like and that might benefit them in some way; what positive effect they personally want does not determine how good—or how bad—their actual argument may be. 3. Tu quoque This type of ad hominem involves pointing out one’s opponent’s hypocrisy. Its Latin name, “tu quoque,” translates roughly as “you, too.” This is the “I know you are but what am I?” and “look who’s talking” fallacy. It’s a technique used in very specific circumstances: your opponent accuses you of doing or advocating something that’s wrong, and, instead of making an argument to defend the rightness of your actions, you simply throw the accusation back in your opponent’s face—they did it too. However, just because they’re being hypocritical does not make the action in question right, and it does not mean the argument they’re making is bad. An example. In February 2016, Supreme Court Justice Antonin Scalia died unexpectedly. President Obama, as is his constitutional duty, nominated a successor. The Senate is supposed to ‘advise and consent’ (or not consent) to such nominations, but instead of holding hearings on the nominee (Merrick Garland), the Republican leaders of the Senate declared that they wouldn’t even consider the nomination. Since the presidential primary season had already begun, they reasoned, they should wait until the voters had spoken and allow the new president to make a nomination. Democrats objected strenuously, arguing that the Republicans were shirking their constitutional duty. The response was classic tu quoque. A conservative writer asked, “Does any sentient human being believe that if the Democrats had the Senate majority in the final year of a conservative president’s second term—and Justice [Ruth Bader] Ginsburg’s seat came open—they would approve any nominee from that president?” David French, National Review, 2/14/16 Senate Majority Leader Mitch McConnell said that he was merely following the “Biden Rule,” a principle advocated by Vice President Joe Biden when he was a Senator, back in the election year of 1992, that then-President Bush should wait until after the election season was over before appointing a new Justice (the rule was hypothetical; there was no Supreme Court vacancy at the time). This is a fallacious argument. Whether or not Democrats would do the same thing if the circumstances were reversed is irrelevant to determining whether that’s the right, constitutional thing to do. Red Herring One final fallacy of relevance, the red herring. Red herring fallacies aren’t trying to get you to accept their conclusion—they instead attempt to distract from the issue altogether. This fallacy gets its name from the actual fish, though there’s some debate about exactly how. Here’s one story that’s told: when herring are smoked, they turn red and are quite pungent. Stinky things can be used to distract hunting dogs, who of course follow the trail of their quarry by scent; if you pass over that trail with a stinky fish and run off in a different direction, the hound may be distracted and follow the wrong trail. Whether or not this practice was ever used to train hunting dogs, as some suppose, the connection to logic and argumentation is clear. One commits the red herring fallacy when one attempts to distract one’s audience from the main thread of an argument, taking things off in a different direction. The diversion is often subtle, with the detour starting on a topic closely related to the original—but gradually wandering off into unrelated territory. The tactic is often (but not always) intentional: one commits the red herring fallacy because one is not comfortable arguing about a particular topic on the merits, often because one’s case is weak; so instead, the arguer changes the subject to an issue about which they feel more confident, making strong points on the new topic, and pretending to have won the original argument. People often offer red herring arguments unintentionally, without the subtle deceptive motivation to change the subject—usually because they’re just parroting a red herring argument they heard from someone else. Sometimes a person’s response will be off-topic, apparently because they weren’t listening to their interlocutor or they’re confused for some reason. I prefer to label such responses as instances of Missing the Point (Ignoratio Elenchi), a fallacy that some books discuss at length, but which I’ve just relegated to a footnote. A fictional example can illustrate the technique. Consider Frank, who, after a hard day at work, heads to the tavern to unwind. He has far too much to drink, and, unwisely, decides to drive home. Well, he’s swerving all over the road, and he gets pulled over by the police. Let’s suppose that Frank has been pulled over in a posh suburb where there’s not a lot of crime. When the police officer tells him he’s going to be arrested for drunk driving, Frank becomes belligerent: Where do you get off? You’re barely even real cops out here in the ’burbs. All you do is sit around all day and pull people over for speeding and stuff. Why don’t you go investigate some real crimes? There’s probably some unsolved murders in the inner city they could use some help with. Why do you have to bother a hard-working citizen like me who just wants to go home and go to bed? Frank is committing the red herring fallacy (and not very subtly). The issue at hand is whether or not he deserves to be arrested for driving drunk. He clearly does. Frank is not comfortable arguing against that position on the merits. So he changes the subject—to one about which he feels like he can score some debating points. He talks about the police out here in the suburbs, who, not having much serious crime to deal with, spend most of their time issuing traffic violations. Yes, maybe that’s not as taxing a job as policing in the city. Sure, there are lots of serious crimes in other jurisdictions that go unsolved. But that’s beside the point! It’s a distraction from the real issue of whether Frank should get a DUI. Politicians use the red herring fallacy all the time. Consider a debate about Social Security—a retirement stipend paid to all workers, financed by a dedicated payroll tax. Suppose a politician makes the following argument: We need to cut Social Security benefits, raise the retirement age, or both. As the baby boom generation reaches retirement age, the amount of money set aside for their benefits will not be enough cover them while ensuring the same standard of living for future generations when they retire. The status quo will put enormous strains on the federal budget going forward, and we are already dealing with large, economically dangerous budget deficits now. We must reform Social Security. Now imagine an opponent of the proposed reforms offering the following reply: Social Security is a sacred trust, instituted during the Great Depression by FDR to ensure that no hard-working American would have to spend their retirement years in poverty. I stand by that principle. Every citizen deserves a dignified retirement. Social Security is a more important part of that than ever these days, since the downturn in the stock market has left many retirees with very little investment income to supplement government support. The second speaker makes some good points, but notice that they do not speak to the assertion made by the first: Social Security is economically unsustainable in its current form. It’s possible to address that point head on, either by making the case that in fact the economic problems are exaggerated or non-existent, or by making the case that a tax increase could fix the problems. The respondent does neither of those things, though; he changes the subject, and talks about the importance of dignity in retirement. I’m sure he’s more comfortable talking about that subject than the economic questions raised by the first speaker, but it’s a distraction from that issue—a red herring. Perhaps the most blatant kind of red herring is evasive: used especially by politicians, this is the refusal to answer a direct question by changing the subject. Examples are almost too numerous to cite; to some degree, no politician ever answers difficult questions straightforwardly (there’s an old axiom in politics, put nicely by Robert McNamara: “Never answer the question that is asked of you. Answer the question that you wish had been asked of you.”). A particularly egregious example of this occurred in 2009 on CNN’s Larry King Live. Michele Bachmann, Republican Congresswoman from Minnesota, was the guest. The topic was “birtherism,” the (false) belief among some that Barack Obama was not in fact born in America and was therefore not constitutionally eligible for the presidency. After playing a clip of Senator Lindsey Graham (R, South Carolina) denouncing the myth and those who spread it, King asked Bachmann whether she agreed with Senator Graham. She responded thus: You know, it's so interesting, this whole birther issue hasn't even been one that's ever been brought up to me by my constituents. They continually ask me, where's the jobs? That's what they want to know, where are the jobs? Bachmann doesn’t want to respond directly to the question. If she outright declares that the “birthers” are right, she’s endorsing a clearly false belief. But if she denounces them, she alienates a lot of her potential voters who believe the falsehood. Tough bind. So, she blatantly, and rather desperately, tries to change the subject. Jobs! Let’s talk about those instead. Please? III. Fallacies of Weak Induction As their name suggests, what these fallacies have in common is that they are bad—that is, weak— inductive arguments. Recall, inductive arguments attempt to provide premises that make their conclusions more probable. We evaluate them according to how probable their conclusions are in light of their premises: the more probable the conclusion (given the premises), the stronger the argument; the less probable, the weaker. The fallacies of weak induction are arguments whose premises do not make their conclusions very probable—but that are nevertheless often successful in convincing people of their conclusions. We will discuss five informal fallacies that fall under this heading. Argument from Ignorance In essence, an argument from ignorance is an inference from premises that directly or implicitly state there’s a lack of knowledge about some topic, to a definite conclusion about that topic. We don’t know; therefore, we know! Of course, put that baldly, it’s plainly absurd; actual instances are more subtle. The fallacy comes in a variety of closely related forms. It will be helpful to state them in bald/absurd schematic fashion first, then elucidate with more subtle real-life examples. The first form can be put like this: Nobody knows how to explain phenomenon X. Therefore, my theory about X is true. That sounds silly, but consider an example: those “documentary” programs on cable TV about aliens. You know, the ones where they suggest that extraterrestrials built the pyramids or something (there are books and websites, too). How do they get you to believe that crazy theory? By creating mystery! By pointing to facts that nobody can explain. The Great Pyramid at Giza is aligned (almost) exactly with the magnetic north pole! On the day of the summer solstice, the sun sets exactly between two of the pyramids! The height of the Great Pyramid is (almost) exactly one one-millionth the distance from the Earth to the Sun! How could the ancient Egyptians have such sophisticated astronomical and geometrical knowledge? Why did the Egyptians, careful record- keepers in (most) other respects, (apparently) not keep detailed records of the construction of the pyramids? Nobody knows. Conclusion: aliens built the pyramids. In other words, there are all sorts of (sort of) surprising facts about the pyramids, and nobody knows how to explain them. From these premises, which establish only our ignorance, we’re encouraged to conclude that we know something: aliens built the pyramids. That’s quite a leap— too much of a leap. Another form this fallacy takes can be put crudely thus: Nobody can PROVE that I’m wrong. Therefore, I’m right. The word ‘prove’ is in all-caps because stressing it is the key to this fallacious argument: the standard of proof is set impossibly high, so that almost no amount of evidence would constitute a refutation of the conclusion. An example will help. There are lots of people who claim that evolutionary biology is a lie: there’s no such thing as evolution by natural selection, and it’s especially false to claim that humans evolved from earlier species, that we share a common ancestor with apes. Rather, the story goes, the Bible is literally true: the Earth is only about 6,000 years old, and humans were created as-is by God just as the Book of Genesis describes. The Argument from Ignorance is one of the favored techniques of proponents of this view. They are especially fond of pointing to “gaps” in the fossil record—the so-called “missing link” between humans and a pre-human, ape-like species—and claim that the incompleteness of the fossil record vindicates their position. But this argument is an instance of the fallacy. The standard of proof—a complete fossil record without any gaps—is impossibly high. Evolution has been going on for a LONG time (the Earth is actually about 4.5 billion years old, and living things have been around for at least 3.5 billion years). So many species have appeared and disappeared over time that it’s absurd to think that we could even come close to collecting fossilized remains of anything but the tiniest fraction of them. It’s hard to become a fossil, after all: a creature has to die under special circumstances to even have a chance for its remains to do anything than turn into compost. And we haven’t been searching for fossils in a systematic way for very long (only since the mid-1800s or so). It’s no surprise that there are gaps in the fossil record, then. What’s surprising, in fact, is that we have as rich a fossil record as we do. Many, many transitional species have been discovered, both between humans and their ape-like ancestors, and between other modern species and their distant forbears (whales used to be land-based creatures, for example; we know this (in part) from the fossils of early proto- whale species with longer and longer rear hip- and leg-bones). We will never have a fossil record complete enough to satisfy skeptics of evolution. But their standard is unreasonably high, so their argument is fallacious. Sometimes they put it even more simply: nobody was around to witness evolution in action; therefore, it didn’t happen. This is patently absurd, but it follows the same pattern: an unreasonable standard of proof (witnesses to evolution in action; impossible, since it takes place over such a long period of time), Although, in the case of short-lived bacteria, where we can study multiple generations of development across the span of a few weeks, we can watch evolution take place in real time, as demonstrated by scientists from the Kishony Lab at Harvard Medical School and Technion: https://hms.harvard.edu/news/bugs-screen. followed by the leap to the unwarranted conclusion. One final note on this fallacy: it’s common for people to mislabel certain bad arguments as arguments from ignorance; namely, arguments made by people who obviously don’t know what they’re talking about. People who are confused or ignorant about the subject on which they’re offering an opinion are liable to make bad arguments, but the fact of their ignorance is not enough to label those arguments as instances of the fallacy. We reserve that designation for arguments that take the forms canvassed above: those that rely on ignorance—and not just that of the arguer, but of the audience as well—as a premise to support the conclusion. Appeal to Unqualified Authority One way of making an inductive argument—of lending more credence to your conclusion—is to point to the fact that some relevant authority figure agrees with you. In law, for example, this kind of argument is indispensable: appeal to precedent (Supreme Court rulings, etc.) is the attorney’s bread and butter. And in other contexts, this kind of move can make for a strong inductive argument. If I’m trying to convince you that fluoridated drinking water is safe and beneficial, I can point to the Center for Disease Control, where a wealth of information supporting that claim can be found. Check it out: https://www.cdc.gov/fluoridation/ Those people are scientists and doctors who study this stuff for a living; they know what they’re talking about. One commits the fallacy when one points to the testimony of someone who’s not a reliable authority on the issue at hand. There are several things that disqualify someone from being a reliable authority; the most blatant form of this is when the person being relied upon is not an expert in that subject at all. This is a favorite technique of advertisers. We’ve all seen celebrity endorsements of various products. In the 2000s, Tiger Woods was in commercials selling Buicks. Tiger Woods is an expert at golf, but not Buicks. Usually, the inappropriateness of the authority being appealed to is obvious, but sometimes it isn’t. A particularly subtle example is AstraZeneca’s hiring of Dr. Phil McGraw in 2016 as a spokesperson for their diabetes outreach campaign. AstraZeneca is a drug manufacturing company. They make a diabetes drug called Bydureon. The aim of the outreach campaign, ostensibly, is to increase awareness among the public about diabetes; but of course the real aim is to sell more Bydureon. A celebrity like Dr. Phil can help. Is he an appropriate authority? That’s a hard question to answer. It’s true that Dr. Phil had suffered from diabetes himself for 25 years, and that he personally takes the medication. So that’s a mark in his favor, authority-wise. But is that enough? We’ll talk about how feeble Phil’s sort of anecdotal evidence is in supporting general claims (in this case, about a drug’s effectiveness) when we discuss the hasty generalization fallacy; suffice it to say, one person’s positive experience doesn’t prove that the drug is effective. But, Dr. Phil isn’t just a person who suffers from diabetes; he’s a doctor! It’s right there in his name (everybody always simply refers to him as ‘Dr. Phil’). Surely that makes him an appropriate authority on the question of drug effectiveness. Or maybe not. Phil McGraw is not a medical doctor; he’s a PhD. He has a doctorate in Psychology. He’s not a licensed psychologist; he cannot legally prescribe medication. He has no relevant professional expertise about drugs and their effectiveness. He is not a qualified medical authority in this case. He looks like one, though, which makes this a very sneaky, but effective, advertising campaign. Of course, even a qualified expert in the relevant subject should be doubted if they appear in an ad. Not being an expert disqualifies someone, but having reasons to distort the truth also disqualifies someone from being a reliable expert. Humans lie and distort the truth for many reason—to protect themselves or someone they love, to get a job or promotion or get out of trouble, and, of course, money is a huge motivation to distort the truth. Michael Jordan, a very talented basketball player, has spent years selling athletic shoes. Even though he’s not a scientist comparing the objective qualities of one shoe versus another, he is a man who has spent decades relying on a good pair of shoes to get him from one side of the court to the other. This qualifies him to at least have an opinion we should listen to. However, the fact that he’s making a great deal of money from selling a particular brand of shoes disqualifies him from being a reliable expert. A third thing disqualifies a person from being a reliable expert, and that’s if there’s reason to believe they somehow can’t perceive the facts clearly. In court, eye witnesses to a crime are considered experts in what happened; it’s up to the lawyers to determine whether they’re reliable or not. We see this in a movie called Twelve Angry Men, a famous courtroom drama from 1957. In the movie, a young man is accused of killing his father, and there are two witnesses who testify, one who heard the crime from the apartment below, and one who saw the crime from across the street through the windows of a passing train. Eleven jurors vote the young man guilty, but one votes not guilty, because there is reasonable doubt. A passing train would make it difficult to hear the crime from another apartment, and the woman who saw the crime not only did so through the windows of the train instead of getting a clear view – but also, as it turns out, she was not wearing her glasses. The inability to clearly perceive the crime disqualifies both jurors from being reliable witnesses. Eleven jurors made the appeal to unqualified authority fallacy. A final note on this fallacy. Remember that ad hominem fallacies are when you turn your attention to the arguer, rather than their argument, and try to dismiss the argument based on some personal fact about themselves. If you are evaluating someone’s argument, you shouldn’t pay attention to the arguer’s personal circumstances, perceived failings, whether they just have something to gain, etc. However, if you’re trusting someone as an expert, instead of evaluating their argument, you do need to turn your attention to them personally, to judge if they’re trustworthy. So if you accuse the lawyer of arguing the young man is guilty just because she’s being paid to make that argument, you’re making an ad hominem circumstantial fallacy. Of course lawyers get paid to do their jobs; it’s the jury’s job to figure out if the argument they gave was good or not. On the other hand, if you accuse the witness of saying the young man is guilty just because he’s being paid to say that, this isn’t an ad hominem fallacy. The witness is not giving an argument to evaluate, he’s stating what he supposedly witnessed, and if someone is bribing him to say that, he’s disqualified from being a reliable authority. False Cause Fallacies False cause fallacies are mistakes humans make in causal reasoning. There are many, but I want to introduce three: post hoc ergo propter hoc, non causa pro causa, and oversimplified cause. These are the fallacies we’re trying to avoid by using Mill’s Methods to provide evidence for (or against) our causal assumptions. The first two completely misunderstand a causal link; the third one is partially correct about a causal link, but in a way that is often misleading. 1. Post Hoc Ergo Propter Hoc This is Latin for “after this, therefore because of this.” It is often shortened to just “post hoc.” The post hoc fallacy is where we see causation where there is none, because one thing happened right after another one. Usually, there are two significant events that caught our attention, and our brain forms a causal link between them. (If two events are insignificant, we usually don’t leap to a causal connection. This morning I brushed my teeth and then a little later, I put my shoes on. I had to stop and think what I did after I brushed my teeth; neither of these events are notable, so I don’t even think about them. I certainly don’t think that brushing my teeth caused me to put shoes on.) Here are a few examples of the post hoc fallacy. In 2013, Beyoncé played the half time show during the Super Bowl. It was a spectacular show, with great music, dance routines, and special effects. About twenty minutes into the second half of the game, the power went out in the Superdome. It took about 40 minutes before the power was fully restored. Fortunately, there was enough power that the commenters could talk about how the lights were still off… Of course the immediate assumption was that Beyoncé’s show caused the power outage. (It did not. Beyoncé’s crew brought their own generators, they didn’t even use the Superdome’s power). See the “after this, therefore because of this” structure? Beyoncé played, and then the lights went out. So people (falsely) assumed that her show caused the lights to go out. Here’s another one: The governor of California gave a speech, and then an earthquake hit Los Angeles. He needs to stop giving speeches; he’s putting people in danger! He spoke, and then the earthquake hit, so the assumption is he caused the earthquake, maybe by bringing bad luck to the residents of Los Angeles. Superstitions are often reinforced by the post hoc fallacy. If a black cat crosses my path, and then I get in a car wreck, I will assume that the black cat crossing my path caused my bad luck. 2. Non Causa Pro Causa The Latin here translates to “not the cause for the effect.” Just like with post hoc, I have completely misidentified a causal link. However, this time it’s not because one event happened before another one, it’s because the two things happen at the same time, or in the same place, or I notice some other correlation between them, and use this correlation to assume there’s a causal link. As the saying goes, correlation is not causation. Here are a few examples. Every time the governor of California gives a speech, a natural disaster happens somewhere in the world. He needs to stop giving speeches; he’s putting people in danger! Notice how this is different from the example in the last section. The post hoc had two events, that happened one right after the other. The non causa finds a correlation – these two things keep happening at the same time. Now, correlation can give you a sense that two things might be causally linked. If every venue Beyonce ever played in suffered a power outage, I’d start to suspect her show did have something to do with it. But simply noticing a correlation is not proof. Here’s another example: This is from Tyler Vigen’s web page; the link is here (it opens a new page): Spurious Correlations. If we drew from this chart the conclusion that eating cheese causes people to die by becoming tangled in their bedsheets (or that people dying in their bedsheets is causing more people to eat cheese), we are making the non causa fallacy. And, a famous example: Every time ice cream sales go up, so do shark attacks! Sharks must really like the taste of ice-cream stuffed humans. This is an interesting one, because there is actually sort of an indirect a causal relationship between ice cream sales and shark attacks—they both share a cause. Buying (and presumably eating) ice cream does not cause sharks to attack; shark attacks do not cause a rise in ice cream sales. Rather, hot weather is responsible for both ice cream sales rising and for humans going swimming in shark-infested water. I taught with this example for years before I finally looked it up to see if the premise was true. It is not. Ice cream sales and shark attacks peak in different months. However, remember that we’re looking at the connection between premises and conclusion when we evaluate logic. IF the premise were true, how much evidence would it give for the conclusion? Answer: not very much at all. 3. Oversimplified cause The fallacy of oversimplified cause is where you’re dealing with a complex phenomenon with many moving parts, but you pick a partial cause and pin the full weight of blame (or praise) on that one thing. It doesn’t completely misidentify a causal connection, like post hoc and non causa; the thing you’ve picked out does play a causal role, but you’ve oversimplified by ignoring all of the other parts of the equation. Here’s an example: The economy of California is thriving. It must be because of the governor’s wise economic policies. Or The economy of California is crashing. It must be because of the governor’s idiotic economic policies. The truth is that the governor’s economic policies do have an effect on the economy of the state. However, one person is never solely responsible for an entire state’s economy. Further, economic policies never have an immediate effect. If government policies do partially cause a change in the economy, we’re unlikely to see that change for several years – usually during the next governor’s term of office. This is the fallacy of oversimplified cause. Slippery Slope Like the false cause fallacies, the slippery slope fallacy is a weak inductive argument to a conclusion about causation. This fallacy involves making an insufficiently supported claim that a certain action or event will set off an unstoppable causal chain-reaction—putting us on a slippery slope— leading to some disastrous effect. This style of argument was a favorite tactic of religious conservatives who opposed gay marriage. They claimed that legalizing same-sex marriage would put the nation on a slippery slope to disaster. Famous Christian leader Pat Robertson, on his television program The 700 Club, puts the case nicely. When asked about gay marriage, he responded with this: We haven’t taken this to its ultimate conclusion. You’ve got polygamy out there. How can we rule that polygamy is illegal when you say that homosexual marriage is legal? What is it about polygamy that’s different? Well, polygamy was outlawed because it was considered immoral according to Biblical standards. But if we take Biblical standards away in homosexuality, well what about the other? […]You mark my words, this is just the beginning of a long downward slide in relation to all the things that we consider to be abhorrent. This is a classic slippery slope fallacy; he even uses the phrase ‘long downward slide’! The claim is that allowing gay marriage will force us to decriminalize polygamy, and ultimately, “all the things that we consider to be abhorrent.” Yikes! That’s a lot of things. Apparently, gay marriage will lead to utter anarchy. There are genuine unstoppable causal chain-reactions out there—but this isn’t one of them. The mark of the slippery slope fallacy is the assertion that the chain can’t be stopped, with reasons that are insufficient to back up that assertion. In this case, Pat Robertson has given us the abandonment of “Biblical standards” as the lubrication for the slippery slope. This is obviously insufficient. Biblical standards are expressly forbidden, by the “establishment clause” of the First Amendment to the U.S. Constitution, from forming the basis of the legal code. The slope is not slippery. As recent history has shown, the legalization of same sex marriage did not lead to the legalization of polygamy; the argument is fallacious. Fallacious slippery slope arguments have long been deployed to resist social change. Those opposed to the abolition of slavery warned of economic collapse and social chaos. Those who opposed women’s suffrage asserted that it would lead to the dissolution of the family, rampant sexual promiscuity, and social anarchy. Of course none of these dire predictions came true; the slopes simply weren’t slippery. Hasty Generalization Many inductive arguments involve an inference from particular premises to a general conclusion; this is a generalization. For example, if you make a bunch of observations every morning that the sun rises in the east, and conclude on that basis that, in general, the sun always rises in the east, this is a generalization. And it’s a good one! With all those particular sunrise observations as premises, your conclusion that the sun always rises in the east has a lot of support; that’s a strong inductive argument. The data you rely on when generalizing is called your “sample.” The sample of the above generalization is the collection of all the times I saw the sun rise in the east. One commits the hasty generalization fallacy when one makes this kind of inference based on sample that is too small, not sufficiently random, or in some way biased. A random sample is one where every member of the group has an equal chance of being selected into your sample; random samples that are large enough have a pretty good chance of being a fair representation of the population. Another method of insuring the sample accurately reflects the population at large is to construct a ”representative sample,” purposefully making sure the sample has the same proportions of key demographics as the population at large. Bias refers to anything which slants your data. Suppose I want to know who will win the next presidential election in the United States, so I take a survey. If I ask two thousand people who they intend to vote for, and all two thousand are from Massachusetts, this sample is not random, and it’s also biased, since Massachusetts is a blue state—you’ve biased your survey in favor of the Democratic candidate. People who deny that global warming is a genuine phenomenon often commit the hasty generalization fallacy. In February of 2015, the weather was unusually cold in Washington, DC. Senator James Inhofe of Oklahoma famously took to the Senate floor wielding a snowball. “In case we have forgotten, because we keep hearing that 2014 has been the warmest year on record, I ask the chair, ‘You know what this is?’ It’s a snowball, from outside here. So it’s very, very cold out. Very unseasonable.” He then tossed the snowball at his colleague, Senator Bill Cassidy of Louisiana, who was presiding over the debate, saying, “Catch this.” Senator Inhofe commits the hasty generalization fallacy. He’s trying to establish a general conclusion—that 2014 wasn’t the warmest year on record, or that global warming isn’t really happening (he’s on the record that he considers it a “hoax”). But the evidence he presents is insufficient to support such a claim. His evidence is an unseasonable coldness in a single place on the planet, on a single day. We can’t derive from that any conclusions about what’s happening, temperature-wise, on the entire planet, over a long period of time. That the earth is warming is not a claim that everywhere, at every time, it will always be warmer than it was; the claim is that, on average, across the globe, temperatures are rising. This is compatible with a couple of cold snaps in the nation’s capital. Many people are susceptible to hasty generalizations in their everyday lives. When we rely on anecdotal evidence to make decisions, we commit the fallacy. Suppose you’re thinking of buying a new car, and you’re considering a Subaru. Your neighbor has a Subaru. So what do you do? You ask your neighbor how he likes his Subaru. He tells you it runs great, hasn’t given him any trouble. You then, fallaciously, conclude that Subarus must all be terrific cars. But one person’s testimony isn’t enough to justify that conclusion; you’d need to look at many, many more drivers’ experiences to reach such a conclusion (this is why the magazine Consumer Reports is so useful). A particularly pernicious instantiation of the Hasty Generalization fallacy is the development of negative stereotypes. People often make general claims about religious or racial groups, ethnicities and nationalities, based on very little experience with them. If you once got mugged by a Puerto Rican, that’s not a good reason to think that, in general, Puerto Ricans are crooks. If a waiter at a restaurant in Paris was snooty, that’s no reason to think that French people are stuck up. And yet we see this sort of faulty reasoning all the time. Weak Analogy A final fallacy of weak induction is the Weak Analogy fallacy. Arguments by Analogy are a fundamental form of inductive reasoning. An argument by analogy draws on a conclusion based on a comparison between two or more things. It’s the form of reasoning that lets us see patterns and extend those patterns to new situations. Many forms of inductive arguments have at least an element of analogy in them. For example, think of the following prediction: On the last 100 days with weather conditions like today, it rained in the evening. Therefore, it will rain this evening. This is a logical prediction – using data from the past to draw a conclusion about the future. But there’s an element of analogy here, too. The reason I feel confident predicting the future here is because I’m looking at days similar to today – I’m comparing today to past days with specific similarities. Because all 101 days (the 100 in the past and today) share similar weather conditions, we predict they will share similar precipitation as well. Or, consider a generalization: Of the 2,000 Americans polled, 55% of them said they prefer Pepsi to Coke. Therefore, 55% of Americans prefer Pepsi to Coke. Like most of my statistics, this is completely made up for the sake of example. You learned above that in order to be a good generalization, your sample needs to be large enough, free of bias, and random, or representative. A random sample is one where every single member of the population has an equal chance of being included in the sample. A representative sample is one where the researcher purposefully constructs a sample that has the same proportion of key demographic groups as the population at large. Both random and representative samples are methods of making sure your sample is like the population at large in key ways. The generalization only works if we can draw a firm analogy between the sample and the population. The basic argument by analogy compares two things, and based on their similarities, transfers a property from one of those things to the other. For example: Stacy’s house has 900 square feet and is all one level. Her electric bill is $150 a month. If I buy that 910 square feet house in her neighborhood, which is also one level, I can expect my electric bill to be about the same. Her house is similar to the one I’m considering buying: same neighborhood, same number of floors, same square footage. Given these similarities, I have a reasonable chance of having a similar electric bill. A good argument by analogy must compare things that are not only similar to each other, but are similar to each other in relevant ways, and there needs to be no significant differences between the two cases. So, the fallacy of weak analogy is one where the similarities between the two things being compared do not warrant the conclusion. Either they’re just not very similar, or their similarities have little to no relevance to the conclusion you’re trying to draw, or you’ve discovered a relevant difference between the two things being compared. Consider the following argument: Stacy’s house has 900 square feet and is all one level. She has a great kitchen. If I buy that 910 square feet house in her neighborhood, which is also one level, I can expect it to also have a great kitchen. This argument commits the fallacy of weak analogy. Square footage, how many floors your house has, and neighborhood, do have an impact on electric bills, because it will take a similar amount of electricity to heat and cool the place. These similarities, however, have nothing to do with whether a house has a good kitchen or not. These similarities do not warrant the conclusion. Or, consider this variation: Stacy’s house has 900 square feet and is all one level. Her electric bill is $150 a month. She’s single and works out of the house; I have three kids and work from home. If I buy that 910 square feet house in her neighborhood, which is also one level, I can expect my electric bill to be about the same. The similarities are still relevant, but I’ve now discovered a significant difference. How many people are using devices, lights, and so on has a big impact on electric bills. So does the fact that her house will be unoccupied (and hopefully then with lights and devices turned off) for a lot of the day, where mine will be occupied more often. So, this too commits the fallacy of weak analogy. IV. Fallacies of Illicit Presumption This is a family of fallacies whose common characteristic is that they (often tacitly, implicitly) presume the truth of some claim that they’re not entitled to. They are arguments with a premise (again, often hidden) that is assumed to be true, but is actually a controversial claim, which at best requires support that’s not provided, which at worst is simply false. We will look at six fallacies under this heading. Accident This fallacy is the reverse of the hasty generalization. That was a fallacious inference from insufficient particular premises to a general conclusion; accident is a fallacious inference from a general premise to a particular conclusion. What makes it fallacious is an illicit presumption: the general rule in the premise is assumed, incorrectly, not to have any exceptions; the particular conclusion fallaciously inferred is one of the exceptional cases. Here’s a simple example to help make that clear: Cutting people with knives is illegal. Surgeons cut people with knives. Therefore, surgeons should be arrested. One of the premises is the general claim that cutting people with knives is illegal. While this is true in almost all cases, there are exceptions—surgery among them. We pay surgeons lots of money to cut people with knives! It is therefore fallacious to conclude that surgeons should be arrested, since they are an exception to the general rule. The inference only goes through if we presume, incorrectly, that the rule is exceptionless. Another example. Suppose I volunteer at my first-grade daughter’s school; I go in to her class one day to read a book aloud to the children. As I’m sitting down on the floor with the kiddies, criss-cross applesauce, as they say, I realize that I can’t comfortably sit that way because of the .44 Magnum revolver that I have tucked into my waistband. That’s Dirty Harry’s gun, “the most powerful handgun in the world.” So I remove the piece from my pants and set it down on the floor in front of me, among the circled-up children. The teacher screams and calls the office, the police are summoned, and I’m arrested. As they’re hauling me out of the room, I protest: “The Second Amendment to the Constitution guarantees my right to keep and bear arms! This state has a ‘concealed carry’ law, and I have a license to carry that gun! Let me go!” I’m committing the fallacy of Accident in this story. True, the Second Amendment guarantees the right to keep and bear arms; but that rule is not without exceptions. Similarly, concealed carry laws also have exceptions—among them being a prohibition on carrying weapons into elementary schools. My insistence on being released only makes sense if we presume, incorrectly, that the legal rules I’m citing are without exception. One more example from real life: After the financial crisis in 2008, the Federal Reserve—the central bank in the United States, whose task it is to create conditions leading to full employment and moderate inflation—found itself in a bind. The economy was in a free-fall, and unemployment rates were skyrocketing, but the usual tool it used to mitigate such problems—cutting the short- term federal funds rate (an interest rate banks charge each other for overnight loans)—was unavailable, because they had already cut the rate to zero (the lowest it could go). So they had to resort to unconventional monetary policies, among them something called “quantitative easing”. This involved the purchase, by the Federal Reserve, of financial assets like mortgage-backed securities and longer-term government debt (Treasury notes). The hope was to push down interest rates on mortgages and government debt, encouraging people to buy houses and spend money instead of saving it—thus stimulating the economy. Now, the nice thing about being the Federal Reserve is that when you want to buy something—in this case a bunch of financial assets—it’s really easy to pay for it: you have the power to create new money out of thin air! That’s what the Federal Reserve does; it controls the amount of money that exists. So if the Fed wants to buy, say, $10 million worth of securities from Bank of America, they just press a button and presto—$10 million dollars that didn’t exist a second ago comes into being as an asset of Bank of America. It’s obviously a bit more complicated than that, but that’s the essence of it. This quantitative easing policy was controversial. Many people worried that it would lead to runaway inflation. Generally speaking, the more money there is, the less each bit of it is worth. So creating more money makes things cost more—inflation. The Fed was creating money on a very large scale—on the order of a trillion dollars. Shouldn’t that lead to a huge amount of inflation? Economist Art Laffer thought so. In June of 2009, he wrote an op-ed in the Wall Street Journal warning that “[t]he unprecedented expansion of the money supply could make the '70s look benign.” Art Laffer, “Get Ready for Inflation and Higher Interest Rates,” June 11, 2009, Wall Street Journal (There was a lot of inflation in the ’70s.) Another famous economist, Paul Krugman, accused Laffer of committing the fallacy of accident. While it’s generally true that an increase in the supply of money leads to inflation, that rule is not without exceptions. He had described such exceptional circumstances in 1998 “But if current prices are not downwardly flexible, and the public expects price stability in the long run, the economy cannot get the expected inflation it needs; and in that situation the economy finds itself in a slump against which short- run monetary expansion, no matter how large, is ineffective.” From Paul Krugman, "It's baack: Japan's Slump and the Return of the Liquidity Trap,” 1998, Brookings Papers on Economic Activity, 2 , and pointed out that the economy of 2009 was in that condition (which economists call a “liquidity trap”): “Let me add, for the 1.6 trillionth time, we are in a liquidity trap. And in such circumstances a rise in the monetary base does not lead to inflation.” Paul Krugman, June 13, 2009, The New York Times It turns out Krugman was correct. The expansion of the monetary supply did not lead to runaway inflation; as a matter of fact, inflation remained below the level that the Federal Reserve wanted, barely moving at all. Laffer had indeed committed the fallacy of accident. Begging the Question (Petitio Principii) First things first: “begging the question” is not synonymous with “raising the question;” this is an extremely common usage, but it is wrong. You might hear a newscaster say, “Today Donald Trump’s private jet was spotted at the Indianapolis airport, which begs the question: ‘Will he choose Indiana Governor Mike Pence as running mate?’” This is a mistaken usage of “begs the question;” the newscaster should have said “raises the question” instead. “Begging the question” is a translation of the Latin ‘petitio principii’, which refers to the practice of asking (begging, petitioning) your audience to grant you the truth of a claim (principle) as a premise in an argument—but it turns out that the claim you're asking for is either identical to, or presupposes the truth of, the very conclusion of the argument you're trying to make. In other words, when you beg the question, you're arguing in a circle: one of the reasons for believing the conclusion is the conclusion itself! It’s a Fallacy of Illicit Presumption where the proposition being presumed is the very proposition you’re trying to demonstrate; that’s clearly an illicit presumption. Here’s a stark example. If I'm trying to convince you that Donald Trump is a dangerous idiot (the conclusion of my argument is ‘Donald Trump is a dangerous idiot’), then I can't ask you to grant me the claim ‘Donald Trump is a dangerous idiot’. The premise can't be the same as the conclusion. Imagine a conversation: Me: “Donald Trump is a dangerous idiot.” You: “Really? Why do you say that?” Me: “Because Donald Trump is a dangerous idiot.” You: “So you said. But why should I agree with you? Give me some reasons.” Me: “Here's a reason: Donald Trump is a dangerous idiot.” And round and round we go. Circular reasoning; begging the question. It's not always so blatant. Sometimes the premise is not identical to the conclusion, but merely presupposes its truth. Why should we believe that the Bible is true? Because it says so right there in the Bible that it’s the infallible Word of God. This premise is not the same as the conclusion, but it can only support the conclusion if we take the Bible's word for its own truthfulness, i.e., if we assume that the Bible is true. But that was the very claim we were trying to prove! Sometimes the premise is just a re-wording of the conclusion. Consider this argument: To allow every man unbounded freedom of speech must always be, on the whole, advantageous to the state; for it is highly conducive to the interests of the community that each individual should enjoy a liberty, perfectly unlimited, of expressing his sentiments. This is a classic example, from Richard Whately’s 1826 Elements of Logic. Replacing synonyms with synonyms, this comes down to “Free speech is good for society because free speech is good for society.” Not a good argument. Though it’s valid! P, therefore P is a valid form: if the premise is true, the conclusion must be; they’re the same. Loaded Questions Loaded questions are questions the very asking of which presumes the truth of some claim. Asking these can be an effective debating technique, a way of sneaking a controversial claim into the discussion without having outright asserted it. The classic example of a loaded question is, “Have you stopped doing drugs?” Notice that this is a yes-or-no question, and no matter which answer one gives, one admits to doing drugs: if the answer is ‘no’, then the person continues to do drugs; if the answer is ‘yes’, then he admits to doing drugs in the past. Either way, he’s done drugs. The question itself presumes the truth of this claim; that’s what makes it “loaded”. Strategic deployment of loaded yes-or-no questions can be an extremely effective debating technique. If you catch your opponent off-guard, they will struggle to respond to your question, since a simple ‘yes’ or ‘no’ commits them to the truth of the illicit presumption, which they want to deny. This makes them look evasive, shifty. And as they struggle to come up with a response, you can pounce on them: “It’s a simple question. Yes or no? Why won’t you answer the question?” It’s a great way to appear to be winning a debate, even if you don’t have a good argument. Imagine the following dialogue: TV Host: “Are you or are you not in favor of the president’s plan to force wealthy business owners to pay their fair share in taxes to protect the vulnerable and aid this nation’s underprivileged?” Guest: “Well, I don’t agree with the way you’ve laid out the question. As a matter of fact…” Host: “It’s a simple question. Should business owners pay their fair share; yes or no?” Guest: “You’re implying that the president’s plan would correct some injustice. But corporate taxes are already very…” Host: “Stop avoiding the question! It’s a simple yes or no!” Combine this with the sort of subconscious appeal to force discussed above—yelling, finger- pointing, etc.—and the host might come off looking like the winner of the debate, with his opponent appearing evasive, uncooperative, and inarticulate. Another use for loaded questions is the particularly sneaky political practice of “push polling”. In a normal opinion poll, you call people up to try to discover what their views are about the issues. In a push poll, you call people up pretending to be conducting a normal opinion poll, pretending only to be interested in discovering their views, but with a different intention entirely: you don’t want to know what their views are; you want to shape their views, to convince them of something. And you use loaded questions to do it. A famous example of this occurred during the Republican presidential primary in 2000: George W. Bush was the front-runner, but was facing a surprisingly strong challenge from the upstart John McCain. After McCain won the New Hampshire primary, he had a lot of momentum. The next state to vote was South Carolina; it was very important for the Bush campaign to defeat McCain there and reclaim the momentum. So they conducted a push poll designed to spread negative feelings about McCain—by implanting false beliefs among the voting public. “Pollsters” called voters and asked, “Would you be more or less likely to vote for John McCain for president if you knew he had fathered an illegitimate black child?” The aim, of course, is for voters to come to believe that McCain fathered an illegitimate black child. But he did no such thing. He and his wife adopted a daughter, Bridget, from Bangladesh. A final note on loaded questions: there’s a minimal sense in which every question is loaded. The social practice of asking questions is governed by implicit norms. One of these is that it’s only appropriate to ask a question when there’s some doubt about the answer. So every question carries with it the presumption that this norm is being adhered to, that it’s a reasonable question to ask, that the answer is not certain. One can exploit this fact, again to plant beliefs in listeners’ minds that they otherwise wouldn’t hold. In a particularly shameful bit of alarmist journalism, the cover of the July 1, 2016, issue of Newsweek asks the question, “Can ISIS Take Down Washington?” The cover is an alarming, eye-catching shade of yellow, and shows four missiles converging on the Capitol dome. The simple answer to the question, though, is ‘no, of course not’. There is no evidence that ISIS has the capacity to destroy the nation’s capital. But the very asking of the question presumes that it’s a reasonable thing to wonder about, that there might be a reason to think that the answer is ‘yes’. The goal is to scare readers (and sell magazines) by getting them to believe there might be such a threat. False Choice This fallacy occurs when someone tries to convince you of something by presenting it as one of limited number of options and the best choice among those options. The illicit presumption is that the options are limited in the way presented; in fact, there are additional options that are not offered. The choice you’re asked to make is a false choice, since not all the possibilities have been presented. Most frequently, the number of options offered is two. In this case, you’re being presented with a false dilemma. I manipulate my kids with false choices all the time. My younger daughter, for example, loves cucumbers; they’re her favorite vegetable by far. We have a rule at dinner: you’ve got to choose a vegetable to eat. Given her ’druthers, she’d choose cucumber every night. Carrots are pretty good, too; they’re the second choice. But I need her to have some more variety, so I’ll sometimes lie and tell her we’re out of cucumbers and carrots, and that we only have two options: broccoli or green beans, for example. That’s a false choice; I’ve deliberately left out other options. I give her the false choice as a way of manipulating her into choosing green beans, because I know she dislikes broccoli. Politicians often treat us like children, presenting their preferred policies as the only acceptable choice among an artificially restricted set of options. We might be told, for example, that we need to raise the retirement age or cut Social Security benefits across the board; the budget can’t keep up with the rising number of retirees. Well, nobody wants to cut benefits, so we have to raise the retirement age. Bummer. But it’s a false choice. There are any number of alternative options for funding an increasing number of retirees: tax increases, re-allocation of other funds, means-testing for benefits, etc. Liberals are often ambivalent about free trade agreements. On the one hand, access to American markets can help raise the living standards of people from poor countries around the world; on the other hand, such agreements can lead to fewer jobs for American workers in certain sectors of the economy (e.g., manufacturing). So what to do? Support such agreements or not? Seems like an impossible choice: harm the global poor or harm American workers. But it may be a false choice, as this economist argues: But trade rules that are more sensitive to social and equity concerns in the advanced countries are not inherently in conflict with economic growth in poor countries. Globalization’s cheerleaders do considerable damage to their cause by framing the issue as a stark choice between existing trade arrangements and the persistence of global poverty. And progressives needlessly force themselves into an undesirable tradeoff. … Progressives should not buy into a false and counter-productive narrative that sets the interests of the global poor against the interests of rich countries’ lower and middle classes. With sufficient institutional imagination, the global trade regime can be reformed to the benefit of both. Dani Rodrik, “A Progressive Logic of Trade,” Project Syndicate, 4/13/2016 When you think about it, almost every election in America is a False Choice. With the dominance of the two major political parties, we’re normally presented with a stark, sometimes unpalatable, choice between only two options: the Democrat or the Republican. But of course, if enough people decided to vote for a third-party candidate, that person could win. Such candidates do exist. But it’s perceived as wasting a vote when you choose someone like that. This fact was memorably highlighted on The Simpsons back in the fall of 1996, before the presidential election between Bill Clinton and Bob Dole. In the episode, the diabolical, scheming aliens Kang and Kodos (the green guys with the tentacles and giant heads who drool constantly) contrive to abduct the two major-party candidates and perform a “bio-duplication” procedure that allows Kang and Kodos to appear as Dole and Clinton, respectively. The disguised aliens hit the campaign trail and give speeches, making bizarre campaign promises. Kodos: “I am Clin-ton. As overlord, all will kneel trembling before me and obey my brutal command. End communication.” When Homer reveals the subterfuge to a horrified crowd, Kodos taunts the voters: “It’s true; we are aliens. But what are you going to do about it? It’s a two-party system. You have to vote for one of us.” When a guy in the crowd declares his intention to vote for a third-party candidate, Kang responds, “Go ahead, throw your vote away!” Then Kang and Kodos laugh maniacally. Later, as Marge and Homer—chained together and wearing neck-collars—are being whipped by an alien slave-driver, Marge complains and Homer quips, “Don’t blame me; I voted for Kodos.” Composition The fallacy of Composition rests on an illicit presumption about the relationship between a whole thing and the parts that make it up. This is an intuitive distinction, between whole and parts: for example, a person can be considered as a whole individual thing; it is made up of lots of parts— hands, feet, brain, lungs, etc., etc. We commit the fallacy of Composition when we mistakenly assume that any property that all of the parts share is also a property of the whole. Schematically, it looks like this: All of the parts of X have property P. Any property shared by all of the parts of a thing is also a property of the whole. Therefore, X has the property P. The second premise is the illicit presumption that makes this argument go through. It is illicit because it is simply false: sometimes all the parts of something have a property in common, but the whole does not have that property. Consider the 1980 U.S. Men’s Hockey Team. They won the gold medal at the Olympics that year, beating the unstoppable-seeming Russian team in the semifinals. (That game is often referred to as “The Miracle on Ice” after announcer Al Michaels’ memorable call as the seconds ticked off at the end: “Do you believe in miracles? Yes!”) Famously, the U.S. team that year was a rag-tag collection of no-name college guys; the average age on the team was 21, making them the youngest team ever to compete for the U.S. in the Olympics. The Russian team, on the other hand, was packed with seasoned hockey veterans with world-class talent. In this example, the team is the whole, and the individual players on the team are the parts. It’s safe to say that one of the properties that all of the parts shared was mediocrity—at least, by the standards of international competition at the time. They were all good hockey players, of course— Division I college athletes—but compared to the Hall of Famers the Russians had, they were mediocre at best. So, all of the parts have the property of being mediocre. But it would be a mistake to conclude that the whole made up of those parts—the 1980 U.S. Men’s Hockey Team—also had that property. The team was not mediocre; they defeated the Russians and won the gold medal! They were a classic example of the whole being greater than the sum of its parts. Division The fallacy of Division is the exact reverse of the fallacy of Composition. It’s an inference from the fact that a whole has some property to a conclusion that a part of that whole has the same property, based on the illicit presumption that wholes and parts must have the same properties. Schematically: X has the property P. Any property of a whole thing is shared by all of its parts. Therefore x, which is a part of X, has property P. The second premise is the illicit presumption. It is false, because sometimes parts of things don’t have the same properties as the whole. George Clooney is handsome; does it follow that his large intestine is also handsome? Of course not. Toy Story 3 is a funny movie. Remember when Mr. Potato Head had to use a tortilla for his body? Or when Buzz gets flipped into Spanish mode and does the flamenco dance with Jessie? Hilarious. But not all of the parts of the movie are funny. When it looks like all the toys are about to be incinerated at the dump? When Andy finally drives off to college? Not funny at all! I admit it: I teared up a bit; I’m not ashamed. V. Fallacies of Linguistic Emphasis Natural languages like English are unruly things. They’re full of ambiguity, shades of meaning, vague expressions; they grow and develop and change over time, often in unpredictable ways, at the capricious collective whim of the people using them. Languages are messy, complicated. This state of affairs can be taken advantage of by the clever debater, exploiting the vagaries of language to make convincing arguments that are nevertheless fallacious. This exploitation involves the manipulation of linguistic forms to emphasize facts, claims, emotions, etc. that favor one’s position, and to de-emphasize those that do not. We will survey four techniques that fall under this heading. Accent This is one of the original 13 fallacies that Aristotle recognized in his Sophistical Refutations. Our usage, however, will depart from Aristotle’s. He identifies a potential for ambiguity and misunderstanding that is peculiar to his language—ancient Greek. That language—in written form—used diacritical marks along with the alphabet, and transposition of these could lead to changes in meaning. English is not like this, but we can identify a fallacy that is roughly in line with the spirit of Aristotle’s accent: it is possible, in both written and spoken English (along with every other language), to convey different meanings by stressing individual words and phrases. The devious use of stress to emphasize contents that are helpful to one’s rhetorical goals, and to suppress or obscure those that are not—that is the fallacy of accent. There are a number of techniques one can use with the written word that fall in the category of accent. Perhaps the simplest way to emphasize favorable contents, and de-emphasize unfavorable ones, is to vary the size of one’s text. We see this in advertising all the time. You drive past a store that’s having a sale, which they advertise with a sign in the window. In the largest, most eye- catching font, you read, “70% OFF!” “Wow,” you might think, “that’s a really steep discount. I should go into the store and get a great deal.” At least, that’s what the store wants you to think. They’re emphasizing the fact of (at least one) steep discount. If you look more closely at the sign, however, you’ll see the things that they’re legally required to say, but that they’d like to de- emphasize. There’s a tiny ‘Up to’ in front of the gigantic ‘70% OFF!’. For all you know, there’s one crappy item that nobody wants, tucked in the back of the store, that’s discounted at 70%; everything else has much smaller discounts, or none at all. Also, if you squint really hard, you’ll see an asterisk after the ‘70% OFF!’, which leads to some text at the bottom of the poster, in the tiniest font possible, that reads, “While supplies last. See store details. Not available in all locations. Offer not valid weekends or holidays. All sales are final.” This is the proverbial “fine print”. It makes the sale look a lot less exciting. So they hide it. Footnotes are generally a good place to hide unfavorable content. We all know that CEOs of big companies—especially banks—get paid ridiculous sums of money. Some of it is just their salary and stock options; those amounts are huge enough to turn most people off. But there are other perks that are so over-the-top, companies and executives feel like it’s best to hide them from the public (and their shareholders) in the footnotes of CEO contracts and SEC reports. Michelle Leder runs a website called footnoted.com, which is dedicated to combing through these documents and exposing outrageous compensation packages. She’s uncovered executives spending over $700,000 to renovate their offices, demanding helicopters in addition to their corporate jets, receiving millions of dollars’ worth of private security services, etc., etc. These additional, extravagant forms of compensation seem excessive to most people, so companies do all they can to hide them from the public. Another abuse of footnotes can occur in academic or legal writing. Legal briefs and opinions and academic papers seek to persuade. If you’re writing such a document, and you relegate a strong objection to your conclusion to a brief mention in the footnotes Or worse, the endnotes: people have to flip all the way to the back to see those., you’re de-emphasizing that point of view and making it less likely that the reader will reject your arguments. That’s a fallacious suppression of opposing content, a sneaky trick to try to convince people you’re right without giving them a forthright presentation of the merits (and demerits) of your position. The fallacy of accent can occur in speech as well as writing. The audible correlate of “fine print” is that guy talking really fast at the end of the commercial, rattling off all the unpleasant side effects and legal disclaimers that, if given a full, deliberate presentation might make you less likely to buy the product they’re selling. The reason, by the way, that we know about such horrors as the possibility of driving while not awake (a side-effect of some sleep aids) and a four-hour erection (side-effect of erectile-dysfunction drugs), is that drug companies are required, by federal law, not to commit the fallacy of accent if they want to market drugs directly to consumers. They have to read what’s called a “major statement” that lists all of these side-effects explicitly, and no fair cramming them in at the end and talking over them really fast. When we speak, how we stress individual words and phrases can alter the meaning that we convey with our utterances. Consider the sentence ‘These pretzels are making me thirsty.’ Now consider various utterances of that sentence, each stressing a different word; different meanings will be conveyed: These pretzels are making me thirsty. [Not those over there, these right here.] These pretzels are making me thirsty. [It’s not the chips, it’s the pretzels.] These pretzels are making me thirsty. [Don’t try to tell me they’re not; they are.] And so on. We can capture the various stresses typographically by using italics (or boldface or all- caps), but if we leave that out, we lose some of the meaning conveyed by the actual, stressed utterance. One can commit the fallacy of accent by transcribing someone’s speech in a way that omits stress-indicators, and thereby obscures or alters the meaning that the person actually conveyed. Suppose a candidate for president says, “I hope this country never has to wage war with Iran.” The stress on ‘hope’ clearly conveys that the speaker doubts that his hopes will be realized; the candidate has expressed a suspicion that there may be war with Iran. This speech might set off a scandal: saying such a thing during an election could negatively affect the campaign, with the candidate being perceived as a war-monger; it could upset international relations. The campaign might try to limit the damage by writing an op-ed in a major newspaper, and transcribing the candidate’s utterance without any indication of stress: “The Senator said, ‘I hope this country never has to wage war with Iran.’ This is a sentiment shared by most voters, and even our opponent.” This transcription, of course, obscures the meaning of the original utterance. Without the stress, there is not additional implication that the candidate suspects that there will in fact be a war. Quoting out of Context Another way to obscure or alter the meaning of what someone actually said is to quote them selectively. Remarks taken out of their proper context might convey a different meaning than they did within that context. Consider a simple example: movie ads. These often feature quotes from film critics, which are intended to convey the impression that the movie was well-liked by them. “Critics call the film ‘unrelenting’, ‘amazing’, and ‘a one-of-a-kind movie experience’”, the ad might say. That sounds like pretty high praise. I think I’d like to see that movie. That is, until I read the actual review from which those quotes were pulled: I thought I’d seen it all at the movies, but even this jaded reviewer has to admit that this film is something new, a one-of-a-kind movie experience: two straight hours of unrelenting, snooze-inducing mediocrity. I find it amazing that not one single aspect of this movie achieves even the level of “eh, I guess that was OK.” The words ‘unrelenting’ and ‘amazing’—and the phrase ‘a one-of-a-kind movie experience’—do in fact appear in that review. But situated in their original context, they’re doing something completely different than the movie ad would like us to believe. Politicians often quote each other out of context to make their opponents look bad. In the 2012 presidential campaign, both sides did it rather memorably. The Romney campaign was trying to paint President Obama as anti-business. In a campaign speech, Obama once said the following: If you’ve been successful, you didn’t get there on your own. You didn’t get there on your own. I’m always struck by people who think, well, it must be because I was just so smart. There are a lot of smart people out there. It must be because I worked harder than everybody else. Let me tell you something: there are a whole bunch of hardworking people out there. If you’ve got a business, you didn’t build that. Somebody else made that happen. Yikes! What an insult to all the hard-working small-business owners out there. They didn’t build their own businesses? The Romney campaign made some effective ads, with these remarks playing in the background, and small-business people describing how they struggled to get their firms going. The problem is, that quote above leaves some bits out—specifically, a few sentences before the last two. Here’s the full transcript: If you’ve been successful, you didn’t get there on your own. You didn’t get there on your own. I’m always struck by people who think, well, it must be because I was just so smart. There are a lot of smart people out there. It must be because I worked harder than everybody else. Let me tell you something: there are a whole bunch of hardworking people out there. If you were successful, somebody along the line gave you some help. There was a great teacher somewhere in your life. Somebody helped to create this unbelievable American system that we have that allowed you to thrive. Somebody invested in roads and bridges. If you’ve got a business, you didn’t build that. Somebody else made that happen. Oh. He’s not telling business owners that they didn’t build their own businesses. The word ‘that’ in “you didn’t build that” doesn’t refer to the businesses; it refers to the roads and bridges—the “unbelievable American system” that makes it possible for businesses to thrive. He’s making a case for infrastructure and education investment; he’s not demonizing small-business owners. The Obama campaign pulled a similar trick on Romney. They were trying to portray Romney as an out-of-touch billionaire, someone who doesn’t know what it’s like to struggle, and someone who made his fortune by buying up companies and firing their employees. During one speech, Romney said: “I like being able to fire people who provide services to me.” Yikes! What a creep. This guy gets off on firing people? What, he just finds joy in making people suffer? Sounds like a moral monster. Until you see the whole speech: I want individuals to have their own insurance. That means the insurance company will have an incentive to keep you healthy. It also means if you don’t like what they do, you can fire them. I like being able to fire people who provide services to me. You know, if someone doesn’t give me the good service that I need, I want to say I’m going to go get someone else to provide that service to me. He’s making a case for a particular health insurance policy: self-ownership rather than employer- provided health insurance. The idea seems to be that under such a system, service will improve since people will be empowered to switch companies when they’re dissatisfied—kind of like with cell phones, for example. When he says he likes being able to fire people, he’s talking about being a savvy consumer. I guess he’s not a moral monster after all. Equivocation Typical of natural languages is the phenomenon of homonymy Greek word, meaning ‘same name’.: when words have the same spelling and pronunciation, but different meanings—like ‘bat’ (referring to the nocturnal flying mammal) and ‘bat’ (referring to the thing you hit a baseball with). This kind of natural-language messiness allows for potential fallacious exploitation: a sneaky debater can manipulate the subtleties of meaning to convince people of things that aren’t true—or at least not justified based on what they say. We call this kind of maneuver the fallacy of equivocation. Here’s an example: Consider a banker; let’s call him Fred. Fred is the president of a bank, a real big-shot. He’s married, but he’s not faithful: he’s carrying on an affair with one of the tellers at his bank, Linda. Fred and Linda have a favorite activity: they take long lunches away from their workplace, having romantic picnics at a beautiful spot they found a short walk away. They lay out their blanket underneath an old, magnificent oak tree, which is situated right next to a river, and enjoy champagne and strawberries while canoodling and watching the boats float by. One day—let’s say it’s the anniversary of when they started their affair—Fred and Linda decide to celebrate by skipping out of work entirely, spending the whole day at their favorite picnic spot. (Remember, Fred’s the boss, so he can get away with this.) When Fred arrives home that night, his wife is waiting for him. She suspects that something is up: “What are you hiding, Fred? Are you having an affair? I called your office twice, and your secretary said you were ‘unavailable’ both times. Tell me this: Did you even go to work today?” Fred replies, “Scout’s honor, dear. I swear I spent all day at the bank today.” See what he did there? ‘Bank’ can refer either to a financial institution or the side of a river—a riverbank. Fred and Linda’s favorite picnic spot is on a riverbank, and Fred did indeed spend the whole day at that bank. He’s trying to convince his wife he hasn’t been cheating on her, and he exploits this little quirk of language to do so. That’s equivocation. A similar linguistic phenomenon can also be exploited to equivocate: polysemy. Greek word, meaning ‘many signs (or meanings)’. This is distinct from, but similar to, homonymy. The meanings of homonyms are typically unrelated. In polysemy, the same word or phrase has multiple, related meanings—different senses. Consider the word ‘law.’ The meaning that comes immediately to mind is the statutory one: “A rule of conduct imposed by authority.” From the Oxford English Dictionary. The state law prohibiting murder is an instance of a law in this sense. There is another sense of ‘law’, however; this is the sense operative when we speak of scientific laws. These are regularities in nature—Newton’s law of universal gravitation, for example. These meanings are similar, but distinct: statutes, human laws, are prescriptive; scientific laws are descriptive. Human laws tell us how we ought to behave; scientific laws describe how things actually do, and must, behave. Human laws can be violated: I could murder someone. Scientific laws cannot be violated: if two bodies have mass, they will be attracted to one another by a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them; there’s no getting around it. A common argument for the existence of God relies on equivocation between these two senses of ‘law’: There are laws of nature. By definition, laws are rules imposed by an Authority. So the laws of nature were imposed by an Authority. The only Authority who could impose such laws is an all-powerful Creator—God. Therefore, God exists. This argument relies on fallaciously equivocating between the two senses of ‘law’—human and natural. It’s true that human laws are by definition imposed by an authority; but that is not true of natural laws. Additional argument is needed to establish that those must be so imposed. A famous instance of equivocation of this sort occurred in 1998, when President Bill Clinton denied having an affair with White House intern Monica Lewinsky by declaring forcefully in a press conference: “I did not have sexual relations with that woman—Ms. Lewinsky.” The president wanted to convince his audience that nothing sexually inappropriate had happened, even though, as was revealed later, lots of sex stuff had been going on. He does this by taking advantage of the polysemy of the phrase ‘sexual relations.’ In the broadest sense, the phrase connotes sexual activity of any kind—including oral sex (which Bill and Monica engaged in). This is the sense the president wants his audience to have in mind, so that they’re convinced by his denial that nothing untoward happened. But a more restrictive sense of ‘sexual relations’—a bit more old-fashioned usage—refers specifically to intercourse (which Bill and Monica did not engage in). It’s this sense that the president can fall back on if anyone accuses him of having lied; he can claim that, strictly speaking, he was telling the truth: he and Monica didn’t have ‘relations’ in the intercourse sense. Clinton later admitted to “misleading” the American people—but, importantly, not to lying. The distinction between lying and misleading is a hard one to draw precisely, but roughly speaking it’s the difference between trying to get someone to believe something false by saying something false (lying) and trying to get them to believe something false by saying something true but deceptive (misleading). Besides homonymy and polysemy, yet another common linguistic phenomenon can be exploited to this end. This phenomenon is implicature, identified and named by the philosopher Paul Grice in the 1960s. See his Studies in the Way of Words, 1989, Cambridge: Harvard University Press. Implicatures are contents that we communicate over and above the literal meaning of what we say—aspects of what we mean by our utterances that aren’t stated explicitly. People listening to us infer these additional meanings based on the assumption that the speaker is being cooperative, observing some unwritten rules of conversational practice. To use one of Grice’s examples, suppose your car has run out of gas on the side of the road, and you stop me as I walk by, explaining your plight, and I say, “There’s a gas station right around the corner.” Part of what I communicate by my utterance is that the station is open and selling gas right now—that you can go there and solve your problem. You can infer this content based on the assumption that I’m being a cooperative conversational partner; if the station is closed or out of gas—and I knew it—then I would be acting unhelpfully, uncooperatively. Notice, though, that this content is not part of what I literally said: all I told you is that there is a gas station around the corner, which would still be true even if it were closed and/or out of gas. Implicatures are yet another subtle aspect of meaning in natural language that can be exploited. So a final technique that we might classify under the fallacy of equivocation is false implication— saying things that are strictly speaking true, but which communicate false implicatures. Grocery stores do this all the time. You know those signs posted under, say, cans of soup that say “10 for $10”? That’s the store’s way of telling us that soup’s on sale for a buck a can; that’s right, you don’t need to buy 10 cans to get the deal; if you buy one can, it’s $1; 2 cans are $2, and so on. So why not post a sign saying “$1 per can”? Because the 10-for-$10 sign conveys the false implicature that you need to buy 10 cans in order to get the sale price. The store’s trying to drive up sales. A striking example of false implicature is featured in one of the most prominent U.S. Supreme Court rulings on perjury law. In the original criminal case, a defendant by the name of Bronston had the following exchange with the prosecuting attorney: Q. Do you have any bank accounts in Swiss Banks, Mr. Bronston? A. No, sir. Q. Have you ever? A. The company had an account there for about six months, in Zurich. Bronston v. United States, 409 US 352 - Supreme Court 1973 As it turns out, Bronston did not have any Swiss bank accounts at the time of the questioning, so his first answer was strictly true. But he did have Swiss bank accounts in the past. However, his second answer does not deny this. All he says is that his company had Swiss bank accounts—an answer that implicates that he himself did not. Based on this exchange, Bronston was convicted of perjury, but the Supreme Court overturned that conviction, pointing out that Bronston had not made any false statements (a requirement of the perjury statute); the falsehood he conveyed was an implicature. The court didn’t use the term ‘implicature’ in its ruling, but this was the thrust of their argument. Manipulative Framing Words are powerful. They can trigger emotional responses and activate associations with related ideas, altering the way we perceive the world and conceptualize issues. The language we use to describe a particular policy, for example, can affect how favorably our listeners are likely to view that proposal. How we frame issues with language can profoundly influence how persuasive our arguments about those issues will be. The technique of choosing words to frame issues intentionally to manipulate your audience is what we will call the fallacy of manipulative framing. The importance of framing in politics has long been recognized, but only in recent decades has it been raised to an art form. One prominent practitioner of the art is Republican consultant Frank Luntz. In a 200-plus page memo he sent to Congressional Republicans in 1997, and later in a book, Frank Luntz, 2007, Words That Work: It’s Not What You Say, It’s What People Hear. New York: Hyperion. Luntz stressed the importance of choosing persuasive language to frame issues so that voters would be more likely to support Republican positions on issues. One of his recommendations illustrates manipulative framing nicely. In the United States, if you leave a fortune to your heirs after you die, then the government taxes it (provided it’s greater than about $5.5 million, or $11 million for a couple, as of 2016). The usual name for this tax is the ‘estate tax’. Luntz encouraged Republicans—who are generally opposed to this tax—to start referring to it instead as the “death tax”. This framing is likelier to cause voters to oppose the tax as well: taxing people for dying? Talk about kicking a man when he’s down! (Polling bears this out: people oppose the tax in higher numbers when it’s called the ‘death tax’ than when it’s called the ‘estate tax’). The linguist George Lakoff has written extensively on the subject of framing. See, e.g., his 2004 book, Don’t Think of an Elephant!, White River Junction, Vermont: Chelsea Green Publishing. His remarks on the subject of “tax relief” nicely illustrate how framing works: On the day that George W. Bush took office, the words tax relief started appearing in White House communiqués to the press and in official speeches and reports by conservatives. Let us look in detail at the framing evoked by this term. The word relief evokes a frame in which there is a blameless Afflicted Person who we identify with and who has some Affliction, some pain or harm that is imposed by some external Cause-of-pain. Relief is the taking away of the pain or harm, and it is brought about by some Reliever-of-pain. The Relief frame is an instance of a more general Rescue scenario, in which there a Hero (The Reliever-of-pain), a Victim (the Afflicted), a Crime (the Affliction), A Villain (the Cause-of-affliction), and a Rescue (the Pain Relief). The Hero is inherently good, the Villain is evil, and the Victim after the Rescue owes gratitude to the Hero. The term tax relief evokes all of this and more. Taxes, in this phrase, are the Affliction (the Crime), proponents of taxes are the Causes-of Affliction (the Villains), the taxpayer is the Afflicted Victim, and the proponents of “tax relief” are the Heroes who deserve the taxpayers’ gratitude. Every time the phrase tax relief is used and heard or read by millions of people, the more this view of taxation as an affliction and conservatives as heroes gets reinforced. George Lakoff, 2/14/2006, “Simple Framing,” Rockridge Institute. Carefully chosen words can trigger all sorts of mental associations, mostly at the subconscious level, that affect how people perceive the issues and have the power to change opinions. That’s why manipulative framing is ubiquitous in public discourse. Consider debates about illegal immigration. Those who are generally opposed to policies that favor such people will often refer to them as “illegal immigrants”. This framing emphasizes the fact that they are in this country illegally, making it likelier that the listener will also oppose policies that favor them. A further modification can further increase this likelihood: “illegal aliens.” The word ‘alien’ has a subtle dehumanizing effect; if we don’t think of them as individual people with hopes and dreams, we’re not likely to care much about them. Even more dehumanizing is a framing one often sees these days: referring to illegal immigrants simply as “illegals”. They are the living embodiment of illegality! Those who advocate on behalf of such people, of course, use different terminology to refer to them: “undocumented workers”, for example. This framing de-emphasizes the fact that they’re here illegally; they’re merely “undocumented”. They lack certain pieces of paper; what’s the big deal? It also emphasizes the fact that they are working, which is likely to cause listeners to think of them more favorably. The use of manipulative framing in the political sphere extends to the very names that politicians give the laws they pass. Consider the healthcare reform act passed in 2010. Its official name is The Patient Protection and Affordable Care Act. Protection of patients, affordability, care—these all trigger positive associations. The idea is that every time someone talks about the law prior to and after its passage, they will use the name with this positive framing and people will be more likely to support it. As you may know, this law is commonly referred to with a different moniker: ‘Obamacare’. This is the framing of choice for the law’s opponents: any negative associations people have with President Obama are attached to the law; and any negative feelings they have about healthcare reform get attached to Obama. Late night talk show host Jimmy Kimmel demonstrated the effectiveness of framing on his show one night in 2013. He sent a crew outside his studio to interview people on the street and ask them which approach to health reform they preferred, the Affordable Care Act or Obamacare. Overwhelmingly, people expressed a preference for the Affordable Care Act over Obamacare, even though those are just two different ways of referring to the same piece of legislation. Framing is especially important when the public is ignorant of the actual content of policy proposals, which is all too often the case.
Chapter 5: Propositional Logic Translations This chapter is based on Fundamental Methods of Logic, by Matthew Knachel. I. Introduction You have seen the basics of how arguments work and the fundamental building blocks of our logical thinking. The focus of this chapter is putting these building blocks together in a more formal fashion, and then using the language and operations of logic to create and analyze deductive arguments. The purpose of deductive logic is to figure out what new information we can obtain from information we already have, and propositional logic is a useful way of doing this. Why Propositional Logic? The fundamental logical unit of the system you’ll be learning in this chapter is the proposition; therefore, we call it “propositional logic.” (Some refer to the same chunks of information as sentences, and call this system “sentential logic” instead I’m using the term “proposition” instead of “sentence” here because a single grammatical sentence can contain multiple propositions, and the same proposition can be communicated by a variety of different sentences. When I do use the term “sentence,” I’m usually using it as a synonym for “proposition,” rather than referring to the grammatical unit that spans from the capital letter up front to the period which ends it.). A proposition is the meaning behind the words; the same proposition can be expressed different ways. For example, “My brother is tall,” “My brother Wade is tall,” and “Wade is tall,” are different sentences, but express the same proposition. In this system of logic, we’ll break complex sentences down into simple propositions and logical terms. This system has its roots in Ancient Greece, but it wasn’t until the 19th century that it was developed into a formal system. It is not the only system of symbolic logic that was developed in the 19th and 20th centuries, but it is the most direct and fundamental. Most systems of symbolic logic begin with propositional logic and develop it into other forms to suit other purposes, depending on the level of logical complexity we wish to analyze. In this chapter, we will discuss the basics of the proposition-centered approach to deductive logic. Propositional logic must accomplish three tasks: 1. Tame natural language. 2. Precisely define logical forms. 3. Develop a way to analyze logical forms for validity. The approach to the first task—taming natural language—will be accomplished by essentially translating natural language (in our case, English) into this symbolic system. A lot of the nuances of natural language are lost, but the logic itself becomes very clear, which is our main goal. We will call our artificial language “PL,” short for ‘Propositional Logic.’ In constructing a language, we must specify its syntax and its semantics. By the syntax of a language, we mean the rules governing what counts as a well-formed construction within that language; that is, syntax is the language’s grammar. Syntax is what tells me that ‘What a handsome poodle you have there.’ is a well-formed English construction, while ‘Poodle a handsome there you what have.’ is not. So, the syntax of PL will tell you in what order we can put the letters, logical symbols, and parentheses so the resulting formula is well-formed, instead of logical nonsense. While learning the symbols and their syntax, we will also learn some rules for how to translate from English into PL. The semantics of a language is an account of the meanings of its well-formed bits. In PL, this is different from the correspondence between the symbols and English. Our semantics section will tell you under what conditions a given proposition is true or false. II. Syntax and Translation First, we cover syntax. This discussion will give us some clues as to the relationship between PL and English, and it will teach us how to translate English sentences into PL. We can distinguish, in English, between two types of (declarative) propositions: simple and compound. A simple proposition is one that does not contain any other sentence as a component part. A compound proposition is one that contains at least one other sentence as a component part. Compound propositions take simple propositions and add logic, often (but not always) combining more than one simple proposition together. ‘Beyoncé is logical’ is a simple proposition; none of its parts is itself a sentence. ‘Beyoncé is logical and James Brown is alive’ is a compound proposition: it contains two simple sentences as component parts—namely, ‘Beyoncé is logical’ and ‘James Brown is alive’—and it combines them using the logical word ‘and.’ Simple propositions, compound propositions In PL, we will use capital letters—‘A’, ‘B’, ‘C’, …, ‘Z’—to stand for simple sentences. You can use any capital letters you want for your simple sentences, as long as you remember two rules: in any particular compound sentence, paragraph, or argument, (1) you cannot use the same letter to refer to different propositions, and (2) you have to make sure you use the same letter for a proposition every time it appears. If I chose the letter ‘B’ to refer to ‘Beyoncé is logical,” and later chose ‘B’ to refer to ‘James Brown is alive,’ that would be confusing; that’s the importance of rule (1). For rule (2), you need to make sure the same proposition gets the same letter, even if we use different words to express it. If I give a ‘J’ to ‘James Brown is alive,’ and later on in the same argument I state ‘he is alive,’ if it is clear the pronoun refers to James Brown, ‘he is alive’ will also get a ‘J.’ When we replace the simple propositions with letters, what ends up being left are logical terms. ‘Beyoncé is logical and James Brown is alive,’ using the letters we chose above, becomes ‘B and J.’ We are very soon going to replace the logical words with symbols, but I suggest you begin by simply replacing the simple propositions with letters, and retaining everything else – including all logical words and, for now, punctuation. Here’s a more complex example: Neither my brother nor sister are tall, unless both my mom and dad are tall. This one has four different simple sentences: ‘my brother is tall,’ ‘my sister is tall,’ ‘my mom is tall,’ and ‘my dad is tall.’ Notice that I only said the word ‘tall’ twice in the original sentence, but I meant it four times—it is meant to apply to both of my siblings, and to both of my parents. We just don’t like repeating ourselves unnecessarily, so in natural language, we shorten things where we can. Let’s assign letters to these four simple sentences: B = My brother is tall. S = My sister is tall. M = My mom is tall. D = My dad is tall. Now, replacing these chunks of information with the letters, and keeping the logical words and punctuation, we end up with: Neither B nor S, unless both M and D. The next step in translation will be to replace the logical words with symbols. We’ll also use parentheses when we need to group things. You’ll notice in the example above that the comma separates ‘neither B nor S’ from ‘both M and D.’ We’re going to drop the comma, so we’ll use parentheses to let us know that the word ‘unless’ is the logical concept that brings the whole sentence together—it’s the main connective. We will have five total logical symbols, each representing a different concept: conjunctions, disjunctions, negations, conditionals, and biconditionals. 1. Conjunctions The first type of compound proposition is one that we’ve already seen. Conjunctions are, roughly, ‘and’ sentences. A conjunction takes two propositions and tells us both of them are true. For the conjunction “Beyoncé is logical and James Brown is alive,” we’ve already decided to let ‘B’ stand for “Beyoncé is logical” and to let ‘J’ stand for “James Brown is alive.” What we need is a symbol that stands for ‘and’. In PL, that symbol is a “dot.”. It looks like this: •. To form a conjunction in PL, we stick the dot between the two component letters: That is the PL version of “Beyoncé is logical and James Brown is alive.” A note on terminology. The dot goes between the two things it connects, so a conjunction has two components, one on either side of the dot. We will refer to these as the “conjuncts” of the conjunction. If we need to be specific, we might refer to the “first conjunct” (‘B’ in this case) or the “second conjunct” (‘J’ in this case). There are a lot of words in the English language that mean, logically, “and.” Here are a few of them: and both … and also moreover however but although furthermore as well as You’ll notice these words and phrases have slightly different nuances. If I were to tell you “My brother and sister are tall,” and I wanted to use a different word for ‘and,’ only some of these would work. I would not say, for example, “My brother is tall but my sister is tall.” We use ‘but,’ ‘however,’ and ‘although’ when the second sentence is significantly different from the first. I would use it here: “My brother is tall but my sister is short.” Logically, though, all of these conjunctions are simply telling us “both of these things are true.” 2. Disjunctions Disjunctions are, roughly, ‘or’ sentences. A disjunction takes two propositions and tells us that at least one of them is true. For example, “Beyoncé is logical or James Brown is alive.” Sometimes, the ‘or’ is accompanied by the word ‘either,’ as in “Either Beyoncé is logical or James Brown is alive.” Again, we let ‘B’ stand for “Beyoncé is logical” and let ‘J’ stand for “James Brown is alive.” What we need is a symbol that stands for ‘or’ (or ‘either/or’). In PL, that symbol is a “wedge.” It looks like this: v. To form a disjunction in PL, we simply stick the wedge between the two component letters, thus: That is the PL version of “Beyoncé is logical or James Brown is alive.” It is also the way we symbolize “Either Beyoncé is logical or James Brown is alive.” We don’t need a separate symbol for the ‘either’ here—though it is a useful word, as we can use it to figure out how a sentence is grouped, if we need to use parentheses in our translation. There are many, many words that are conjunctions, but only a few that are true disjunctions. or either … or unless The last one is a weird one; you might not expect it to be equivalent to an ‘or’ statement, but it turns out that it is. For those more interested in why this is the case, here you go! Most of us want to view ‘unless’ as a conditional—an if … then statement. Conditionals get their own symbol as we’ll see in a minute. It’s really a conditional and a negation. If I want to say “Amir will go to the movies, unless Jodie goes,” what this means is “If Jodie does not go to the movies, then Amir will.” It’s easy to remember that ‘unless’ is an if … then statement, and you add a ‘not’ to one side, but it’s hard to remember which side to add the ‘not’ to, and which side is the ‘if’ part of your conditional. It turns out that it’s logically equivalent to “Amir will go to the movies, or Jodie will go,” so we’ll translate ‘unless’ using a wedge. A note on terminology. A disjunction has two components, one on either side of the wedge. We will refer to these as the “disjuncts” of the disjunction. If we need to be specific, we might refer to the “first disjunct” (‘B’ in this case) or the “second disjunct” (‘J’ in this case). 3. Negations Negations are, roughly, ‘not’ sentences—sentences like “James Brown is not alive.” A negation takes one proposition and tells us that it’s false. You may find it surprising that this would be considered a compound sentence. Remember that a simple proposition is one that does not contain any other propositions within it. “James Brown is not alive” does contain another proposition inside it: “James Brown is alive.” Another way to think of it is that “James Brown is not alive” is a compound proposition because it takes a simple proposition and combines it with a logical concept. Any time we add logic, we’re forming compound proposition. We have ‘J’ to stand for the simple proposition; we need a symbol for ‘it is not the case that.’ In PL, that symbol is a “tilde.” It looks like this: ~. To form a negation in PL, we simply prefix a tilde to the simpler component being negated: This is the PL version of “James Brown is not alive.” There are many, many words and phrases that are negations. Here are a few: not it is false that it is not the case that never nowhere nothing no one neither … nor n’t That last one, ‘n’t,’ usually sticks onto another word: isn’t, wasn’t, won’t. Actually, the last two rows have negations that are stuck onto another word. ‘Isn’t’ is short for ‘is not.’ ‘Never’ is short for ‘not ever.’ ‘Nothing’ is short for ‘not anything’ or ‘not something.’ Importantly for us, ‘neither … nor’ is short for ‘not either … or.’ That’s both a disjunction and a negation. Watch out for negations; they hide. Sometimes, we don’t even express the negation: “James Brown is not alive” means the same thing as “James Brown is dead.” If you’ve already given a letter to “James Brown is alive,” then both of these should be translated the same way – with a tilde. 4. Conditionals Conditionals are, roughly, ‘if … then’ sentences—sentences like “If Beyoncé is logical, then James Brown is alive.” Again, we let ‘B’ stand for “Beyoncé is logical” and let ‘J’ stand for “James Brown is alive.” What we need is a symbol that stands for the ‘if/then’ part. In PL, that symbol is a “horseshoe.” It looks like this: ⊃. To form a conditional in PL, we simply stick the horseshoe between the two component letters (where the word ‘then’ occurs), thus: That is the PL version of “If Beyoncé is logical, then James Brown is alive.” Conditionals are tricky, for two reasons. First, there are again a lot of words and phrases that express conditionals. Second, order matters, and humans do not always present their conditionals in if … then order. Imagine I give my child a promise: if she eats her veggies, then she’ll eat dessert. It means something different if I switch the order and say “If you eat dessert, then you’ll eat your veggies.” The dessert is supposed to be her reward, the consequence of her eating her veggies as asked. But I might switch the order I express them in, and say something like “You can have dessert, provided that you eat your veggies,” dangling the reward first, and then telling her the condition she must meet to earn it. Since the horseshoe always means ‘if … then’ in that order, I need to switch the two sides here so the veggies come before the horseshoe, and dessert comes last. Here are some words and phrases that should be horseshoes, and you get to keep them in the same order: If A then B If A, B A is a sufficient condition for B A implies B A only if B All of these should be translated as: And here are some words and phrases that should be horseshoes, but you need to switch the order: A, if B A provided that B A on the condition that B A given that B A is a necessary condition for B All of these should be translated as: A note on terminology. Unlike our treatment of conjunctions and disjunctions, we will distinguish between the two components of the conditional, since order matters. The component that comes before the horseshoe will be called the “antecedent” of the conditional; the component after the horseshoe is its “consequent.” 5. Biconditionals Biconditionals are, roughly, ‘if and only if’ sentences—sentences like “Beyoncé is logical if and only if James Brown is alive.” Again, we let ‘B’ stand for “Beyoncé is logical” and let ‘J’ stand for “James Brown is alive.” What we need is a symbol that stands for the ‘if and only if’ part. In PL, that symbol is a “triple-bar.” It looks like this: ≡. To form a biconditional in PL, we simply stick the triple-bar between the two component letters, thus: That is the PL version of “Beyoncé is logical if and only if James Brown is alive.” There are very few phrases that translate as biconditionals. We’ve got three: if and only if necessary and sufficient condition is logically equivalent to ‘If’ by itself is a conditional and gets a horseshoe; ‘only if’ by itself is a conditional. ‘If and only if’ is a biconditional. A biconditional has two conditionals, like a bicycle has two wheels. Similarly, ‘necessary’ by itself is a conditional; ‘sufficient’ by itself is a conditional. ‘Necessary and sufficient condition’ is a biconditional. There are no special names for the components of the biconditional, and order does not matter. Punctuation: Parentheses and Brackets Our language, PL, is quite austere: so far, we have only 31 different symbols—the 26 capital letters, and the five symbols for the five different types of compound sentence. We will now add parentheses and brackets. And that’ll be it. We use parentheses in PL for one reason (and one reason only): to remove ambiguity. To see how this works, it will be helpful to draw an analogy between PL and the language of simple arithmetic. The latter has a limited number of symbols as well: numbers, signs for the arithmetical operations (addition, subtraction, multiplication, division), and parentheses. The parentheses are used in arithmetic for disambiguation. Consider this combination of symbols: As it stands, this formula is ambiguous. If I’ve forgotten my order of operations, I don’t know whether this is a sum or a product; that is, I don’t know which operator—the addition sign or the multiplication sign—is the main operator. We can use parentheses to disambiguate, and we can do so in two different ways: or And of course, where we put the parentheses makes a big difference. The first formula is a product; the multiplication sign is the main operator. It comes out to 25. The second formula is a sum; the addition sign is the main operator. It comes out to 17. Different placement of parentheses, different results. This same sort of thing is going to arise in PL. Our logical operators are the dot, wedge, tilde, horseshoe, and triple-bar, and we need to be able to tell what’s the main operator. There are ways of combining PL symbols into compound formulas with more than one operator; and just as is the case in arithmetic, without parentheses, these formulas would be ambiguous. Let’s look at an example. Consider this sentence: “If Beyoncé is logical and James Brown is alive, then I’m the Queen of England.” This is a compound proposition, but it contains both the word ‘and’ and the ‘if … then’ construction. It has three simple components: the two that we’re used to by now about Beyoncé and James Brown, which we’ve been symbolizing with ‘B’ and ‘J,’ respectively, and a new one—“I’m the Queen of England”—which we may as well symbolize with a ‘Q.’ Based on what we already know about how PL symbols work, we would render the sentence like this: But just as was the case with the arithmetical example above, this formula is ambiguous. I don’t know what kind of compound proposition this is, a conjunction or a conditional. That is, I don’t know which of the two operators—the dot or the horseshoe—is the main operator. In order to disambiguate, we need to add some parentheses. There are two ways this can go, and we need to decide which of the two options correctly captures the meaning of the original sentence: or The first formula is a conditional; horseshoe is its main operator, and its antecedent is a compound sentence. The second formula is a conjunction; dot is its main operator, and its second conjunct is a compound sentence. We need to decide which of these two formulations correctly captures the meaning of the English sentence “If Beyoncé is logical and James Brown is alive, then I’m the Queen of England.” We have two clues that the first translation, that groups B and J together, is the correct one. First, we’ve expressed our conditional with two words: if … then. Everything between the ‘if’ and the ‘then’ is the antecedent. B and J come between ‘if’ and ‘then,’ so they’re grouped together as the antecedent. The second clue is that comma—remember I told you to pay attention to punctuation? A comma often tells you where the sentence splits. In this case, it separates B and J from the Q. If a comma is the strongest form of punctuation inside a sentence, you will notice that often, the logical word or phrase right after the comma is the main operator. So, the correct translation is: Again, in PL, parentheses have one purpose: to remove ambiguity. We only use them for that. This kind of ambiguity arises in formulas, like the one just discussed, involving multiple instances of the operators dot, wedge, horseshoe, and triple-bar. Let’s translate a more complex sentence—the one I gave you above, before we started learning the logical symbols. Remember it was: Neither my brother nor sister are tall, unless both my mom and dad are tall. See that comma in the middle? That splits our sentence into “Neither my brother nor sister are tall” on the one side and “Both my mom and dad are tall” on the other. ‘Unless’ is the word that will end up being the main connective. Let’s take one side at a time. Remember that ‘neither … nor’ is short for ‘not either … or’—it’s both a negation and a disjunction. More specifically, it’s the negation OF a disjunction. So, I want to use both a tilde and a wedge. Here’s my first (incorrect) try: What’s wrong with this? Tildes are sticky—they stick to whatever comes directly after them and negate only that. What I’ve ended up saying here is “Either my brother is not tall, or my sister is tall.” That’s not what the original said. Neither B nor S means the entire disjunction, B or S, is false. So, I need to collect that disjunction in parentheses, and then stick the tilde on the front of that, like this: Now let’s tackle the second side: “Both my mom and dad are tall.” That’s easy; that’s just a dot put between M and D. Now, if that was a stand-alone sentence, that’s all we have to do, but I want to attach this compound sentence to another sentence, so I need to group it in parentheses: Last thing! I need to connect these two sides together. The word used here is ‘unless.’ Remember, ‘unless’ should be translated like a disjunction. The symbol for disjunction is the wedge, and our final product is: Before moving on, I want to say a quick word about “neither … nor.” You’ll notice we put the tilde on front of parentheses, with the wedge in the middle—it is not the case that (B or S). We know we can’t just stick the tilde to the B, that says something different. But can’t we just put a tilde in front of both B and S, ending up with: NO YOU CAN NOT. (Sorry for the all-caps, I really want to emphasize that). This says “Either my brother is not tall, or my sister is not tall.” That means something different. Think about it. If I tell you “Neither my brother nor sister are tall,” how many tall siblings do I have? Zero. If I tell you “Either my brother is not tall, or my sister is not tall,” how many tall siblings do I have? Well, I don’t know. My brother could be the short one, my sister could be the short one, or both of them could be short. I can have one tall sibling and make that ‘either … or’ sentence true—it just tells me someone is not tall, but doesn’t tell me who. The ‘neither … nor’ sentence is more specific. I know exactly who is tall: no one. In addition to parentheses, you’ll also see (and use) brackets. They mean exactly the same thing as parentheses do—they just group things together so you can find the main operative of the sentence, and the main operative of each part of the sentence. We use them when we need to group something that already has used parentheses. For example, consider this: No, I don’t know what it means in English, but I do know it’s a conjunction—the main operator is the dot, and the dot connects two compound statements to each other. Suppose, though, that this whole conjunction is false? I need to add a tilde, but if I just put it on the front without further grouping, it sticks to that first parenthesis, and negates just the first half of the sentence, instead of the whole thing. I need to group the entire sentence together and put a tilde on the front of that. Since this sentence already has parentheses, I’m going to step it up to square brackets, just so it’s easier to see how things are grouped: Translation Tips Here’s some advice for how to approach translating sentences from English into PL. Assigning letters to propositions You need to start with this step, assigning letters to propositions. When you’re translating more complicated sentences into PL, take the time to rewrite the sentence using the letters instead of the propositions they represent. Make sure you keep all logical words and punctuation. So, if you see a sentence like this: The Mandalorian will take his helmet off in the Grogu movie only if Pedro Pascal finishes filming Fantastic Four in time; but him taking his helmet off is a necessary condition for getting a lot of Pascal’s fans to see it in the theater, and a lot of his fans seeing it in the theater is sufficient for the movie to become a huge success. Take the time to rewrite it like this: M only if P; but M is a necessary condition for F, and F is sufficient for S. Now that you’ve gotten rid of the content and retained the logic, you can stop thinking about Pedro Pascal, and you can focus on picking the right connectives and figuring out where the parentheses are going to go. Next tip: make sure you give capital letters to full propositions. “Neil Armstrong walked on the moon” should be given a single letter representing the whole sentence. A lot of students have the strong impulse to give letters to all noun terms and have something like “NA walked on M.” The whole thing just gets a single letter, so this is just translated as “N.” Only give capital letters to simple propositions. Look especially for negations, and extract them as a logical symbol. If I see “My brother is not tall,” I want to give the letter to the positive simple proposition “My brother is tall” and add the negation back later as a logical symbol. Don’t use the same letters for different simple propositions. We can get in the habit of picking a letter related to any names we see in the sentence, like I did above with Neil Armstrong. But if you get a sentences like this, “Jodie went to the movies but Johnny did not,” it’s tempting to give both propositions a ‘J.’ If we use ‘J’ for Jodie, we need to pick a different letter for Johnny’s movie-going habits. Make sure you use the same letter for the same proposition. Look for the meaning of the sentence; if it means the same as a previous sentence, even though it’s worded differently, it needs to get the same letter. In the Mando example above, I gave the M to both “The Mandalorian will take his helmet off in the Grogu movie,” and to “him taking his helmet off.” We understand from context that the second string of words means the same thing as the first, so they get the same letter. Picking the right connective Keep in mind what each symbol means. The words representing each connective I’ve listed above are only a partial list, especially for negations, conjunctions, and conditionals. So, ask yourself: Are they trying to tell you that both things are true? That’s a conjunction and gets a dot. Are they saying at least one of these things are true? That’s a disjunction and gets the wedge. Are they saying something is false? That’s a negation and gets a tilde. Are they pointing out a connection between two things, such as a promise, or a causal connection? That’s a conditional and gets a horseshoe UNLESS… Is it a very strong connection? That’s probably a biconditional, and gets a triple-bar. Some advice on conditionals Every time you see a conditional, STOP and ask yourself if you can keep the two terms in the same order or if you need to switch them. Re-read the section above on conditionals several times until you get a good feel for it. The Mandalorian example above has three conditionals: M only if P; but M is a necessary condition for F, and F is sufficient for S. Remember, “only if” and “sufficient for” do not switch order. But if you have “necessary condition for,” then you have to switch the order. So the three conditionals will be translated as: Grouping Remember, we need to use parentheses so we can find the main operator of each sentence, and the main operator of each part of the sentence. A good rule of thumb is that if you are doing logic on a compound statement, you’ll need to group that statement into parentheses or brackets first. So, negating a compound statement means you put the statement in parentheses, and then put the tilde on front. Hooking two compound statements together means each of them needs to be grouped before you connect them. Figuring out how to group things is more of an art form than a science; you really have to read the sentences carefully to figure it out. Look for clues in the language and punctuation. Commas and semi-colons give you clues of how to group things. How we express ourselves also gives clues to how we think things should be grouped. For example, look at these two sentences: My mom and dad will go to dinner unless my uncle does. My mom is going to dinner, and my dad will go unless my uncle does. Without parentheses, both of these look the same. Giving ‘M’ to the proposition “My mom is going to dinner,” ‘D’ to “my dad is going to dinner,” and ‘U’ to “my uncle is going to dinner,” they both look like this, pre-grouping: Yet they mean different things. #1 means if my uncle does not go, then both my mom dad will go—their dinner plans depend on what my uncle does. #2 means my mom is going for sure, but my dad’s going depends on what my uncle does. #2 has that nice comma telling us where the sentence breaks. For #1, we focus on how we’ve grouped mom and dad together linguistically. Instead of “My mom is going to dinner and my dad is going to dinner,” we’ve got “my mom and dad are going to dinner.” These two simple propositions are grouped so closely we’ve smushed their sentences together. So, the translations for these two should be: And, just because I can’t leave an example unfinished, let’s finish translating the Mandalorian sentence. Here’s a reminder of the sentence we’re working with: M only if P; but M is a necessary condition for F, and F is sufficient for S. We figured out that the first and third conditionals need to keep the letters in this same order, while the middle conditional switches the order. We know how things are grouped because of the comma and semi-colon. The semi-colon is stronger than the comma, so that tells me the whole sentence breaks there, and the last two conditionals are grouped together. And finally, both “but” and “and” are conjunctions, so they get dots. Here’s the complete translation: I had to use square brackets, because that side of the sentence groups two compound sentences together; since I’d already used parentheses, the square brackets help me keep track of how the whole sentence comes together. Now it’s your turn! Good luck! III. Semantics of Propositional Logic Our task is to give precise meanings to all of the well-formed formulas of PL. Some of this task is already complete. We know something about the meanings to the 26 capital letters: they stand for simple English sentences of our choosing. We know roughly how each symbol corresponds to English logical words and phrases. We need more precise semantics, however. We want to know what makes a compound sentence true. We need the truth conditions for our operators. A sentence in PL can have one of two semantic values: true or false. That’s it. This is one of the ways in which the move to PL is a taming of natural language. In PL, every sentence has a determinate truth-value; and there are only two choices: true or false. English and other natural languages are more complicated than this. Of course, there’s the issue of nondeclarative sentences (such as questions or commands), which don’t express propositions and don’t have truth-values at all. But even if we restrict ourselves to declarative English sentences, things don’t look quite as simple as they are in PL. Consider the sentence “Napoleon was short.” You may not be aware that the popular conception of the French Emperor as diminutive in stature has its roots in British propaganda at the time. As a matter of fact, he was about 5’ 7”. Is that short? Well, not at the time (late 18th, early 19th centuries); Napoleon was about average or slightly above for that time period. People are taller now, though, so is 5’ 7” short from today’s perspective? The average height for a modern Frenchman is 5’ 9.25”. Napoleon is 2.25 inches shorter than average. How much shorter than average do you have to be to qualify as ‘short?’ Heck, I don’t know! The problem here is that relative terms like ‘short’ have borderline cases; they’re vague. It’s not clear how to assign a truth-value to sentences like “Napoleon was short.’ So, in English, we might say that they lack a truth-value (at least until we define our terms more precisely). Some systems of logic that are more sophisticated than our PL have developed ways to deal with these sorts of cases. Instead of just two truth-values, some systems add more. There are three-value systems, where you have true, false, and neither. There are systems with infinitely many truth-values between true and false (where false is zero and true is 1, and every real number in between is a degree of truth). The point is, English and other natural languages are messy when it comes to truth-value. We’re taming them in PL by assuming that every PL sentence has a determinate truth-value, and that there are only two truth-values: true and false—which we will indicate, by the way, with the letters ‘T’ and ‘F.’ Our task here is to provide truth conditions for the five operators: dot, wedge, tilde, horse shoe, and triple-bar, and horseshoe (we start with the dot because it’s the most intuitive). We will specify the meanings of these symbols in terms of their effects on truth-value: what is the truth-value of a compound sentence featuring them as the main operator, given the truth-values of the components? The semantic values of the operators will be truth functions: systematic accounts of the truth-value outputs (of the compound proposition) resulting from the possible truth-value inputs (of the simpler components). Another way to put this—we will specify the truth conditions for each connective; under what conditions would this compound sentence be true, and under what conditions would it be false? 1. Conjunctions (DOT) Our rough-and-ready characterization of conjunctions was that they are ‘and’ sentences— sentences like “Beyoncé is logical and James Brown is alive.” Since these sorts of compound sentences involve two simpler components, we say that dot is a two-place operator. It takes two sentences and hooks them together. So, when we specify the general form of a conjunction using generic variables, we need two of them. The general form of a conjunction in PL is: We’re using lower-case p and q here, because capital letters represent specific simple sentences. Here we want to talk about any sentence that has a dot as a main connective, so p and q are variables which could be replaced with any sentence, simple or compound. To figure out the truth conditions of a conjunction, the questions we need to answer are these: Under what circumstances is the entire conjunction true, and under what circumstances false? And how does this depend on the truth-values of the component parts? We remarked earlier that when someone utters a conjunction, they’re committing themselves to both of the conjuncts—they’re telling you both conjuncts are true. If I tell you “Beyoncé is wise and James Brown is alive,” I’m committing myself to the truth of both of those alleged facts; so, if even one of them turns out false, I’ve have not told you the truth. This is how conjunctions work, then: they’re true just in case both conjuncts are true; false otherwise. We can represent this graphically, using what we’ll call a “truth-table”: p q p • q T T T T F F F T F F F F Since the dot is a two-place operator, we need columns for each of the two variables in its general form—p and q. Each of these is a generic PL sentence that can be either true or false. That gives us four possibilities for their truth-values as a pair: both are true, p is true but q is false, p is false but q is true, or both false. These four possibilities give us the four rows of the table. For each of these possible inputs to the truth-function, we get an output, listed under the dot. T is the output when both inputs are Ts; F is the output in every other circumstance. In other words, a conjunction is only true when both sides are true, and false in every other circumstance, which is exactly what we know ‘and’ statements mean. If I told you my brother and sister are tall, you would expect me to have two tall siblings. 2. Disjunctions (WEDGE) Our rough characterization of disjunctions was that they are ‘or’ sentences—sentences like “Beyoncé is logical or James Brown is alive.” In PL, the general form of a disjunction is: where p and q are variables representing any sentence in PL, simple or compound. We need to figure out the circumstances in which such a ‘or’ statements are true; we need the truth-function represented by the wedge. While ‘and’ statements have only one way of making them true, ‘or’ statements have more. If I tell you either my brother or sister are tall, I’ve told you someone is tall, but I haven’t told you who. At this point we face a complication. Wedge is supposed to capture the essence of ‘or’ in English, but the word ‘or’ has two distinct senses. This is one of those cases where natural language needs to be tamed: our wedge can only have one meaning, so we need to choose between the two alternative senses of the English word ‘or.’ ‘Or’ can be used exclusively or inclusively. The exclusive sense of ‘or’ is expressed in a sentence like this: “Candidate A will win the election, or Candidate B will win.” The two disjuncts present exclusive possibilities: one or the other will happen, but not both. The inclusive sense of ‘or,’ however, allows the possibility of both. If I told you I was having trouble deciding what to order at a restaurant, and said, “I’ll order lobster or steak,” and then I ended up deciding to get both, you wouldn’t say I had lied to you when I said I’d order lobster or steak. The inclusive sense of ‘or’ allows for one or the other—or both. We will use the inclusive sense of ‘or’ for our wedge. There are arguments for choosing the inclusive sense over the exclusive one, but we will not dwell on those here. As we will see later, the exclusive sense will not be lost to us because of this choice: we will be able to symbolize exclusive ‘or’ within PL, using a combination of operators. So, wedge is an inclusive disjunction. It’s true whenever one or the other—or both—conjuncts is true; false otherwise. This is its truth-table definition: p q p v q T T T T F T F T T F F F If there are any trues at all, a disjunction is true; it is only false where both disjuncts are false. If I told you either my brother or sister are tall, that sentence is true if the tall one is my brother, my sister, or both of them. 3. Negations (TILDE) Tilde is a one-place operator. It does not connect two sentences together, like two-place operators; instead, it attaches to just one sentence. The general form of a negation is: where ‘p’ is a variable standing for any generic PL sentence, simple or compound. We need to give an account of the meaning of the tilde in terms of its effect on truth-value. Tilde, as we said, is the PL equivalent of ‘not’ or ‘it is not the case that.’ Let’s think about what happens in English when we use those terms. If we take a true sentence, say “Edison invented the light bulb,” and form a compound with it and ‘not,’ we get “Edison did not invent the light bulb”—a falsehood. Ok, to be clear, Edison leaned heavily on the electrical inventions of scientists before him, and an English scientist named Joseph Swan independently invented a successful light bulb, around the same time Edison did. This happens in science more than you’d think it would. With inventions, the first person to get a patent gets credit. The light bulb case is complicated because Edison got his patent in the U.S., and Swan got one in the U.K. My fifteen minutes of googling turned up a wide variety of patent dates for both men. We still give the credit to Edison, so we’re going to call “Edison invented the light bulb” a true statement. If we take a false sentence, like “James Brown is alive,” and negate it, we get “James Brown is not alive”—a truth. Evidently, the effect of negation on truth-value is to turn a truth into a falsehood, and a falsehood into a truth. We can represent this graphically, using what we’ll call a “truth-table.” The following table gives a complete specification of the semantics of tilde: p ~ p T F F T Because tilde is a one-place operator, our table only needs two lines to represent all possible truth values: a single sentence in PL is either true or false, those are all of your options. We can compute the truth-value of the negation based on the truth-value of the sentence being negated: if the original sentence is true, then its negation is false; if the original sentence is false, then the negation is true. The logical function of negations is to flip the truth value of whatever sentence to which they attach. 4. Conditionals (HORSESHOE) Our rough characterization of conditionals was that they are ‘if … then’ sentences—sentences like “If Beyoncé is logical, then James Brown is alive.’ We use such sentences all the time in everyday speech, but is surprisingly difficult to pin down the precise meaning of the conditional, especially within the constraints imposed by PL. There are in fact many competing accounts of the conditional—many different conditionals to choose from—in a literature dating back all the way to the Stoics of ancient Greece. Whole books can be, and have been, written on the topic of conditionals. In the course of our discussion of the semantics for horseshoe, we will get a sense of why this is such a vexed topic; it’s complicated. The general form of a conditional in PL is: We need to decide for which values of p and q the conditional turns out true and false. To help us along let’s consider a conditional claim with a little story to go along with it. Suppose Barb is suffering from joint pain; she doesn’t know what is causing it and hasn’t been to the doctor to find out. She’s complaining about her pain to her neighbor, Sally. After hearing a brief description of the symptoms, Sally is ready with a prescription, which she delivers to Barb in the form of a conditional claim: “If you drink this herbal tea every day for a week, then your pain will go away.” She hands over a packet of tea leaves and instructs Barb in their proper preparation. We want to evaluate Sally’s conditional claim—that if Barb drinks the herbal tea daily for a week, then her pain will go away—for truth/falsity. To do so, we will consider various scenarios, the details of which will bear on that evaluation. Scenario #1: Barb does in fact drink the tea every day for a week as prescribed, and, after doing so, lo and behold, her pain is gone. Sally was right! Has she proved that the tea works? Scenario #2: Barb does as Sally said and drinks the tea every day for a week, but, after the week is finished, the pain remains, the same as ever. In this scenario, we would say that Sally was wrong: her conditional advice was false. Scenario #3: Barb doesn’t drink the tea for a week; the antecedent is false. But in this scenario, it turns out that after the week is up, Barb’s pain has gone away; the consequent is true. What do we say about Sally’s advice—if you drink the tea, the pain will go away—in this set of circumstances? Scenario #4: Again Barb does not drink the tea (false antecedent), and after the week is up, the pain remains (false consequent). What do we say about the Sally’s conditional advice in this scenario? Perhaps you can see what I’m doing here. Each of the scenarios represents one of the rows in the truth-table definition for the horseshoe. Sally’s conditional claim has an antecedent—Barb drinks the tea every day for a week—and a consequent—Barb’s pain goes away. These are p and q, respectively, in the conditional. Each scenario corresponds to a line in the truth table for conditional, below. The difficulty with conditional statements in a truth-functional language, is that we have to assign either true or false to each statement, and the only clear case we have here is Scenario #2. When the antecedent is true (Barb drinks the tea) but the consequent never happens (her pain did not go away), that clearly breaks the conditional, and proves that Sally was wrong. The problem with the first scenario, where she drinks the tea and the pain goes away, is that causation is very hard to prove (another chapter in this book goes over causation in detail). It is possible that the pain went away for an entirely different reason and the tea had no causal effect. The problem with the third and forth scenarios, where Barb didn’t drink the tea, is that we haven’t even tested the tea, so how do we come to a conclusion about its connection to getting rid of pain? With our truth-functional conditional, we go with an “innocent until proven guilty” approach. If the condition is met, but the consequent never happens, we’ve proved that the conditional was “guilty”—or false. We haven’t proved a causal connection in any of the other lines—in the first one, one test of the tea is not proof, and in the third and fourth lines we didn’t even test the tea—but we haven’t disproved it either, so we call it “true” in all of these cases. The table for the conditional looks like this: p q p ⊃ q T T T T F F F T T F F T This works rather well with conditionals where we already know their truth value. Consider this one: “If I live in Macon, then I live in Georgia.” If a person lived in Macon but did not live in Georgia, then this would clearly be false—someone got their geography wrong. There are four cities named Macon in the U.S., so it is possible to live in a Macon without living in Georgia. I am talking about the one in Georgia, though, so if you do live there (true antecedent), then you do live in Georgia (true consequent). Consider, though, a person who does not live in Macon (false antecedent). They could live elsewhere in Georgia (true consequent), or they could live in, say, Tennessee (false consequent). The false antecedent does not tell us anything about the consequent—but someone not living in Macon doesn’t suddenly mean Macon is not a town in Georgia. The conditional remains true in these cases. Another example: consider a promise. “If you give me gas money, then I’ll give you a ride to school.” Now consider the four scenarios, the four lines of the truth table, and consider when you have broken your promise and when you have not. Scenario #1: Your friend gives you gas money, and you give them a ride. You did not break your promise, this was a true promise. Scenario #2: Your friend gives you gas money; you take their money and run. You broke your promise—this was a false promise. Scenario #3: Your friend tells you they’re completely broke (false antecedent), and they beg you to drive them to school anyway. You decide to be nice and drive them anyway (true consequent). This does not mean the promise was false, and you definitely haven’t broken your promise. The promise is still true, even though they couldn’t come up with the money this week. Scenario #4: Your friend tells you they’re broke (false antecedent) and ask for a ride anyway; you decide you don’t feel like it (false consequent). You haven’t broken your promise here, either. The promise was still true, but they didn’t meet the condition, so you don’t have to give them a ride. The point is, the conditional only kicks in when the antecedent is met. When they give you money, you have to give them a ride, or you’ve broken your promise. When they don’t give you money, you have no obligation either way, and you can make a choice without breaking anything. You told them what you’d do if they gave you gas money; you did not tell them what you’d do if they did not give you gas money. The take-away: a conditional is only false when the antecedent is met but the consequent never happened – when the antecedent is true, but the consequent is false. 5. Biconditionals (TRIPLE-BAR) As we said, biconditionals are, roughly, ‘if and only if’ sentences—sentences like “Beyoncé is logical if and only if James Brown is alive.” They’re called biconditionals because they contain two conditionals in them. Remember two of the phrases we symbolize with the triple bar: ‘if and only if’ and ‘necessary and sufficient condition.’ We can treat these phrases as a conjunction of two conditionals. So, “A if and only if B” can be broken down into “A if B, and A only if B.” If you remember from the syntax section above, when you have an ‘if’ in the middle, as in “A if B,” you have to switch the order of the two sides. When you have an ‘only if’ in the middle, you need to keep them in the same order. So we can translate it like this: Two conditionals, going in either direction. The triple bar is essentially short for this. So, how do we figure out the truth conditions of this one? I think it helps if we stick with our “promise” example from the conditional section above. Suppose I tell my friend “I will give you a ride if and only if you pay me gas money.” A logically equivalent For those interested in how this is logically equivalent: I’ve repeated my second conjunct “if A then B.” The first conjunct is “if B then A,” which is logically equivalent to its contraposition: “if not A then not B.”” way to say the same thing is this: “If you pay me gas money, then I will give you a ride; and if you don’t pay me gas money, I will not give you a ride.” In symbols this would be: Now you have told your friend what you would do if they give you that money, but you’ve also told them what you’ll do if they fail to pay you. If they don’t give you money, you no longer have a choice, you have to refuse the ride. Take a look at the triple-bar truth table: p q p ≡ q T T T T F F F T F F F T “If and only if”—two conditionals going in each direction. “Necessary and sufficient condition”—two conditionals, going in each direction. Remember the third phrase we use the triple-bar for: “logically equivalent to.” Notice that the triple-bar is true where both sides are true, and where both sides are false. It’s true when the two sides match. If there is a mismatch, either FT or TF, the biconditional is false. IV. Computing Truth-Values of Compound PL Sentences With the truth-functional definitions of the five PL operators in hand, we can compute the truth-values of compound PL sentences, given the truth-values of their simplest parts (the simple sentences—capital letters). To do so, we must first determine what type of compound sentence we’re dealing with: negation, conjunction, disjunction, conditional, or biconditional. This involves deciding which of the operators in the PL sentence is the main operator. We then compute the truth-value of the compound by taking the values of the simpler components, and using the truth table for the operator to figure out the value of the whole sentence. If these components are themselves compound, we need to first determine their main operators and compute accordingly, in terms of their simpler components, and so on. A few examples will make the process clear. Let’s suppose that A and B are true sentences. Consider this compound: What is its truth value? This is pretty easy to figure out intuitively—it’s a disjunction, and I know that means at least one disjunct has to be true to make the whole sentence true. Both disjuncts are true, so this is a true sentence. Let’s walk through the steps of using our truth tables, though. You always want to start by filling in the values of the simple sentences. Write them directly under each letter. We’ve been told that both A and B are true, so we write that: Now we take the values of each side, and plug them into our disjunction table. (If you don’t have all five tables written out on a piece of paper you can reference easily, now is a good time to do so). Both sides are Ts; this puts us on the top row of our truth table, and we can see the answer is T—the whole sentences is true. Fill that in, and you have your answer: Let’s look at a slightly more complicated one: Now we have three connectives. We need to figure out the main connective before we begin. The main connective is always the last thing you compute, and that will give you the answer for the whole sentence. This is still a disjunction; the wedge is the main connective. So, before I figure out the value of the wedge, I need to first figure out the value of each side. I start the same way as before; I put the values of the simple sentences directly under each letter. Both A and B are true, so I fill that in: I’m not done calculating each side, yet. Remember that a negation switches the value of whatever it attaches to. A and B are true, but NOT A and NOT B are both false. Fill that in: All that is left is to calculate the main connective. I will still use my wedge table, but I want to make sure the values I plug into it are the main connectives of each side. In this case, the tildes have the values I want to use. So, both not A and not B are false; this puts me on the fourth line of the wedge truth table, and that tells me the whole sentence is false. Fill that in under the wedge, and you have your answer: Once these compound sentences get complicated, you’ll have a long string of Ts and Fs. Use any tools you can think of to help you navigate. Different colors, circling things, whatever helps you focus on the correct T or F at the correct step. Let’s walk through a similar one, only instead of a disjunction, this sentence is a negation: In the previous example, the main connective was the wedge. The tildes attached directly to the letters, and the wedge brought the whole sentence together. Now, I’ve put the tilde in front of parentheses. That means it negates the whole disjunction instead of each individual letter, so it is now the main connective. I need to solve the problem inside parentheses before I apply the negation. Start the same way as before: put the T under each letter, and then use the wedge table to figure out the value of the disjunction. Since both sides are true, the disjunction will be true: I’ve solved the problem inside parentheses: the disjunction is true. Now I apply the negation. I know that the tilde switches the value of whatever it attaches to. It attaches to a disjunction, and the disjunction is true, so the negation turns that to false, and that’s our final answer: It will perhaps be useful to look at one more example, this time of a more complex PL sentence. Suppose again that A and B are true simple sentences, and that X and Y are false. Let’s compute the truth-value of the following compound sentence: As a first step, it’s useful to mark the truth-values of the simple sentences: Now, we need to figure out what kind of compound sentence this is; what is the main operator? This sentence is a conditional; the main operator is the horseshoe. The tilde at the far left negates the first half of the sentence only. We need to compute the truth-values of both the antecedent and consequent before we can figure out the value of the conditional. Let’s take the antecedent first. The tilde negates the conjunction, so before we can know what the tilde does, we need to know the truth-value of the conjunction inside the parentheses. Conjunctions are true just in case both conjuncts are true; in this case, A is true but X is false, so the conjunction is false. If you forget that, plug the T and F into the conjunction table; it will tell you the answer is false. Now apply the tilde: it switches the value of the conjunction. Since the conjunction is false, its negation must be true: So, the antecedent of our conditional is true. Let’s look at the consequent. There is a tilde directly in front of a letter, so I deal with that first. Y is false, so not Y must be true. That means both disjuncts are true. When I plug two trues into the wedge table, the answer is true. Let’s fill that in: When I figure out the main connective—the horseshoe—I need to make sure I’m looking at the main connective of each side—the tilde on the first side, and the wedge in the second side. Both the antecedent and consequent of the conditional are true, and looking at the horseshoe table, that makes the whole conditional true: One final note: sometimes you only need partial information to make a judgment about the truth-value of a compound sentence. Look again at the truth table definitions of the two-place operators: For three of these operators—the dot, wedge, and horseshoe—one of the rows is not like the others. For the dot: it only comes out true when both p and q are true, in the top row. For the wedge: it only comes out false when both p and q are false, in the bottom row. For the horseshoe: it only comes out false when p is true and q is false, in the second row. Noticing this allows us, in some cases, to compute truth-values of compounds without knowing the truth-values of both sides. Suppose again that A is true and X is false, and let Q be a simple sentence with an unknown truth-value (it has one, like all of them must; I’m just not telling you what it is). Consider this compound: We know one of the disjuncts is true; we don’t know the truth-value of the other one. But we don’t need to! A disjunction is only false when both of its disjuncts are false; it’s true when even one of its disjuncts is true. A being true is enough to tell us the disjunction is true; the value of Q doesn’t matter. Consider the conjunction: We only know the truth-value of one of the conjuncts: X is false. That’s all we need to know to compute the truth-value of the conjunction. Conjunctions are only true when both of their conjuncts are true; they’re false when even one of them is false. X being false is enough to tell us that this conjunction is false. Finally, consider these conditionals: They are both true. Conditionals are only false when the antecedent is true and the consequent is false; so they’re true whenever the consequent is true (as is the case in Q ⊃ A) and whenever the antecedent is false (as is the case in X ⊃ Q).
Chapter 3: Categorical Propositions/Statements This chapter is based on For All X, The Lorain County Remix, remixed by J. Robert Loftis. I. Categorical Statements Earlier we saw that a statement was a unit of language that could be true or false. In this chapter and the next we are going to look at a particular kind of statement, called a quantified categorical statement, and begin to develop a formal theory of how to create arguments using these statements. This kind of logic is generally called “categorical” or “Aristotelian” logic, because it was originally invented by the great logician and philosopher Aristotle in the fourth century BCE BCE (before the common era) and CE (common era) are the academic way of saying BC (before Christ) and AD (anno domini – the year of our Lord).. This kind of logic dominated the European and Islamic worlds for 20 centuries afterward, and was expanded in all kinds of fascinating ways, some of which we will look at here. Consider the following propositions: (a) All dogs are mammals. (b) Most physicists are smart. (c) Few teachers are rock climbers. (d) No dogs are cats. (e) Some Americans are doctors. (f) Some adults are not logicians. (g) Thirty percent of Canadians speak French. (h) One chair is missing. These are all examples of quantified categorical statements. A quantified categorical statement is a statement that makes a claim about a certain quantity of the members of a class or group. (Sometimes we will just call these “categorical statements”). Statement (a), for example, is about the class of dogs and the class of mammals. These statements make no mention of any particular members of the categories or classes or types they are about. The propositions are also quantified in that they state how many of the things in one class are also members of the other. For instance, statement (b) talks about most physicists, while statement (c) talks about few teachers. Categorical statements can be broken down into four parts: the quantifier, the subject term, the predicate term, and the copula: The quantifier is the part of a categorical sentence that specifies a portion of a class. It is the “how many” term. The quantifiers in the sentences above are all, most, few, no, some, thirty percent, and one. Notice that the “no” in sentence (d) counts as a quantifier, the same way zero counts as a number. The subject and predicate terms are the two classes the statement talks about. The subject class is the first class mentioned in a quantified categorical statement, and the predicate class is the second. In sentence (e), for instance, the subject class is the class of Americans, and the predicate class is the class of doctors. The copula is simply the form of the verb “to be” that links subject and predicate. Notice that the quantifier is always referring to the subject. The statement “Thirty percent of Canadians speak French” is saying something about a portion of Canadians, not about a portion of French speakers. Sentence (g) is a little different than the others. In sentence (g) the subject is the class of Canadians, and the predicate is the class of people who speak French. That’s not quite the way it is written, however. There is no explicit copula, and instead of giving a noun phrase for the predicate term, like “people who speak French,” it has a verb phrase, “speak French.” If you are asked to identify the copula and predicate for a sentence like this, you should say that the copula is implicit and transform the verb phrase into a noun phrase, for example, “people who speak French.” You would do something similar for sentence (h): the subject term is “chair,” and the predicate term is “things that are missing.” Sometimes we will replace the classes referred to in a quantified categorical statement with capital letters that act as variables. Typically, we will use the letter S when referring to the class in the subject term and P when referring to the predicate term, although sometimes more letters will be needed. Thus, the sentence “Some Americans are doctors,” above, will sometimes become “Some S are P.” The sentence “No dogs are cats” will sometimes become “No S is P.” II. Categorical Propositions and Standard Form A categorical proposition that expresses the relationships between its terms with complete clarity is said to be in standard form. “Standard form” just means that the proposition has all the right parts in the right order, and from here on when we deal with categorical propositions, or (later) categorical syllogisms, we will always put the statements in standard from. To be in standard form, a proposition be written as follows: Quantifier (All, No, or Some), Subject Term (plural noun or plural noun phrase), Copula (“are” or “are not”), Predicate Term (plural noun or plural noun phrase). It is essential that what comes between the Quantifier and the Copula, and after the Copula, MUST be plural nouns or plural noun phrases. If a proposition doesn’t have those things, in that order, it’s not in “standard form.” For example: “Everyone likes coffee,” while it is a categorical statement, is not a “standard form” categorical proposition because it doesn’t have a proper quantifier, subject term, copula, or predicate term. You can make it a “standard form” categorical proposition, though, by changing it to: “All persons who exist are persons who like coffee.” Clunky, but standard form. Here we will show ordinary English sentences can be transformed into logically structured English. Logically structured English is English that has been put into a “standard form” that allows us to see its logical structure more clearly and removes ambiguity. Transforming English sentences into logically structured English is fundamentally a matter of understanding the meaning of the English sentence and then finding the logically structured English statements with the same or similar meaning. Sometimes this will require judgment calls. English, like any natural language, is fraught with ambiguity. One of our goals with logically structured English is to reduce the amount of ambiguity. Clarifying ambiguous sentences will always require making judgments that can be questioned. For example, does the sentence “Bananas are yellow” state that all bananas are yellow or that only some are? The statement “All bananas are yellow” is false, for instance, “Some bananas are green” is true. However, if someone tells me that all bananas are yellow, I might assume that the person is really speaking about ripe, ready-to-eat bananas and say that the statement is true. To transform a quantified categorical statement into standard form, we have to put all of its elements in a fixed order and be sure they are all of the right type. All statements must begin with the quantifiers “All” or “Some” or the negated quantifier “No.” Next comes the subject term, which must be a plural noun, a noun phrase, or a variable that stands for any plural noun or noun phrase. Then comes the copula “are” or the negated copula “are not.” Last is the predicate term, which must also be a plural noun or noun phrase. We also specify that you can only say “are not” with the quantifier “some,” that way the universal negative statement is always phrased “No S are P,” instead of “All S are not P.” Taken together, these criteria define the standard form for a categorical statement in logically structured English. The subsections below identify different kinds of changes you might need to make to put a statement into standard form. Sometimes translating a sentence will require using multiple changes. Change the Predicate into a Noun Phrase Above we saw that “Some Canadians speak French” has a verb phrase “speaks French” instead of a copula and a plural noun phrase. To transform these sentences into standard form, you need to add the copula and turn all the terms into plural nouns or plural noun phrases. Adding a plural noun phrase means you have to come up with some category, like “people” or “animals.” When in doubt, you can always use the most general category, “things.” The table below gives some examples English Standard form No cats bark. No cats are animals that bark. All birds can fly. All birds are animals that can fly. Some thoughts should be left unsaid. Some thoughts are things that should be left unsaid. Sometimes English sentences will have a copula and an adjective or adjective phrase as the predicate. These need to be changed to noun phrases, just as the verb phrases did. The following table gives examples. English Standard form Some roses are red. Some roses are red flowers. Football players are strong. All football players are strong persons. Some names are hurtful. Some names are hurtful things. Again, you will have to come up with a category for the predicate, and when it doubt, you can just use “things.” Standardize the Quantifier English has a wide variety of ways to express quantity. We need to reduce all of these to either “all” or “some,” plus negations. The following table has some examples: English Logically Structured English Most people with a PhD in psychology are female. Some people with a PhD in psychology are female. Among the things that Sylvia inherited was a large mirror. Some things that Sylvia inherited were large mirrors. There are Americans that are doctors. Some Americans are doctors. At least a few Americans are doctors. Some Americans are doctors. A man is walking down the street. Some men are things that are walking down the street. Every day is a blessing. All days are blessings. Whatever is a dog is not a cat. No dogs are cats. Take nothing for granted. No things are things that should be taken for granted. Something is rotten in Denmark. Some things are things that are rotten in Denmark. Everything is coming up roses. All things are things that are coming up roses. “What does not destroy me, makes me stronger.” (Friedrick Nietzsche) All things that do not destroy me are things that make me stronger. Most Americans are doctors. Some Americans are doctors. Notice in the last case we are losing quite a bit of information when we transform the sentence into logically structured English. “Most” means more than fifty percent, while “some” could be any percentage less than a hundred. This is simply a price we have to pay in creating a standard logical form. No logical language has the expressive richness of a natural language. Sometimes universal statements in English don’t have an explicit quantifier. Instead they use a plural noun or indefinite article to express generality. English Logically Structured English Boots are footwear. All boots are footwear. Giraffes are tall. All giraffes are tall things. A dog is not a cat. No dogs are cats. A lion is a fierce creature. All lions are fierce creatures. Notice that in the second sentence we had to make two changes, adding both the words “All” and “things.” In the last two sentences, the indefinite article “a” is being used to create a kind of generic sentence. Not all sentences using the indefinite article work this way. The list before this one included the example “A man is walking down the street.” This sentence is not talking about all men generically. It is talking about a specific man whose identity is unknown. Here the indefinite article is being used like a nonstandard version of the quantifier “some,” which is why it appeared in the earlier list. You will have to use your good judgment and understanding of context to know when the indefinite article is being used like the word “all” and when it is being used like the word “some.” English also uses specialized adverbial phrases as quantifiers for people, places and times. If we want to talk about all people, we use a specialized quantifier like “everyone,” “someone” or “no one.” We use “everywhere,” “somewhere,” and “nowhere” for places, and “always,” “sometimes,” and “never” for times. All of these need to be transformed into logically structured English by using the simple quantifiers “all” or “some,” plus negations. English Logically Structured English Someone in America is a doctor. Some Americans are doctors. Not everyone who is an adult is a logician. Some adults are not logicians. “Whenever you need me, I’ll be there.” (Michael Jackson) All times that you need me are times that I will be there. “We are never, ever, ever getting back together.” (Taylor Swift) No times are times when we will get back together. “Whoever fights with monsters should be careful lest he thereby become a monster.” (Friedrich Nietzsche) All persons who fight with monsters are persons who should be careful lest they become a monster. Standardize Alternative Universal Forms Many constructions in English can be represented as universal statements in Logically Structured English, either affirmative (A) or negative (E) For instance, it turns out that statements about individual people or specific objects can be represented by A or E statements. This is not something Aristotle originally noticed. For him a statement like “Socrates is mortal,” were neither universal nor particular. They were a third class he called “singular.” The power of categorical logic was expanded considerably when it was realized singular statements can converted into universal statements. The trick is to add a phrase like “All things identical to ...” to our singular sentence. Essentially we are adding a universal quantifier that only picks out one specific object. English Logically Structured English Socrates is mortal. All persons identical with Socrates are persons who are mortal. The Empire State Building is tall. All things identical to the Empire State Building are things that are tall. Ludwig was not happy. No people identical with Ludwig are people who were happy. Another kind of statement that can be transformed into a universal statement is a conditional. A conditional is a statement of the form “If ... then ...” They will become a big focus of our attention when we begin introducing modern formal languages. Aristotle, owing to his interest in sound arguments, does not spend much time considering conditional statements, which begin with ‘if’ and accordingly might not be true. However, later thinkers worked to include conditional statements into a broadly Aristotelian logic.” In this chapter, where we can, we just treat them as categorical generalizations: English Logically Structured English If something is a cat, then it is a feline. All cats are felines. If something is a dog, then it is not a cat. No dogs are cats. The word “only” is used in a couple of different constructions in English that can be represented as universal statements. The first kind are called “exclusive propositions.” These are statements that say the subject excludes everything except what is in the predicate. For instance the sentence “Only people over 21 may drink” says that the class of people who may drink excludes everyone except those who are over 21. In English exclusive propositions are created using the words “only,” “none but,” or “none except.” These statements become A statements when translated into logically structured English. So “Only people over 21 may drink” becomes “If you may drink, you are over 21.” It is important to see that in each case these words are used to introduce the predicate, not the subject. In the sentence “Only people over 21 may drink,” the term “people over 21” is actually the predicate, and “people who may drink” is the subject. English Logically Structured English Only people over 21 may drink. All people who drink are over 21. No one, except those with a ticket, may enter the theater. All people who enter the theater have a ticket. None but the strong survive. All people who survive are strong people. Sentences with “The only” are a little different than regular exclusive propositions, which just have “only” in them. The sentence “Humans are the only animals that talk on cell phones” should be translated as “All animals who talk on cell phones are humans.” In this sentence, “the only” introduces the subject, rather than the predicate. The statement still asserts that the subject excludes everything except what is in the predicate, and we still represent them using mood A statements. English Logically Structured English Humans are the only animals who talk on cell phones. All animals who talk on cell phones are human. Shrews are the only venomous mammal in North America. All venomous mammals in North America are shrews. Transforming sentences into Logically Structured English requires judgment and attention to the nuances of meaning in English. You must be able to recognize which of the transformations describe above needs to be applied and apply it correctly. One frequent mistake by people starting out is to overgeneralize. We saw at the start of the subsection on alternative universal forms that singular propositions can be turned into universal propositions by adding the phrase “Things identical to ...” Once you get in the habit of doing this, it becomes tempting to add the phrase “things identical to ...” to everything, even when it isn’t necessary or doesn’t make sense. The sentence “Fido is a dog” should become “all things identical to Fido are dogs” in logically structured English, because “Fido” is a singular term referring to an individual dog. But with the sentence “dogs are mammals,” you do not need to add the phrase “All things identical to. . . ”, because “dogs” is already a collective noun, not an individual. The same is true for the phrases we use to transform adjective and verb phrases into noun phrases. The sentence “No cats bark” has to be changed, because “bark” is a verb, so it becomes “No cats are animals that bark” in Logically Structured English. But the sentence “No cats are reptiles” already has a noun, “reptiles,” for a predicate, so you do not need to transform it into “No cats are animals that are reptiles.” The key is not only knowing when to use the transformations we describe, but knowing when not to use them. II. Quantity, Quality, and Distribution Ordinary English contains all kinds of quantifiers, including the counting numbers themselves. In this chapter and the next, however, we are only going to deal with two quantifiers: “all,” and “some.” We are restricting ourselves to the quantifiers “all” and “some” because they are the ones that can easily be combined to create valid arguments using the system of logic that was invented by Aristotle. The quantifier used in a statement is said to give the quantity of the statement. Statements with the quantifier “all” are said to be “universal” and those with the quantifier “some” are said to be “particular.” Here “some” will just mean “at least one.” So, “some people in the room are standing” will be true even if there is only one person standing. Also, because “some” means “at least one,” it is compatible with “all” statements. If I say “some people in the room are standing” it might actually be that all people in the room are standing, because if all people are standing, then at least one person is standing. This can sound a little weird, because in ordinary circumstances, you wouldn’t bother to point out that something applies to some members of a class when, in fact, it applies to all of them. It sounds odd to say “some dogs are mammals,” when in fact they all are. Nevertheless, when “some” means “at least one” it is perfectly true that some dogs are mammals. In addition to talking about the quantity of statements, we will talk about their quality. The quality of a statement refers to whether the statement is negated. Statements that include the words “no” or “not” are negative, and other statements are affirmative. Combining quantity and quality gives us four basic types of quantified categorical statements, which we call the statement moods or just “moods.” The four moods are labeled with the letters A, E, I, and O. Statements that are universal and affirmative are mood-A statements. Statements that are universal and negative are mood-E statements. Particular and affirmative statements are mood-I statements, and particular and negative statements are mood-O statements. The following table shows the four moods of a categorical statement. Mood Form Example A All S are P All dogs are mammals. E No S are P No dogs are reptiles. I Some S are P Some birds can fly. O Some S are not P Some birds cannot fly. Aristotle didn’t actually use those letters to name the kinds of categorical propositions. His later followers writing in Latin came up with the idea. They remembered the labels because the “A” and the “I” were in the Latin word “affirmo,” (“I affirm”) and the “E” and the “O” were in the Latin word “nego” (“I deny”). The distribution of a categorical statement refers to how the statement describes its subject and predicate class. A term in a sentence is said to be distributed if a claim is being made about the whole class. In the sentence “All dogs are mammals,” the subject class, dogs, is distributed, because the quantifier “all” refers to the subject. The sentence is asserting that every dog out there is a mammal. On the other hand, the predicate class, mammals, is not distributed, because the sentence isn’t making a claim about all the mammals. We can infer that at least some of them are dogs, but we can’t infer that all of them are dogs. So in mood-A statements, only the subject is distributed. On the other hand, in an I sentence like “Some birds can fly,” the subject is not distributed. The quantifier “some” refers to the subject, and indicates that we are not saying something about all of that subject. We also aren’t saying anything about all flying things, either. So in mood-I statements, neither subject nor predicate is distributed. Even though the quantifier always refers to the subject, the predicate class can be distributed as well. This happens when the statement is negative. The sentence “No dogs are reptiles” is making a claim about all dogs: they are all not reptiles. It is also making a claim about all reptiles: they are all not dogs. So mood-E statements distribute both subject and predicate. Finally, negative particular statements (mood-O) have only the predicate class distributed. The statement “some birds cannot fly” does not say anything about all birds. It does, however say something about all flying things: the class of all flying things excludes some birds. The quantity, quality, and distribution of the four forms of a categorical statement are given in the following table. The general rule to remember here is that universal statements distribute the subject, and negative statements distribute the predicate. Mood Form Quantity Quality Terms Distributed A All S are P Universal Affirmative S E No S are P Universal Negative S and P I Some S are P Particular Affirmative None O Some S are not P Particular Negative P III. Venn Diagrams In 1880, English logician John Venn published two essays on the use of diagrams with circles to represent categorical propositions. The first one is Venn, John. “On the diagrammatic and mechanical representation of propositions and reasonings.” Philosophical Magazine and Journal of Science 10, Fifth Series (59), 1880, 1–17. The second one is “On the employment of geometrical diagrams for the sensible representation of logical propositions.” Proceedings of the Cambridge Philosophical Society, 4, 1880, 47–59. Venn noted that the best use of such diagrams so far had come from the brilliant Swiss mathematician Leonhard Euler, but they still had many problems, which Venn felt could be solved by bringing in some ideas about logic from his fellow English logician George Boole. Although Venn only claimed to be building on the long logical tradition he traced, since his time these kinds of circle diagrams have been known as Venn diagrams. In this section we are going to learn to use Venn diagrams to represent our four basic types of categorical statement. Later in this chapter, we will find them useful in evaluating arguments. Let us start with a statement in mood A: “All S are P.” We are going to use one circle to represent S and another to represent P. There are a couple of different ways we could draw the circles if we wanted to represent “All S are P.” One option would be to draw the circle for S entirely inside the circle for P, as in this picture: It is clear from this picture that all S are in fact P. And outside of college logic classes, you may have seen people use a diagram like this to represent a situation where one group is a subclass of another. You may have even seen people call concentric circles like this a Venn diagram. But Venn did not think we should put one circle entirely inside the other if we just want to represent “All S are P.” Technically speaking, the picture above shows Euler circles. Venn pointed out that these concentric circles don’t just say that “All S are P.” They also say that “All P are S” is false. But we don’t necessarily know that if we have only asserted “All S are P.” The statement “All S are P” leaves it open whether the S circle should be smaller than or the same size as the P circle. Here is Venn’s original diagram for an A-mood statement: Venn suggested that to represent just the content of a single proposition, we should always begin by drawing partially overlapping circles. This means that we always have spaces available to represent the four possible ways the terms can combine: Area 1 represents things that are S but not P; area 2, things that are S and P; area 3, things that are just P; and area 4 represents things that are neither S nor P. We can then mark up these areas to indicate whether something is there or could be there. We shade a region of the diagram to represent the claim that nothing can exist in that region. For instance, if we say “All S are P,” we are asserting that nothing can exist that is in the S circle unless it is also in the P circle. So we shade out the part of the S circle that doesn’t overlap with P. If we want to say that something does exist in a region, we put an “x” in it. This is the diagram for “Some S are P”: If a region of a Venn diagram is blank, if it is neither shaded nor has an x in it, it could go either way. Maybe such things exist, maybe they do not. “Some S are not P” says that something exists in the S region, but outside of the P region. Here is the diagram for “Some S are not P”: And finally, if I say “No S are P,” I’m saying that nothing exists in the area where the two circles overlap, so I want to shade that out. Notice that when we draw diagrams for the two universal forms, A and E, we do not draw any x’s. For these forms we are only ruling out possibilities, not asserting that things actually exist. This is part of what Venn learned from Boole, and we will see its importance later on. Finally, notice that so far, we have only been talking about categorical statements involving the variables S and P. Sometimes, though, we will want to represent statements in regular English. To do this, we will include a key saying what the variables S and P represent in this case. We will call a list that assigns English phrases or sentences to variable names a translation key. These are sometimes also called “symbolization keys” or simply just “dictionaries.” As our logical systems get more complicated, the symbolization keys will get more complicated. For now, though, they just consist of a note saying what the S and P stand for. For instance, this is the diagram for “No dogs are reptiles.” S: Dogs P: Reptiles Here are all four two-circle Venn diagrams: IV. Conversion, Obversion, and Contraposition Now that we have shown the wide range of statements that can be represented in our four standard logical forms A, E, I, and O, it is time to begin constructing arguments with them. The arguments we are going to look at are sometimes called “immediate inferences” because they only have one premise. We are going to learn to identify some valid forms of these one-premise arguments by looking at ways you can transform a sentence so that a true sentence will stay true and a false sentence will stay false. Earlier in the book we said that the truth value of a sentence is simply whether the sentence is true or false. So we can say that the transformations we will be looking at here preserve the truth values of the sentences. Consider the statements, “No dogs are reptiles” and “No reptiles are dogs.” They have the same truth value and basically mean the same thing. On the other hand, if you change “All dogs are mammals” into “All mammals are dogs” you turn a true sentence into a false one. In this section we are going to look at three ways of transforming categorical statements—conversion, obversion, and contraposition—and use Venn diagrams to determine whether these transformations also lead to a change in truth value. From there we can identify valid argument forms. Conversion The two examples in the last paragraph are examples of conversion. Conversion is the process of transforming a categorical statement by switching the subject and the predicate. When you convert a statement, it keeps its form—an A statement remains an A statement, an E statement remains an E statement—however it might change its truth value. The Venn diagrams in the table below illustrate this. Original Converse A: All S are P A: All P are S E: No S are P E: No P are S I: Some S are P I: Some P are S O: Some S are not P O: Some P are not S As you can see, the Venn diagram for the converse of an E statement is exactly the same as the original E statement, and likewise for I statements. This means that the two statements are logically equivalent. Two statements are logically equivalent if they always have the same truth value. In this case, that means that if an E statement is true, then its converse is also true, and if an E statement is false, then its converse is also false. For instance, “No dogs are reptiles” is true, and so is “No reptiles are dogs.” On the other hand, “No dogs are mammals” is false, and so is “No mammals are dogs.” Likewise, if an I statement is true, its converse is true, and if an I statement is false, than its converse is false. “Some dogs are pets” is true, and so is “Some pets are dogs.” On the other hand, “Some dogs can fly” is false and so is “Some flying things are dogs.” The converses of A and O statements are not so illuminating. As you can see from the Venn diagrams, these statements are not identical to their converses. They also don’t contradict their converses. If we know that an A or O statement is true, we still don’t know anything about their converses. We say their truth value is undetermined. Because E and I statements are logically equivalent to their converses, we can use them to construct valid arguments. An argument is valid if it is impossible for its conclusion to be false whenever its premises are true. Because E and I are logically equivalent to their converses, the following two argument forms are valid: No S are P. Therefore, no P are S. Some S are P. C: Therefore, some P are S. Notice that these are argument forms, with variables in the place of the key terms. This means that these arguments will be valid no matter what; S and P could be people, or squirrels, or the Gross Domestic Product of industrialized nations, or anything, and the arguments are still valid. While these particular argument forms may seem trivial and obvious, we are beginning to see some of the power of formal logic here. We have uncovered a very general truth about the nature of validity with these two argument forms. The truth value of the converses of A and O statements, on the other hand, are undetermined by the truth value of the original statements. This means we cannot construct valid arguments from them. Imagine you have an argument with an A or O statement as its premise and the converse of that statement as the conclusion. Even if the premise is true, we know nothing about the truth of the conclusion. So there are no valid argument forms to be found here. Obversion Obversion is a more complex process. To understand what an obverse is, we first need to define the complement of a class. The complement of a class is everything that is not in the class. So the complement of the class of dogs is everything that is not a dog, including not just cats, but battleships, pop songs, and black holes. In English we can easily create a name for the complement of any class using the prefix “non-”. So the complement of the class of dogs is the class of non-dogs. We will use complements in defining both obversion and contraposition. The obversion of a categorical proposition is a new proposition created by changing the quality of the original proposition and switching its predicate to its complement. Obversion is thus a two-step process. Take, again, the proposition “All dogs are mammals.” For step 1, we change its quality, in this case going from affirmative to negative. That gives us “No dogs are mammals.” For step 2, we take the complement of the predicate. The predicate in this case is “mammals” so the complement is “non-mammals.” That gives us the obverse “No dogs are non-mammals.” We can map this process out using Venn diagrams. Let’s start with an A statement. A: All S are P. Changing the quality turns it into an E statement. E: No S are P. Now what happens when we take the complement of P ? That means we will shade in all the parts of S that are non-P, which puts us back where we started. We still have an E statement, but it is now equivalent to the A statement. E: No S are non-P. The final statement is logically equivalent to the original A statement. It has the same form as an E statement, but because we have changed the predicate, it is not logically equivalent to an A statement. As you can see from the following table, this is true for all four forms of categorical statement. Original Obverse A: All S are P E: No S are non-P E: No S are P A: All S are non-P I: Some S are P O: Some S are not non-P O: Some S are not P I: Some S are non-P This in turn gives us four valid argument forms: All S are P Therefore, no S are non-P No S are P Therefore, all S are non-P Some S are P Therefore, some S are not non-P Some S are not P Therefore, some S are non-P One further note on complements. We don’t just use complements to describe sentences that come out of obversion and contraposition. We can also perform these operations on statements that already have complements in them. Consider the sentence “Some S are non-P.” This is its Venn diagram. Some S are non-P. How would we take the obverse of this statement? Step 1 is to change the quality, making it “Some S are not non-P.” Now how do we take the complement of the predicate? We could write “non-non-P,” but if we think about it for a second, we’d realize that this is the same thing as P. So we can just write “Some S are not P.” This is logically equivalent to the original statement, which is what we wanted. Taking the converse of “Some S are non-P” also takes a moment of thought. We are supposed to reverse subject and predicate. But does that mean that the “non-” moves to the subject position along with the “P”? Or does the “non-” now attach to the S? We saw that E and I statements kept their truth value after conversion, and we want this to still be true when the statements start out referring to the complement of some class. This means that the “non-” has to travel with the predicate, because “Some S are non-P” will always have the same truth value as “Some non-P are S.” Another way of thinking about this is that the “non-” is part of the name of the class that forms the predicate of “Some S are non-P.” The statement is making a claim about a class, and that class happens to be defined as the complement of another class. So, the bottom line is when you take the converse of a statement where one of the terms is a complement, move the “non-” with that term. Contraposition Contraposition is a two-step process, like obversion, but it doesn’t always lead to results that are logically equivalent to the original sentence. The contrapositive of a categorical sentence is the sentence that results from reversing subject and predicate and then replacing them with their complements. Thus “All S are P” becomes “All non-P are non-S.” The table below shows the corresponding Venn diagrams. In this case, the shading around the outside of the two circles in the contraposed form of E is meant to indicate that nothing can lie outside the two circles. Everything must be S or P or both. Like conversion, applying contraposition to two of the forms gives us statements that are logically equivalent to the original. This time, though, it is forms A and O that come through the process without changing their truth value. Original Contrapositive A: All S are P A: All non-P are non-S E: No S are P E: No non-P are non-S I: Some S are P I: Some non-P are non-S O: Some S are not P O: Some non-P are not non-S This then gives us two valid argument forms, as follows: All S are P Therefore, all non-P are non-S Some S are not P Therefore, some non-P are not non-S If you have an argument with an A or O statement as its premise and the contrapositive of that statement as the conclusion, you know it must be valid. Whenever the premise is true, the conclusion must be true, because the two statements are logically equivalent. On the other hand, if you had an E or an I statement as the premise, the truth of the conclusion is undetermined, so these arguments would not be valid. Evaluating Short Arguments So far we have seen eight valid forms of argument with one premise: two arguments that are valid by conversion, four that are valid by obversion, and two that are valid by contraposition. As we said, short arguments like these are sometimes called “immediate inferences,” because your brain just flits automatically from the truth of the premises to the truth of the conclusion. Now that we have identified these valid forms of inference, we can use this knowledge to see whether some of the arguments we encounter in ordinary language are valid. We can now tell in a few cases if our brain is right to flit so seamlessly from the premise to the conclusion. In the real world, the inferences we make are messy and hard to classify. Right now we are just going to deal with a limited subset of inferences: immediate inferences that might be based on conversion, obversion, or contraposition. Let’s start with the uncontroversial premise “All dogs are mammals.” Can we infer from this that all non-mammals are non-dogs? In canonical form, the argument would look like this. All dogs are mammals Therefore, all non-mammals are non-dogs. Evaluating an immediate inference like this is a four-step process. First, identify the subject and predicate classes. Second, draw the Venn diagram for the premise. Third, see if the Venn diagram shows that the conclusion must be true. If it must be, then the argument is valid. Finally, if the argument is valid, identify the process that makes it valid. (You can skip this step if the argument is invalid.) For the argument above, the result of the first two steps would look like this: S: Dogs P: Mammals The Venn diagram for the premise shades out the possibility that there are dogs that aren’t mammals. For step three, we ask, does this mean the conclusion must be true? In this case, it does. The same shading implies that everything that is not a mammal must also not be a dog. In fact, the Venn diagram for the premise and the Venn diagram for the conclusion are the same. So the argument is valid. This means that we must go on to step four and identify the process that makes it valid. In this case, the conclusion is created by reversing subject and predicate and taking their complements, which means that this is a valid argument by contraposition. Now, remember what it means for an argument to be valid. As we said in Chapter 2, an argument is valid if it is impossible for the premises to be true and the conclusion false. This means that we can have a valid argument with false premises, so long as it is the case that if the premises were true, the conclusion would have to be true. So if the argument above is valid, then so is this one: All dogs are reptiles. Therefore, all non-reptiles are non-dogs. The premise is now false: all dogs are not reptiles. However, if all dogs were reptiles, then it would also have to be true that all non-reptiles are non-dogs. The Venn diagram works the same way. S: Dogs P: Reptiles The Venn diagram for the premise still matches the Venn diagram for the conclusion. Only the labels have changed. So arguments by transposition, just like any argument, can be valid even if they have false premises. The same is true for arguments by conversion and obversion. Arguments like these can also be invalid, even if they have true premises and a true conclusion. Remember that A statements are not logically equivalent to their converse. So this is an invalid argument with a true premise and a false conclusion: All dogs are mammals. Therefore, all mammals are dogs. Our Venn diagram test shows that this is invalid. Steps one and two give us this for the premise: S: Dogs P: Mammals But this is the Venn diagram for the conclusion: S: Dogs P: Mammals This is an argument by conversion on a mood-A statement, which is invalid. The argument remains invalid, even if we substitute in a predicate where the conclusion happens to be true. For instance this argument is invalid. All dogs are Canis familiaris. Therefore, all Canis familiaris are dogs. The Venn diagrams for the premise and conclusion of this argument will be just like the ones for the previous argument, just with different labels. So even though the argument has a true premise and a true conclusion, it is still invalid, because it is possible for an argument of this form to have a true premise and a false conclusion. This is an unreliable argument form that just happened, in this instance, not to lead to a false conclusion. This again is just a variation on a theme we saw in Chapter 2, when we saw an invalid argument for the conclusion that Paris was in France. V. The Traditional Square of Opposition We have seen that conversion, obversion, and contraposition allow us to identify some valid one-premise arguments. There are actually more we can find out there, but investigating them is a bit more complicated. The original investigation made by the Aristotelian philosophers made an assumption that logicians no longer make. To help you understand all sides of the issue, we will begin by looking at things in the traditional Aristotelian fashion, and then in the next section move on to the modern way of looking at things. When Aristotle was first investigating these four kinds of categorical statements, he noticed that they conflicted with each other in different ways. If you are just thinking casually about it, you might say that “No S are P” is somehow “the opposite” of “All S are P.”But isn’t the real “opposite” of “All S are P” actually “Some S are not P”? Aristotle, in his book On Interpretation, notes that the real opposite of A is O, because one must always be true and the other false. If we know that “All dogs are mammals” is true, then we know “some dog is not a mammal” is false. On the other hand, if “All dogs are mammals” is false then “some dog is not a mammal” must be true. We said earlier that when two propositions must have opposite truth values they are called contradictories. Aristotle noted that A and O sentences are contradictory in this way. Forms E and I also form a contradictory pair. If “Some dogs are mammals” then “No dogs are mammals” is false, and if “Some dogs are mammals” is false, then “No dogs are mammals” is true. Mood-A and mood-E statements are opposed to each other in a different way. Aristotle claimed that they can’t both be true, but could both be false. Take the statements “All dogs are strays” and “No dogs are strays.” We know that they are both false, because some dogs are strays and others aren’t. However, it is also clear that they could not both be true. When a pair of statements cannot both be true, but might both be false, the Aristotelian tradition says they are contraries. Aristotle’s idea of a pair of contraries is really just a specific case of a set of sentences that are inconsistent. These distinctions, plus a few other comments from Aristotle, were developed by his later followers into an idea that came to be known as the square of opposition. The square of opposition is simply the diagram you see below. It is a way of representing the four basic propositions and the ways they relate to one another. As we said before, this way of picturing the proposition turned out to make a problematic assumption. To emphasize that this is no longer the way logicians view things, we will call this diagram the traditional square of opposition. The traditional square of opposition begins by picturing a square with A, E, I, and O at the four corners. The lines between the corners then represent the ways that the kinds of propositions can be opposed to each other. The diagonal lines between A and O and between E and I represent contradiction. These are pairs of propositions where one has to be true and the other false. The line across the top represents contraries. These are propositions that Aristotle thought could not both be true, although they might both be false. In our diagram of the traditional square of opposition, we have actually drawn each relationship as a pair of lines, representing the kinds of inferences you can make in that relationship. Contraries cannot both be true. So we know that if one is true, the other must be false. This is represented by the two lines going from a T to an F. Notice that there aren’t any lines here that point from an F to something else. This is because you can’t infer anything about contrary statements if you just know that one is false. For the contradictory statements, on the other hand, we have drawn double-headed arrows. This is because we know both that the truth of one statement implies that the other is false and that the falsity of one statement implies the truth of the other. Contraries and contradictories just give us the diagonal lines and the top line of the square. There are still three other sides to investigate. Form I and form O are called subcontraries. In the traditional square of opposition, their situation is reversed from that of A and E. Statements of forms A and E cannot both be true, but they can both be false. Statements of forms I and O cannot both be false, but they can both be true. Consider the sentences “Some people in the classroom are paying attention” and “Some people in the classroom are not paying attention.” It is possible for them both to be true. Some people are paying attention and some aren’t. But the two sentences couldn’t both be false. That would mean that everyone in the room was neither paying attention nor not paying attention. But they have to be doing one or the other! This means that there are two inferences we can make about subcontraries. We know that if I is false, O must be true, and vice versa. This is represented in the traditional square of opposition by arrows going from Fs on one side to Ts on the other. This is reversed from the way things were on the top of the square with the contraries. Notice that this time there are no arrows going away from a T. This is because we can’t infer anything about subcontraries if all we know is that one is true. The trickiest relationship is the one between universal statements and their corresponding particulars. We call this subalternation. Both of the statements in these pairs could be true, or they could both be false. However, in the traditional square of opposition, if the universal statement is true, its corresponding particular statement must also be true. For instance, “All dogs are mammals” implies that some dogs are mammals. Also, if the particular statement is false, then the universal statement must also be false. Consider the statement “Some dinosaurs had feathers.” If that statement is false, if no dinosaurs had feathers, then “All dinosaurs have feathers” must also be false. Something like this seems to be true on the negative side of the diagram as well. If “No dinosaurs have feathers” is true, then you would think that “some dinosaurs do not have feathers” is true. Similarly, if “some dinosaurs do not have feathers” is false, then “No dinosaurs have feathers” cannot be true either. In our diagram for the traditional square of opposition, we represent subalternation by a downward arrow for truth and an upward arrow for falsity. We can infer something here if we know the top is true, or if we know the bottom is false. In other situations, there is nothing we can infer. Note, by the way, that the language of subalternation works a little differently than the other relationships. With contradiction, we say that each sentence is the “contradictory” of the other. The relationship is symmetrical. With subalternation, we say that the particular sentence is the “subaltern” of the universal one, but not the other way around. As with the processes of conversion, obversion, and contraposition, we can use the traditional square of opposition to evaluate arguments written in canonical form. It will help us here to introduce the phrase “It is false that” to some of our statements, so that we can make inferences from the truth of one proposition to the falsity of another. This, for instance, is a valid argument, because A and O statements are contradictories. All humans are mortal. Therefore, it is false that some humans are not mortal. The argument above is an immediate inference, like the arguments we saw in the previous section, because it only has one premise. It is also similar to those arguments in that the conclusion is actually logically equivalent to the premise. This will not be the case for all immediate inferences based on the square of opposition, however. This is a valid argument, based on the subaltern relationship, but the premise and the conclusion are not logically equivalent. It is false that some humans are dinosaurs. Therefore, it is false that all humans are dinosaurs. VI. Existential Import and the Modern Square of Opposition The traditional square of opposition seems straightforward and fairly clever. Aristotle made an interesting distinction between contraries and contradictories, and subsequent logicians developed it into a nifty little diagram. So why did we have to keep saying things like “Aristotle thought” and “according to the traditional square of opposition.” What is wrong here? The traditional square of opposition goes awry because it makes assumptions about the existence of the things being talked about. Remember that when we drew the Venn diagram for “All S are P,” we shaded out the area of S that did not overlap with P to show that nothing could exist there. We pointed out, though, that we did not put a little x in the intersection between S and P. Statements of the form A ruled out the existence of one kind of thing, but they did not assert the existence of another. The A proposition, “All dogs are mammals,” denies the existence of any dog that is not a mammal, but it does not assert the existence of some dog that is a mammal. But why not? Dogs obviously do exist. The problem comes when you start to consider categorical statements about things that don’t exist, for instance “All unicorns have one horn.” This seems like a true statement, but unicorns don’t exist. Perhaps what we mean by “All unicorns have one horn” is that if a unicorn existed, then it would have one horn. But if we interpret the statement about unicorns that way, shouldn’t we also interpret the statement about dogs that way? Really all we mean when we say “All dogs are mammals” is that if there were dogs, then they would be mammals. It takes an extra assertion to point out that dogs do, in fact, exist. The issue we are discussing here is called existential import. A sentence is said to have existential import if it asserts the existence of the things it is talking about. The diagrams below show the two ways you could draw Venn diagrams for an A statement, with the x, as in the traditional interpretation, and without, as in our interpretation. This is “all S are P” without existential import (Modern). This is “all S are P” with existential import (Traditional). If you interpret A statements in the traditional way, they are always false when you are talking about things that don’t exist. So, “All unicorns have one horn” is false in the traditional interpretation. On the other hand, in the modern interpretation all statements about things that don’t exist are true. “All unicorns have one horn” simply asserts that there are no multi-horned unicorns, and this is true because there are no unicorns at all. We call this vacuous truth. Something is vacuously true if it is true simply because it is about things that don’t exist. Note that all statements about nonexistent things become vacuously true if you assume they have no existential import, even a statement like “All unicorns have more than one horn.” A statement like this simply rules out the existence of unicorns with one horn or fewer, and these don’t exist because unicorns don’t exist. This is a complicated issue that will come up again in later chapters when we consider conditional statements. For now just assume that this makes sense because you can make up any stories you want about unicorns. Any statement can be read with or without existential import, even the particular ones. Consider the statements “Some unicorns are rainbow colored” and “Some unicorns are not rainbow colored.” You can argue that both of these statements are true, in the sense that if unicorns existed, they could come in many colors. If you say these statements are true, however, you are assuming that particular statements do not have existential import. As Terence Parsons (2012) points out, you can change the wording of particular categorical statements in English to make them seem like they do or do not have existential import. “Some unicorns are not rainbow colored” might have existential import, but “Not every unicorn is rainbow colored” doesn’t seem to. So what does this have to do with the square of opposition? A lot of the claims made in the traditional square of opposition depend on assumptions about which statements have existential import. For instance, Aristotle’s claim that contrary statements cannot both be true requires that A statements have existential import. Think about the sentences “All dragons breathe fire” and “no dragons breathe fire.” If the first sentence has no existential import, then both sentences could actually be true. They are both ruling out the existence of certain kinds of dragons and are correct because no dragons exist. In fact, the entire traditional square of opposition may seem to fall apart if you assume that all four forms of a categorical statement have existential import. There is an interesting piece on this here: https://philpapers.org/rec/KLIEIA. Terrence Parsons shows how we can derive a contradiction in this situation. Parsons, Terence. “Things that are right with the traditional square of opposition.” Logica Universalis 2 (1), 2008, 3–11. Consider the I statement “Some dragons breathe fire.” If you interpret it as having existential import, it is false, because dragons don’t exist. But then its contradictory statement, the E statement “No dragons breathe fire” must be true. And if that statement is true, and has existential import, then its subaltern, “Some dragon does not breathe fire” is true. But if it has existential import, it can’t be true, because dragons don’t exist. In logic, the worst thing you can ever do is contradict yourself, but that is what we have just done. So we have to change the traditional square of opposition. The way some textbooks talk about the problem, you’d think that for two thousand years logicians were simply ignorant about the problem of existential import and thus woefully confused about the square of opposition, until finally George Boole wrote The Laws of Thought (1854) and found the one true solution to the problem. In fact, there was an extensive discussion of existential import from the 12th to the 16th centuries, mostly under the heading of the “supposition” of a term. Very roughly, we can say that the supposition of a term is the way it refers to objects, or what we now call the “denotation” of the term (Read 2015). So in “All people are mortal” the supposition of the subject term is all of the people out there in the world. Or, as the medievals sometimes put it, the subject term “supposits” all the people in the world. At least some medieval thinkers had a theory of supposition that made the traditional square of opposition work. Terrance Parsons has argued for the importance of one solution, found most clearly in the writings of William of Ockham. Under this theory, affirmative forms A and I had existential import, but the negative forms E and O did not. We would say that a statement has existential import if it would be false whenever the subject or predicate terms refer to things that don’t exist. To put the matter more precisely, we would say that the statement would be false whenever the subject or predicate terms “fail to refer.” Linguistic philosophers these days prefer say that a term “fails to refer” rather than saying that it “refers to something that doesn’t exist,” because referring to things that don’t exist seems impossible. In any case, Ockham describes the supposition of affirmative propositions a manner that is quite similar to way we would describe the reference of terms in those propositions. For medievals, terms have multiple types of supposition and it is generally agreed that modern equivalents are just that, equivalents, not perfect matches. Again, if the proposition supposes the existence of something in the world, the medievals would say it “supposits.” Ockham says “In affirmative propositions a term is always asserted to supposit for something. Thus, if it supposits for nothing the proposition is false.” Ockham, William of. 1974. Ockham’s theory of terms: Part I of the Summa logicae. Translated by Michael J. Loux. University of Notre Dame Press, 1974, 206. On the other hand, failure to refer or to supposit actually supports the truth of negative propositions: “in negative propositions the assertion is either that the term does not supposit for something or that it supposits for something of which the predicate is truly denied. Thus a negative proposition has two causes of truth.” Ibid. So, for Ockham, affirmative statements about nonexistent objects are false. “All unicorns have one horn” and “Some unicorns are rainbow colored” are false, because there are no unicorns. Negative statements, on the other hand, are vacuously true. “No unicorns are rainbow colored” and “No unicorns have one horn” are both true. There are no rainbow-colored unicorns out there, and no one horned unicorns out there, because there are no unicorns out there. The O statement “Some unicorns are not rainbow colored” is also vacuously true. This might be harder to see, but it helps to think of the statement as saying “It is not the case that every unicorn is rainbow colored.” This way of thinking about existential import leaves the traditional square of opposition intact, even in cases where you are referring to nonexistent objects. Contraries still cannot both be true when you are talking about nonexistent objects, because the A proposition will be false, and the E vacuously true. “All dragons breathe fire” is false, because dragons don’t exist, and “No dragons breathe fire” is vacuously true for the same reason. Similarly, subcontraries cannot both be false when talking about dragons and whatnot, because the I will always be false and the O will always be true. You can go through the rest of the relationships and show that similar arguments hold. Boole proposed a different solution, which is now taken as the standard way to do things. Instead of looking at the division between positive and negative statements, Boole looked at the division between singular and universal propositions. The universal statements A and E do not have existential import, but the particular statements I and O do have existential import. Thus all particular statements about nonexistent things are false and all universal statements about nonexistent things are vacuously true. John Venn was building on the work of George Boole. His diagrams avoided the problems that Euler had by using a Boolean interpretation of mood-A statements, where they really just assert that something is impossible. In fact, the whole system of Venn diagrams embodies Boole’s assumptions about existential import. The particular forms I and O have you draw an x, indicating that something exists. The other two forms just have us shade in regions to indicate that certain combinations of subject and predicate are impossible. Thus A and E statements like “All dragons breathe fire” or “No dragons are friendly” can be true, even though no dragons exist. Venn diagrams don’t even have the capacity to represent Ockham’s understanding of existential import. We can represent A statements as having existential import by adding an x, as we did above. However, we have no way to represent the O form without existential import. We have to draw the x, indicating existence. We don’t have a way of representing O form statements about nonexistent objects as vacuously true. The Boolean solution to the question of existential import leaves us with a greatly restricted form of the square of opposition. Contrary statements are both vacuously true when you refer to nonexistent objects, because neither have existential import. Subcontrary statements are both false when you refer to nonexistent objects, because they do have existential import. Finally, the subalterns of vacuously true statements are false, while on the traditional square of opposition they had to be true. The only thing remaining from the traditional square of opposition is the relationship of contradiction, as you can see in the modern square of opposition, below:
Chapter 4: Categorical Syllogisms This chapter is based on For All X, The Lorain County Remix, remixed by J. Robert Loftis. I. Standard Form, Mood, and Figure So far we have just been looking at very short arguments using categorical statements. The arguments just had one premise and a conclusion that was often logically equivalent to the premise. For most of the history of logic in the West, however, the focus has been on arguments that are a step more complicated called categorical syllogisms. A categorical syllogism is a two-premise argument composed of categorical statements. Aristotle began the study of this kind of argument in his book the Prior Analytics. This work was refined over the centuries by many thinkers in the Pagan, Christian, Jewish, and Islamic traditions until it reached the form it is in today. There are actually all kinds of two-premise arguments using categorical statements, but Aristotle only looked at arguments where each statement is in one of the moods A, E, I, or O. The arguments also had to have exactly three terms, arranged so that any two pairs of statements will share one term. Let’s call a categorical syllogism that fits this narrower description an Aristotelian syllogism Here is a typical Aristotelian syllogism using only mood-A sentences: Premise 1: All mammals are vertebrates. Premise 2: All dogs are mammals. Conclusion: Therefore, All dogs are vertebrates. Notice how the statements in this argument overlap each other. Each statement shares a term with the other two. Premise 2 shares its subject term with the conclusion and its predicate with Premise 1. Thus, there are only three terms spread across the three statements. Aristotle dubbed these the major, middle, and minor premises, but there was initially some confusion about how to define them. In the 6th century, the Christian philosopher John Philoponus, drawing on the work of his pagan teacher Ammonius, decided to arbitrarily designate the major term as the predicate of the conclusion, the minor term as the subject of the conclusion, and the middle term as the one term of the Aristotelian syllogism that does not appear in the conclusion. So, in the argument above, the major term is “vertebrate,” the middle term is “mammal,” and the minor term is “dog.” We can also define the major premise as the one premise in an Aristotelian syllogism that names the major term, and the minor premise as the one premise that names the minor term. So, in the argument above, Premise 1 is the major premise and Premise 2 is the minor premise. With these definitions in place, we can now define the standard form for an Aristotelian syllogism in logically structured English. Recall that in the Categorical Statements chapter, we started standardizing our language into something we called “logically structured English” in order to remove ambiguity and to make its logical structure clear. The first step was to define the standard form for a categorical statement. Now we do the same thing for an Aristotelian syllogism. We say that an Aristotelian syllogism is in standard form for logically structured English if and only if these criteria have been met: (1) all of the individual statements are in standard form, (2) each instance of a term is in the same format and is used in the same sense, (3) the major premise appears first, followed by the minor premise, and then the conclusion. Once we standardize things this way, we can actually catalog every possible form of an Aristotelian syllogism. To begin with, each of the three statements can take one of four forms: A, E, I, or O. This gives us 4 × 4 × 4, or 64 possibilities. These 64 possibilities are called the syllogism mood, and we designate it just by writing the three letters of the moods of the statements that make it up. So, the mood of the argument above is simply AAA. In addition to varying the kind of statements we use in an Aristotelian syllogism, we can also vary the placement of the major, middle, and minor terms. There are four ways we can arrange them that fit the definition of an Aristotelian syllogism in standard form, called the four figures of the Aristotelian Syllogism. The following table demonstrates them all: Figure 1: P1: M P P2: S M C: S P Figure 2: P1: P M P2: S M C: S P Figure 3: P1: M P P2: M S C: S P Figure 4: P1: P M P2: M S C: S P Here P stands for the major term, S for the minor term, and M for the middle. The thing to pay attention to is the placement of the middle terms. In figure 1, the middle terms form a line slanting down to the right. In figure 2, the middle terms are both pushed over to the right. In figure 3, they are pushed to the left, and in figure 4, they slant in the opposite direction from figure 1. The combination of 64 moods and 4 figures gives us a total of 256 possible Aristotelian syllogisms. We can name them by simply giving their mood and figure. So, this is OAO-3: Some M are not P. All M are S. Some S are not P. For most of the arguments in this chapter, we will state the premises first, and the conclusion last, separating the premises from the conclusion with a line. Syllogism OAO-3 is a valid argument. We will be able to prove this with Venn diagrams in the next section. For now, just read it over and try to see intuitively why it is valid. Most of the 256 possible syllogisms, however, are not valid. In fact, most of them, like IIE-2, are quite obviously invalid: Some P are M. Some S are M. No S are P. Given an Aristotelian syllogism in ordinary English, we can transform it into standard form in logically structured English and identify its mood and figure. Consider the following: No geckos are cats. I know this because all geckos are lizards, but cats aren’t lizards. The first step is to identify the conclusion, using the basic skills you acquired earlier. In this case, you can see that “because” is a premise indicator word, so the statement before it, “No geckos are cats,” must be the conclusion. Step two is to identify the major, middle, and minor terms. Remember that the major term is the predicate of the conclusion, and the minor term is the subject. So here the major term is “cats,” the minor term is “geckos.” The leftover term, “lizards,” must then be the middle term. We show that we have identified the major, middle, and minor terms by writing a translation key. A translation key is just a list that assigns English phrases or sentences to variable names. For categorical syllogisms, this means matching the English phrases for the terms with the variables S, M, and P. S: Geckos M: Lizards P: Cats Step three is to write the argument in canonical form using variables for the terms. The last statement, “cats aren’t lizards,” is the major premise, because it has the major term in it. We need to change it to standard form, however, before we substitute in the variables. So first we change it to “No cats are lizards.” Then we write “No S are M.” For the minor premise and the conclusion we can just substitute in the variables, so we get this: No P are M. All S are M. No S are P. Step four is to identify mood and figure. We can see that this is figure 2, because the middle term is in the predicate of both premises. Looking at the form of the sentences tells us that this is EAE. II. Testing Validity We have seen that there are 256 possible categorical arguments that fit Aristotle’s requirements. Most of them are not valid, and as you probably saw in the exercises, many don’t even make sense. In this section, we will learn to use Venn diagrams to sort the good arguments from the bad. The method we will use will simply be an extension of what we did in the last chapter, except with three circles instead of two. Venn Diagrams for Single Propositions In the previous chapter, we drew Venn diagrams with two circles for arguments that had two terms. The circles partially overlapped, giving us four areas, each of which represented a way an individual could relate to the two classes. So area 1 represented things that were S but not P, etc. . Now that we are considering arguments with three terms, we will need to draw three circles, and they need to overlap in a way that will let us represent the eight possible ways an individual can be inside or outside these three classes. So, in this diagram, area 1 represents the things that are S but not M or P, area 2 represents the things that are M but not S or P, etc. As before, we represent universal statements by filling in the area that the statement says cannot be occupied. The only difference is that now there are more possibilities. So, for instance, there are now four mood-A propositions that can occur in the two premises. The major premise can either be “All P are M” or “All M are P,” and the minor premise can be either “All S are M” or “All M are S.” The Venn diagrams for those four sentences are given in the table below: Mood A statement, Major Premise All P are M Mood A statement, Major Premise All M are P Mood A statement, Minor Premise All S are M Mood A statement, Minor Premise All M are S Similarly, there are four mood-E propositions that can occur in the premises of an Aristotelian syllogism: “No P are M,” “No M are P,” “No S are M,” and “No M are S.” And again, we diagram these by shading out overlap between the two relevant circles. In this case, however, the first two statements are equivalent by conversion, as are the second two. Thus, we only have two diagrams to worry about, as seen on the table below: Mood E statements, major premise No M are P, or No P are M Mood E statements, minor premise No M are S, or no S are M Particular propositions are a bit trickier. Consider the statement “Some M are P.” With a two-circle diagram, you would just put an x in the overlap between the M circle and the P circle. But with the three-circle diagram, there are now two places we can put it. It can go in either area 6 or area 7: The solution here will be to put the x on the boundary between areas 6 and 7, to represent the fact that it could go in either location: Sometimes, however, you won’t have to draw the x on a border between two areas, because you will already know that one of those areas can’t be occupied. Suppose, for instance, that you want to diagram “Some M are P,” but you already know that all M are S. You would diagram “All M are S” like this: Then, when it comes time to add the x for “Some M are P,” you know that it has to go in the exact center of the diagram: Venn Diagrams for Full Syllogisms In the last chapter, we used Venn diagrams to evaluate arguments with single premises. It turned out that when those arguments were valid, the conclusion was logically equivalent to the premise, so they had the exact same Venn diagram. This time we have two premises to diagram, and the conclusion won’t be logically equivalent to either of them. Nevertheless, we will find that for valid arguments, once we have diagrammed the two premises, we will also have diagrammed the conclusion. First, we need to specify a rule about the order to diagram the premises in: if one of the premises is universal and the other is particular, diagram the universal one first. This will allow us to narrow down the area where we need to put the x from the particular premise, as in the example above where we diagrammed “Some M are P” assuming that we already knew that all M are S. Let’s start with a simple example, an argument with the form AAA-1. All M are P. All S are M. All S are P. Since both premises are universal, it doesn’t matter what order we do them in. Let’s do the major premise first. The major premise has us shade out the parts of the M circle that don’t overlap the P circle, like this: The second premise, on the other hand, tells us that there is nothing in the S circle that isn’t also in the M circle. We put that together with the first diagram, and we get this: From this we can see that the conclusion must be true. All S are P, because the only space left in S is the area in the exact center, area 7. Now let’s look at an argument that is invalid. One of the interesting things about the syllogism AAA-1 is that if you change the figure, it ceases to be valid. Consider AAA-2. All P are M. All S are M. All S are P. Again, both premises are universal, so we can do them in any order, so we will do the major premise first. This time, the major premise tells us to shade out the part of P that does not overlap M. The second premise adds the idea that all S are M, which we diagram like this: Now we ask if the diagram of the two premises also shows that the conclusion is true. Here the conclusion is that all S are P. If this diagram had made this true, we would have shaded out all the parts of S that do not overlap P. But we haven’t done that. It is still possible for something to be in area 5. Therefore, this argument is invalid. Now let’s try an argument with a particular statement in the premises. Consider the argument IAI-1: Some M are P. All S are M. Some S are P. Here, the second premise is universal, while the first is particular, so we begin by diagramming the universal premise. Then we diagram the particular premise “Some M are P.” This tells us that something is in the overlap between M and P, but it doesn’t tell us whether that thing is in the exact center of the diagram or in the area for things that are M and P but not S. Therefore, we place the x on the border between these two areas. Now we can see that the argument is not valid. The conclusion asserts that something is in the overlap between S and P. But the x we drew does not necessarily represent an object that exists in that overlap. There is something out there that could be in area 7, but it could just as easily be in area 6. The second premise doesn’t help us, because it just rules out the existence of objects in areas 1 and 4. For a final example, let’s look at a case of a valid argument with a particular statement in the premises. If we simply change the figure of the argument in the last example from 1 to 3, we get a valid argument. This is the argument IAI-3: Some M are P. All M are S. Some S are P. Again, we begin with the universal premise. This time it tells us to shade out part of the M circle. But now, since we filled in the parts of M that don’t overlap with S, when we add the particular premise “Some M are P,” we have to put the x in the exact center of the diagram. And now this time we see that the conclusion, “Some S are P,” has to be true based on the premises, because the X has to be in area 7. So, this argument is valid. Using this method, we can show that 15 of the 256 possible syllogisms are valid. Remember, however, that the Venn diagram method uses Boolean assumptions about existential import. If you make other assumptions about existential import, you will allow more valid syllogisms, as we will see in the next section. The additional syllogisms we will be able to prove valid in the next section will be said to have conditional validity because they are valid on the condition that the objects talked about in the universal statements actually exist. The 15 syllogisms that we can prove valid using the Venn diagram method have unconditional validity. These syllogisms are given in the following table: Figure 1 Figure 2 Figure 3 Figure 4 Barbara (AAA) Camestres (AEE) Disamis (IAI) Calemes (AEE) Celarent (EAE) Cesare (EAE) Bocardo (OAO) Dimatis (IAI) Ferio (EIO) Festino (EIO) Ferison (EIO) Fresison (EIO) Darii (AII) Baroco (AOO) Datisi (AII) The names in this table come from the Christian part of the Aristotelian tradition, where thinkers were writing in Latin. Students in that part of the tradition learned the valid forms by giving each one a name. The vowels in the name represented the mood of the syllogism. So, Barbara has the mood AAA, Fresison has the mood EIO, etc. The consonants in each name were also significant: they related to a process the Aristotelians were interested in called reduction, where arguments in the later figures were shown to be equivalent to arguments in the first figure, which was taken to be more self-evident. We won’t worry about reduction in this textbook, however. The columns in this table represent the four figures. Syllogisms with the same mood also appear in the same row. So, the EIO sisters—Ferio, Festino, Ferison, and Fresison—fill up row 3. Camestres and Calemes share row 1; Celarent and Cesare share row 2; and Darii and Datisi share row 4. The names of the valid syllogisms were often worked into a mnemonic poem. The oldest known version of the poem appears in a late 13th century book called Introduction to Logic by William of Sherwood. Below is an image of the oldest surviving manuscript of the poem, digitized by the Bibliothèque Nationale de France (ms. Lat. 16617). III. Existential Import and Conditionally Valid Forms In the last section, we mentioned that you can prove more syllogisms valid if you make different assumptions about existential import. Recall that a statement has existential import if, when you assert the statement, you are also asserting the existence of the things the statement talks about. So, if you interpret a mood-A statement as having existential import, it not only asserts “All S is P,” it also asserts “S exists.” Thus, the mood-A statement “All unicorns have one horn” is false, if it is taken to have existential import, because unicorns do not exist. It is probably true, however, if you do not imagine the statement as having existential import. If anything is true of unicorns, it is that they would have one horn if they existed. We saw in the last chapter that before Boole, Aristotelian thinkers had all sorts of opinions about existential import, or, as they put it, whether a term “supposits.” This generally led them to recognize additional syllogism forms as valid. You can see this pretty quickly if you just remember the traditional square of opposition. The traditional square allowed for many more valid immediate inferences than the modern square. It stands to reason that traditional ideas about existential import will also allow for more valid syllogisms. Our system of Venn diagrams can’t represent all of the alternative ideas about existential import. For instance, it has no way of representing Ockham’s belief that mood-O statements do not have existential import. Nevertheless, it would be nice if we could expand our system of Venn diagrams to show that some syllogisms are valid if you make additional assumptions about existence. Consider the argument Barbari (AAI-1). All M are P. All S are M. Some S are P. You won’t find this argument in the list of unconditionally valid forms. This is because under Boolean assumptions about existence it is not valid. The Venn diagram, which follows Boolean assumptions, shows this. This is essentially the same argument as Barbara, but the mood-A statement in the conclusion has been replaced by a mood-I statement. We can see from the diagram that the mood-A statement “All S are P” is true. There is no place to put an S other than in the overlap with P. But we don’t actually know that the mood-I statement “Some S is P” is true, because we haven’t drawn an x in that spot. Really, all we have shown is that if an S existed, it would be P. But by the traditional square of opposition (see the previous chapter), we know that the mood-I statement is true. The traditional square, unlike the modern one, allows us to infer the truth of a particular statement given the truth of its corresponding universal statement. This is because the traditional square assumes that the universal statement has existential import. It is really two statements, “All S is P” and “Some S exists.” Because the mood-A statement is actually two statements on the traditional interpretation, we can represent it simply by adding an additional line to our argument. It is always legitimate to change an argument by making additional assumptions. The new argument won’t have the exact same impact on the audience as the old argument. The audience will now have to accept an additional premise, but in this case all we are doing is making explicit an assumption that the Aristotelian audience was making anyway. The expanded argument will look like this: All M are P. All S are M. Some S exists.* Some S are P Here the asterisk after “Some S exists” indicates that we are looking at an implicit premise that has been made explicit. Now that we have an extra premise, we can add it to our Venn diagram. Since there is only one place for the S to be, we know where to put our x. In this argument S is what we call the “critical term.” The critical term is the term that names things that must exist in order for a conditionally valid argument to be actually valid. In this argument, the critical term was S, but sometimes it will be M or P. We have used Venn diagrams to show that Barbari is valid once you include the additional premise. Using this method we can identify nine more forms, on top of the previous 15, that are valid if we add the right existence assumptions. Below, we repeat the table of unconditionally valid forms, and add a second table, representing the conditionally valid forms, along with what condition must be satisfied to make them valid. Unconditionally Valid Forms Figure 1 Figure 2 Figure 3 Figure 4 Condition Barbara (AAA) Camestres (AEE) Disamis (IAI) Calemes (AEE) None Celarent (EAE) Cesare (EAE) Bocardo (OAO) Dimatis (IAI) None Ferio (EIO) Festino (EIO) Ferison (EIO) Fresison (EIO) None Darii (AII) Baroco (AOO) Datisi (AII) None Conditionally Valid Forms Figure 1 Figure 2 Figure 3 Figure 4 Condition Barbari (AAI) Camestros (AEO) Calemos (AEO) S exists Celaront (EAO) Cesaro (EAO) S exists Felapton (EAO) Fesapo (EAO) M exists Darapti (AAI) M exists Bamalip (AAI) P exists Thus, we now have an expanded method for evaluating arguments using Venn diagrams. To evaluate an argument, we first use a Venn diagram to determine whether it is unconditionally valid. If it is, then we are done. If it is not, then we see if adding an existence assumption can make it conditionally valid. If we can add such an assumption, add it to the list of premises and put an x in the relevant part of the Venn diagram. If we cannot make the argument valid by including additional existence assumptions, we say it is completely invalid. Let’s run through a couple of examples. Consider the argument EAO-3. No M are P. All M are S. Some S are not P. First, we use the regular Venn diagram method to see whether the argument is unconditionally valid. We can see from this that the argument is not valid. The conclusion says that some S are not P, but we can’t tell that from this diagram. There are three possible ways something could be S, and we don’t know if any of them are occupied. Simply adding the premise S exists won’t help us, because we don’t know whether to put the x in the overlap between S and M, the overlap between S and P, or in the area that is just S. Of course, we would want to put it in the overlap between S and M, because that would mean that there is an S that is not P. However, we can’t justify doing this simply based on the premise that S exists. The premise that P exists will definitely not help us. The P would either go in the overlap between S and P or in the area that is only P. Neither of these would show “Some S is not P.” The premise “M exists” does the trick, however. If an M exists, it has to also be S but not P. And this is sufficient to show that some S is not P. We can then add this additional premise to the argument to make it valid. No M are P. All M are S. M exists.* Some S are not P. Checking it against the tables of unconditionally and conditionally valid forms, we see that we were right: this is a conditionally valid argument named Felapton. Now consider the argument EII-3: No M are P. Some M are S. Some S are P. First, we need to see if it is unconditionally valid. So, we draw the Venn diagram. The conclusion says that some S are P, but we obviously don’t know this from the diagram above. There is no x in the overlap between S and P. Part of that region is shaded out, but the rest could go either way. What about conditional validity? Can we add an existence assumption that would make this valid? Well, the x we have already drawn lets us know that both S and M exist, so it won’t help to add those premises. What about adding P? That won’t help either. We could add the premise “P exists” but we wouldn’t know whether that P is in the overlap between S and P or in the area to the right, which is just P. Therefore, this argument is invalid. And when we check the argument against the tables of unconditionally and conditionally valid forms, we see that it is not present. VI. Rules and Fallacies In this section, we are going to identify rules that all valid syllogisms amongst the 256 Aristotelian syllogisms must obey. Seeing these rules will help you understand the structure of this part of logic. We aren’t just assigning the labels “valid” and “invalid” to arguments randomly. Each of the rules we will identify is associated with a fallacy. If you violate the rule, you commit the fallacy. In the next subsection we are going to outline five basic rules and the fallacies that go with them, along with an addition rule/fallacy pair that can be derived from the initial five. All standard logic textbooks these days use some version of these rules, although they might divide them up differently. Some textbooks also include rules that we have built into our definition of an Aristotelian syllogism in standard form. For instance, other textbooks might have a rule here saying valid syllogisms can’t have four terms, or have to use terms in the same way each time. All of this is built into our definitions of an Aristotelian syllogism and standard form for such a syllogism, so we don’t need to discuss them here. Six Rules and Fallacies Rule 1: The middle term in a valid Aristotelian syllogism must be distributed at least once. Consider these two arguments: Argument 1 All M are P. All S are M. All S are P. Argument 2 All P are M. All S are M. All S are P. Argument 1 is Barbara, and is obviously valid, but if you change it to figure 2, you get Argument 2, which is obviously invalid. What causes this change? The premises in Argument 2 say that S and P are both parts of M, but they no longer tell us anything about the relationship between S and P. To see why this is the case, we need to bring back a term we saw earlier, distribution. A term is distributed in a statement if the statement makes a claim about every member of that class. So, in “All M are P” the term M is distributed, because the statement tells us something about every single M. They are all also P. The term P is not distributed in this sentence, however. We do not know anything about every single P. We know that M is in P, but not vice versa. In general, mood-A statements distribute the subject, but not the predicate. This means that when we reverse P and M in the first premise, we create an argument where S and P are distributed, but M is not. This means that the argument is always going to be invalid. This short argument can show us that arguments with an undistributed middle are always invalid: The conclusion of an Aristotelian syllogism tries to say something about the relationship between S and P. It does this using the relationship those two terms have to the third term M. But if M is never distributed, then S and P can be different, unrelated parts of M. Therefore, arguments with an undistributed middle are invalid. Syllogisms that violate this rule are said to commit the fallacy of the undistributed middle. Rule 2: If a term is distributed in the conclusion of a valid Aristotelian syllogism, then it must also be distributed in one of the premises. Suppose instead of changing Barbara from a figure 1 to a figure 2 argument, we changed it to a figure 4 argument. This is what we’d get. All P are M. All M are S. All S are P. When we changed the argument from figure 1 to figure 2, it ceased to be valid because the middle became undistributed. But this time the middle is distributed in the second premise, and the argument still doesn’t work. You can see this by filling in “animals,” “mammals,” and “dogs,” for S, M, and P. All dogs are mammals. ⇐ True All mammals are animals. ⇐ True All animals are dogs. ⇐ False This version of the argument has true premises and a false conclusion, so you know the argument form must be invalid. A valid argument form should never be able to take true premises and turn them into a false conclusion. What went wrong here? The conclusion is a mood-A statement, which means it tries to say something about the entire subject class, namely, that it is completely contained by the predicate class. But that is not what these premises tell us. The premises tell us that the subject class, animals, is actually the broadest class of the three, containing within it the classes of mammals and dogs. As with the previous rule, the problem here is a matter of distribution. The conclusion has the subject class distributed. It wants to say something about the entire subject class, animals. But the premises do not have “animals” as a distributed class. Premise 1 distributes the class “dogs” and premise 2 distributes the class “mammals.” Here is another argument that makes a similar mistake: All M are P. Some S are not M. Some S are not P. This time the conclusion is a mood-O statement, so the predicate term is distributed. We are trying to say something about the entire class P. But again, the premises do not say something about the entire class P. P is undistributed in the major premise. These examples illustrate rule 2: If a term is distributed in the conclusion, it must also be distributed in the corresponding premise. Arguments that violate this rule are said to commit the fallacy of illicit process. This fallacy has two versions, depending on which term is not distributed. If the subject term is the one that is not distributed, we say that the argument commits the fallacy of an illicit minor. If the predicate term isn’t distributed, we say that the argument commits the fallacy of the illicit major. Some particularly silly arguments commit both. The justification for this rule is easy enough to see. If the conclusion makes a claim about all members of a class, but the premises only make a claim about some members of the class, the conclusion clearly says more than what the premises justify. Rule 3: A valid Aristotelian syllogism cannot have two negative premises. Consider the argument “No P are M, and no M are S, therefore _________.” Try to find a conclusion about S and P that you can draw from this pair of premises. Hopefully you have convinced yourself that there is no conclusion to be drawn from the premises above using standard Aristotelian format. No matter what mood you put the conclusion in, it will not follow from the premises. The same thing would be true of any syllogism with two negative premises. We could show this conclusively by running through the 16 possible combinations of negative premises and figures. A more intuitive proof of this rule goes like this: The conclusion of an Aristotelian syllogism must tell us about the relationship between subject and predicate. But if both premises are negative then the middle term must be disjoint, either entirely or partially, from the subject and predicate terms. An argument that breaks this rule is said to commit the fallacy of exclusive premises. Rule 4: A valid Aristotelian syllogism can have a negative conclusion if and only if it has exactly one negative premise. Again, let’s start with examples, and try to see what is wrong with them. Argument 1: All M are P. All P are M. Some S are not P. Argument 2: No P are M. All S are M. All S are P. These arguments are so obviously invalid, you might look at them and say, “Sheesh, is there anything right about them?” Actually, these arguments obey all the rules we have seen so far. Look at Argument 1. Premise 1 ensures that the middle term is distributed. The conclusion is mood O, which means the predicate is distributed, but P is also distributed in the second premise. The argument does not have two negative premises. A similar check will show that Argument 2 also obeys the first three rules. These arguments illustrate an important premise that is independent of the previous three. You can’t draw a negative conclusion from two affirmative premises, and you cannot drawn an affirmative conclusion if there is a negative premise. Because the previous rule tells us that you can never have two negative premises, we can actually state this rule quite simply: an argument can have a negative conclusion if and only if it has exactly one negative premise. (The phrase “if and only if” means that the rule goes both ways. If you have a negative conclusion, then you must have one negative premise, and if you have one negative premise, you must have a negative conclusion.) To see why this rule is justified, you need to look at each part of it separately. First, consider the case with the affirmative conclusion. An affirmative conclusion tells us that some or all of S is contained in P. The only way to show this is if some or all of S is in M, and some or all of M is in P. You need a complete chain of inclusion. Therefore, if an argument has a negative premise, it cannot have an affirmative conclusion. On the other hand, if an argument has a negative conclusion, it is saying that S and P are at least partially separate. But if you have all affirmative premises, you are never separating classes. Also, a valid argument cannot have two negative premises. Therefore, a valid argument with a negative conclusion must have exactly one negative premise. There is not a succinct name for the fallacy that goes with violating this rule, because this is not a mistake people commonly make. We will call it the negative-affirmative fallacy. Rule 5: A valid Aristotelian syllogism cannot have two universal premises and a particular conclusion. This rule is a little different than the previous ones, because it really only applies if you take a Boolean approach to existential import. Consider Barbari, the sometimes maligned step-sister of Barbara: All M are P. All S are M. Some S are P. This syllogism is not part of the core 15 valid syllogisms we identified with the Venn diagram method using Boolean assumptions about existential import. The premises never assert the existence of something, but the conclusion does. And this is something that is generally true under the Boolean interpretation. Universal statements never have existential import and particular statements always do. Therefore, you cannot derive a particular statement from two universal statements. Some textbooks act as if the ancient Aristotelians simply overlooked this rule. They say things like “the traditional account paid no attention to the problem of existential import” which is simply false. As we have seen in chapter three, the Latin part of the Aristotelian tradition engaged in an extensive discussion of the issue from the 12th to the 16th centuries, under the heading “supposition of terms.” And at least some people, like William of Ockham, had consistent theories that show why syllogisms like Barbari were valid. In this textbook, we handle the existential import of universal statements by adding a premise, where appropriate, which makes the existence assumption explicit. So, Barbari should look like this. All M are P. All S are M. Some S exist.* Some S are P. Adding this premise merely gives a separate line in the proof for an idea that Ockham said was already contained in premise 2. And if we make it a practice of adding existential premises to arguments like these, Rule 5 still holds true. You cannot conclude a particular statement from all universal premises. However, in this case, we do have a particular premise, namely, P3. So, if we provide this reasonable accommodation, we can see that syllogisms like Barbari are perfectly good members of the valid syllogism family. We will say, however, that an argument like this that does not provide the extra premise commits the existential fallacy. Proving the Rules For each rule, we have presented an argument that any syllogism that breaks that rule is invalid. It turns out that the reverse is also true. If a syllogism obeys all five of these rules, it must be valid. In other words, these rules are sufficient to characterize validity for Aristotelian syllogisms. It is good practice to actually walk through a proof that these five rules are sufficient for validity. After all, that sort of proof is what formal logic is really all about. Imagine we have a syllogism that obeys the five rules above. We need to show that it must be valid. There are four possibilities to consider: the conclusion is either mood A, mood E, mood I, or mood O. If the conclusion is in mood A, then we know that S is distributed in the conclusion. If the syllogism obeys rules 1 and 2, then we know that S and M are distributed in the premises. Rule 4 tells us that both premises must be affirmative, so the premises can’t be I or O. They can’t be E, either, because E does not distribute any terms, and we know that terms are distributed in the premises. Therefore, both premises are in mood A. Furthermore, we know that they are in the first figure, because they have to distribute S and M. Therefore, the syllogism is Barbara, which is valid. Now suppose the conclusion is in mood E. By rule 4, we have one negative and one affirmative premise. Because mood-E statements distribute both subject and predicate, rules 1 and 2 tell us that all three terms must be distributed in the premises. Therefore, one premise must be E, because it will have to distribute two terms. Since E is negative, the other premise must be affirmative, and since it has to distribute a term, it can’t be I. So we know one premise is A and the other E. If all the terms are distributed, this leaves us four possibilities: EAE-1, EAE-2, AEE-2, and AEE-4. These are the valid syllogisms Celarent, Cesare, Camestres, and Calemes. Next up, consider the case where the conclusion is in mood I. By rule 4, it has two affirmative premises, and by rule 5 both premises cannot be universal. This means that one premise must be an affirmative particular statement, that is, mood I. But we also know that by rule 1 some premise must distribute the middle term. Since this can’t be the mood-I premise, it must be the other premise, which then must be in mood A. Again we are reduced to four possibilities: AII-1, AII-2, IAI-3, and IAI-4, which are the valid syllogisms Darii, Datisi, Disamis, and Dimatis. Finally, we need to consider the case where the conclusion is mood O. Rule 4 tells us that one premise must be negative and the other affirmative, and rule 5 tells us that they can’t both be universal. Rules 1 and 2 tell us that M and P are distributed in the premises. This means that the premises can’t both be particular, because then one would be I and one would be O, and only one term could be distributed. So one premise must be negative and the other affirmative, and one premise must be particular and the other universal. In other words, our premises must be a pair that goes across the diagonal of the square of opposition, either an A and an O or an E and an I. With the AO pair, there are two possibilities that distribute the right terms: OAO-3 and AOO-II. These are the valid syllogisms Bocardo and Baroco. With the EI pair, there are four possibilities, which are all valid. They are the EIO sisters: Ferio, Festino, Ferison, and Fresison. So, there you have it. Those five rules completely characterize the possible valid Aristotelian syllogisms. Any other patterns you might notice among the valid syllogisms can be derived from these five rules. For instance, Problem (1) in exercise set E of Section II asked if you could have a valid Aristotelian syllogism with two particular premises. If you did that problem, hopefully you saw that the answer was “no.” We could, in fact, make this one of our five rules above. But we don’t need to. When we showed that these five rules were sufficient to characterize validity, we also showed that any other rule characterizing validity that we care to come up with can be derived from the rules we already set out. So, let’s state the idea that a syllogism cannot have two particular premises as a rule, and show how it can be derived. This will be our statement of the rule: Derived Rule 1: A valid Aristotelian syllogism cannot have two particular premises. And let’s call the associated fallacy the fallacy of particular premises. To show that this rule can be derived from the previous five, it is sufficient to show that any syllogism that violates this rule will also violate one of the previous five rules. Thus, there will always be a reason, independent of this rule, that can explain why that syllogism is false. So, suppose we have a syllogism with two particular premises. If we want to avoid violating rule 1, we need to distribute the middle term, which means that both premises cannot be mood I, because mood-I statements don’t distribute any term. We also know that both statements can’t be mood O, because rule 3 says we can’t have two negative premises. Therefore, our syllogism has one premise that is I and one premise that is O. It thus has exactly one negative premise, and by rule 4, must have a negative conclusion, either an E or an O. But an argument with premises I and O can only have one term distributed: if the conclusion is mood O, then two terms are distributed; and if it is mood E then all three terms are distributed. Thus, any syllogism that manages to avoid rules 1, 3, and 4 will fall victim to rule 2. Therefore, any syllogism with two particular premises will violate one of the five basic rules.
Chapter 6: Natural Deduction This chapter is based on Introduction to Logic and Critical Thinking, by Matthew J. Van Cleave. I. Introduction In the last chapter, we learned how to translate English language sentences into a system of symbolic logic. In this chapter, we will learn a system called “natural deduction,” a formal method of proving Predicate Logic (PL) arguments valid. Although you cannot construct a proof to show that an argument is invalid, you can construct proofs to show that an argument is valid. Learning natural deduction will teach you several valid reasoning patterns. It will also help you follow and create complex chains of reasoning. And, proofs are an efficient way to show an argument is indeed valid. Here’s how we’ll proceed. The valid forms of inference you will learn are the rules you’ll use in your proofs. Each line of the proof will be justified by citing one of these rules, with the last line of the proof being the conclusion that we are trying to ultimately establish. I will introduce eight valid forms of inference in groups. II. The First Four Valid Forms After learning these four forms, we’ll begin constructing proofs using just these, before we introduce more rules. The first three rules involve conditionals, and the fourth is a disjunction rule. 1. Modus Ponens The first form of inference is “modus ponens” which is Latin for “way that affirms.” Modus ponens has the following form: The three dots before the q is a symbol that means “therefore.” It’s the conclusion indicator we will use in this chapter. What this form says, in words, is that if we have asserted a conditional statement and we have also asserted the antecedent of that conditional statement, then we are entitled to infer the consequent of that conditional statement. For example, if I asserted the conditional, “if it is raining, then the ground is wet,” and I also asserted, “it is raining” (the antecedent of that conditional), then I (or anyone else, for that matter) am entitled to assert the consequent of the conditional, “the ground is wet.” The lowercase p and q are sentence variables; they stand for any sentence, simple or compound. Thus, any argument that has this same form is valid. For example, the following argument also has this same form (modus ponens): In this argument we can assert C according to the rule, modus ponens. This is so even though the antecedent of the conditional is itself complex (i.e., it is a conjunction). That doesn’t matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the antecedent of that conditional statement. Modus ponens says that if we have matched the antecedent of the conditional, we are entitled to infer the consequent. Here's another example of modus ponens: Notice that the first line is a conditional statement (its main operator is a horseshoe), and the second line is the antecedent of that conditional statement. This allows us to infer the consequent of the conditional. In this case, the consequent is itself a compound statement, so my conclusion should be the entire consequent, not just part of it. However, I can (and should) drop the parentheses. They’re needed in our first conditional, so we can see which horseshoe is the main connective. Our conclusion only has one connective, so it doesn’t need the parentheses to tell us what the main connective is. So, modus ponens tells us that if we have a conditional, and we have matched the antecedent exactly, then we get exactly whatever the consequent is. 2. Modus Tollens The next form of inference is called “modus tollens,” which is Latin for “the way that denies.” Modus tollens has the following form: What this form says, in words, is that if we have asserted a conditional statement and we have also asserted the negated consequent of that conditional, then we are entitled to infer the negated antecedent of that conditional statement. For example, if I asserted the conditional, “if it is raining, then the ground is wet,” and I also asserted, “the ground is not wet” (the negated consequent of that conditional), then I am entitled to assert the negated antecedent of the conditional, “it is not raining.” It is important to see that any argument that has this same form is a valid argument. For example, the following argument is also an argument with this same form: In this argument we can assert not-C according to the rule, modus tollens. This is so even though the consequent of the conditional is itself complex (i.e., it is a disjunction). That doesn’t matter. The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the negated consequent of that conditional statement. The rule modus tollens says that if we have that much, we are entitled to infer the negated antecedent of the conditional. Here’s another example of modus tollens: Both the antecedent and consequent are compound here. That’s ok, it still follows the form. As long as we have the negation of the consequent, we can infer the negation of the antecedent. Notice this time we need to keep the parentheses, because we’re adding a negation. We want to make sure we negate the entire compound statement, so the parentheses need to remain. With modus ponens, make sure you match the antecedent and infer the consequent. With modus tollens, make sure you negate the consequent and infer the negation of the antecedent. Trying to do this in another order will lead you to do something invalid. 3. Hypothetical Syllogism The next form of inference is called “hypothetical syllogism. Strictly speaking, any two premise deductive argument that uses a conditional to get its conclusion is a hypothetical syllogism; so, modus ponens and modus tollens are also hypothetical syllogisms. In this chapter, however, we’ll reserve the term “hypothetical syllogism” just for this chain-argument rule.” This is what ancient philosophers called “the chain argument” and it should be obvious why in a moment. Here is the form of the rule: If p leads to q and q leads to r, we are allowed to infer that p will lead to r. The “link” in this chain argument must be on opposite sides of the horseshoe—the matching term must be the antecedent of one conditional and the consequent of the other. We could construct a longer chain, if r led to s and s led to t and so on, and it would be a valid argument, but the rule that we will cite in our proofs only connects two different conditional statements together. As before, it is important to realize that any argument with this same form is a valid argument. For example, Notice that the consequent of the first premise and the antecedent of the second premise are exactly the same term. That is what allows us to “link” the antecedent of the first premise and the consequent of the second premise together in a “chain” to infer the conclusion. Being able to recognize the forms of these inferences is an important skill that you will have to become proficient at in order to do proofs. 4. Disjunctive Syllogism The next form of inference we will introduce is called “disjunctive syllogism” and it has the following form: In words, this rule states that if we have asserted a disjunction and we have asserted the negation of one of the disjuncts, then we are entitled to assert the other disjunct. Once you think about it, this inference should be pretty obvious. If we are taking for granted the truth of the premises—that either p or q is true; and that p is not true—then it has to follow that q is true in order for the original disjunction to be true. (Remember that we must assume the premises are true when evaluating whether an argument is valid). If it is true that either Bob or Linda stole the diamond, and we find out that Bob did not steal the diamond, then it has to follow that Linda did. That is a disjunctive syllogism. As before, any argument that has this same form is a valid argument. For example: This is a valid inference because it has the same form as disjunctive syllogism. The first premise is a disjunction (since the wedge is the main operator), the second premise is simply the negation of the left disjunct, and the conclusion is the right disjunct of the original disjunction. Notice that the second premise contains a double negation. Your English teacher may tell you never to use double negatives, but as far as logic is concerned, there is absolutely nothing wrong with a double negation. In this case, our left disjunct in premise 1 is itself a negation, while premise 2 is simply a negation of that negation. Here's one more example of disjunctive syllogism: We can perform disjunctive syllogism here because we have a sentence where the main connective is a wedge, and we have the negation of one of the disjuncts. That lets us conclude that the other disjunct must be true. Notice that again we can drop the parentheses here when we draw our conclusion. The negation of the first disjunct had to keep the parentheses, however, to make sure that tilde negates the whole sentence. III. Constructing proofs We now have enough rules to start proving some arguments. I want to start with a longer chain argument: If hypothetical syllogism is valid (and it is), this should be too, right? A leads to B, B leads to C, and C leads to D. So if we start with A, it should lead to D. I can’t call this an instance of hypothetical syllogism, though. Hypothetical syllogism has exactly two premises and then tells us what conclusion we get; this has three premises. However, I can use hypothetical syllogism to prove this is also valid. Think of natural deduction like a game. The rules tell us how we can move. The goal of the game is to start with the premises, and apply our valid rules one at a time until we’re able to generate the conclusion. Not every proof requires you to use every rule, and you can use the same rule more than once. You may use any of the rules—as along as your use of the rule is correct. Like most games, people can be better or worse at the “game” of constructing proofs. Better players will be able to a) make fewer mistakes, b) construct the proofs more quickly, and c) construct the proofs more efficiently. And, like most games, you get better with practice! Let’s set up our game board. First, I want to number my premises. I’m going to move the conclusion—I want to put it on the same line as the last premise. I don’t want to put the conclusion on a numbered line at this point (assuming your conclusion as a premise is a fallacy called “begging the question”) but I do want to keep my goal in view. The gameboard looks like this: Now, I look at my premises to see if any of them can be used as input for any of my rules. I see that I have two places where I could use hypothetical syllogism. Lines 1 and 2 are both conditionals, and they have that matching, overlapping link: B. We can say the same thing about 2 and 3, they share the term C. I’m going to apply the rule to lines 1 and 2, though. I write the result of applying the rule on the next numbered line, and then I need to write down which rule I used and which lines I applied it to, like this: Here’s the cool part—the new line is now a new ingredient I can use. I want to see if I can combine it with any of the other lines as input for one of my rules. I see that line 4 has that matching, overlapping term (in this case C). Get rid of the link, and hook the remaining antecedent and consequent together. Put the result on the next line: Notice that the last line of the proof is the conclusion that we are supposed to derive, and that each statement that I have derived (i.e., lines 4 and 5) has a rule to the right. The rule, plus the numbers of the lines I applied the line to, are called “justification.” The premises do not need justification, because we’re assuming they’re true. Every line I add to the proof after the premises, though, needs justification. Once I derive the conclusion I was going for, I’ve succeeded. This is what is called a proof. A proof is a series of statements, starting with the premises and ending with the conclusion, where each additional statement after the premises is derived from some previous line(s) of the proof using one of the valid forms of inference. Let’s look at another one. The first step is to look at our premises, and see if any two of them can be used with any of the rules we know so far. We can only work with the main operators, though. Any other connective cannot be worked with until you can get it as the main operators on a further line. Looking at lines 1 and 3, I recognize the modus ponens we saw earlier in the chapter. Line 1 has a horseshoe as the main connective, and line 3 matches the antecedent of that conditional exactly. That allows me to write the consequent of the conditional on the next line. Don’t forget to record what rule you used, and which lines you applied it to. My new line gives me a new ingredient. Its main connective is a horseshoe, and on line 2, I have the negation of its consequent. This allows me to use modus tollens to put the negation of its antecedent on the next line. The last line is the conclusion we were trying to derive. That means we have succeeded in proving our argument valid. We won the game! Here is some general advice for approaching your proofs. At any stage of your proof, you can use any two lines that form suitable ingredients for the rule you wish to apply. The lines do not have to be in order, they do not have to be next to each other, and you can use a line more than once if it comes in handy later in your proof as well. Any two lines—as long as they fit the form of one of our rules. Right now you only have four rules. A good strategy when you’re starting is to just pick two lines and see if they fit the form of one of our rules. Every time you see you cannot use a rule, ask yourself “why not?” This will help you learn how to apply the rules. Every time you use a rule to derive a new line, look specifically to see if you can use that line with another line and one of the rules. Don’t give up. You may not see the general strategy of your proof, but you don’t have to. Every time you successfully apply a rule, you generate a new line you can use, that might give you inspiration. IV Four More Rules The next four forms of inference we will introduce utilize conjunction, disjunction and negation in different ways. In the exercises at the end of this section, you’ll use all eight rules in your proofs. 5. Simplification We will start this section with the rule called “simplification,” which has the following form: What this rule says, in words, is that if we have asserted a conjunction, then we are entitled to infer either one of the conjuncts. It is a pretty “obvious” rule—a conjunction is an and-type statement, and it means that both conjuncts are true. This rule lets us break apart a conjunction so we can use each conjunct separately in our proofs. As before, it is important to realize that any inference that has the same form as simplification is a valid inference. For example (and now I’m going to put these applications of rules into the form of a proof): is a valid inference because it has the same form as simplification. That is, line 1 is a conjunction (since the dot is the main operator of the sentence) and line 2 is inferring one of the conjuncts. 6. Conjunction The next rule we will introduce is called “conjunction” and is almost the reverse of simplification. Simplification is how we get rid of a conjunction, and the rule conjunction is how we introduce one into our proof. Conjunction has the following form: What this rule says, in words, is that if you have asserted two different propositions, then you are entitled to assert the conjunction of those two propositions. If you know p is true, and you know q is true, then you know ‘p and q’ is also a true sentence. As before, it is important to realize that any inference that has the same form as conjunction is a valid inference. For example, is a valid inference because it has the same form as conjunction. We are simply conjoining two propositions together; it doesn’t matter whether those propositions are simple or compound. In this case, of course, the propositions we are conjoining together are complex, but as long as those propositions have already been asserted as premises in the argument, or derived by some other valid form of inference, we can conjoin them together. Notice, though, that before we combine them with a dot, we have to collect each compound sentence in parentheses, to make sure the dot we just added is the main connection of the proposition on that line. 7. Addition The next rule we’ll introduce is called “addition.” It is not quite as “obvious” a rule as the ones we’ve introduced above. However, once you understand the conditions under which a disjunction is true, then you should be able to understand why this form of inference is valid. Addition has the following form: What this rule says, in words, is that that if we have asserted some proposition, p, then we are entitled to assert the disjunction of that proposition p and any other proposition q we wish. Here’s the simple justification of the rule. If we know that p is true, and a disjunction is true if at least one of the disjuncts is true, then we know that ‘p or q’ is true even if we don’t know whether q is true or false. Why? Because it doesn’t matter whether q is true or false, since we already know that p is true. I only need one true disjunct to make the whole disjunction true. Another way to think about it is to compare it with the conjunction rule you just learned. If I want to prove an ‘and’ statement, I have to prove both sides. Conjunction means ‘both of these things are true,’ so I need to prove both things before I can put a dot between them. Disjunctions, on the other hand, mean ‘at least one of these things is true.’ So, to prove an ‘or’ statement, I only have to prove one side of the disjunction before I can introduce a wedge. The cool thing about this rule is that you can introduce anything you want to as your second disjunct. It doesn’t even have to be anything that already occurs in the proof. You’ll want to make sure you introduce whatever you need to finish your proof. As before, is it important to realize that any argument that has this same form, is a valid argument. For example, is a valid inference because it has the same form as addition. The first premise asserts a statement (which in this case is compound—a disjunction) and the conclusion is a disjunction of that statement and some other statement. In this case, that other statement is itself compound (also a disjunction). But an argument or inference can have the same form, regardless of whether the components of those sentences are simple or compound. As with conjunction, if the disjuncts are compound, you need to use parentheses to group them together, so the wedge you introduced is the main connective of your new line. 8. Constructive Dilemma The eighth valid form of inference is called “constructive dilemma” and is the most complicated of them all. It may be most helpful to introduce it using an example. Suppose I reasoned thus: The killer is either in the attic or the basement. If the killer is in the attic, then he is above me. If the killer is in the basement, then he is below me. Therefore, the killer is either above me or below me. That this argument is valid should be obvious (can you imagine a scenario where all the premises are true and yet the conclusion is false?). What might not be as obvious is the form that this argument has. However, you should be able to identify that form if you utilize the tools that you have learned so far. The first premise is a disjunction. The second premise is a conditional statement whose antecedent is the left disjunct of the disjunction in the first premise. And the third premise is a conditional statement whose antecedent is the right disjunct of the disjunction in the first premise. The conclusion is the disjunction of the consequents of the conditionals in premises 2 and 3. Here is this form of inference using symbols: Constructive dilemma is almost like a double modus ponens. I have two conditionals, and I know that at least one of the antecedents is true (even though I don’t know which one), so I can derive that at least one of the consequents is true (even though I don’t know which one). Constructive dilemmas are often used in decision-making reasoning. Suppose I need to decide between p and q. I know p leads to r, and q leads to s, so I know I will end up with either r or s as a consequence of my decision. Now all I need to do is figure out which outcome looks more appealing to me. We have now introduced eight forms of inference. In the next section I will walk you through some basic proofs that utilize these eight rules. V. More help in how to construct proofs The introduction of rules 5 through 8 complicate proof construction. With just rules 1 through 4, you can just apply any rule you have the ingredients for, and you will usually get to the conclusion eventually. With all eight rules, it often helps to work out a strategy, so here are a few tips on that process. In order to construct proofs, it is imperative that you internalize the eight valid forms of inference. By “internalize” I mean that you have memorized them so well that you can see those forms manifest in various sentences almost without even thinking about it. If you internalize the rules in this way, constructing proofs will be a pleasant diversion, rather than a frustrating activity. In addition to memorizing your rules, there are a couple of different strategies that can help when you’re stuck and can’t figure out what to do next. The first is the strategy of working backwards. When we work backwards in a proof, we ask ourselves what rule we can use to derive the sentence(s) we need to derive. Here is an example: The conclusion, which is to the right of the second premise and follows the ‘therefore’, symbol, is a conjunction (since the dot is the main operator). If we are trying to “work backwards,” the relevant question to ask is: What rule can we use to derive a conjunction? If you know the rules, you should know the answer to that question. There is only one rule that allows us to introduce a conjunction; that rule is called “conjunction.” The form of the rule conjunction says that in order to derive a conjunction, we need to have each conjunct on a separate line. We already have the second conjunct on its own line, so the only other thing we need to derive is first conjunct. Once we have that on a separate line, then we can use the rule conjunction to connect those two sentences with a dot to get the conclusion. The next question we have to ask is: How can I derive the sentence “T v L”? Again, if we are working backwards, the relevant question to ask here is: What rule allows me to introduce a disjunction? There are only two: constructive dilemma and addition. However, we know that we won’t be using constructive dilemma since none of the premises are conditional statements, and constructive dilemma requires conditional statements as premises. That leaves addition. Addition allows us to disjoin any statement we like to an existing statement. Since we have “T” as the second premise, the rule addition allows us to disjoin “L” to that statement. The first new line of the proof should thus look like this: The next step of the proof should be clear since we have already talked through it above. All we have to do now is go directly to the conclusion, since the conclusion is a conjunction and we now have (on separate lines of the proof) each conjunct. Thus, the final line of this (quite simple) proof should look like this: The order in which you cite the lines doesn’t matter, as along as you have cited the correct lines (that is, I could have equally well have written, “Conjunction, lines 3 and 1” as the justification). The complete proof should look like this: The last line of the proof is the conclusion to be derived: check. Each line of the proof follows by the rule and the line(s) cited: check. Since both of those requirements check out, our proof is complete and correct. I have just walked you through a simple proof using the strategy of working backwards. This strategy works well as long as the conclusion we are trying to derive is complex—that is, if it’s a compound statement. However, sometimes our conclusion will be a simple statement—a single letter with no connectives. In that case, we will not as easily be able to utilize the strategy of working backwards. So, you can use the strategy we learned with the first four rules: working forward. To remind you of how this strategy works, we ask ourselves what rules we can apply to the existing premises to derive something, even if it isn’t the conclusion we are ultimately trying to derive. We look to see if any of our lines are good ingredients for any of our rules. As a part of this strategy, here are some suggestions: If you see a conjunction as the main connective, break it apart using simplification. If you see a disjunction, see if you can find the negation of one of its disjunct. If you see a conditional, see if you can find the antecedent on a separate line for modus ponens, or the negation of the consequent for modus tollens. If you see two conditionals, see if they have an overlapping term, for hypothetical syllogism, or a disjunction on another line for constructive dilemma. If you see a negation as the main connective, look to see if you can do a modus tollens or a disjunctive syllogism. Here is an example of a proof where we should utilize the strategy of working forward: Notice that since the conclusion is a simple statement, it doesn’t give us any clues to help us work backward. So, we’ll work forward. The first line is a conjunction: every time you see a dot as the main connective, you should use the rule simplification to break it apart. If you can see which conjunct you need, just take that one. If you’re not sure, apply simplification twice, so you’ll have both conjuncts as new ingredients in your proof, like this: The first two lines of the proof are the result of breaking down the conjunction in line 1, where line 3 is the left conjunct and line 4 is the right conjunct. We then ask the question, do any of these four lines look like ingredients for one of my rules? I have a conditional on line 2 (and it’s the only conditional, so I won’t be using hypothetical syllogism or constructive dilemma), so following the third tip in my list above, you want to look for its antecedent or the negation of its consequent on a different line. We now have that antecedent on line 4, and that that means we can apply the rule modus ponens: The line that we have just derived is in fact the conclusion of the argument. So, our proof is finished. Some proofs you’ll want to work backwards, some proofs you’ll want to work forwards. With some longer proofs, you might work both backwards and forwards and meet somewhere in the middle—see what ingredients you can use, but also look for what ingredients you’re going to need, based on your conclusion. Remember: any proof, long or short, is the same process and utilizes the same strategy. It’s just a matter of keeping track of where you are in the proof and what you’re ultimately trying to derive. So here is a bit more complex proof: Before you read any further, take a minute and try this proof on your own, then come back and compare yours to mine. Your proof might not be exactly like mine—it could be longer, shorter, or do the steps in a different order—but as long as you’ve used all of the rules correctly, and your last line is the conclusion you wanted, you’ve done your proof correctly. Using the strategy of working backwards, we see the conclusion is a conjunction, so we know that if we can get each of those conjuncts on a separate line, then we can use the rule conjunction to derive the conclusion. This is our long-term strategy. However, we cannot see how to get there from here at this point, so we’ll begin working forward. The first thing we’ll do is simplify the conjunction on line 5, putting each conjunct on a separate line: Look at lines 2 and 6: they are both negated simple propositions, so I want to see if I can use them with other lines, to do a modus tollens or disjunctive syllogism. Looking at lines 2 and 3, I see the ingredients for a modus tollens. That will be our next step: The next step of this proof can be a bit tricky. There are a couple different ways we could go. One would be to utilize the rule “addition.” Can you see how we might helpfully utilize this rule using either line 6 or 8? If not, I’ll give you a hint: what if we were to use addition on line 8, adding a wedge B? We’d end up with the antecedent of the conditional in line 1, so we could then use modus ponens to derive the consequent. Thus, let’s try starting with the addition on line 8: Next, we’ll utilize line 9 and line 1 with modus ponens to derive the next line: Notice at this point that what we have derived on line 10 is “L,” and this is very close to one of the conjuncts I wanted to derive for my conclusion. The first conjunct in the conclusion is a disjunction, and I now have one of those disjuncts—I can use “addition” again to add the other. That will be the next line of the proof: We have one conjunct of our conclusion. At this point, our strategy should be to try to derive the other conjunct, not-R. Notice that it’s contained within the sentence on line 4, but it is embedded. How can we “get it free”? Line 4 has a horseshoe as its main operator; I first want to get rid of the antecedent and the horseshoe. I’ve matched the antecedent on line 10, so I can do a modus ponens, and write the consequent on the next line: This is a disjunction, so I’ll want to get rid of the wedge and the D. Looking at line 6, and combining it with our new line 12, I see I have the ingredients for a disjunctive syllogism: The final step is simply to conjoin lines 11 and 13 to get the conclusion (make sure you gather line 11 in parentheses, so the new dot ends up being the main connective): Here is the completed proof: Constructing proofs is a skill that takes practice. Don’t give up, just keep trying.
Chapter 10: Causal Reasoning I. Introduction Human beings see causation everywhere. It’s how our brains are hard-wired; it’s not just a habit of ours, it’s a survival technique. When we touch a hot stove and it hurts, we immediately form the causal connection in our brains, “hot stoves cause pain.” Thereafter, we’ll try to avoid touching them when we can. We see causation so often, however, that we over-see it – we make causal connections when there really isn’t any connection between the two things. In this chapter, we’ll accomplish three things. First, we’ll go over the false cause fallacies – typical mistakes people make in causal reasoning. Next, we’ll go over different kinds of causal connections – specifically necessary conditions and sufficient conditions. Then, we’ll go over Mill’s Methods – John Stuart Mill’s methods that we can use to provide evidence for our causal assumptions. II. False Cause Fallacies False cause fallacies are mistakes humans make in causal reasoning. There are many, but I want to introduce three: post hoc ergo propter hoc, non causa pro causa, and oversimplified cause. These are the fallacies we’re trying to avoid by using Mill’s Methods to provide evidence for (or against) our causal assumptions. The first two completely misunderstand a causal link; the third one is partially correct about a causal link, but in a way that is often misleading. 1. Post Hoc Ergo Propter Hoc This is Latin for “after this, therefore because of this.” It is often shortened to just “post hoc.” The post hoc fallacy is where we see causation where there is none, because one thing happened right after another one. Usually, there are two significant events that caught our attention, and our brain forms a causal link between them. (If two events are insignificant, we usually don’t leap to a causal connection. This morning I brushed my teeth and then a little later, I put my shoes on. I had to stop and think what I did after I brushed my teeth; neither of these events are notable, so I don’t even think about them. I certainly don’t think that brushing my teeth caused me to put shoes on.) Here are a few examples of the post hoc fallacy. In 2013, Beyoncé played the half time show during the Super Bowl. It was a spectacular show, with great music, dance routines, and special effects. About twenty minutes into the second half of the game, the power went out in the Superdome. It took about 40 minutes before the power was fully restored. Fortunately, there was enough power that the commenters could talk about how the lights were still off… Of course the immediate assumption was that Beyoncé’s show caused the power outage. (It did not. Beyoncé’s crew brought their own generators, they didn’t even use the Superdome’s power). See the “after this, therefore because of this” structure? Beyoncé played, and then the lights went out. So people (falsely) assumed that her show caused the lights to go out. Here’s another one: The governor of California gave a speech, and then an earthquake hit Los Angeles. He needs to stop giving speeches; he’s putting people in danger! He spoke, and then the earthquake hit, so the assumption is he caused the earthquake, maybe by bringing bad luck to the residents of Los Angeles. Superstitions are often reinforced by the post hoc fallacy. If a black cat crosses my path, and then I get in a car wreck, I will assume that the black cat crossing my path caused my bad luck. 2. Non Causa Pro Causa The Latin here translates to “not the cause for the effect.” Just like with post hoc, I have completely misidentified a causal link. However, this time it’s not because one event happened before another one, it’s because the two things happen at the same time, or in the same place, or I notice some other correlation between them, and use this correlation to assume there’s a causal link. As the saying goes, correlation is not causation. Here are a few examples. Every time the governor of California gives a speech, a natural disaster happens somewhere in the world. He needs to stop giving speeches; he’s putting people in danger! Notice how this is different from the example in the last section. The post hoc had two events, that happened one right after the other. The non causa finds a correlation – these two things keep happening at the same time. Now, correlation can give you a sense that two things might be causally linked. If every venue Beyonce ever played in suffered a power outage, I’d start to suspect her show did have something to do with it. But simply noticing a correlation is not proof. Here’s another example: This is from Tyler Vigen’s web page; the link is here (it opens a new page): Spurious Correlations. If we drew from this chart the conclusion that eating cheese causes people to die by becoming tangled in their bedsheets (or that people dying in their bedsheets is causing more people to eat cheese), we are making the non causa fallacy. And, a famous example: Every time ice cream sales go up, so do shark attacks! Sharks must really like the taste of ice-cream stuffed humans. This is an interesting one, because there is actually sort of an indirect a causal relationship between ice cream sales and shark attacks—they both share a cause. Buying (and presumably eating) ice cream does not cause sharks to attack; shark attacks do not cause a rise in ice cream sales. Rather, hot weather is responsible for both ice cream sales rising and for humans going swimming in shark-infested water. I taught with this example for years before I finally looked it up to see if the premise was true. It is not. Ice cream sales and shark attacks peak in different months. However, remember that we’re looking at the connection between premises and conclusion when we evaluate logic. IF the premise were true, how much evidence would it give for the conclusion? Answer: not very much at all. 3. Oversimplified cause The fallacy of oversimplified cause is where you’re dealing with a complex phenomenon with many moving parts, but you pick a partial cause and pin the full weight of blame (or praise) on that one thing. It doesn’t completely misidentify a causal connection, like post hoc and non causa; the thing you’ve picked out does play a causal role, but you’ve oversimplified by ignoring all of the other parts of the equation. Here’s an example: The economy of California is thriving. It must be because of the governor’s wise economic policies. Or The economy of California is crashing. It must be because of the governor’s idiotic economic policies. The truth is that the governor’s economic policies do have an effect on the economy of the state. However, one person is never solely responsible for an entire state’s economy. Further, economic policies never have an immediate effect. If government policies do partially cause a change in the economy, we’re unlikely to see that change for several years – usually during the next governor’s term of office. This is the fallacy of oversimplified cause. III. Necessary and Sufficient Conditions Now that we’ve reviewed the fallacies involved in causation we’re trying to avoid, we need to look at what we mean by “causation” in any particular case. A cause can be a necessary condition for the effect, a sufficient condition, both necessary and sufficient, or neither necessary nor sufficient. 1. Necessary conditions A necessary condition is something that needs to happen in order to get the result. If you don’t have the condition, you’re not going to get the result. This does not mean that you’re guaranteed the result every time you have the condition in place, however. It just means that if you don’t have the condition, you’re not going to get the result. For example: water is a necessary condition of plant growth. The appropriate amount of water is needed to get your plants to grow. It’s not ALL you need though; the plant is also going to need sunlight, nutrients, access to air, and so forth. Another example: getting enough sleep is necessary to have energy. If you don’t get any sleep, you’re not going to have a very good day the next day, and you’ll struggle to pay attention. The human body absolutely needs sleep. It’s not the only thing we need to give us energy, however; we also need to eat enough food, and to not have the flu, among other things. 2. Sufficient conditions A sufficient condition is a condition that is enough to bring about the result. If the sufficient conditions are in place, you will get the result. It doesn’t mean the condition is needed, though. For example: beheading is a sufficient condition for death. That’s enough, that will get the job done. It’s not at all necessary, though; there are other ways to die. You know it’s not necessary because you can get the result without this condition being present. Another example: throwing a brick through a glass window is sufficient to break it. That will get the job done. It’s not necessary though, because there are other ways to get the same result. You can throw a rock through it, you can throw a person through it (I’m thinking of those old Western movies with bar fights, where they usually throw someone right through a window out onto the street), an earthquake could break it, and so on. 3. Both necessary and sufficient conditions Sometimes a condition is both necessary and sufficient. To identify these, you just ask “is it necessary?” and “is it sufficient?” If you say “yes” both times, it’s necessary and sufficient. It’s not a third category, really, just a conjunction of the two types of conditions. It is a special sort of condition that is both necessary and sufficient, though. If you’ve identified one of these connections, you’ve identified the total cause. Scientific phenomenon, thoroughly explored, can turn up necessary and sufficient conditions. Really good definitions will give you the necessary and sufficient conditions of when that term can be used. For example, being between the ages of 13 and 19 (inclusive) is necessary and sufficient for being a teenager. You have to be one of those ages to be a teenager, so it’s necessary. And, turning 13 is all you need to do. You don’t need to register someplace or sign a contract; turning 13 is sufficient for being a teenager until you’re 20. So, being between 13 and 19 is a great definition of teenager; I have provided necessary and sufficient conditions for when the word applies. Here’s an example of a bad definition, because it fails to be necessary and sufficient. Consider this: “A human is a tool-using mammal.” Using tools is not necessary to be a human. Babies count as human, long before they are able to use tools. And, using tools is not sufficient for being a human – many primates and even some birds use tools. If tool-using mammal was sufficient to count as human, those primates would be human (but not the birds). Let’s return to the plant example from the necessary condition section. Water is necessary, but not sufficient, for plant growth. What if I thoroughly explored what plants need to grow, though, and put them all together? What if I found all of the necessary conditions? I’m thinking about water, sunlight, nutrients, air, and anything else that plants might need. Once you lay all of this out, you’d have the necessary and sufficient conditions for plant growth. Each individual element is necessary; the group together is sufficient. This is called “individually necessary and jointly sufficient.” And then you’d understand how to make plants grow. 4. Contributing causes Some causes are actual causes of a phenomenon without being necessary or sufficient. We can call these “contributing causes.” Think of the governor of California’s economic policies, from the false cause section above. The governor of California’s policies are not necessary for the state’s economy to do what it’s going to do; and they’re by no means sufficient. They do add something, though; they will eventually contribute to how well the state’s economy does. Or, thinking back to the example of what I need in order to have energy – I absolutely need enough sleep, and to consume enough calories, and to avoid having an illness that saps my energy. What about exercise, though? Exercise is absolutely not sufficient for having energy – you still need sleep and food. Is it necessary? Do you need to exercise regularly to have energy? Nope, but it does help. Exercise is a contributing cause. So is caffeine. You don’t have to drink coffee to have energy – caffeine is not necessary. It’s also not sufficient. If you’re seriously sleep deprived, no amount of coffee can fix that. IV. Mill’s Methods Mill’s methods were laid out by British philosopher John Stuart Mill in 1843, in his book A System of Logic. He was specifically looking into scientific inquiry of causes, but his methods can be applied outside of laboratories. You’ve probably used similar techniques to try to investigate causes yourself. He by no means invented these methods; he did, however, formalize them. Taking a more formal approach to providing evidence for (or against) a causal hypothesis can help you avoid making a leap in logic that is unwarranted, and committing a false cause fallacy. They do have some drawbacks that you need to keep in mind: This is solidly inductive logic. We can never prove with absolutely certainty what, in particular, caused an effect. Each of these methods has a different level of strength, and the more research we put into it can make our arguments stronger, but we can never be guaranteed we are correct. Mill’s methods are not great at telling you what the cause is if you have absolutely no idea. You need to come up with some hypotheses, some theories, and then Mill’s methods can help you test those theories. The methods can help you establish a causal connection, but can’t tell you if you’ve found the full cause or not. Beware of committing the fallacy of oversimplified cause! That said, these methods help you systematically organize your data and testing of a hypothesis, to help you avoid mistakes. They can absolutely add evidence to your theory. And, they’re really good at eliminating causal suspects – they’re better at refuting your hypothesis than they are at proving it. There are five methods: the method of agreement, the method of difference, the joint method of agreement and difference, concomitant variations, and the method of residues. 1. The Method of Agreement Suppose you went to a picnic with a bunch of friends, and everyone brought a different dish to share. You had a great time, tried some of everyone’s food, and about twelve hours later, you got hit with a nasty bout of food poisoning. Once you recover, you’re curious exactly what caused your food poisoning – maybe we don’t want to let the friend who made it cook for you anymore. How would you go about figuring out what poisoned you? You’d call your friends and ask them if they got sick, right? And then you’d probably ask them what they ate. The method of agreement does exactly this, only a little more systematically. When we use the method of agreement, we want to focus on cases where the effect is present, and see if they all have something in common. So in this case, we want to talk to all the friends who also got sick, and ask them what they ate, to see if they all ate at least some of the same food. Mill described the Method of Agreement as a way to gather evidence to explain a phenomenon (an effect) “by comparing together different instances in which the phenomenon occurs.” John Stuart Mill, A System of Logic, Vol. 1, Project Gutenberg, 2008, p. 394. Available online at this link: A System Of Logic, Ratiocinative And Inductive (Vol. 1 of 2) We see something has happened multiple times, so we gather instances where we know the effect occurred, and look to see if there’s something they all had in common. Let’s break the method down into steps. Identify the effect you’re trying to explain. List all potential causes for that effect. Identify which case experienced which of the causes. If you find one potential cause that all cases had in common, you have evidence that this is the cause. For our poisoned picnic: (1) the effect we’re trying to explain is “food poisoning,” (2) all potential causes are all the foods present at the picnic. For step (3) we want to ask our friends what they ate. (4) If we can find one and only one food that everyone ate, we have evidence that this is the food causing people to get sick. A good chart can help us track our data. I’m going to put the friends who got sick up at the top of the chart, and the foods present down the side. Then I’m going to check off who ate what. Here’s my chart: Andrew Betty Cesar Dominique Elaine Sick Sick Sick Sick Sick Fried chicken * * * * Potato salad * * * Coleslaw * * * * * Rolls * * Fruit salad * * * Cake * * * Notice that I’ve specified that each of these cases has the effect present – all five of these people got sick. It should stick out to you that the only thing everyone who got sick ate is the coleslaw. Coleslaw is now our main suspect. I haven’t proved it with certainty – it’s possible I’m leaving off a potential cause, or that someone misremembered what they ate, or my whole “food poison” hypothesis is wrong and we really have the stomach flu, or something else. But it’s a pretty good start. Also, I’ve got great evidence that the rolls are safe. Three people avoided the rolls and still got sick. Except someone misremembered what they ate. Cesar calls back and says “whoops, I also had fried chicken. I forgot.” So we amend our chart: Andrew Betty Cesar Dominique Elaine Sick Sick Sick Sick Sick Fried chicken * * * * * Potato salad * * * Coleslaw * * * * * Rolls * * Fruit salad * * * Cake * * * Now everyone who got sick had two things in common. They all ate the chicken, and they all ate the coleslaw. (The rolls are still looking pretty safe, though). This leaves us with a few different possibilities: (1) the chicken was bad. (2) the coleslaw was bad. (3) BOTH of them were bad. (4) some strange reaction between these two foods is what made people sick. (5) Neither of them were part of the cause; the real cause is something we didn’t look for (like the stomach flu). Fortunately, we have other methods we can use to try to figure out what exactly was making people sick. To sum up: the method of agreement gathers instances where the effect is present, and looks for one (and hopefully only one) thing they all have in common. It is most often used where the data is already available, so you can, for example, go through a database and retrieve only examples where the effect happened to analyze further. Or, after the picnic happened, you can call your friends, find out who else got sick, and find out if there is something they all ate in common. It is also important that you brainstorm as many possible causes as you can; you want to be thorough with this part of the procedure to make sure you didn’t overlook the actual cause. So, the method of agreement: (1) gathers instances with the effect present; (2) lists all elements which are potential causes; (3) looks for one element which was present every time the effect is present. If you find one, that is likely the cause. 2. The Method of Difference The method of difference is very different indeed from the method of agreement. The method of difference cannot be performed by analyzing existing data; you need to set up an experiment to test your hypothesis. This is because you need to control for as many factors as you can, to make sure you’re only testing your hypothesis. Mill described the method of difference as a means of providing evidence to explain a phenomenon “by comparing instances in which the phenomenon does occur, with instances in other respects similar in which it does not.” Ibid. So, you want to compare a case where the effect happened, and a case that is quite similar in which the effect did not happen. Or, to put it another way, you want to try to make the effect happen – and then take away the cause you’re testing and see if the effect also goes away. How the method of difference works: After identifying the effect you want to explain, and the element you want to test for a causal relationship, you want to set up two trials, one with the element present, and one with the element absent. You want to see if you get the effect when the element is present, and fail to get the effect with the element absent. When you set up your two trials, you want to keep everything else as similar in both instances as you can, to make sure you’re only testing your hypothesis, and not accidentally testing something else. To break it down into steps: Identify the effect you want to explain. Identify the causal hypothesis you want to test. Set up an experiment, with two trials. Control as many variables as you can. Vary only the element you want to test as the cause. One trial should receive that element, one trial should not receive it. If you get the effect when that element is present, and do not get it when that effect is absent, you have evidence that it is playing a causal role. If we go back to our poisoned picnic where we were left with two suspects, fried chicken and coleslaw, we can now set up a trial. We can’t test both of these at once, so we have to choose one to test at a time. If we choose the fried chicken, we want to take two friends, and try to poison one of them. (Don’t try this at home. Or in the lab. There are ethical regulations on what kinds of tests you can do on humans.) To formalize it: find two friends who are as similar as possible, especially in relevant ways. You want them to be in similar health, and have the same food allergies. They should be around the same age, and so forth. Make sure neither of them have the stomach flu. Then, control everything they eat for 24 hours, to make sure they’re eating the same amount of the same thing at the same times. Once you have your two trials set up, perform the test: give one friend the potentially bad chicken and let the other friend get off without eating it. Examine the results: If the friend who ate the chicken gets sick, and the one who avoided it does not get sick, that’s some evidence that the chicken was bad. If both of them get sick, that’s evidence that something you controlled for might be a suspect. Did they both eat that coleslaw? Did you feed them both milk, and both are lactose intolerant? Note, the chicken still could be bad in this case. If neither of them get sick, that’s evidence that the chicken is fine, and the cause is something different that they haven’t consumed in the last 24 hours. Let’s look at another example, which doesn’t violate obvious ethical regulations. Suppose I want to test to see if yeast actually has a causal role in helping bread rise. Going through steps 1 through 5 above, here’s what I’ll do: Identify the effect: bread rising. Identify the element you want to test as a potential cause: yeast. Set up your trials, controlling for what you can: this means you want to bake two loaves of bread. You want to make sure that other than the yeast, both breads have exactly the same amounts of all the same ingredients. You want to make sure you prepare the doughs in exactly the same ways. You want to make sure you bake them at the same temperature for the same amount of time in the same type/size/shape of pan. The only thing that should differ between them is that one gets yeast, and one does not. The result is going to be the one with yeast shows the effect (it rises), and the one that lacks yeast did not show the effect (it did not rise). The method of agreement is called that because you find cases that agree in that they all have the effect, and you look to see if they also all agree on an element that is a potential cause. The method of difference is called that because you want two cases to differ on whether they include the potential cause, and you hope they will also differ in whether they show the effect. With the method of agreement, you’re analyzing cases for similarities; with the method of difference, you’re purposefully trying to make the effect happen, and having a control case to make sure the thing you tested is really the cause. 3. Joint Method of Agreement and Difference The joint method combines some elements from each of the above two methods. But really, it’s an expansion of the method of agreement. John Stuart Mill said this method could also be called the indirect method of difference, but also said it is essentially applying the method of agreement, but twice. Here’s how to lay it out: Identify the effect you want to explain. Collect several examples with the effect present, and several examples with the effect absent. Brainstorm all elements which could be potential causes for that effect. If you find one element which is present every time the effect is present, and absent every time the effect is absent, you have evidence that this is the cause. Informally, if we go back to the poisoned picnic attendees, we already have data on what everyone who got sick ate, and what they have in common. Now we want to phone the rest of the people who attended the picnic, and ask them what they ate. You’re looking for something the people who got sick ate, and the people who dodged food poisoning all avoided. So, let’s put up our last chart from the method of agreement, where we ended with a mixed result: Andrew Betty Cesar Dominique Elaine Sick Sick Sick Sick Sick Fried chicken * * * * * Potato salad * * * Coleslaw * * * * * Rolls * * Fruit salad * * * Cake * * * Now, let’s call the people who did not get sick back, and ask what they ate: Andrew Betty Cesar Dominique Elaine Franco George Haily Isaac Jacob Sick Sick Sick Sick Sick Not sick Not sick Not sick Not sick Not sick Fried chicken * * * * * * * Potato salad * * * * * * Coleslaw * * * * * Rolls * * * * * * Fruit salad * * * * * Cake * * * I’ve put a dark line to divide the sick from the not sick people. Notice that everyone who was sick ate the coleslaw and the chicken; everyone who did not get sick avoided the coleslaw. I’ve ruled out the chicken; George and Isaac ate chicken and remained healthy. Mill thought this was doing the method of agreement twice – once on the group who was sick, and once on the group who did not get sick. For the group who was sick, we want to see agreement on what element is present – one food they all ate. For the group who did not get sick, we want to see agreement on what element is absent. One food they all managed to avoid. Summary of the First Three Methods The methods of agreement, difference, and the joint method are the three main methods for attempting to prove a causal link, and to rule out suspects that have no causal effect. Let’s take a minute and summarize them, highlighting their various features, before moving on to the last two methods. Agreement: Identify the effect you want to explain. Collect several cases with the effect present. List all of your suspected causes for this effect. If there is one element that is present in all of these cases, you have evidence that it could be a cause. Usually done by analyzing data you already have, so you can pick out the cases that have the effect. Example: if I want to use agreement to find evidence for the yeast/rising cause and effect, I would collect several loafs of bread that have the desired effect, and see if they all share any ingredients. Difference: Identify the effect you want to explain, and the cause you want to test. You will set up two trials; one will have the potential cause present, one will not. Everything else must be kept as identical as possible, so that you’re only testing the one factor you are interested in. If you get the effect where the element is present, and do not get the effect when the element is absent, you have evidence that this could be a cause. Cannot be done by gathering pre-existing data; you must set up a controlled experiment. The example we saw above with the yeast had us bake two identical loaves of bread, only one left out the yeast, and one made sure it had yeast in it. Joint: Identify the effect you want to explain, and all of your suspected causes for this effect. Collect several cases where the effect is present, and several cases where the effect is absent. If you find one element that is present wherever the effect is, and absent wherever the effect is absent, you have evidence that this is your cause. This method can be used to analyze pre-existing data, where you sort the cases into those with the effect and those without the effect. It also can be done with a series of experiments, where you see what combinations get you the effect and what combinations don’t. Example: suppose my method of agreement gave me a long list of ingredients, and four of them were present in every loaf: flour, sugar, salt, and yeast. I can bake several loaves of bread using a variety of combinations of ingredients, making sure I leave one of these out for at least one loaf, and playing with the other ingredients listed as well. I’ll get a variety of results; some will rise, and some will not. Then I can see which ingredients were present wherever it rose and absent wherever the loaf did not rise. 4. Concomitant Variations This method can be very helpful in gathering further evidence for a suspected causal connection. And, unlike the other methods, it can sometimes be used to identify causal connections you hadn’t suspected. However, it must never be your only method of proof. “Variations” means changes, and “concomitant” roughly means “together.” You’re looking for two things that change together. For example, you may have noticed that when you push the gas pedal when you’re driving, the car speeds up. The more you push the pedal, the more speed you get. If you back off on the gas, the car will slow down. These two things – how much you’re pushing the gas pedal, and how fast the car is going – are changing together. We can represent this with a line graph. Below, the solid orange line will be the speed of the car in miles per hour, and the blue line with dashes will be how much I pushed the gas pedal, in centimeters (because this is science, and centimeters sound more “sciencey” than inches). Concomitant Variations usually never end up with such a smoothly correlated data set, but I just made this data up and I wanted the chart to be pretty. When the two things you’re measuring on your chart rise at the same time, and fall at the same time, we call this a “positive correlation.” Negative, or inverse, correlations exist as well; this is where the two things change together, but in opposite directions from each other. For example: the brake pedal has an effect on the speed of the car too, but in this case, the more I push the pedal, the less speed I get. Let’s graph this out too, assuming I start by coasting along at 60 miles an hour and not touching the brake pedal at all. If I don’t touch the brake at all, the speed remains at 60; the more I push it, the less speed I get, and if I floor the brake pedal, the car will come to a full stop. Notice the two lines mirror each other, even though they head in opposite directions. This method can add data to our attempt to prove causation, but it also can help us discover unsuspected cause/effect relationships. If we find a correlation we never knew existed, we might have discovered a new cause/effect relationship. Ok cool, two things change together so they might be causally related. But why only “might be?” This is because concomitant variations establishes a correlation. And, as we learned in the false cause fallacy section above, correlation does not prove causation. Let me repeat a chart from that section. Tyler Vigen has used the method of concomitant variations to establish a correlation between the amount of cheese Americans eat and the number of people who died by becoming tangled in their bedsheets. If we assume that one is causing the other, we’re committing the non causa pro causa false cause fallacy. Our other example of a non causa fallacy was: Every time ice cream sales go up, so do shark attacks. Sharks must really like ice-cream stuffed humans! Here, I’ve used the method of concomitant variations as well, but in this case, the fallacy occurred because I didn’t use the method correctly. Here’s what John Stuart Mill had to say about this method: Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation. John Stuart Mill, A System of Logic, Ratiocinative, and Inductive, originally published in 1843. You can find it at the following link (opens in a new window): A System of Logic In other words, if I find a correlation between A and B, that suggests there could be a causal relationship, but there are three possible causal relationships here: A causes B, B causes A, or A and B are in some other way causally connected. So, let’s rephrase our original argument to be less fallacious. Every time ice cream sales go up, so do shark attacks. Shark attacks and ice cream sales must be causally related! And this conclusion is much closer to the truth! Ice cream sales are not causing the sharks to attack; shark attacks are not causing humans to buy more ice cream. Rather, these two phenomenon share a common cause: hotter weather causes humans to eat more ice cream, and it causes us to go swimming in shark-infested water more often. They are causally related. Bed sheet deaths and cheese consumption are not causally related (to my knowledge), however, even though they’re well correlated. So, here’s the take-away lesson for this section: The method of concomitant variations establishes that there is a correlation between two things, A and B. One of four possibilities is responsible for this correlation: A causes B. B causes A. A and B are otherwise causally connected (such as sharing a common cause). The correlation is a complete coincidence and there is no causal connection. Ok, we only have one more method to explore! Of all five methods, the last one gives us the least amount of evidence – except for in very particular circumstances. 5. Method of Residues The word “residues” means, roughly, “leftovers.” If you bake a pan of brownies, and remove the brownies, those crumbs still sticking to the pan are the brownie residue. Here, we’re looking for a left-over cause, and a left-over effect. For a complicated cause-effect relationship, with many elements, you want to eliminate the parts of the effect that you can explain, by identifying what caused each part. This ends up taking a large problem and making it smaller, but it also can give you some hints as to where to look next. Here’s an example: Suppose a restaurant manager is looking at the budget at the end of the month and realizes they have $1,000 less than they should have. Where did this extra money go? She goes through the accounts books and discovers the following causes for parts of the missing money: $500: replacing a broken freezer $100: replacing the food that went bad due to a broken freezer $200: training a new employee We have systematically accounted for the parts of the effect that we can explain, by hooking each part of the effect up with its corresponding cause. We have not, however, explained the entire phenomenon. We’ve only explained $800 worth; there’s a $200 residue effect – the portion we were unable to explain. This is useful, however, because we took a $1,000 problem and turned it into a $200 problem. But also, it can give us hints of where to look next. The manager systematically went through the accounts books, and explained every part of the extra expense that she could. That means that whatever happened to that $200, it’s the sort of thing that didn’t get written in the books! Theft, of course, is one explanation. No thief ever has pocketed $200, and then has gone back to the office to record the theft in the accounts books. However, while I want to investigate theft, I also want to look for other explanations. Human error is another thing that can account for that $200 gap. Maybe some of the math has been done wrong. Maybe one of the managers made a legitimate purchase for the restaurant, and simply forgot to record it; they’re still carrying the receipt around in their wallet. So, the method of residues can make a large problem smaller, and can give you clues of where to look next, but it can not prove that your suspected cause is the actual cause of the residue effect – except in one particular circumstance. Mill wants us to imagine that we’ve established a causal link between a complicated cause and a complicated effect. So suppose we’ve used our previous methods to get evidence that ABC together produce the effect abc. Now, if we can show that A by itself can produce the effect a, and B can produce the effect b, it follows from this that the remaining part of the cause, C, must be responsible for the remaining part of the effect, c. The restaurant example is using the method of residues to try to figure out a cause. Here, though, we already know the cause, we’re just trying to untangle which part of the cause is responsible for which part of the effect. Either way you work it, though, the method of residues requires more work proving a causal link. If you’ve established the complete cause earlier, you can use the method of residues to link up each part of the cause with each part of the effect. If you’re searching for the cause, once you have hints of where the residue might lie (theft, human error, and so on), then you have to use other methods to prove that your newly suspected cause is in fact responsible for the residue effect. There you have it, folks, all five of Mill’s Methods for proving causation. Now instead of leaping to causal conclusions and making false cause fallacies, you have some tools at your disposal to help you gather evidence for your suspected causes.
Chapter 11: Basic Statistical Concepts and Techniques This chapter is based on Fundamental Methods of Logic, by Matthew Knachel. I. Introduction In this chapter, the goal is to equip ourselves to understand, analyze, and criticize arguments using statistics. Such arguments are extremely common; they’re also frequently manipulative and/or fallacious. As Mark Twain once said, “There are three kinds of lies: lies, damned lies, and statistics.” Misused statistics are very convincing lies, because for some reason or another, putting numbers into an inductive argument makes it look more official, and we’re more willing to trust the argument. It is possible, however, with a minimal understanding of some basic statistical concepts and techniques, along with an awareness of the various ways these are commonly misused (intentionally or not), to see the “lies” for what they are: bad arguments that shouldn’t persuade us. We will first provide a foundation of basic statistical knowledge, and then we will look at various statistical fallacies. II. Basic Statistical Knowledge Here we’ll look over averages, testing hypotheses, and statistical sampling. The goal here is not to deep-dive into the math behind these concepts; this is a logic book, not a math book. The goal, rather, is to help non-mathematicians be less deceived when confronted with arguments using statistics. A. Averages: Mean, Median, Mode The word ‘average’ is slippery: it can be used to refer to one of three things: the arithmetic mean, the median, or the mode of a set of values. The mean, median, and mode for the same set are often different, and when this is the case, use of the word ‘average’ is equivocal. A clever person can use this fact to her rhetorical advantage. We hear the word ‘average’ thrown around quite a bit in arguments: the average family has such-and-such an income, the average student carries such-and-such in student loan debt, and so on. It is important to recognize that there is not an actual family which is The Average One—averages are collective properties, not individual ones. That is, they are a property of the group as a whole—all families, all students. There may be a family in the group whose income matches the average—or there may not be one. However, audiences are supposed to take this fictional average entity to be representative of all (or at least most) of the actual members of the group. So, depending on the conclusion she’s trying to convince people of, the person making the argument will choose between mean, median, and mode, picking the number that best serves her rhetorical purpose. It’s important, therefore, for the critical listener to ask, every time the word ‘average’ is used, “What type of average are they talking about? What’s the difference between the three averages for this group? How would using another type of average affect the argument?” To define our terms: a mean is calculated by adding all the values together, and dividing by the total number of values. So, if we’re finding the average age of a group of people, to find the mean, you add all the ages together, and divide by the number of people in the group. A median is the exact middle number of a group. To find the median age, we’d line the group members up from youngest to oldest (not physically lining up the people, unless you really want to), and find the age of the person in the middle. If there is an even number of people in the group, and thus no member who is in the exact middle, you find the two members in the middle, add their ages together and divide by 2, to find the age in the middle of them. A mode refers to the most common value in a set. So, the modal age would be the most popular age. A simple example can demonstrate both how to calculate the three different averages, and how an unscrupulous person can use the different results to their advantage. Inspiration for this example, as with much that follows, comes from Darrell Huff, How to Lie with Statistics, New York: Norton, 1954. Suppose I run a masonry contracting business on the side—Logical Constructions (a wholly owned subsidiary of LogiCorp). Including myself, 22 people work at Logical Constructions. This is how much they’re paid per year: $350,000 for me (I’m the boss) $75,000 each for two foremen $70,000 for my accountant $50,000 each for five stone masons $30,000 for the office secretary $25,000 each for two apprentices $20,000 each for ten laborers. To calculate the mean salary at Logical Constructions, we add up all the individual salaries (my $350,000, $75,000 twice since there are two foremen, $70,000 once, $50,000 five times since there are five stone masons, and so on). This gives us $1,100,000. We next divide by the number of employees—there are 22 employees total. The result is $50,000. To calculate the median salary, we put all the individual salaries in numerical order (ten entries of $20,000 for the laborers, then two entries of $25,000 for the apprentices, and so on) and find the middle number—or, as is the case with our set, which has an even number of entries, the mean of the middle two numbers. The middle two numbers are both $25,000, so the median salary is $25,000. The median tells me that half of my employees get that much or more, and half of my employees get that much or less. The mode is often the easiest to calculate; here it is the most common salary. There are more laborers than any other type of employee, so the mode is what each of them get: $20,000. Now, you may have noticed, a lot of my workers don’t get paid particularly well. Suppose one day, as I’m driving past our construction site (in the back of my limo, naturally), I notice some union organizers commiserating with my laborers during their lunch break. They’re trying to convince my employees to bargain collectively for higher wages. Now we have a debate: should the workers at Logical Constructions be paid more? I take one side of the issue; the workers and organizers take the other. In the course of making our arguments, we might both refer to the “average worker” at Logical Constructions. I’ll want to do so in a way that makes it appear that this mythical worker is doing pretty well, and so we don’t need to change anything; the organizers will want to do so in such a way that makes it appear that the average worker isn’t doing very well at all. If you were the boss, would you present the mean, median, or mode as your average? What if you were the union organizers? In this case, the mean is higher, so I will use it: “The average worker at Logical Constructions makes $50,000 per year. That’s a pretty good wage!” My opponents, the union organizers, will counter, using either the median, or even better, the mode: “The average worker at Logical Constructions makes a mere $20,000 per year. Try raising a family on such a pittance!” The mean is very different from the other two averages because my salary is so much higher than everyone else’s. An outlier (someone significantly different from the rest of the group) will pull the mean toward them. A single outlier, however, will not affect the mean very much, and won’t affect the mode at all. So, a lot hangs on which sense of ‘average’ we pick. This is true in lots of real-life circumstances. For example, household income in the United States is distributed much as salaries are at my fictional Logical Constructions company: those at the top of the range fare much better than those at the bottom. In 2014, the richest fifth of American households accounted for over 51% of income; the poorest fifth, 3%. Because the minority at the top have salaries so much higher than everyone else, that pulls the mean up, so it’s higher than the median or the mode. In 2014, the mean household income in the U.S. was $72,641. The median, however, was $53,657. That’s a big difference! “The average family makes about $72,000 per year” sounds a lot better than “The average family makes about $53,000 per year.” B. Normal Distributions: Standard Deviation, Confidence Intervals If you gave IQ tests to a whole bunch of people, and then graphed the results on a histogram or bar chart—so that every time you saw a particular score, the bar for that score would get higher— you’d end up with a picture like this: This kind of distribution is called a “normal” or “Gaussian” distribution “Gaussian” because the great German mathematician Carl Friedrich Gauss made a study of such distributions in the early 19th century (in connection with their relationship to errors in measurement).; because of its shape, it’s also often called a “bell curve.” Many phenomena in nature are (approximately) distributed along a bell curve: height, blood pressure, motions of individual molecules in a collection, lifespans of industrial products, measurement errors, and so on. And even when traits are not normally distributed, it can be useful to treat them as if they were. This is because the bell curve provides an extremely convenient starting point for making certain inferences. Because the curve is symmetrical, the mean is the same as the median. Because the highest point is in the middle, the mode is also the same as the median. A bell curve is thus convenient because one can know everything about such a curve by specifying two of its features: its mean and its standard deviation. We already understand the mean. Let’s get a grip on standard deviation. We don’t need to learn how to calculate it (though that can be done); we just want a qualitative (as opposed to quantitative) understanding of what it signifies. Roughly, it’s a measure of the spread of the data represented on the curve; it’s a way of indicating how far, on average, values tend to stray from the mean. An example can make this clear. Consider two cities: Milwaukee, Wisconsin, and San Diego, California. These two cities are different in a variety of ways, not least in the kind of weather their residents experience. Setting aside precipitation, let’s focus just on temperature. If you recorded the high temperatures every day in each town over a long period of time and made a histogram for each (with temperatures on the x-axis, number of days on the y-axis), you’d get two very different-looking curves. Maybe something like these: The average high temperatures for the two cities—the peaks of the curves—would of course be different: San Diego is warmer on average than Milwaukee. But the range of temperatures experienced in Milwaukee is much greater than that in San Diego: some days in Milwaukee, the high temperature is below zero, while on some days in the summer it’s over 100°F. San Diego, on the other hand, is basically always perfect: right around 70° or so. This is an exaggeration, of course, but not much of one. The average high in San Diego in January is 65°; in July, it’s 75°. Meanwhile, in Milwaukee, the average high in January is 29°, while in July it’s 80°. While both of them have normal distributions, the standard deviation of temperatures in Milwaukee is much greater than in San Diego. This is reflected in the shapes of the respective bell curves. Milwaukee’s is shorter and wider, with a non-trivial number of days at the temperature extremes and a wide spread for all the other days. San Diego’s is taller and narrower, with temperatures hovering in a tight range all year, and hence more days at each temperature recorded (which explains the relative heights of the curves). When you encounter a standard deviation, the take-away is that a small number means there is very little variety in the group, like San Diego temperatures. A large standard deviation means a lot of variety in the group, like Milwaukee. If we’re dealing with a normal distribution, once we know the mean and standard deviation, we know everything we need to know about it. There are three very useful facts about these curves that can be stated in terms of the mean and standard deviation (SD). As a matter of mathematical fact, 68.3% of the population depicted on the curve (whether they’re people with certain IQs, days on which certain temperatures were reached, measurements with a certain amount of error) falls within a range of one standard deviation on either side of the mean. So, for example, the mean IQ is 100; the standard deviation is 15. It follows that 68.3% of people have an IQ between 85 and 115—15 points (one SD) on either side of 100 (the mean). Another fact: 95.4% of the population depicted on a bell curve will fall within a range two standard deviations from the mean. So, 95.4% of people have an IQ between 70 and 130—30 points (2 SDs) on either side of 100. Finally, 99.7% of the population falls within three standard deviations of the mean; 99.7% of people have IQs between 55 and 145. These ranges are called confidence intervals. Pick a person at random. How confident are you that they have an IQ between 70 and 130? 95.4%, that’s how confident. They are convenient reference points commonly used in statistical inference. As a matter of fact, in current practice, other confidence intervals are more often used: 90%, (exactly) 95%, 99%, etc. These ranges lie on either side of the mean within non-whole-number multiples of the standard deviation. For example, the exactly-95% interval is 1.96 SDs to either side of the mean. The convenience of calculators and spreadsheets to do our math for us makes these confidence intervals more practical. But we’ll stick with the 68.3/95.4/99.7 intervals for simplicity’s sake. This only works if we’re dealing with a normal distribution pattern, though. Yearly income in the U.S., or the yearly salary of employees at my fictional company, do not follow a normal distribution. In these cases, it’s often much more helpful to know the median and mode than it is to know the mean. The median gives us a number and tells us that half of the population is at or above that number, and half of the population is at or below it. The mode tells us the most common number in the group. Looking at all three types of average, in addition to the standard deviation, gives us a much broader picture of the population as a whole. C. Statistical Inference: Hypothesis Testing If we start with knowledge of the properties of a given normal distribution, we can test claims about the world to which that information is relevant. Starting with a bell curve—information of a general nature—we can draw conclusions about particular hypotheses. These are conclusions of inductive arguments; they are not certain, but more or less probable. When we use knowledge of normal distributions to draw them, we can be precise about how probable they are. This is inductive logic. The basic pattern of the kinds of inferences we’re talking about is this: one formulates a hypothesis, then runs an experiment to test it; the test involves comparing the results of that experiment to what is known (some normal distribution); depending on how well the results of the experiment comport with what would be expected given the background knowledge represented by the bell curve, we draw a conclusion about whether or not the hypothesis is true. Though they are applicable in a very wide range of contexts, it’s perhaps easiest to explain the patterns of reasoning we’re going to examine using examples from medicine. These kinds of cases are vivid; they aid in understanding by making the consequences of potential errors more real. Also, in these cases the hypotheses being tested are relatively simple: claims about individuals’ health—whether they’re healthy or sick, whether they have some condition or don’t—as opposed to hypotheses dealing with larger populations and measurements of their properties. Examining these simpler cases will allow us to see more clearly the underlying patterns of reasoning that cover all such instances of hypothesis testing, and to gain familiarity with the vocabulary statisticians use in their work. The knowledge we start with is how some trait relevant to the particular condition is distributed in the population generally—a bell curve. The experiment we run is to measure the relevant trait in the individual whose health we’re assessing. The result of a comparison with the result of this measurement and the known distribution of the trait tells us something about whether or not the person is healthy. Suppose we start with information about how a trait is distributed among people who are healthy. Hematocrit, for example, is a measure of how much of a person’s blood is taken up by red blood cells, expressed as a percentage of total blood volume. Lower hematocrit levels are associated with anemia; higher levels are associated with dehydration, certain kinds of tumors, and other disorders. Among healthy men, the mean hematocrit level is 47%, with a standard deviation of 3.5%. We can draw the curve, noting the boundaries of the confidence intervals: Because of the fixed mathematical properties of the bell curve, we know that 68.3% of healthy men have hematocrit levels between 43.5% and 50.5%; 95.4% of them are between 40% and 54%; and 99.7% of them are between 36.5% and 57.5%. Let’s consider a man whose health we’re interested in evaluating. Call him Larry. We take a sample of Larry’s blood and measure the hematocrit level. We compare it to the values on the curve to see if there might be some reason to be concerned about Larry’s health. Remember, the curve tells us the levels of hematocrit for healthy men; we want to know if Larry’s one of them. The hypothesis we’re testing is that Larry’s healthy. Statisticians often refer to the hypothesis under examination in such tests as the “null hypothesis”—a default assumption, something we’re inclined to believe unless we discover evidence against it. Anyway, we’re measuring Larry’s hematocrit; what kind of result should he be hoping for? Clearly, he’d like to be as close to the middle, fat part of the curve as possible; that’s where most of the healthy people are. The further away from the average healthy person’s level of hematocrit he strays, the more he’s worried about his health. That’s how these tests work: if the result of the experiment (measuring Larry’s hematocrit) is sufficiently close to the mean, we have no reason to reject the null hypothesis (that Larry’s healthy); if the result is far away, we do have reason to reject it. How far away from the mean is too far away? It depends. A typical cutoff is two standard deviations from the mean—the 95.4% confidence interval. Actually, the typical level is now exactly 95%, or 1.96 standard deviations from the mean. From now on, we’re just going to pretend that the 95.4% and 95% levels are the same thing. That is, if Larry’s hematocrit level is below 40% or above 54%, then we might say we have reason to doubt the null hypothesis that Larry is healthy. The language statisticians use for such a result—say, for example, if Larry’s hematocrit came in at 38%—is to say that it’s “statistically significant.” In addition, they specify the level at which it’s significant—an indication of the confidence-interval cutoff that was used. In this case, we’d say Larry’s result of 38% is statistically significant at the .05 level. (95% = .95; 1 - .95 = .05) Either Larry is unhealthy (anemia, most likely), or he’s among the (approximately) 5% of healthy people who fall outside of the two standard-deviation range. If he came in at a level even further from the mean—say, 36%—we would say that this result is significant at the .003 level (99.7% = .997; 1 - .997 = .003). That would give us all the more reason to doubt that Larry is healthy. So, when we’re designing a medical test like this, the crucial decision to make is where to set the cutoff. Again, typically that’s the 95% confidence interval. If a result falls outside that range, the person tests “positive” for whatever condition we’re on the lookout for. (Of course, a “positive” result is hardly positive news—in the sense of being something you want to hear). But these sorts of results are not conclusive: it may be that the null hypothesis (this person is healthy) is true, and that they’re simply one of the relative rare 5% who fall on the outskirts of the curve. In such a case, we would say that the test has given the person a “false positive” result: the test indicates sickness when in fact there is none. Statisticians refer to this kind of mistake as “type I error.” We could reduce the number of mistaken results our test gives by changing the confidence levels at which we give a positive result. Returning to the concrete example above: suppose Larry has a hematocrit level of 38%, but that he is not in fact anemic; since 38% is outside of the two standard-deviation range, our test would give Larry a false positive result if we used the 95% confidence level. However, if we raised the threshold of statistical significance to the three standard-deviation level of 99.7%, Larry would not get flagged for anemia; there would be no false positive, no type I error. So, we should always use the wider range on these kinds of tests to avoid false positives, right? Not so fast. There’s another kind of mistake we can make: false negatives, also called “type II errors.” Increasing our range increases our risk of this second kind of foul-up. Down there at the skinny end of the curve there are relatively few healthy people. Sick people are the ones who generally have measurements in that range; they’re the ones we’re trying to catch. When we issue a false negative, we’re missing them. A false negative occurs when the test tells you there’s no reason to doubt the null hypothesis (that you’re healthy), when as a matter of fact you are sick. If we increase our range from two to three standard deviations—from the 95% level to the 99.7% level—we will avoid giving a false positive result to Larry, who is healthy despite his low 38% hematocrit level. But we will end up giving false reassurance to some anemic people who have levels similar to Larry’s; someone who has a level of 38% and is sick will get a false negative result if we only flag those outside the 99.7% confidence interval (36.5% - 57.5%). This is a perennial dilemma in medical screening: how best to strike a balance between the two types of errors—between needlessly alarming healthy people with false positive results and failing to detect sickness in people with false negative results. The terms clinicians use to characterize how well diagnostic tests perform along these two dimensions are “sensitivity” and “specificity.” A highly sensitive test will catch a large number of cases of sickness—it has a high rate of true positive results. Of course, this comes at the cost of increasing the number of false positive results as well. A test with a high level of specificity will have a high rate of true negative results— correctly identifying healthy people as such. The cost of increased specificity, though, is an increase in the number of false negative results—sick people that the test misses. Since every false positive is a missed opportunity for a true negative, increasing sensitivity comes at the cost of decreasing specificity. And since every false negative is a missed true positive, increasing specificity comes at the cost of decreasing sensitivity. A final bit of medical jargon: a screening test is accurate to the degree that it is both sensitive and specific. Given sufficiently thorough information about the distributions of traits among healthy and sick populations, clinicians can rig their diagnostic tests to be as sensitive or specific as they like. But since those two properties pull in opposite directions, there are limits to degree of accuracy that is possible. And depending on the particular case, it may be desirable to sacrifice specificity for more sensitivity, or vice versa. To see how a screening test might be rigged to maximize sensitivity, let’s consider an abstract hypothetical example. Suppose we knew the distribution of a certain trait among the population of people suffering from a certain disease. (Contrast this with our starting point above: knowledge of the distribution among healthy individuals). This kind of knowledge is common in medical contexts: various so-called biomarkers—gene mutations, proteins in the blood, etc.—are known to be indicative of certain conditions; often, one can know how such markers are distributed among people with the condition. Again, keeping it abstract and hypothetical, suppose we know that among people who suffer from Disease X, the mean level of a certain biomarker β for the disease is 20, with a standard deviation of 3. We can sum up this knowledge with a curve: Now, suppose Disease X is very serious indeed. It would be a benefit to public health if we were able to devise a screening test that could catch as many cases as possible—a test with a high sensitivity. Given the knowledge we have about the distribution of β among patients with the disease, we can make our test as sensitive as we like. We know, as a matter of mathematical fact, that 68.3% percent of people with the disease have β-levels between 17 and 23; 95.4% of people with the disease have levels between 14 and 26; 99.7% have levels between 11 and 29. Given these facts, we can devise a test that will catch 99.7% of cases of Disease X like so: measure the level of biomarker β in people, and if they have a value between 11 and 29, they get a positive test result; a positive result is indicative of disease. This will catch 99.7% of cases of the condition, because the range chosen is three standard deviations on either side of the mean, and that range contains 99.7% of unhealthy people; if we flag everybody in that range, we will catch 99.7% of cases. Of course, we’ll probably end up catching a whole lot of healthy people as well if we cast our net this wide; we’ll get a lot of false positives. We could correct for this by making our test less sensitive, say by lowering the threshold for a positive test to the two standard-deviation range of 14 – 26. We would now only catch 95.4% of cases of sickness, but we would reduce the number of healthy people given false positives; instead, they would get true negative results, increasing the specificity of our test. Notice that the way we used the bell curve in our hypothetical test for Disease X was different from the way we used the bell curve in our test of hematocrit levels above. In that case, we flagged people as potentially sick when they fell outside of a range around the mean; in the new case, we flagged people as potentially sick when they fell inside a certain range. This difference corresponds to the differences in the two populations the respective distributions represent: in the case of hematocrit, we started with a curve depicting the distribution of a trait among healthy people; in the second case, we started with a curve telling us about sick people. In the former case, sick people will tend to be far from the mean; in the latter, they’ll tend to cluster closer. D. Statistical Inference: Sampling When we were testing hypotheses, our starting point was knowledge about how traits were distributed among a large population—e.g., hematocrit levels among healthy men. We now ask a pressing question: how do we acquire such knowledge? How do we figure out how things stand with a very large population? The difficulty is that it’s usually impossible to check every member of the population. Instead, we have to make an inference. This inference involves sampling: instead of testing every member of the population, we test a small portion of the population—a sample— and infer from its properties to the properties of the whole. Reasoning from part of a group to a group as a whole is a generalization. Reasoning from statistical samples is a simple inductive argument: The sample has property X. Therefore, the general population has property X. The argument is clearly inductive: the premise does not guarantee the truth of the conclusion; it merely makes it more probable. As was the case in hypothesis testing, we can be precise about the probabilities involved, and our probabilities come from the good-old bell curve. Let’s take a simple example. I am indebted for this example in particular (and for much background on the presentation of statistical reasoning in general) to John Norton, 1998, How Science Works, New York: McGraw-Hill, pp. 12.14 – 12.15. Suppose we were trying to discover the percentage of men in the general population; we survey 100 people, and it turns out there are 55 men in our sample. So, the proportion of men in our sample is .55 (or 55%). We’re trying to make an inference from this premise to a conclusion about the proportion of men in the general population. What’s the probability that the proportion of men in the general population is .55? This isn’t exactly the question we want to answer in these sorts of cases, though. Rather, we ask, what’s the probability that the true proportion of men in the general population is in some range on either side of .55? We can give a precise answer to this question, and the answer depends on the size of the range you’re considering in a familiar way. Given that our sample’s proportion of men is .55, it is relatively more likely that the true proportion in the general population is close to that number, less likely that it’s far away. For example, it’s more likely, given the result of our survey, that in fact 50% of the population is men than it is that only 45% are men. And it’s still less likely that only 40% are men. The same pattern holds in the opposite direction: it’s more likely that the true percentage of men is 60% than 65%. Generally speaking, the further away from our survey results we go, the less probable it is that we have the true value for the general population. The drop off in probabilities described takes the form of a bell curve: The standard deviation of .05 is a function of our sample size of 100. We can use the usual confidence intervals—again, with 2 standard deviations, 95.4% being standard practice—to interpret the findings of our survey: we’re pretty sure—to the tune of 95%—that the general population is between 45% and 65% male. That’s a pretty wide range. Our result is not that impressive (especially considering the fact that we know the actual number is very close to 50%). But that’s the best we can do given the limitations of our survey. The main limitation, of course, was the size of our sample: 100 people just isn’t very many. We could narrow the range within which we’re 95% confident if we increased our sample size; doing so would likely (though not certainly) give us a proportion in our sample closer to the true value of (approximately) .5. The relationship between the sample size and the width of the confidence intervals is a purely mathematical one. As sample size goes up, standard deviation goes down—the curve narrows. The pattern of reasoning on display in our toy example is the same as that used in sampling generally. Perhaps the most familiar instances of sampling in everyday life are public opinion surveys. Rather than trying to determine the proportion of people in the general population who are men (not a real mystery), opinion pollsters try to determine the proportion of a given population who, say, intend to vote for a certain candidate, or approve of the job the president is doing, or believe in Bigfoot. Pollsters survey a sample of people on the question at hand, and end up with a result: 29% of Americans believe in Bigfoot, for example. Here’s an actual survey with that result: http://angusreidglobal.com/wp-content/uploads/2012/03/2012.03.04_Myths.pdf But the headline number, as we have seen, doesn’t tell the whole story. 29% of the sample (in this case, about 1,000 Americans) reported believing in Bigfoot; it doesn’t follow with certainty that 29% of the general population (all Americans) have that belief. Rather, the pollsters have some degree of confidence (again, 95% is standard) that the actual percentage of Americans who believe in Bigfoot is in some range around 29%. You may have heard the “margin of error” mentioned in connection with such surveys. This phrase refers to the very range we’re talking about. In the survey about Bigfoot, the margin of error is 3%. That’s the distance from the mean (the 29% found in the sample) and the ends of the two standard-deviation confidence interval—the range in which we’re 95% sure the true value lies. Again, this range is just a mathematical function of the sample size: if the sample size is around 100, the margin of error is about 10% (see the toy example above: 2 SDs = .10); if the sample size is around 400, you get that down to 5%; at 600, you’re down to 4%; at around 1,000, 3%; to get down to 2%, you need around 2,500 in the sample, and to get down to 1%, you need 10,000. Interesting mathematical fact: these relationships hold no matter how big the general population from which you’re sampling (as long as it’s above a certain threshold). It could be the size of the population of Wisconsin or the population of China: if your sample is 600 Wisconsinites, your margin of error is 4%; if it’s 600 Chinese people, it’s still 4%. This is counterintuitive, but true—at least, in the abstract. We’re omitting the very serious difficulty that arises in actual polling (which we will discuss in a minute): finding the right 600 Wisconsinites or Chinese people to make your survey reliable; China will present more difficulty than Wisconsin due to the size of the population. So the real upshot of the Bigfoot survey result is something like this: somewhere between 26% and 32% of Americans believe in Bigfoot, and we’re 95% sure that’s the correct range; or, to put it another way, we used a method for determining the true proportion of Americans who believe in Bigfoot that can be expected to determine a range in which the true value actually falls 95% of the time, and the range that resulted from our application of the method on this occasion was 26% - 32%. That last sentence, we must admit, would make for a pretty lousy newspaper headline (“29% of Americans believe in Bigfoot!” is much sexier), but it’s the most honest presentation of what the results of this kind of sampling exercise actually show. Sampling gives us a range, which will be wider or narrower depending on the size of the sample, and not even a guarantee that the actual value is within that range. That’s the best we can do; these are inductive, not deductive, arguments. Finally, on the topic of sampling, we should acknowledge than in actual practice, polling is hard. The mathematical relationships between sample size and margin of error/confidence that we’ve noted all hold in the abstract, but real-life polls can have errors that go beyond these theoretical limitations on their accuracy. As the 2016 U.S. presidential election—and the so-called “Brexit” vote in the United Kingdom that same year, and many, many other examples throughout the history of public opinion polling—showed us, polls can be systematically in error. The kinds of facts we’ve been stating—that with a sample size of 600, a poll has a margin of error of 4% at the 95% confidence level—hold only on the assumption that there’s a systematic relationship between the sample and the general population it’s meant to represent; namely, that the sample is representative. A representative sample mirrors the general population; in the case of people, this means that the sample and the general population have the same demographic make-up—same percentage of old people and young people, white people and people of color, rich people and poor people, etc., etc. Polls whose samples are not representative are likely to misrepresent the feature of the population they’re trying to capture. Suppose I wanted to find out what percentage of the U.S. population thinks favorably of Donald Trump. If I asked 1,000 people in, say, rural Oklahoma, I’d get one result; if I asked 1,000 people in midtown Manhattan, I’d get a much different result. Neither of those two samples is representative of the population of the United States as a whole. To get such a sample, I’d have to be much more careful about whom I surveyed. A famous example from the history of public polling illustrates the difficulties here rather starkly: in the 1936 U.S. presidential election, the contenders were Republican Alf Landon of Kansas, and the incumbent President Franklin D. Roosevelt. A (now-defunct) magazine, Literary Digest conducted a poll with 2.4 million (!) participants, and predicted that Landon would win in a landslide. Instead, he lost in a landslide; FDR won the second of his four presidential elections. What went wrong? With a sample size so large, the margin of error would be tiny. The problem was that their sample was not representative of the American population. They chose participants randomly from three sources: (a) their list of subscribers; (b) car registration forms; and (c) telephone listings. The problem with this selection procedure is that all three groups tended to be wealthier than average. This was 1936, during the depths of the Great Depression. Most people didn’t have enough disposable income to subscribe to magazines, let alone have telephones or own cars. The survey therefore over-sampled Republican voters and got a skewed result. Even a large and seemingly random sample can lead one astray. This is what makes polling so difficult: finding representative samples is hard. It’s even harder than this paragraph makes it out to be. It’s usually impossible for a sample—the people you’ve talked to on the phone about the president or whatever—to mirror the demographics of the population exactly. So pollsters have to weight the responses of certain members of their sample more than others to make up for these discrepancies. This is more art than science. Different pollsters, presented with the exact same data, will make different choices about how to weight things, and will end up reporting different results. See this fascinating piece for an example: http://www.nytimes.com/interactive/2016/09/20/upshot/the-error-the-polling-world-rarely-talks- about.html?_r=0 Other practical difficulties with polling are worth noting. First, the way your polling question is worded can make a big difference in the results you get. The framing of an issue—the words used to specify a particular policy or position—can have a dramatic effect on how a relatively uninformed person will feel about it. If you wanted to know the American public’s opinion on whether or not it’s a good idea to tax the transfer of wealth to the heirs of people whose holdings are more than $5.5 million or so, you’d get one set of responses if you referred to the policy as an “estate tax,” a different set of responses if you referred to it as an “inheritance tax,” and a still different set if you called it the “death tax.” A poll of Tennessee residents found that 85% opposed “Obamacare,” while only 16% opposed “Insure Tennessee” (they’re the same thing, of course). Source: http://www.nbcnews.com/politics/elections/rebuke-tennessee-governor-koch-group-shows-its-power- n301031 Even slight changes in the wording of questions can alter the results of an opinion poll. This is why the polling firm Gallup hasn’t changed the wording of its presidential-approval question since the 1930s. They always ask: “Do you approve or disapprove of the way [name of president] is handling his job as President?” A deviation from this standard wording can produce different results. The polling firm Ipsos found that its polls were more favorable than others’ for the president. They traced the discrepancy to the different way they worded their question, giving an additional option: “Do you approve, disapprove, or have mixed feelings about the way Barack Obama is handling his job as president?” http://spotlight.ipsos-na.com/index.php/news/is-president-obama-up-or-down-the-effect-of-question-wording-on- levels-of-presidential-support/ Another difficulty with polling is that some questions are harder to get reliable data about than others, simply because they involve topics about which people tend to be untruthful. Asking someone whether they approve of the job the president is doing is one thing; asking them whether or not they’ve ever cheated on their taxes, say, is quite another. They’re probably not shy about sharing their opinion on the former question; they’ll be much more reluctant to be truthful on the latter (assuming they’ve ever fudged things on tax returns). There are lots of things it would be difficult to discover for this reason: how often people floss, how much alcohol they drink, whether or not they exercise, their sexual habits, and so on. Sometimes this reluctance to share the truth about oneself is quite consequential: some experts think that the reason polls failed to predict the election of Donald Trump as president of the United States in 2016 was that some of his supporters were “shy”—unwilling to admit that they supported the controversial candidate. See here, for example: https://www.washingtonpost.com/news/monkey-cage/wp/2016/12/13/why-the-polls- missed-in-2016-was-it-shy-trump-supporters-after-all/?utm_term=.f20212063a9c They had no such qualms in the voting booth, however. Finally, who’s asking the question—and the context in which it’s asked—can make a big difference. People may be more willing to answer questions in the relative anonymity of an online poll, slightly less willing in the somewhat more personal context of a telephone call, and still less forthcoming in a face-to-face interview. Pollsters use all of these methods to gather data, and the results vary accordingly. Of course, these factors become especially relevant when the question being polled is a sensitive one, or something about which people tend not to be honest or forthcoming. To take an example: the best way to discover how often people truly floss is probably with an anonymous online poll. People would probably be more likely to lie about that over the phone, and still more likely to do so in a face-to-face conversation. The absolute worst source of data on that question, perversely, would probably be from the people who most frequently ask it: dentists and dental hygienists. Every time you go in for a cleaning, they ask you how often you brush and floss; and if you’re like most people, you lie, exaggerating the assiduity with which you attend to your dental-health maintenance (“I brush after every meal and floss twice a day, honest.”). As was the case with hypothesis testing, the logic of statistical sampling is relatively clear. Things get murky, again, when straightforward abstract methods confront the confounding factors involved in real-life application. VI. How to Lie with Statistics The title of this section, a lot of the topics it discusses, and even some of the examples it uses, are taken from Darrell Huff’s book, How to Lie with Statistics. The basic grounding in fundamental statistical concepts and techniques provided in the last section gives us the ability to understand and analyze statistical arguments. Since real-life examples of such arguments are so often manipulative and misleading, our aim in this section is to build on the foundation of the last by examining some of the most common statistical fallacies—the bad arguments and deceptive techniques used to try to bamboozle us with numbers. 1. Impressive Numbers without Context I’m considering buying a new brand of shampoo. The one I’m looking at promises “85% more body.” That sounds great to me (I’m pretty bald; I can use all the extra body I can get). But before I make my purchase, maybe I should consider the fact that the shampoo bottle doesn’t answer this simple follow-up question: 85% more body than what? The bottle does mention that the formulation inside is “new and improved.” So maybe it’s 85% more body than the unimproved shampoo? Or possibly they mean that their shampoo gives hair 85% more body than their competitors’. Which competitor, though? The one that does the best at giving hair more body? The one that does the worst? The average of all the competing brands? Or maybe it’s 85% more body than something else entirely. I once had a high school teacher who advised me to massage my scalp for 10 minutes every day to prevent baldness (I didn’t take the suggestion; maybe I should have). Perhaps this shampoo produces 85% more body than daily 10-minute massages. Or maybe it’s 85% more body than never washing your hair at all. And just what is “body” anyway? How is it quantified and measured? Did they take high-precision calipers and systematically gauge the widths of hairs? Or is it more a function of coverage—hairs per square inch of scalp surface area? The sad fact is, answers to these questions are not forthcoming. The claim that the shampoo will give my hair 85% more body sounds impressive, but without some additional information for me to contextualize that claim, I have no idea what it means. This is a classic rhetorical technique: throw out a large number to impress your audience, without providing the context necessary for them to evaluate whether or not your claim is actually all that impressive. Usually, on closer examination, it isn’t. Advertisers and politicians use this technique all the time. In the spring of 2009, the economy was in really bad shape (the fallout from the financial crisis that began in the fall of the year before was still being felt; stock market indices didn’t hit their bottom until March 2009, and the unemployment rate was still on the rise). Barack Obama, the newly inaugurated president at the time, wanted to send the message to the American people that he got it: households were cutting back on their spending because of the recession, and so the government would do the same thing. After his first meeting with his cabinet (the Secretaries of Defense, State, Energy, etc.), he held a press conference in which he announced that he had ordered each of them to cut $100 million from their agencies’ budgets. He had a great line to go with the announcement: “$100 million there, $100 million here—pretty soon, even here in Washington, it adds up to real money.” Funny. And impressive-sounding. $100 million is a hell of a lot of money! At least, it’s a hell of a lot of money to me. I’ve got—give me a second while I check—$64 in my wallet right now. I wish I had $100 million. But of course my personal finances are the wrong context in which to evaluate the president’s announcement. He’s talking about cutting from the federal budget; that’s the context. How big is that? In 2009, it was a little more the $3 trillion. There are fifteen departments that the members of the cabinet oversee. The cut Obama ordered amounted to $1.5 billion, then. That’s .05% of the federal budget. That number’s not sounding as impressive now that we put it in the proper context. 2009 provides another example of this technique. Opponents of the Affordable Care Act (“Obamacare”) complained about the length of the bill: they repeated over and over that it was 1,000 pages long. That complaint dovetailed nicely with their characterization of the law as a boondoggle and a government takeover of the healthcare system. 1,000 pages sure sounds like a lot of pages. That’s up there with notoriously long books like War and Peace, Les Miserables, and Infinite Jest. It’s long for a book, but is it a lot of pages for a piece of federal legislation? Well, it’s big, but certainly not unprecedented. That year’s stimulus bill was about the same length. President Bush’s 2007 budget bill was just shy of 1,500 pages. His No Child Left Behind bill clocks in at just shy of 700. The fact is, major pieces of legislation have a lot of pages. The Affordable Care Act was not especially unusual. 2. Misunderstanding Error As we discussed, built in to the logic of sampling is a margin of error. It is true of measurement generally that random error is unavoidable: whether you’re measuring length, weight, velocity, or whatever, there are inherent limits to the precision and accuracy with which our instruments can measure things. Measurement errors are built into the logic of scientific practice generally; they must be accounted for. Failure to do so—or intentionally ignoring error—can produce misleading reports of findings. This is particularly clear in the case of public opinion surveys. As we saw, the results of such polls are not the precise percentages that are often reported, but rather ranges of possible percentages (with those ranges only being reliable at the 95% confidence level, typically). And so to report the results of a survey, for example, as “29% of Americans believe in Bigfoot,” is a bit misleading since it leaves out the margin of error and the confidence level. A worse sin is committed (quite commonly) when comparisons between percentages are made and the margin of error is omitted. This is typical in politics, when the levels of support for two contenders for an office are being measured. A typical newspaper headline might report something like this: “Smith Surges into the Lead over Jones in Latest Poll, 44% to 43%.” This is a sexy headline: it’s likely to sell papers (or, nowadays, generate clicks), both to (happy) Smith supporters and (alarmed) Jones supporters. But it’s misleading: it suggests a level of precision, a definitive result, that the data simply do not support. Let’s suppose that the margin of error for this hypothetical poll was 3%. What the survey results actually tell us, then, is that (at the 95% confidence level) the true level of support for Smith in the general population is somewhere between 41% and 47%, while the true level of support for Jones is somewhere between 40% and 46%. Those data are consistent with a Smith lead, to be sure; but they also allow for a commanding 46% to 41% lead for Jones. The best we can say is that it’s slightly more likely that Smith’s true level of support is higher than Jones’s (at least, we’re pretty sure; 95% confidence interval and all). When differences are smaller than the margin of error (really, twice the margin of error when comparing two numbers), they just don’t mean very much. That’s a fact that headline-writers typically ignore. This gives readers a misleading impression about the certainty with which the state of the election can be known. Early in their training, scientists learn that they cannot report values that are smaller than the error attached to their measurements. If you weigh some substance, say, and then run an experiment in which it’s converted into a gas, you can plug your numbers into the ideal gas law and punch them into your calculator, but you’re not allowed to report all the numbers that show up after the decimal place. The number of so-called “significant digits” (or sometimes “figures”) you can use is constrained by the size of the error in your measurements. If you can only know the original weight to within .001 grams, for example, then even though the calculator spits out .4237645, you can only report a result using three significant digits—.424 after rounding. The more significant digits you report, the more precise you imply your measurement is. This can have the rhetorical effect of making your audience easier to persuade. Precise numbers are impressive; they give people the impression that you really know what you’re talking about, that you’ve done some serious quantitative analytical work. Suppose I ask 1,000 college students how much sleep they got last night. I add up all the numbers and divide by 1,000, and my calculator gives me 7.037 hours. If I went around telling people that I’d done a study that showed that the average college student gets 7.037 hours of sleep per night, they’d be pretty impressed: my research methods were so thorough that I can report sleep times down to the thousandths of an hour. They’ve probably got a mental picture of my laboratory, with elaborate equipment hooked up to college students in beds, measuring things like rapid eye movement and breathing patterns to determine the precise instants at which sleep begins and ends. But I have no such laboratory. I just asked a bunch of people. Ask yourself: how much sleep did you get last night? I got about 9 hours (it’s the weekend). The key word in that sentence is ‘about.’ Could it have been a little bit more or less than 9 hours? Could it have been 9 hours and 15 minutes? 8 hours and 45 minutes? Sure. The error on any person’s report of how much they slept last night is bound to be something like a quarter of an hour. That means that I’m not entitled to those 37 thousandths of an hour that I reported from my little survey. The best I can do is say that the average college student gets about 7 hours of sleep per night, plus or minus 15 minutes or so. 7.037 is precise, but the precision of that figure is spurious (not genuine, false). Ignoring the error attached to measurements can have profound real-life effects. Consider the 2000 U.S. presidential election. George W. Bush defeated Al Gore that year, and it all came down to the state of Florida, where the final margin of victory (after recounts were started, then stopped, then started again, then finally stopped by order of the Supreme Court of the United States) was 327 votes. There were about 6 million votes cast in Florida that year. The margin of 327 is about .005% of the total. Here’s the thing: counting votes is a measurement like any other; there is an error attached to it. You may remember that in many Florida counties, they were using punch-card ballots, where voters indicate their preference by punching a hole through a perforated circle in the paper next to their candidate’s name. Sometimes, the circular piece of paper—a so-called “chad”—doesn’t get completely detached from the ballot, and when that ballot gets run through the vote-counting machine, the chad ends up covering the hole and a non-vote is mistakenly registered. Other types of vote-counting methods—even hand-counting—have their own error. And whatever method is used, the error is going to be greater than the .005% margin that decided the election. As one prominent mathematician put it, “We’re measuring bacteria with a yardstick.” John Paulos, “We’re Measuring Bacteria with a Yardstick,” November 22, 2000, The New York Times. That is, the instrument we’re using (counting, by machine or by hand) is too crude to measure the size of the thing we’re interested in (the difference between Bush and Gore). He suggested they flip a coin to decide Florida. It’s simply impossible to know who won that election. In 2011, newly elected Wisconsin Governor Scott Walker, along with his allies in the state legislature, passed a budget bill that had the effect, among other things, of cutting the pay of public sector employees by a pretty significant amount. There was a lot of uproar. People who were against the bill made their case in various ways. One of the lines of attack was economic: depriving so many Wisconsin residents of so much money would damage the state’s economy and cause job losses (state workers would spend less, which would hurt local businesses’ bottom lines, which would cause them to lay off their employees). One newspaper story at the time quoted a professor of economics who claimed that the Governor’s bill would cost the state 21,843 jobs. Steven Verburg, “Study: Budget Could Hurt State’s Economy,” March 20, 2011, Wisconsin State Journal. Not 21, 844 jobs; it’s not that bad. Only 21,843. This number sounds impressive; it’s very precise. But of course that precision is spurious. Estimating the economic effects of public policy is an extremely uncertain business. I don’t know what kind of model this economist was using to make his estimate, but whatever it was, it’s impossible for its results to be reliable enough to report that many significant digits. My guess is that at best the 2 in 21,843 has any meaning at all. 3. Tricky Percentages Statistical arguments are full of percentages, and there are lots of ways you can fool people with them. The key to not being fooled by such figures, usually, is to keep in mind what it’s a percentage of. Inappropriate, shifting, or strategically chosen numbers can give you misleading percentages. When the numbers are very small, using percentages instead of fractions is misleading. Johns Hopkins Medical School, when it opened in 1893, was one of the few medical schools that allowed women to enroll. Not because the school’s administration was particularly enlightened. They could only open with the financial support of four wealthy women who made this a condition for their donations. In those benighted times, people worried about women enrolling in schools with men for a variety of silly reasons. One of them was the fear that the impressionable young ladies would fall in love with their professors and marry them. Absurd, right? Well, maybe not: in the first class to enroll at the school, 33% of the women did indeed marry their professors! The sexists were apparently right. That figure sounds impressive, until you learn that the denominator is 3. Three women enrolled at Johns Hopkins that first year, and one of them married her anatomy professor. Using the percentage rather than the fraction exaggerates in a misleading way. Another made up example: I live in a relatively safe little town. If I saw a headline in my local newspaper that said “Armed Robberies are Up 100% over Last Year” I would be quite alarmed. That is, until I realized that last year there was one armed robbery in town, and this year there were two. That is a 100% increase, but using the percentage of such a small number is misleading. You can fool people by changing the number you’re taking a percentage of mid-stream. Suppose you’re an employee at my aforementioned LogiCorp. You evaluate arguments for $10.00 per hour. One day, I call all my employees together for a meeting. The economy has taken a turn for the worse, I announce, and we’ve got fewer arguments coming in for evaluation; business is slowing. I don’t want to lay anybody off, though, so I suggest that we all share the pain: I’ll cut everybody’s pay by 20%; but when the economy picks back up, I’ll make it up to you. So you agree to go along with this plan, and you suffer through a year of making a mere $8.00 per hour evaluating arguments. But when the year is up, I call everybody together and announce that things have been improving and I’m ready to set things right: starting today, everybody gets a 20% raise. First a 20% cut, now a 20% raise; we’re back to where we were, right? Wrong. I changed numbers mid- stream. When I cut your pay initially, I took twenty percent of $10.00, which is a reduction of $2.00. When I gave you a raise, I gave you twenty percent of your reduced pay rate of $8.00 per hour. That’s only $1.60. Your final pay rate is a mere $9.60 per hour. Often, people make a strategic decision about what number to take a percentage of, choosing the one that gives them a more impressive-sounding, rhetorically effective figure. Suppose I, as the CEO of LogiCorp, set an ambitious goal for the company over the next year: I propose that we increase our productivity from 800 arguments evaluated per day to 1,000 arguments per day. At the end of the year, we’re evaluating 900 arguments per day. We didn’t reach our goal, but we did make an improvement. In my annual report to investors, I proclaim that we were 90% successful. That sounds good; 90% is really close to 100%. But it’s misleading. I chose to take a percentage of 1,000: 900 divided by 1,000 give us 90%. But is that the appropriate way to measure the degree to which we met the goal? I wanted to increase our production from 800 to 1,000; that is, I wanted a total increase of 200 arguments per day. How much of an increase did we actually get? We went from 800 up to 900; that’s an increase of 100. Our goal was 200, but we only got up to 100. In other words, we only got to 50% of our goal. That doesn’t sound as good. Another case of strategic choices. Opponents of abortion rights might point out that 97% of gynecologists in the United States have had patients seek abortions. This creates the impression that there’s an epidemic of abortion-seeking, that it happens regularly. Someone on the other side of the debate might point out that only 1.25% of women of childbearing age get an abortion each year. That’s hardly an epidemic. Each of the participants in this debate has chosen a convenient number to take a percentage of. For the anti-abortion activist, that is the number of gynecologists. It’s true that 97% have patients who seek abortions; only 14% of them actually perform the procedure, though. The 97% exaggerates the prevalence of abortion (to achieve a rhetorical effect). For the pro-choice activist, it is convenient to take a percentage of the total number of women of childbearing age. It’s true that a tiny fraction of them get abortions in a given year; but we have to keep in mind that only a small percentage of those women are pregnant in a given year. As a matter of fact, among those that actually get pregnant, something like 17% have an abortion. The 1.25% minimizes the prevalence of abortion (again, to achieve a rhetorical effect). 4. The Base-Rate Fallacy The base rate is the frequency with which some kind of event occurs, or some kind of phenomenon is observed. When we ignore this information, or forget about it, we commit a fallacy and make mistakes in reasoning. Most car accidents occur in broad daylight, at low speeds, and close to home. So does that mean I’m safer if I drive really fast, at night, in the rain, far away from my house? Of course not. Then why are there more accidents in the former conditions? The base rates: much more of our driving time is spent at low speeds, during the day, and close to home; relatively little of it is spent driving fast at night, in the rain and far from home. Consider a woman formerly known as Mary (she changed her name to Moon Flower). She’s a committed pacifist, vegan, and environmentalist; she volunteers with Green Peace; her favorite exercise is yoga. Which is more probable: that she’s a best-selling author of new-age, alternative- medicine, self-help books—or that she’s a waitress? If you answered that she’s more likely to be a best-selling author of self-help books, you fell victim to the base-rate fallacy. Granted, Moon Flower fits the stereotype of the kind of person who would be the author of such books perfectly. Nevertheless, it’s far more probable that a person with those characteristics would be a waitress than a best-selling author. Why? Base rates. There are far, far (far!) more waitresses in the world than best-selling authors (of new-age, alternative-medicine, self-help books). The base rate of waitressing is higher than that of best-selling authorship by many orders of magnitude. Sometimes people will ignore base rates on purpose to try to fool you. Did you know that marijuana is more dangerous than heroin? Neither did I. But look at this chart: That graphic was published in a story in USA Today under the headline “Marijuana poses more risks than many realize.” Liz Szabo, “Marijuana poses more risks than many realize,” July 27, 2014, USA Today. The following link opens in a new window: Marijuana article The chart/headline combo create an alarming impression: if so many more people are going to the emergency room because of marijuana, it must be more dangerous than I realized. Look at that: more than twice as many emergency room visits for pot than heroin; it’s almost as bad as cocaine! Or maybe not. What this chart ignores is the base rates of marijuana, cocaine, and heroin use in the population. Far (far!) more people use marijuana than use heroin or cocaine. A truer measure of the relative dangers of the various drugs would be the number of emergency room visits per user. That gives you a far different chart: From German Lopez, “Marijuana sends more people to the ER than heroin. But that's not the whole story.” August 2, 2014, Vox.com. The following link opens in a new window: second marijuana article. 5. Lying with Pictures Speaking of charts, they are another tool that can be used (abused) to make dubious statistical arguments. We often use charts and other pictures to graphically convey quantitative information. But we must take special care that our pictures accurately depict that information. There are all sorts of ways in which graphical presentations of data can distort the actual state of affairs and mislead our audience. Consider, once again, my fictional company, LogiCorp. Business has been improving lately, and I’m looking to get some outside investors so I can grow even more quickly. So I decide to go on that TV show Shark Tank. You know, the one with Mark Cuban and panel of other rich people, where you make a presentation to them and they decide whether or not your idea is worth investing in. Anyway, I need to plan a persuasive presentation to convince one of the sharks to give me a whole bunch of money for LogiCorp. I’m going to use a graph to impress them with company’s potential for future growth. Here’s a graph of my profits over the last decade: Not bad. But not great, either. The positive trend in profits is clearly visible, but it would be nice if I could make it look a little more dramatic. I’ll just tweak things a bit: Better. All I did was adjust the y-axis. No reason it has to go all the way down to zero and up to 240. Now the upward slope is accentuated; it looks like LogiCorp is growing more quickly. But I think I can do even better. Why does the x-axis have to be so long? If I compressed the graph horizontally, my curve would slope up even more dramatically: Now that’s explosive growth! The sharks are gonna love this. Well, that is, as long as they don’t look too closely at the chart. Profits on the order of $1.80 per year aren’t going to impress a billionaire like Mark Cuban. But I can fix that: There. For all those sharks know, profits are measured in the millions of dollars. Of course, for all my manipulations, they can still see that profits have increased 400% over the decade. That’s pretty good, of course, but maybe I can leave a little room for them to mentally fill in more impressive numbers: That’s the one. Soaring profits, and it looks like they started close to zero and went up to—well, we can’t really tell. Maybe those horizontal lines go up in increments of 100, or 1,000. LogiCorp’s profits could be unimaginably high. People manipulate the y-axis of charts for rhetorical effect all the time. In their “Pledge to America” document of 2010, the Republican Party promised to pursue various policy priorities if they were able to achieve a majority in the House of Representatives (which they did). They included the following chart in that diagram to illustrate that government spending was out of control: Writing for New Republic, Alexander Hart pointed out that the Republicans’ graph, by starting the y-axis at 17% and only going up to 24%, exaggerates the magnitude of the increase. That bar on the right is more than twice as big as the other two, but federal spending hadn’t doubled. He produced the following alternative presentation of the data Alexander Hart, “Lying With Graphs, Republican Style (Now Featuring 50% More Graphs),” December 22, 2010, New Republic. Link opens in new window: Lying with Graphs article.: One can make mischief on the x-axis, too. In an April 2011 editorial entitled “Where the Tax Money Is”, The Wall Street Journal made the case that President Obama’s proposal to raise taxes on the rich was a bad idea. The article is at the following link, which opens a new page: Wall Street Journal article. If he was really serious about raising revenue, he would have to raise taxes on the middle class, since that’s where most of the money is. To back up that claim, they produced this graph: This one is subtle. What they present has the appearance of a histogram, but it breaks one of the rules for such charts: each of the bars has to represent the same portion of the population. That’s not even close to the case here. To get their tall bars in the middle of the income distribution, the Journal’s editorial board groups together incomes between $50 and $75 thousand, $75 and $100 thousand, then $100 and $200 thousand, and so on. There are far (far!) more people (or probably households; that’s how these data are usually reported) in those income ranges than there are in, say, the range between $20 and $25 thousand, or $5 to $10 million—and yet those ranges get their own bars, too. That’s just not how histograms work. Each bar in an income distribution chart would have to contain the same number of people (or households). When you produce such a histogram, you see what the distribution really looks like (these data are from a different tax year, but the basic shape of the graph didn’t change during the interim): The lesson: don’t just glance at a chart and come away with a (potentially false) picture of the data. Charts can be manipulated to present whatever picture a person wants. People try to fool you in so many different ways. The only defense is a little logic, and a whole lot of skepticism. Be vigilant!
Chapter 2: Recognizing Arguments This chapter is based on For All X, The Lorain County Remix, remixed by J. Robert Loftis. We just saw that arguments are made of statements. However, there are lots of other things you can do with statements. Part of learning what an argument is involves learning what an argument is not, so in this section and the next we are going to look at some other things you can do with statements besides making arguments. The list below of kinds of non-arguments is not meant to be exhaustive. There are all sorts of things you can do with statements that are not discussed here, nor are the items on this list meant to be exclusive. One passage may function as both, for instance, a narrative and a statement of belief. Right now we are looking at real world reasoning, so you should expect a lot of ambiguity and imperfection. I. Simple Statements of Belief An argument is an attempt to support a conclusion, with reasons. Often, though, when people try to persuade others to believe something, they skip the reasons, and give a simple statement of belief instead. This is a kind of nonargumentative passage where the speaker simply asserts what they believe without giving reasons. Sometimes simple statements of belief are prefaced with the words “I believe,” and sometimes they are not. A simple statement of belief can be a profoundly inspiring way to change people’s hearts and minds. Consider this passage from Dr. Martin Luther King’s Nobel acceptance speech. I believe that even amid today’s mortar bursts and whining bullets, there is still hope for a brighter tomorrow. I believe that wounded justice, lying prostrate on the blood-flowing streets of our nations, can be lifted from this dust of shame to reign supreme among the children of men. I have the audacity to believe that peoples everywhere can have three meals a day for their bodies, education and culture for their minds, and dignity, equality and freedom for their spirits. King, Martin Luther. “Acceptance speech at Nobel Peace Prize ceremony.” A Call to Conscience: The Landmark Speeches of Martin Luther King, Jr. Edited by Clayborne Carson. Grand Central Publishing, 2001. This actually is a part of a longer passage that consists almost entirely of statements that begin with some variation of “I believe.” It is incredibly powerful oration, because the audience, feeling the power of King’s beliefs, comes to share in those beliefs. The language King uses to describe how he believes is important, too. He says his belief in freedom and equality requires audacity, making the audience feel his courage and want to share in this courage by believing the same things. These statements are moving, but they do not form an argument. None of these statements provide evidence for any of the other statements. In fact, they all say roughly the same thing, that good will triumph over evil. So the study of this kind of speech belongs to the discipline of rhetoric, not of logic. II. Expository Passages Perhaps the most basic use of a statement is to convey information. Often if we have a lot of information to convey, we will sometimes organize our statements around a theme or a topic. Information organized in this fashion can often appear like an argument, because all of the statements in the passage relate back to some central statement. However, unless the other statements are given as reasons to believe the central statement, the passage you are looking at is not an argument. Consider this passage: From a college psychology textbook. Eysenck advocated three major behavior techniques that have been used successfully to treat a variety of phobias. These techniques are modeling, flooding, and systematic desensitization. In modeling phobic people watch nonphobics cope successfully with dreaded objects or situations. In flooding, clients are exposed to dreaded objects or situations for prolonged periods of time in order to extinguish their fear. In contrast to flooding, systematic desensitization involves gradual, client-controlled exposure to the anxiety eliciting object or situation. Adapted from Ryckman, Richard. Theories of personality. Cengage Learning, 2007. We call this kind of passage an expository passage. In an expository passage, statements are organized around a central theme or topic statement. The topic statement might look like a conclusion, but the other statements are not meant to be evidence for the topic statement. Instead, they elaborate on the topic statement by providing more details or giving examples. In the passage above, the topic statement is “Eysenck advocated three major behavioral techniques...” The statements describing these techniques elaborate on the topic statement, but they are not evidence for it. Although the audience may not have known this fact about Eysenck before reading the passage, they will typically accept the truth of this statement instantly, based on the textbook’s authority. Subsequent statements in the passage merely provide detail. Deciding whether a passage is an argument or an expository passage is complicated by the fact that sometimes people argue by example: Steve: Kenyans are better distance runners than everyone else. Monica: Oh come on, that sounds like an exaggeration of a stereotype that isn’t even true. Steve: What about Dennis Kimetto, the Kenyan who set the world record for running the marathon? And you know who the previous record holder was? Emmanuel Mutai, also Kenyan. Here Steve has made a general statement about all Kenyans. Monica clearly doubts this claim, so Steve backs it up with some examples that seem to match his generalization. This isn’t a very strong way to argue: moving from two examples to statement about all Kenyans is probably going to be a kind of bad argument known as hasty generalization. The point here, however, is that Steve is offering these examples as an argument. The key to telling the difference between expository passages and arguments by example is whether there is a conclusion that the audience needs to be convinced of. In the passage from the psychology textbook, “Eysenck advocated three major behavioral techniques” doesn’t really work as a conclusion for an argument. The audience, students in an introductory psychology course, aren’t likely to challenge this assertion, the way Monica challenges Steve’s overgeneralizing claim. III. Narratives Statements can also be organized into descriptions of events and actions, as in this snippet from Book V of Harry Potter. But she [Hermione] broke off; the morning post was arriving and, as usual, the Daily Prophet was soaring toward her in the beak of a screech owl, which landed perilously close to the sugar bowl and held out a leg. Hermione pushed a Knut into its leather pouch, took the newspaper, and scanned the front page critically as the owl took off again. Rowling, J. K. Harry Potter and the Order of the Phoenix. Scholastic Press, 2003. We will use the term “narrative” loosely to refer to any passage that gives a sequence of events or actions. A narrative can be fictional or nonfictional. It can be told in regular temporal sequence or it can jump around, forcing the audience to try to reconstruct a temporal sequence. A narrative can describe a short sequence of actions, like Hermione taking a newspaper from an owl, or a grand sweep of events, like this passage about the rise and fall of an empire in the ancient near east: The Guti were finally expelled from Mesopotamia by the Sumerians of Erech (c. 2100), but it was left to the kings of Ur’s famous third dynasty to re-establish the Sargonoid frontiers and write the final chapter of the Sumerian History. The dynasty lasted through the twenty first century at the close of which the armies of Ur were overthrown by the Elamites and Amorites. McEvedy, Colin, and John Woodcock. The Penguin Atlas of Ancient History. Penguin Books, 1967. This passage does not feature individual people performing specific actions, but it is still united by character and action. Instead of Hermione at breakfast, we have the Sumerians in Mesopotamia. Instead of retrieving a message from an owl, the Guti are conquered by the Elamites and Amorites. The important thing is that the statements in a narrative are not related as premises and conclusion. Instead, they are all events which are united—common characters acting in specific times and places. III. Explanations Explanations are not arguments, but they share important characteristics with arguments, so we should devote a separate section to them. Both explanations and arguments are parts of reasoning, because both feature statements that act as reasons for other statements. The difference is that explanations are not used to convince an audience of a conclusion. Let’s start with a workplace example. Suppose you see your co-worker, Henry, removing a computer from his office. You think to yourself “Gosh, is he stealing from work?” But when you ask him about it later, Henry says, “I took the computer because I believed that it was scheduled for repair.” Henry’s statement looks like an argument. It has the indicator word “because” in it, which would mean that the statement “I believed it was scheduled for repairs” would be a premise. If it was, we could put the argument in canonical form, like this: I believed the computer was scheduled for repair. I took the computer from the office. However, this would make a strange argument. If it were an argument, it would be trying to convince us of the conclusion that Henry took the computer from the office. But you don’t need to be convinced of this. You already know it—that’s why you were talking to him in the first place. Henry is giving reasons here, but they aren’t reasons that try to prove something. They are reasons that explain something. When you explain something with reasons, you increase your understanding of the world by placing something you already know in a new context. You already knew that Henry took the computer, but now you know why Henry took the computer, and can see that his action was completely innocent (if his story checks out). Both arguments and explanations involve giving reasons, but the reasons function differently in each case. An explanation is defined as a kind of reasoning where reasons are used to provide a greater understanding of something that is already known. Because both arguments and explanations are parts of reasoning, we will use parallel language to describe them. In the case of an argument, we called the reasons “premises.” In the case of an explanation, we will call them explainers. Instead of a “conclusion,” we say that the explanation has an explainee. We can use the generic term “reasons” to refer to either premises or explainers and the generic term “target proposition” to refer to either conclusions or explainees. This figure shows this relationship: We can put explanations in canonical form, just like arguments, but to distinguish the two, we will simply number the statements, and we will separate the explainers and explainee with an E, like this: 1. Henry believed the computer was scheduled for repair E 2. Henry took the computer from the office. Cases where the target proposition is something that is completely common sense are clearcut cases of explanation. Consider the following passage. From Livescience, a science education website, under the headline “Why is grass green?” Like many plants, most species of grass produce a bright pigment called chlorophyll. Chlorophyll absorbs blue light (high energy, short wavelengths) and red light (low energy, longer wavelengths) well, but mostly reflects green light, which accounts for your lawn’s color. Mauk, Ben. “Why is grass green?” Livescience.com, February 20, 2013. https://www.livescience.com/32496-why-is-grass-green.html. The passage contains reasoning. The nature of chlorophyll “accounts for” the color of grass. But in this case the audience does not need to be convinced that grass is green. Everyone knows that. The audience went to the Livescience website because they wanted an explanation for why grass was green. Often the same piece of reasoning can work as either an argument or an explanation, depending on the situation where it is used. Consider this short dialogue Monica visits Steve’s cubical. Monica: All your plants are dead. Steve: It’s because I never water them. In the passage above, Steve uses the word “because,” which we’ve seen in the past is a premise indicator word. But if it were a premise, the conclusion would be “All Steve’s plants are dead.” But Steve can’t possibly be trying to convince Monica that all his plants are dead. It is something that Monica herself says, and that they both can see. The “because” here indicates a reason, but here Steve is giving an explanation, not an argument. He takes something that Steve and Monica already know—that the plants are dead—and puts it in a new light by explaining how it came to be. In this case, the plants died because they didn’t get water, rather than dying because they didn’t get enough light or were poisoned by a malicious co-worker. The reasoning is best represented like this: 1. Steve never waters his plants. E 2. All the plants are dead. The same piece of reasoning can change from an explanation into an argument simply by putting it into a new situation: Monica and Steve are away from the office. Monica: Did you have someone water your plants while you were away? Steve: No. Monica: I bet they are all dead. Here Steve and Monica do not know that Steve’s plants are dead. Monica is inferring this idea based on the premise which she learns from Steve, that his plants are not being watered. This time “Steve’s plants are not being watered” is a premise and “The plants are dead” is a conclusion. We represent the argument like this: Steve never waters his plants. All the plants are dead. In the example of Steve’s plants, the same piece of reasoning can function either as an argument or an explanation, depending on the context where it is given. This is because the reasoning in the example of the plants is causal: the causes of the plants dying are given as reasons for the death, and we can appeal to causes either to explain something that we know happened or to predict something that we think might have happened. Not all kinds of reasoning are flexible like that, however. Reasoning from authority can be used in some kinds of argument, but often makes a lousy explanation. Consider another conversation between Steve and Monica: Monica: I saw on a documentary last night that the universe is expanding and probably will keep expanding for ever. Steve: Really? Monica: Yeah, Steven Hawking said so. There aren’t any indicator words here, but it looks like Monica is giving an argument. She states that the universe is expanding, and Steve gives a skeptical “really?” Monica then replies by saying that she got this information from the famous physicist Steven Hawking. It looks like Steve is supposed to believe that the universe will expand indefinitely because Hawking, an authority in the relevant field, said so. This makes for an ok argument: Steven Hawking said that the universe is expanding and will continue to do so indefinitely. The universe is expanding and will continue to do so indefinitely. Arguments from authority aren’t very reliable, but for very many things they are all we have to go on. We can’t all be experts on everything. But now try to imagine this argument as an explanation. What would it mean to say that the expansion of the universe can be explained by the fact that Steven Hawking said that it should expand? It would be as if Hawking were a god, and the universe obeyed his commands! Arguments from authority are acceptable, but not ideal. Explanations from authority, on the other hand, are completely illegitimate. In general, arguments that appeal to how the world works are more satisfying than ones which appeal to the authority or expertise of others. Compare the following pair of arguments: (a) Jack says traffic will be bad this afternoon. So, traffic will be bad this afternoon. (b) Oh no! Highway repairs begin downtown today. And a bridge lift is scheduled for the middle of rush hour. Traffic is going to be terrible Even though the second passage is an argument, the reasons used to justify the conclusion could be used in an explanation. Someone who accepts this argument will also have an explanation ready to offer if someone should later ask, “Traffic was terrible today! I wonder why?” This is not true of the first passage: bad traffic is not explained by saying “Jack said it would be bad.” The argument that refers to the drawbridge going up is appealing to a more powerful sort of reason, one that works in both explanations and arguments. This simply makes for a more satisfying argument, one that makes for a deeper understanding of the world, than one that merely appeals to authority. Although arguments based on explanatory premises are preferred, we must often rely on other people for our beliefs, because of constraints on our time and access to evidence. But the other people we rely on should hopefully hold the belief on the basis of an empirical understanding. If those people are just relying on authority, then we should hope that at some point the chain of testimony ends with someone who is relying on something more than mere authority. Later on in this book, we’ll look more closely at sources and how much you should trust them. We just have seen that the same set of statements can be used as an argument or an explanation, depending on the context. This can cause confusion between speakers as to what is going on. Consider the following case: Bill and Henry have just finished playing basketball. Bill: Man, I was terrible today. Henry: I thought you played fine. Bill: Nah. It’s because I have a lot on my mind from work. Bill and Henry disagree about what is happening—arguing or explaining. Henry doubts Bill’s initial statement, which should provoke Bill to argue. But instead, he appears to plough ahead with his explanation. What Henry can do in this case, however, is take the reason that Bill offers as an explanation (that Bill is preoccupied by issues at work) and use it as a premise in an argument for the conclusion “Bill played terribly.” Perhaps Henry will argue (to himself) something like this: “It’s true that Bill has a lot on his mind from work. And whenever a person is preoccupied, his basketball performance is likely to be degraded. So, perhaps he did play poorly today (even though I didn’t notice).” In other situations, people can switch back and forth between arguing and explaining. Imagine that Jones says, “The reservoir is at a low level because of several releases to protect the down-stream ecology.” Jones might intend this as an explanation, but since Smith does not share the belief that the reservoir’s water level is low, he will first have to be given reasons for believing that it is low. The conversation might go as follows: Jones: The reservoir is at a low level because of several releases to protect the down-stream ecology. Smith: Wait. The reservoir is low? Jones: Yeah. I just walked by there this morning. You haven’t been up there in a while? Smith: I guess not. Jones: Yeah, it’s because they’ve been releasing a lot of water to protect the ecology lately. When challenged, Smith offers evidence from his memory: he saw the reservoir that morning. Once Smith accepts that the water level is low, Jones can restate his explanation. Some forms of explanation overlap with other kinds of nonargumentative passages. We are dealing right now with thinking in the real world, and as we mentioned above, the real world is full of messiness and ambiguity. One effect of this is that all the categories we are discussing will wind up overlapping. Narratives and expository passages, for instance, can also function as explanations. Consider this passage: From the sports section. Duke beat Butler 61-59 for the national championship Monday night. Gordon Hayward’s half-court, 3-point heave for the win barely missed to leave tiny Butler one cruel basket short of the Hollywood ending. Based on Associated Press. “Butler ends 2009-10 season as National Runner-Up.” April 6, 2010. On the one hand, this is clearly a narrative—retelling a sequence of events united by time, place, and character. But it also can work as an explanation about how Duke won, if the audience immediately accepts the result. “The last shot was a miss and then Duke won” can be understood as “The last shot was a miss and so Duke won.”
Chapter 8: Scientific Reasoning This chapter is based on Logical Reasoning by Bradley H. Dowden. Because the contributions of modern science are our culture's best examples of advances in knowledge, it is important for everyone to have some appreciation of how scientists reason. This chapter deeply examines the nature of scientific reasoning, showing how to assess the scientific claims we encounter in our daily lives, how to do good scientific reasoning, and how to distinguish science from mere pseudoscience. We begin with a description of science and a review of some of the methods of doing science. I. What is Science? Science creates machinery of destruction. It spawns mutants. It spews radiation. It murders in order to dissect. Its apparently objective pose is a cover for callous indifference. Its consequence will be the annihilation of us all. Edward Rothstein said this…. Oops. Wait a minute, a message just arrived, and I need to make an announcement: “Dr. Frankenstein, please call your office.” OK, where was I? Oh, yes, well, enough of this glowing praise of science. "Science" is the Latin term for knowledge. Latin philosophers used the term to refer to an organized body of knowledge, tracing this understanding to the fourth century BCE philosopher Aristotle, who sought to understand the nature of scientific inquiry. https://plato.stanford.edu/entries/scientific-method/. Note, Aristotle’s work went well past considerations of the nature of scientific inquiry. He likewise wrote on logic, biology, astronomy, physics, economics, political theory, metaphysics and ethics (amongst other things). Aristotle was a member of Plato’s Academy; founded in 387 BCE, the Academy broke ground in a diverse number of scientific fields, including geometry. In fact, Euclid’s geometry is a codification of the work done in Plato’s institute. By "science" we will mean pure empirical science, the kind of science that makes observations and runs experiments trying to make predictions, create explanations and produce theoretical understanding of the physical world. This contemporary understanding of science does not match perfectly with Aristotle’s, for instance, Aristotle does not incorporate experimentation into his work. Nevertheless, the practice of formulating theories based upon careful observation and rigorous thinking has a long history predating even Aristotle and we see scientific developments across ancient cultures, e.g. Babylon, Africa and China. Our use of the term “science” in this text will rule out mathematics and formal logic, and it rules in physics, chemistry, and biology. At any particular time in history, science has what it claims is a body of knowledge, but actually science as we use the term in this text is more a way of getting knowledge than it is a body of knowledge. Creating science is not what doctors, engineers and inventors do. These people apply science, but usually they do not do science in the sense of create science. Consider engineering. Unlike scientists, the engineers primarily want to improve existing things that have been made by humans, such as tractors and X-ray machines, or they want to improve human beings’ abilities to move faster and to communicate more easily with people who are far away. Scientists often make use of advances in engineering, but they have different primary concerns. Pure science is concerned primarily with understanding, explaining, and predicting. Engineering (or applied science) focuses rather on creating technology and controlling it, on getting machines to function as we want them to in a particular situation. That is how pure (or theoretical) scientists are different from engineers. Inventors and doctors are more like the engineers than like the scientists. Proposing precise questions and seeking precise answers is one of the keys to successful science. With precision comes sophistication. Although the scientist's vocabulary is often so technical that the rest of us cannot read a scientific research paper, science is not as distant from common sense as many people imagine. Science isn't the only way to know the world around us. They don't have a "lock" on knowledge. But scientists, like the rest of us, do look around at the world, try to explain what they observe, and are careful to back up what they say. Science is a slowed-down and more open and accountable image of what we normally do in coming to know about the world around us. Nevertheless, science isn't just common sense. Science is more cautious about what it claims to know, and it often overthrows traditional common sense in favor of new beliefs that can better stand up to testing. Everybody agrees that science is important, even Edward Rothstein whose sarcastic remarks inspired the paragraph above about science spawning mutants and spewing radiation. But some people think science is much more important and valuable than others do. According to the distinguished historian of science Herbert Butterfield, the rise of European science in the seventeenth and eighteenth centuries is a turning point in our history; however, this view downplays the significance of developments across the centuries in a variety of civilizations. The European scientific revolution was noteworthy for promoting the notion that scientific knowledge should be produced by the process that we now call the scientific method, which emerges out of the work of medieval thinkers, whose attention to observation, theory and experimentation led to developments in fields such as medicine, agriculture, optics, harmonics, metallurgy, and physics. Theorizing regarding the nature of scientific inquiry was its own field of study in the middle ages, whose insights proved essential to contemporary science (L. Laudan, L., “Theories of scientific method from Plato to Mach”, History of Science (1968): 7(1), 1–63. At its heart, the contemporary scientific method is the method of testing hypotheses. However, it is a mistake to suppose that the incorporation of testing into scientific inquiry didn’t occur prior to the work of European scientists in the seventeenth and eighteenth centuries. For instance, attention to the importance of experimentation enabled Ibn al-Haytham (born c. 965, Basra, Iraq—died c. 1040, Cairo, Egypt) to make significant contribution to the science of optics. The idea is that the true hypotheses will stand up to repeated testing while the false hypotheses eventually will get refuted. In addition to biology, chemistry, and physics, which are the more commonly known sciences, another lesser-known science is stamp collecting. Here is why. Stamp collectors are careful; they use tools; they explain; they predict; and they make generalizations. These are marks of good science. Stamp collectors are careful, like scientists. They measure and use tools such as rulers. They can explain why stamps have perforations and why they aren’t cubical. They can predict that most collections will have more three-cent stamps from 1944 than seventy-four cent stamps from 1944. They make generalizations, such as “There are more European stamps than Egyptian stamps.” So that's why stamp collecting is a science. No, think again. Don’t believe everything you read. Stamp collecting is definitely not a science. It’s a hobby. All that reasoning I just performed was making the same kind of error as if I’d argued like this: A woman has two legs, one nose, and breathes air. Mr. Dowden has two legs, one nose, and breathes air. Therefore, Mr. Dowden is a woman. More is involved in being a woman, right? Similarly, more is involved in being a science. The difficulty is in being more specific about just what else is involved. Here is an attempt to specify what else. Many philosophers of science would say that in addition to being precise, careful, using tools, explaining phenomena, predicting observations, and making generalizations, science also: (1) requires using the scientific method to justify its claims. More on this later. (2) Science assumes a background of no miracles and no supernatural causes. It is unscientific to say there was a hurricane in the Philippine Islands because God was angry with the people there. (3) Science has theories that are held tentatively and are falsifiable. That means science is opposed to dogma, and it requires science’s claims to be true or false depending on what the evidence is. If you have a theory that couldn’t be shown to be incorrect no matter what happens, then you aren’t doing science. Freud's theory of psychoanalysis has that defect. II. Reviewing the Principles of Scientific Reasoning One fairly significant aspect of scientific reasoning distinguishes it from other reasoning: Its justification process can be more intricate. For example, you and I might look back over our experience of gorillas, seeing them in zoos and seeing pictures of them in books, and draw the conclusion that all gorillas are black. A biological scientist interested in making a statement about gorilla color would not be so quick to draw this conclusion; he or she would contact gorilla experts and would systematically search through information from all the scientific reports about gorillas to check whether the general claim about gorilla color has even one counterexample. Only if none were found would the scientist then say, "Given all the evidence so far, all gorillas are black." The scientific community as a whole is even more cautious. It would wait to see whether any other biologists disputed the first biologist's claim. If not, only then would the community agree that all gorillas are black. This difference between scientific reasoning and ordinary reasoning can be summed up by saying that scientific reasoning has higher standards of proof. Scientists don't rummage around the world for facts just so they can accumulate more facts. They gather specific facts to reach general conclusions, the "laws of science." Why? Because a general conclusion encompasses a great variety of specific facts, and because a general claim is more useful for prediction, understanding and explanation, which are the three primary goals of science. Scientists aren't uninterested in specifics, but they usually view specific data as a steppingstone to a broader or more general overview of how the world works. This point can be expressed by saying that scientists prefer laws to facts. Although there is no sharp line between laws and facts, facts tend to be more specific; laws, more general. The power that generality provides is often underestimated. At the zoo, suppose you spot a cage marked "Margay" although the margay is out of sight at the moment. You have never heard of a margay, yet you can effortlessly acquire a considerable amount of knowledge about the margay, just by noticing that the cage is part of your zoo's new rare-feline center. It’s cat-like. If so, then you know it cannot survive in an atmosphere of pure nitrogen, that it doesn't have gills, and that it was not hatched from an egg. You know this about the unseen margay because you know on scientific authority that no cat-like beings can survive in nitrogen, that no cats have gills, and that no cats are hatched from eggs. You don’t know all this first-hand, but you’ve heard it indirectly from scientists, and you’ve never heard of any serious disagreement. Of course, scientific generalizations can be wrong. And maybe no experiment has ever been performed to test whether margays can live on pure nitrogen. But you are confident that if there were serious suspicions, the scientists would act quickly to run the tests. Knowing this about how scientists act, you rest comfortably with the generalizations and with your newly acquired knowledge about margays. Definitions: A test is an observation or an experiment intended to provide evidence about a claim. A law of science is a sufficiently well-tested general claim. A theory is a proposed explanation or a comprehensive, integrated system of laws that can be used in explaining a wide variety of phenomena. Testability, Accuracy, and Precision If a proposed hypothesis (a claim) cannot be tested even indirectly, it is not scientific. This point is expressed by saying that scientists highly value testability. For example, suppose someone suggests, “The current laws of chemistry will hold true only as long as the Devil continues to support them. After all, the Devil made the laws, and he can change them on a whim. Luckily he doesn't change his mind too often.” Now, what is a chemist to make of this extraordinary suggestion? Even if a chemist were interested in pursuing the suggestion further, there would be nothing to do. There is no way to test whether the Devil is or isn't the author of the laws of chemistry. Does the Devil show up on any scientific instrument, even indirectly? Therefore, the Devil theory is unscientific. Testability is a key ingredient of any truly scientific claim. Scientists value accuracy and precision. An accurate measurement is one that agrees with the true state of things. A precise measurement is one of a group of measurements that agree with each other and cluster tightly together near their average. However, precision is valuable to science more than in the area of measurement. Precise terminology has helped propel science forward. Words can give a helpful push. How? There are two main ways. A bird may go by one name in the Southeastern United States but by a different name in Central America and by still a different name in Africa. Yet scientists the world over have a common Latin name for it. Thus, the use of precise terminology reduces miscommunication among scientists. Second, a precise claim is easier to test than an imprecise one. How do you test the imprecise claim that "Vitamin C is good for you"? It would be easier to run an experiment to check the more precise claim "Taking 400 milligrams of vitamin C per day will reduce the probability of getting a respiratory infection by fifty percent." If you can test a claim, you can do more with it scientifically. Testability is a scientific virtue, and precision is one path to testability. Because the claims of social science are generally vaguer than the claims of physical science, social scientists have a tougher time establishing results. When a newspaper reports on biology by saying, "Vitamin C was shown not to help prevent respiratory infections," and when the paper reports on social science by saying, "Central America is more politically unstable than South America," we have a better understanding of the former, as "help prevent" can readily be given an unproblematic operational definition, whereas "politically unstable" is more difficult to define operationally. That is, the operation the biologist performs to decide whether something helps prevent respiratory infections can be defined more precisely, and easily, and accurately than the operation to be performed to decide whether one country is more politically stable than another. Reliability of Scientific Reporting Almost every piece of scientific knowledge we have, we justify on the authority of what some scientist has said or is reported to have said. Because scientists are authorities on science, we usually take their word for things scientific. But chemists are not authorities on geology, and chemists who are experts in inorganic chemistry usually are not authorities on organic chemistry. Thus, when we are told that something is so because scientists believe it to be so, we should try to determine whether the proper authorities are being appealed to. Also, we know that scientists disagree on some issues but not on others, and we know that sometimes only the experts know which issues the experts disagree about. Is the reporter reporting the view of just one scientist, unaware that other scientists disagree? Scientists have the same moral failings as the rest of us, so we should also worry about whether a scientist might be biased on some issue or other. If a newspaper reporter tells us that the scientist's research on cloth diapers versus disposable diapers was not financed by the manufacturer of either diaper, we can place more confidence in the report. Scientific journals are under greater pressure than daily newspapers to report the truth. A scientific journal will lose its reputation and its readers faster when there is a slipup than will the daily newspaper. So the stakes in reporting the truth are higher for journals. That is one reason the editors of scientific journals demand that authors provide such good evidence in their articles. If we read a report of a scientific result in a mainstream scientific journal, we can assume that the journal editor and the reviewers demanded good evidence. But if we read the report in a less reputable source, we have to worry that sloppy operational definitions, careless data collection, inaccurate instruments, or misunderstandings by the reporter may have colored the result. When the stakes are high and we are asked to take an authority's word for something, we want independent verification. That means doing something more than merely buying a second copy of the newspaper to check whether what our first copy says is true. In medicine, it means asking for a second opinion from a different doctor. When the doctor says he wants to cut off your leg, you want some other doctor who is independent of the first doctor to verify that your leg really needs to be amputated. The term independent rules out your going to a partner in the first doctor's practice. Ordinarily, though, we can't be bothered to take such pains to find good evidence. When we nonscientists read in the newspaper that some scientist has discovered something or other, we don't have enough time to check out the details for ourselves; we barely have enough time to read the reporter's account, let alone read his or her sources. So, we have to absorb what we can. In doing so, though, we who are critical thinkers are not blank slates willing to accept anything told to us. We are sensitive enough to ask ourselves: Does the report sound silly? Are any scientists protesting the result? What is the source of the report? We know that a reputable scientific journal article about some topic is more reliable than a reporter's firsthand interview with the author; we trust the science reporters for the national news magazines over those for a small, daily newspaper; and we know that daily newspapers are more reliable than independent bloggers and grocery store tabloids. But except for this, we nonscientists have severe difficulties in discriminating among the sources of information. Suppose you were to read the following passage in a magazine: "To ensure the safety of raw fish, it should be frozen for at least five days at minus 4 degrees Fahrenheit (-20°C). That temperature kills all relevant parasitic worms so far tested." Should you believe what you read? It depends. First, ask yourself, "Where was it published and who said it?" In fact, the passage appeared in Science News, a well-respected, popular scientific publication. The magazine in turn was reporting on an article in an authoritative scientific publication, the New England Journal of Medicine. The journal in turn attributed the comment to Peter M. Schantz of the Centers for Disease Control in Atlanta, Georgia, a well-respected U.S. federal research laboratory. The magazine merely reported that Schantz said this. If you learned all this about the source of the passage in Science News, then you should probably accept what is said and add it to your knowledge. You should accept it, but to what degree? You should still have some doubts based on the following concerns. The magazine did not say whether any other scientists disagreed with what Schantz said or even whether Schantz made this comment speculatively rather than as the result of a systematic study of the question. The occurrence of the word tested in the quote would suggest the latter, but you can't be sure. Nevertheless, you can reasonably suppose that the comment by Schantz was backed up by good science or the magazine wouldn't have published it the way it did— that is, with no warning that the claims by Schantz were not well supported. So, you can give Schantz's claims a high degree of belief, but you could be surer of what Schantz said if you had gotten direct answers to your concerns. Hearing from another scientific expert that Schantz's claims about fish are correct should considerably increase your degree of belief in his claims. Causal Explanations vs. Causal Arguments Scientists and reporters of science present us with descriptions, explanations, and arguments. Scientists describe, for example, how ballistic missiles fall through the sky. In addition to description, scientists might also explain the phenomenon, saying why it occurs the way it does. The explanation will give the causes, and in doing so it will satisfy the following principle: Explanations should be consistent with well-established results (except in extraordinary cases when the well-established results are being overthrown with extraordinarily good evidence). Scientists who publicly claim to have the correct explanation for some phenomenon have accepted a certain burden of proof. It is their obligation to back up their explanation with an argument that shows why their explanation is correct. We readers of scientific news usually are more interested in the description and the explanation than in the argument behind it, and we often assume that other scientists have adequately investigated the first scientist's claim. This is usually a good assumption. Thus, reporters rarely include the scientific proof in their report, instead sticking to describing the phenomenon, explaining it, and saying that a certain scientist has proved that the phenomenon should be explained that way. Scientific proofs normally do not establish their conclusions as firmly as mathematical proofs do. Scientific proofs are inductive; mathematical proofs are deductive. So, one scientific proof can be stronger than another scientific proof even though both are proofs. In any inductive scientific proof, there is never a point at which the conclusion has been proved beyond a shadow of all possible doubt. Nevertheless, things do get settled in science. Scientists proved that the Earth is round, not flat; and even though this result is not established beyond all possible doubt, it is established well enough that the scientific community can move on to examine other issues confident that new data will not require any future revision. In fact, you haven't a prayer of getting a research grant to double-check whether the Earth is flat. One scientific proof can be stronger than another scientific proof even though both are proofs. Good Evidence Many persons view science as some vast storehouse of knowledge. That is an accurate view, but we also should view science as a way of getting to that knowledge. This latter way of looking at science is our primary concern in this chapter. In acquiring knowledge, a good scientist adopts a skeptical attitude that says, "I won't believe you unless you show me some good evidence." Why do scientists have this attitude? Because it is so successful. Scientists who are so trusting that they adopt beliefs without demanding good evidence quickly get led astray; they soon find themselves believing what is false, which is exactly what science is trying to avoid. What constitutes good evidence? How do you distinguish good from bad evidence? It’s not like the evidence appears with little attached labels of “good” and “bad.” Well, if a scientist reports that tigers won't eat vegetables, the report is about a phenomenon that is repeatable—namely, tiger meals. If the evidence is any good, and the phenomenon is repeatable, the evidence should be, too. That is, if other scientists rerun the first scientist's tests, they should obtain the same results. If not, the evidence was not any good. The moral here is that reproducible evidence is better than evidence that can't be reproduced. The truth is able to stand up to repeated tests, but falsehood can eventually be exposed. That is one of the major metaphysical assumptions of contemporary science. A scientist who appreciates good evidence knows that having anecdotal evidence isn't as good as having a wide variety of evidence. For example, suppose a scientist reads an article in an engineering journal saying that tests of 300 randomly selected plastic ball bearings showed the bearings to be capable of doing the job of steel ball bearings in the electric windows of Honda cars. The photo below by Solaris 2006. The journal article reports on a wide variety of evidence, 300 different ball bearings. If a scientist were to hear from one auto mechanic that plastic bearings didn't hold up on the car he repaired last week, the scientist won't be a good logical reasoner if he (or she) immediately discounted the wide variety of evidence and adopted the belief of the one auto mechanic. We logical reasoners should trust the journal article over the single anecdote from the mechanic, although the mechanic's report might alert us to be on the lookout for more evidence that would undermine the findings of the journal article. One lemon does not mean that Honda’s electric windows need redesigning. If you discount evidence arrived at by systematic search, or by testing, in favor of a few firsthand stories, you’ve committed the fallacy of overemphasizing anecdotal evidence. A Cautious Approach with an Open Mind The scientific attitude is also a cautious one. If you are a good scientist, you will worry initially that perhaps your surprising new evidence shows only that something is wrong somewhere. You won't claim to have revolutionized science until you’ve made sure that the error isn't in the faulty operation of your own measuring apparatus. If a change of beliefs is needed, you will try to find a change with minimal repercussions; you won't recommend throwing out a cherished fundamental law when you can just as easily revise it by changing that constant from 23 to 24 so that it is consistent with all data, given the margin of error in the experiments that produced the data. The cautious scientific attitude recognizes these principles: Don't make a broader claim than the evidence warrants, and don't reject strongly held beliefs unless the evidence is very strong. In short, don't be wildly speculative. Scientists are supposed to think up reasonable explanations, but what counts as a reasonable explanation? An explanation that conflicts with other fundamental beliefs that science has established is initially presumed to be unreasonable, and any scientist who proposes such an explanation accepts a heavier than usual burden of proof. A related principle of good explanation is to not offer supernatural explanations until it is clear that more ordinary, natural explanations won't work. In assessing potential new beliefs—candidates for new knowledge—scientists actively use what they already believe. They don't come into a new situation with a mental blank. When scientists hear a report of a ghost sighting in Amityville, they will say that the report is unlikely to be true. The basis for this probability assessment is that everything else in the scientists' experience points to there being no ghosts anywhere, and so not in Amityville, either. Because of this background of prior beliefs, a scientist will say it is more probable that the reporter of the Amityville ghost story is confused or lying than that the report is correct. Better evidence, such as multiple reports or a photograph, may prompt a scientist to actually check out the report, if Amityville isn't too far away or if someone provides travel expenses. Good scientists don't approach new data with the self-assurance that nothing will upset their current beliefs. Scientists are cautious, but they are also open to new information, and they don't suppress counterevidence, relevant evidence that weighs against their accepted beliefs. They do search for what is new; finding it is how they get to be famous. So the scientific attitude requires a delicate balance. Keep an open mind, but don't be so open that you spend most of your valuable time on wild goose chases. Discovering Causes, Creating Explanations, and Solving Problems Contrary to what Francis Bacon recommended in 1600, clearing your head of the B.S. and viewing nature with an open mind is not a reliable way to discover the causes behind what you see. Unfortunately, there is no error-free way. Nevertheless, the discovery process is not completely chaotic. There are rules of thumb. For example, to discover a solution to a problem, scientists can often use a simple principle: Divide the problem into manageable components. This principle was used by the space program in solving the problem of how to travel to the moon. The manager of the moon program parceled out the work. Some scientists and engineers concentrated on creating a more powerful rocket engine; others worked on how to jettison the heavy, empty lower stages of the rocket; others designed the communication link between the Earth and the spaceship's computer; and still others created the robot mechanisms that could carry out the computer's commands during flight and after landing on the moon. In short: Divide and conquer. Another principle of scientific discovery says to assume that similar effects are likely to have similar causes. The history of medicine contains many examples of using this principle effectively. Several times before 1847, Doctor Ignaz Semmelweis of the General Hospital in Vienna, Austria had tried but failed to explain the alarming death rate of so many women who gave birth in his maternity ward. They were dying of puerperal fever, a disease with gruesome symptoms: pus discharges, inflammation throughout the body, chills, fever, delirious ravings. One day, a Dr. Kolletschka, who worked with Semmelweis, was performing an autopsy on a puerperal fever victim when a clumsy medical student nicked Kolletschka's arm with a scalpel. A few days later Kolletschka died with the same symptoms as the women who died of puerperal fever. Semmelweis suspected a connection. Perhaps these were similar effects due to a similar cause. And perhaps whatever entered Kolletschka via the student's scalpel was also being accidentally introduced into the women during delivery. Then Semmelweis suddenly remembered that the doctors who delivered the babies often came straight from autopsies of women who had died of puerperal fever. Maybe they were bringing infectious material with them and it somehow entered the bodies of women during delivery of their babies. Semmelweis's suggestion of blaming the doctors was politically radical for his day, but he was in fact correct that this disease, which we now call blood poisoning, was caused by doctors transferring infectious matter from the dead mothers on the dissecting tables to the living mothers in the delivery rooms. Semmelweis's solution was straightforward. Doctors must be required to wash their hands in disinfectant before delivering babies. That is one reason that today doctors wash their hands between visits to patients. A good method to use when trying to find an explanation of some phenomenon is to look for the key, relevant difference between situations in which the phenomenon occurs and situations in which it doesn't. Semmelweis used this method of discovery. You can use the same method to make discoveries about yourself. Suppose you were nauseous, then you vomited. You want to know why. The first thing to do is to check whether prior to your symptoms you ate something you'd never eaten before. If you discover there was something, it is likely to be the cause. Did you get those symptoms after eating raw tuna, but not after eating other foods? If so, you have a potentially correct cause of your problem. To find a cause, look for a relevant difference between situations where the effect occurs and situations where it does not. The rules of thumb we have just discussed can help guide scientific guessing about what causes what. There are a few other rules, some of which are specific to the kind of problem being worked on. Guessing is only the first stage of the discovery process. Before the guess can properly be called a discovery, it needs to be confirmed. This is the second stage, and one that is more systematic than the first, as we shall see. Confirming by Testing To prove your hypothesis about tuna scientifically, you would need to run some tests. One test would be to eat the tuna again and see whether it causes the symptoms again. That sort of test might be dangerous to your health. Here is a better test: acquire a sample of the tuna and examine it under a microscope for bacteria known to cause the symptoms you had. Suppose you do not have access to the tuna. What can you do? You might ask other people who ate the tuna: "Did you get sick, too?" Yes answers would make the correlation more significant. Suppose, however, you do not know anybody to ask. Then what? The difficulty now is that even if you did eat tuna before you got your symptoms, was that the only relevant difference? You probably also ate something else, such as french fries with catsup. Could this have been the problem instead? You would be jumping to conclusions to blame the tuna merely on the basis of the tuna eating being followed by the symptoms; that sort of jump commits the post hoc (or “after this”) fallacy. At this point you simply do not have enough evidence to determine the cause of your illness. Let's reexamine this search for the cause, but at a more general level, one that will provide an overview of how science works in general. When scientists think about the world in order to understand some phenomenon, they try to discover some pattern or some causal mechanism that might be behind it. They try out ideas the way the rest of us try on clothes in a department store. They don't adopt the first idea they have, but instead are willing to try a variety of ideas and to compare them. Suppose you, a scientist, have uncovered what appears to be a suspicious, unexplained correlation between two familiar phenomena, such as vomiting and tuna eating. Given this observed correlation, how do you go about explaining it? You have to think of all the reasonable explanations consistent with the evidence and then rule out as many as you can until the truth remains. One way an explanation is ruled out is when you collect reliable data inconsistent with it. Another way is if you notice that the explanation is inconsistent with accepted scientific laws. If you are unable to refute the serious alternative explanations, you will be unable to find the truth; knowledge of the true cause will elude you. This entire cumbersome process of searching out explanations and trying to refute them is called the scientific method of justifying a claim. There is no easier way to get to the truth. People have tried to take shortcuts by gazing into crystal balls, taking drugs, or contemplating how the world ought to be, but those methods have turned out to be unreliable. Observation is passive; experimentation is active. Experimentation is a poke at nature. It is an active attempt to create the data needed to rule out a hypothesis. Unfortunately, scientists often cannot test the objects they are most interested in. For example, experimenters interested in whether some potential drug might be harmful to humans would like to test humans but must settle for other species of animal. Scientists get into serious disputes with each other about whether the results of testing on rats, rabbits, and dogs carry over to humans. This dispute is really a dispute about analogy; is the animal's reaction analogous to the human's reaction? Scientists often collect data from a population in order to produce a general claim about that population. The goal is to get a representative sample, and this goal is more likely to be achieved if the sample size is large, random, diverse, and stratified. Nevertheless, nothing you do with your sampling procedure will guarantee that your sample will be representative. If you are interested in making some claim about the nature of polar bears, even capturing every living polar bear and sampling it will not guarantee that you know the characteristics of polar bears that roamed the Earth 2,000 years ago. Relying on background knowledge about the population's lack of diversity can reduce the sample size needed for the generalization, and it can reduce the need for a random sampling procedure. If you have well-established background knowledge that electrons are all alike, you can run your experiment with any old electron; don't bother getting Egyptian electrons as well as Japanese electrons. Aiming to Disconfirm In the initial stages of a scientific investigation, when a scientist has an idea or two to try out, it is more important to find evidence in favor of the idea than to spend time looking for disconfirming evidence. However, in the later stages, when a scientist is ready to seriously test the idea, the focus will turn to ways to shoot it down. Confirming evidence—that is, positive evidence or supporting evidence—is simply too easy to find. That is why the scientist designs an experiment to find evidence that would refute the idea if it were false. Scientists want to find the truth, but the good scientist knows that the proper way to determine the truth of some idea is to try to find negative, not positive, evidence. A scientific generalization, at least a universal one of the form "All X are Y," will have all sorts of confirming instances (things that are both X and Y), but it takes just one X that is not Y to refute the whole generalization. So disconfirming evidence is more valuable than confirming evidence at this later stage of scientific investigation. Failure to find the disconfirming evidence is ultimately the confirming evidence. When a hypothesis can stand up to many and varied attempts to rule it out, the hypothesis is tentatively accepted as true or proved. Although scientific reasoning is not so different from other kinds of logical reasoning, it is special in that its claims tend to be more precise, and the evidence backing up the claims is gathered more systematically. This completes our review of what earlier chapters have said about scientific reasoning. Let's now probe deeper into the mysteries of science. Looking for Alternative Explanations Suppose you receive a letter asking you to invest your money with Grover Hallford and Associates (GHA), a new stock brokerage firm. You do have a little extra cash, Some textbook authors make some fantastic assumptions, don't they? so you don't immediately shut the idea out of your mind. The new stockbrokers charge the same rates as other major brokers who offer investment advice. GHA is unusual, though, in that it promises to dramatically increase your investment because, according to the letter, it has discovered a special analytic technique for predicting the behavior of the stock market. Normally you would have to pay for any stock advice from a broker, but to show good faith, the GHA letter offers a free prediction for you. It predicts that the price of IBM stock will close lower next Tuesday from where it closed at the end of trading on the previous day, Monday. You place the letter in file 13, the circular file (the garbage can). However, the following week you happen to notice that IBM stock did perform as predicted. Hmmm. What is going on? A few days later you receive a second letter from GHA. It says that GHA is sorry you have not yet become a client, but, to once again show its good faith, the company asks you to consider its prediction that Standard Oil of New Jersey stock will close up next Tuesday from where it was at the end of Monday. Again you decline to let GHA invest your savings, but you do keep an eye on the stock price of Standard Oil of New Jersey during the next week. Surprisingly, the prediction turns out to be correct. A few days later you receive a third letter suggesting that you invest with GHA, containing yet another free stock tip, but warning that there is a limit to how much free advice you will receive. Are you now ready to invest with GHA? If not, how many more letters would you have to receive before you became convinced that the brokers truly do understand the logic of the stock market? If you demand thirty letters, aren't you being foolish and passing up the chance of a lifetime? Surely GHA is on to something, isn't it? Other brokers cannot perform this well for you. How often do you get a chance to make money so easily? Isn't GHA's unknown technique causing them to be able to make correct predictions? And even if GHA is cheating and somehow manipulating the market, you can still take advantage of this and make money, too. Think about what you would do if you were faced with this decision about investing. You may not have been able to find a reasonable alternative explanation to GHA's claim that it understands the causal forces shaping the stock market. Many people cannot. That's why the swindle works so well. However, it is a swindle, and it is illegal. What GHA did is to get a long mailing list and divide it in half. For their first letter, half of the people get a letter with the prediction that IBM stock will close higher next Tuesday; the other half get a letter making the opposite prediction—that IBM will not close higher. Having no ability to predict the stock market, GHA merely waits until next Tuesday to find out who received a letter with the correct prediction. Only that half then gets a second letter. Half of the second letters say Standard Oil of New Jersey stock will go up; the other half say it won't. After two mailings, GHA will have been right two times in a row with one-fourth of the people it started with. The list of names in the lucky fourth is divided in half and GHA generates a new letter. Each new mailing cuts down by 50 percent the number of people GHA has given good advice to, but if the company starts with a long enough list, a few people will get many letters with correct predictions. You are among those few. This explains why you have received the letters. Along the way, many people will have sent their hard-earned money to GHA, money that will never be returned. This swindle is quite effective. Watch out for it. And don't use it yourself on anybody else. Once again we draw a familiar moral. The degree of belief you should give to a claim that A causes B (that GHA's insight into the stock market causes its correct predictions) is improved or lessened depending on whether you can be more or less sure that reasonable alternative explanations can be ruled out. Thinking up these alternatives is crucial to logical reasoning. Without this creativity you can be more easily led away from the truth, that is, conned. III. Creating Scientific Explanations The power to explain is a mark of your having discovered the truth. Those who can explain more know more. Since at least the fourth century BCE, we have known that the Earth is round, not flat. Plato proposes the idea in his book Phaedo, dated around 380 BCE. How did we draw this conclusion? Not by gathering many positive reports from people declaring that the Earth is round while failing to receive any negative reports declaring it to be flat or simply by photographing the Earth from space as we first did in 1946. The evidence was more indirect: the hypothesis that the Earth is round enabled so many things to be explained that otherwise were unexplainable. By assuming that the Earth is round we can explain why Magellan's ship could keep sailing west from Spain yet return to Spain. By assuming that the Earth is round we can make sense of the shape of eclipses of the moon (they are round shadows of our round Earth). By assuming that the Earth is round we can explain why, when we look away from port with our telescope at a ship sailing toward the horizon, the top of the ship disappears after the bottom, not before. By assuming that the Earth is round we can explain why the sun can shine at midnight in the arctic. All these facts would be deep mysteries without the round-Earth hypothesis, and it would be nearly a miraculous coincidence if all these facts fit so well with an assumption that was false; therefore, the assumption is a fact. The moral is that science is propelled forward by its power to explain. Probabilistic and Deterministic Explanations The best explanations of an event usually give us a good reason to have expected the event. Suppose you want to explain why apples fall off from the apple tree and hit the ground. One untestable explanation would be that it was the apple's "time" to leave the tree. That explanation appeals to a supernatural notion of fate or destiny. A scientific explanation is that the apple fell because it absorbed enough water through its stem that its weight increased above the maximum downward force that the brittle stem could resist. Because explaining people's behavior is harder than explaining the behavior of apples, the current principles of psychology are less precise than the principles of physics. Psychologists depend on rules of thumb; physical scientists have deterministic laws that indicate what will happen rather than what might happen. For example, why did Sarah decide not to go out with Wayne when he mentioned he had an extra ticket to the concert? After talking with her, a psychologist might explain her action this way: Wayne suggested that Sarah spend her time doing something she believed wouldn't be interesting to her. People will not usually do what they have little interest in doing, nor what they perceive to be against their self-interest. Sentence 1 states the relevant initial facts of the situation, and sentence 2 expresses the relevant law of psychology. This law is less precise than the law of gravity. It is only probabilistic, not deterministic, because it doesn't say what will happen but only what probably will happen. Using 1 and 2 in advance, we could predict only what Sarah probably would do, not what she will do. Psychology can't give a deterministic explanation. Such is the current state of that science. Suppose you asked why you can see through glass but not through concrete, and you were told: “Because glass is transparent.” That answer is appropriate for an elementary school student, but not for a more sophisticated audience. After all, transparent merely means being able to be seen through. The explanation is trivial. Up until 1926, however, no one had a better explanation. Glass’s being transparent was just one of the brute facts of nature. It was accepted, but no deeper explanation could show why. Then, in 1926, the theory of quantum mechanics was discovered. From the principles of quantum mechanics, it was possible to deduce that anything made of glass should permit light to pass through. Similarly, quantum mechanics allowed us to find out why water is wet. These examples illustrate two main points: (1) General theories are more valuable than mere collections of specific facts, because with a general theory you can explain a large variety of individual facts. (2) If you can deduce a phenomenon from some well- accepted principles, you have a much deeper explanation of the phenomenon than if you can't carry out this deduction. Fruitful and Unfruitful Explanations Untestable explanations are avoided by good scientists, but fruitful explanations are highly valued. To appreciate this virtue of fruitfulness, consider the scientists' favorite explanation of what caused the demise of the dinosaurs 65 million years ago. Four explanations or specific theories have been proposed in the scientific literature: the sex theory, the drugs theory, the violence theory, and the crime theory. According to the sex theory, 65 million years ago the world's temperature increased a few degrees. This increase warmed the male dinosaurs' testes to the point that they became infertile. According to the drug theory, 65 million years ago the world's first psychoactive (mind altering) plants evolved. Dinosaurs ate these plants, overdosed, and died. According to the violence theory, 65 million years ago some violent global event—perhaps caused by an asteroid or volcano—led to the dinosaur extinctions. According to the crime theory, 65 million years ago the first small mammals got braver and more clever. Some mammals learned to steal dinosaur eggs, which caused the dinosaur extinctions. Of all four theories, current science favors the violence theory. Why? There are two reasons: it has been successfully tested, and it has been fruitful. The other three theories are testable in principle, but they are too hard to test in practice. The soft parts of male dinosaurs don't leave fossils, so the sex theory cannot be tested by looking for fossil remains. The drug theory is too hard to test because nothing much is known about which drugs were in which plants so long ago. The crime theory is too hard to test because there is no practical way to check whether little mammals did or didn't steal the dinosaur eggs. On the other hand, the violence theory can be. Suppose a violent global event threw dust into the air, darkening the Earth, leading to cold weather and the end of most plant photosynthesis. Digging down to the 65-million-year layer should reveal a thin layer of dust, no matter where in the world the scientists dig down. And indeed, scientists have discovered a layer of dust there containing a high concentration of a very rare element, iridium. Although naturally scarce on the Earth's surface, the element is relatively abundant both in asteroids and deep inside volcanoes. In addition to its having stood up to this observational test, the violence theory is favored because it is so fruitful. That is, scientists can imagine many interesting and practical ways in which the theory can be tested. They can search satellite photos looking for 65-million-year-old asteroid craters. At suspected crater sites, they can analyze rocks for signs of impact—tiny fractures in shocked quartz. Digging might reveal pieces of an asteroid. A large speeding asteroid would ionize the surrounding air, making it as acidic as the acid in a car battery, so effects of this acidity might be discovered. Imagine what that rain would do to your car's paint. Scientists can also examine known asteroids and volcanoes for unusual concentrations of other chemical elements in addition to iridium. Ancient beaches can be unearthed to look for evidence of a huge tidal wave having hit them 65 million years ago. All these searches and examinations are under way today, and there has been much success in finding data consistent with the violence theory and little uncontested counterevidence. Thus, the violence theory is the leading contender for explaining the dinosaur extinctions not because the alternative explanations have been refuted but because of its being successfully tested (so far) and its being so fruitful. This brings us to the edge of a controversy about scientific methodology. The other alternative theories of dinosaur extinctions have not been refuted; they have not even been tested. But if they have not been refuted, and if proving the violence theory requires refuting all the alternative theories, doesn't it follow that the violence theory will never be proved, no matter how much new positive evidence is dug up by all those searches and examinations mentioned above? This question cannot be answered easily. We will end our discussion of this problem about scientific reasoning with the comment that not only is there much more to be learned about nature, but there are also unsolved problems about the nature of the science itself. IV. Testing Scientific Explanations If you don’t test the claim, you don’t know it’s true. Designing a Scientific Test It is easy to agree that scientific generalizations should be tested before they are proclaimed as true, and it is easy to agree that the explanations based on those generalizations also should be tested. However, how do you actually go about testing them? The answer is not as straightforward as one might imagine. The way to properly test a generalization differs dramatically depending on whether the generalization is universal (all A are B) or non-universal (some but not all A are B). When attempting to confirm a universal generalization, it is always better to focus on refuting the claim than on finding more examples consistent with it. That is, look for negative evidence, not positive evidence. For example, if you are interested in whether all cases of malaria can be cured by drinking quinine, it would be a waste of research money to seek confirming examples. Even 20,000 such examples would be immediately shot down by finding just one person who drank quinine but was not cured. On the other hand, suppose the generalization were non-universal instead of universal, that is, that most cases of malaria can be cured by drinking quinine. Then the one case in which someone drinks quinine and is not cured would not destroy the generalization. With a non-universal generalization the name of the game would be the ratio of cures to failures. In this case, 20,000 examples would go a long way toward improving the ratio. There are other difficulties with testing. For example, today's astronomers say that all other galaxies on average are speeding away from our Milky Way galaxy because of the Big Bang explosion. This explosion occurred 13.7 billion years ago, when the universe was smaller than the size of a pea. Can this explanation be tested to see whether it is correct? You cannot test it by rerunning the birth of the universe. But you can test its predictions. One prediction that follows from the Big Bang hypothesis is that microwave radiation of a certain frequency will be bombarding Earth from all directions. This test has been run successfully, which is one important reason why today's astronomers generally accept the Big Bang as the explanation for their observations that all the galaxies on average are speeding away from us. There are several other reasons for the Big Bang theory having to do with other predictions it makes of phenomena that do not have good explanations by competing theories. We say a hypothesis is confirmed or proved if several diverse predictions are tested and all are found to agree with the data while none disagree. Similarly, a hypothesis gets refuted if any of the actual test results do not agree with the prediction. However, this summary is superficial—let's see why. Retaining Hypotheses Despite Negative Test Results If a scientist puts a hypothesis to the test, and if the test produces results inconsistent with the hypothesis, there is always some way or other for the researcher to hold onto the hypothesis and change something else. For example, if the meter shows “7” when your hypothesis would have predicted “5,” you might rescue your hypothesis by saying that your meter wasn't working properly. However, unless you have some good evidence of meter trouble, this move to rescue your hypothesis in the face of disconfirming evidence commits the fallacy of ad hoc rescue. If you are going to hold on to your hypothesis no matter what, you are in the business of propaganda and dogma, not science. Psychologically, it is understandable that you would try to rescue your cherished belief from trouble. When you are faced with conflicting data, you are likely to mention how the conflict will disappear if some new assumption is taken into account. However, if you have no good reason to accept this saving assumption other than that it works to save your cherished belief, your rescue is an ad hoc rescue. In 1790 the French scientist Lavoisier devised a careful experiment in which he weighed mercury before and after it was heated in the presence of air. The remaining mercury, plus the red residue that was formed, weighed more than the original. Lavoisier had shown that heating a chemical in air can result in an increase in weight of the chemical. Today, this process is called oxidation. But back in Lavoisier’s day, the accepted theory on these matters was that a posited substance, “phlogiston,” was driven off during any heating of a chemical. If something is driven off, then you would expect the resulting substance to weigh less. Yet Lavoisier’s experiments clearly showed a case in which the resulting substance weighed more. To get around this inconsistency, the chemists who supported the established phlogiston theory suggested their theory be revised by assigning phlogiston negative weight. The negative-weight hypothesis was a creative suggestion that might have rescued the phlogiston theory. It wasn't as strange then as it may seem today because the notion of mass was not well understood. Although Isaac Newton had believed that all mass is positive, the negative-weight suggestion faced a more important obstacle. There was no way to verify it independently of the phlogiston theory. So, the suggestion appeared to commit the fallacy of ad hoc rescue. A new hypothesis can avoid the charge of committing the fallacy of ad hoc rescue if it can meet two conditions: (1) The hypothesis must be shown to be fruitful in successfully explaining phenomena that previously did not have an adequate explanation. (2) The hypothesis's inconsistency with previously accepted beliefs must be resolved There is an exception to this, however. A new hypothesis might be inconsistent with previously accepted beliefs because it is correct and the old views were not. In this case, we wouldn’t want to resolve the inconsistency, we want to set about finding more evidence for the new hypothesis. without reducing the explanatory power of science. Because the advocates of the negative-weight hypothesis were unable to do either, it is appropriate to charge them with committing the fallacy. As a result of Lavoisier's success, and the failure of the negative-weight hypothesis, today's chemists do not believe that phlogiston exists. And Lavoisier’s picture gets a prominent place in history: Portrait of Antoine-Laurent Lavoisier And His Wife (1788) by Jacques-Louis David Three Conditions for a Well-Designed Test As a good rule of thumb, three definite conditions should hold in any well-designed test. First, if you use an experiment or observation to test some claim, you should be able to deduce the predicted test result from the combination of the claim plus a description of the relevant aspects of the test's initial conditions. That is, if the claim is really true, the predicted test result should follow. Second, the predicted test result should not be expected no matter what; instead, the predicted result should be unlikely if the claim is false. For example, a test that predicts water will flow downhill is a useless test because water is expected to do so no matter what. Third, it should be practical to check on whether the test did or did not come out as predicted, and this checking should not need to presume the truth of the claim being tested. It does no good to predict something that nobody can check. These criteria for a good test are well described by Ronald Giere in Understanding Scientific Reasoning (New York: Holt, Rinehart and Winston, 1979), pp. 101-105. To summarize, ideally a good test requires a prediction that meets these three conditions; it is deducible or at least probable, given that the claim is true, improbable, given that the claim is false, and verifiable. A good test of a claim will be able to produce independently verifiable data that should occur if the claim is true but shouldn't occur if the claim is false. Deducing Predictions for Testing Condition 1, the deducibility condition, is somewhat more complicated than a first glance might indicate. Suppose you suspect that one of your co-workers named Philbrick has infiltrated your organization to spy on your company's chief scientist, Oppenheimer. To test this claim, you set a trap. Philbrick is in your private office late one afternoon when you walk out declaring that you are going home. You leave a file folder labeled "Confidential: Oppenheimer's Latest Research Proposal” on your desk. You predict that Philbrick will sneak a look at the file. Unknown to him, your office is continually monitored on closed-circuit TV, so you will be able to catch him in the act. Let's review this reasoning. Is condition 1 satisfied for your test? It is, if the following reasoning is deductively valid: Philbrick has the opportunity to be alone in your office with the Oppenheimer file folder (the test's initial conditions). Philbrick is a spy (the claim to be tested). So, Philbrick will read the Oppenheimer file while in your office, (the prediction) This reasoning might or might not be valid depending on a missing premise. It would be valid if a missing premise were the following: If Philbrick is a spy, then he will read the Oppenheimer file while in your office if he has the opportunity and believes he won’t be detected doing it (background assumption). Is that premise acceptable? No. You cannot be that sure of how spies will act. The missing premise is more likely to be the following hedge: If Philbrick is a spy, then he will probably read the Oppenheimer file while in your office if he has the opportunity and believes he won’t be detected doing it (new background assumption). Although it is more plausible that this new background assumption is the missing premise used in the argument for the original prediction, now the argument isn't deductively valid. That is, the prediction doesn't follow with certainty, and condition 1 fails. Because the prediction follows inductively, it would be fair to say that condition 1 is "almost" satisfied. Nevertheless, it is not satisfied. Practically, though, you cannot expect any better test than this; there is nothing that a spy must do that would decisively reveal the spying. Practically, you can have less than ideal tests about spies or else no tests at all. In response to this difficulty with condition 1, should we alter the definition of the condition to say that the prediction should follow either with certainty or probability? No. The reason why we cannot relax condition 1 can be appreciated by supposing that the closed-circuit TV does reveal Philbrick opening the file folder and reading its contents. Caught in the act, right? Your conclusion: Philbrick is a spy. This would be a conclusion many of us would be likely to draw, but it is not one that the test justifies completely. Concluding with total confidence that he is a spy would be drawing a hasty conclusion because there are alternative explanations of the same data. For example, if Philbrick were especially curious, he might read the file contents yet not be a spy. In other words, no matter whether the prediction comes out to be true or false, you cannot be sure the claim is true or false. So, the test is not decisive because its result doesn't settle which of the two alternatives is correct. Yet being decisive is the mark of an ideally good test. We would not want to alter condition 1 so that this indecisive test can be called decisive. Doing so would encourage hasty conclusions. So the definition of condition 1 must stay as it is. However, we can say that if condition 1 is almost satisfied, then when the other two conditions for an ideal test are also satisfied, the test results will tend to show whether the claim is correct. In short, if Philbrick snoops, this tends to show he is a spy. More testing is needed if you want to be surer. This problem about how to satisfy condition 1 in the spy situation is analogous to the problem of finding a good test for a non-universal generalization. If you suspect that most cases of malaria can be cured with quinine, then no single malaria case will ensure that you are right or that you are wrong. Finding one case of a person whose malaria wasn't cured by taking quinine doesn't prove your suspicion wrong. You need many cases to adequately test your suspicion. The bigger issue here in the philosophy of science is the problem of designing a test for a theory that is probabilistic rather than deterministic. To appreciate this, let’s try another scenario. Suppose your theory of inheritance says that, given the genes of a certain type of blue-eyed father and a certain type of brown-eyed mother, their children will have a 25 percent chance of being blue-eyed. Let's try to create a good test of this probabilistic theory by using it to make a specific prediction about one couple's next child. Predicting that the child will be 25 percent blue-eyed is ridiculous. On the other hand, predicting that the child has a 25 percent chance of being blue-eyed is no specific prediction at all about the next child. Specific predictions about a single event can't contain probabilities. What eye color do you predict the child will have? You should predict it will not be blue-eyed. Suppose you make this prediction, and you are mistaken. Has your theory of inheritance been refuted? No. Why not? Because the test was not decisive. The child's being born blue-eyed is consistent with your theory's being true and also with its being false. The problem is that with a probabilistic theory you cannot make specific predictions about just one child. You can predict only that, if there are many children, then 25 percent of them will have blue eyes and 75 percent won't. A probabilistic theory can be used to make predictions only about groups, not about individuals. The analogous problem for the spy in your office is that when you tested your claim that Philbrick is a spy you were actually testing a probabilistic theory because you were testing the combination of that specific claim about Philbrick with the general probabilistic claim that spies probably snoop. They don’t always snoop. Your test with the video camera had the same problem with condition 1 as your test with the eye color. Condition 1 was almost satisfied in both tests, but strictly speaking it wasn't satisfied in either. Our previous discussion should now have clarified why condition 1 is somewhat more complicated than a first glance might indicate. Ideally, we would like decisive tests or, as they are also called, crucial tests. Practically, we usually have to settle for tests that only tend to show whether one claim or another is true. The stronger the tendency, the better the test. If we arrive at a belief on the basis of these less than ideal tests, we are always in the mental state of not being absolutely sure. We are in the state of desiring data from more tests of the claim so that we can be surer of our belief, and we always have to worry that someday new data might appear that will require us to change our minds. Such is the human condition. Science cannot do better than this. IV. Detecting Pseudoscience The word science has positive connotations, the word pseudoscience has negative connotations. Science gets the grant money; pseudoscience doesn't. Calling some statement, theory, or research program “pseudoscientific” suggests that it is silly or a waste of time. It is pseudoscientific to claim that the position of the planets at the time a person is born determines the person's personality and major life experiences. It is also pseudoscientific to claim that spirits of the dead can be contacted by mediums at seances. Astrology and spiritualism may be useful social lubricants, but they aren't scientific. Despite a few easily agreed-upon examples such as these two, defining pseudoscience is difficult. One could try to define science and then use that to say pseudoscience is not science, or one could try to define pseudoscience directly. A better approach is to try to find many of the key features of pseudosciences. A great many of the scientific experts will agree that pseudoscience can be detected by getting a “no” answer to the first two questions or a “yes” answer to any of the remaining three: Do the "scientists" have a theory to test? Do the "scientists" have reproducible data that their theory explains better than the alternatives? Do the "scientists" seem content to search around for phenomena that are hard to explain by means of current science; that is, do the scientists engage in mystery mongering? Are the "scientists" quick to recommend supernatural explanations rather than natural explanations? Do the "scientists" use the method of ad hoc rescue while treating their own views as unfalsifiable? The research program that investigates paranormal phenomena is called parapsychology. What are the paranormal phenomena we are talking about here? They include astral travel, auras, psychokinesis (moving something without touching it physically), plant consciousness, psychic healing, speaking with the spirits, witchcraft, and ESP—that is telepathy (mind reading), clairvoyance (viewing things at a distance), and precognition (knowing the future). None of the parapsychologists' claims to have found cases of cancer cures, mind reading, or foretelling the future by psychic powers have ever stood up to a good test. Parapsychologists cannot convincingly reproduce any of these phenomena on demand; they can only produce isolated instances in which something surprising happened. Parapsychologists definitely haven't produced repeatable phenomena that they can show need to be explained in some revolutionary way. Rarely do parapsychologists engage in building up their own theories of parapsychology and testing them. Instead, nearly all are engaged in attempts to tear down current science by searching for mysterious phenomena that appear to defy explanation by current science. Perhaps this data gathering is the proper practice for the prescientific stage of some enterprise that hopes to revolutionize science, but the practice does show that the enterprise of parapsychology is not yet a science. Regarding point 1, scientists attack parapsychologists for not having a theory-guided research program. Even if there were repeatable paranormal phenomena, and even if parapsychologists were to quit engaging in mystery mongering, they have no even moderately detailed theory of how the paranormal phenomena occur. They have only simplistic theories such as that a mysterious mind power caused the phenomenon or that the subject tapped a reserve of demonic forces or that the mind is like a radio that can send and receive signals over an undiscovered channel. Parapsychologists have no more-detailed theory that permits testable predictions. Yet, if there is no theory specific enough to make a testable prediction, there is no science. V. Paradigms and Possible Causes Your car's engine is gummed up today. This has never happened before. Could it be because at breakfast this morning you drank grapefruit juice rather than your usual orange juice? No, it couldn't be. Current science says this sort of explanation is silly. OK, forget the grapefruit juice. Maybe the engine is gummed up because today is Friday the 13th. No, that is silly, too. A scientist wouldn't even bother to check these explanations. Let's explore this intriguing notion of what science considers "silly" versus what it takes seriously. What causes the pain relief after swallowing a pain pill? Could it be the favorite music of the inventor? No, that explanation violates medical science's basic beliefs about what can count as a legitimate cause of what. Nor could the pain relief be caused by the point in time when the pill is swallowed. Time alone causes nothing, says modern science. The pain relief could be caused by the chemical composition of the pill, however, or perhaps by a combination of that with the mental state of the person who swallowed the pill. The general restrictions that a science places on what can be a cause and what can't are part of what is called the paradigm of the science. Every science has its paradigm. That is, at any particular time, each science has its own particular problems that it claims to have solved; and, more important, it has its own accepted ways of solving problems that then serve as a model for future scientists who will try to solve new problems. These ways of solving problems, including the methods, standards, and generalizations generally held in common by the community of those practicing the science, is, by definition, the paradigm of that science. The paradigm in medical science is to investigate what is wrong with sick people, not what is right with well people. For a second example, biological science can explain what causes tigers to like meat rather than potatoes in terms of evolutionary factors that give rise to this preference, but biologists would turn to chemists to determine how the preference involves the chemical makeup of the meat, and not the history of zipper manufacturing or the price of rice in China. The paradigm for biological science limits what counts as a legitimate biological explanation. When we take a science course or read a science book, we are slowly being taught the paradigm of that science and, with it, the ability to distinguish silly explanations from plausible ones. Silly explanations do not meet the basic requirement for being a likely explanation, namely coherence with the paradigm. Sensitivity to this consistency requirement was the key to understanding the earlier story about Brother Bartholomew. Scientists today say that phenomena should not be explained by supposing that Bartholomew or anybody else could see into the future; this kind of "seeing" is inconsistent with the current paradigm. It is easy to test whether people can foresee the future if you can get them to make specific predictions rather than vague ones. Successfully testing a claim that someone can foresee the future would be a truly revolutionary result, upsetting the whole scientific world-view, which explains why many people are so intrigued by tabloid reports of people successfully foretelling the future. Suppose a scientist wants to determine whether adding solid carbon dioxide to ice water will cool the water below 32 degrees Fahrenheit (0 degrees Celsius). The scientist begins with two glasses containing equal amounts of water at the same temperature. The glasses touch each other. Solid carbon dioxide is added to the first glass, but not the second. The scientist expects the first glass to get colder but the second glass not to. This second glass of water is the control because it is just like the other glass except that the causal factor being tested—the solid carbon dioxide—is not present in it. After twenty minutes, the scientist takes the temperature of the water in both glasses. Both are found to have cooled, and both are at the same temperature. A careless scientist might draw the conclusion that the cooling is not caused by adding the carbon dioxide, because the water in the control glass also got cooler. A more observant scientist might draw another conclusion, that the experiment wasn’t any good because the touching is contaminating the control. The two glasses should be kept apart during the experiment to eliminate contamination. The paradigm of the science dictates that the glasses not touch because it implies that glasses in contact will reach a common temperature in much faster than glasses not in contact. For a second example of the contamination of experimental controls, suppose a biologist injects some rats with a particular virus and injects control rats with a placebo—some obviously ineffective substance such as a small amount of salt water. The biologist observes the two groups of rats to determine whether the death rate of those receiving the virus is significantly higher than the death rate of those receiving the placebo. If the test is well run and the data show such a difference, there is a correlation between the virus injection and dying. Oh, by the way, the injected rats are kept in the same cages with the control rats. Oops. This contamination will invalidate the entire experiment, won't it? Reputable scientists know how to eliminate contamination, and they actively try to do so. They know that temperature differences and disease transmission can be radically affected by physical closeness. This background knowledge that guides experimentation constitutes another part of the paradigm of the sciences of physics and biology. Without a paradigm helping to guide the experimenter, there would be no way of knowing whether the control group was contaminated. There would be no way to eliminate experimenter effects, that is, the unintentional influence of the experimenter on the outcome of the experiment. There would be no way of running a good test. That fact is one more reason that so much of a scientist's college education is spent learning the science's paradigm. VI. Review of Major Points When scientists are trying to gain a deep understanding of how the world works, they seek general patterns rather than specific facts. The way scientists acquire these general principles about nature is usually neither by deducing them from observations nor by inductive generalization. Instead, they think about the observations, then guess at a general principle that might account for them, then check this guess by testing. When a guess or claim is being tested, it is called a hypothesis. Testing can refute a hypothesis. If a hypothesis does not get refuted by testing, scientists retain it as a prime candidate for being a general truth of nature, a law. Hypotheses that survive systematic testing are considered to be proved, although even the proved statements of science are susceptible to future revision, unlike the proved statements of mathematics. Scientific reasoning is not discontinuous from everyday reasoning, but it does have higher standards of proof. This chapter reviewed several aspects of scientific reasoning from earlier chapters, including general versus specific claims, testing by observation, testing by experiment, accuracy, precision, operational definition, pseudoprecision, the role of scientific journals, independent verification, consistency with well-established results, reproducible evidence, anecdotal evidence, a scientist’s cautious attitude and open mind, attention to relevant evidence, the scientific method of justifying claims, disconfirming evidence, and the methods of gaining a representative sample. Deterministic explanations are preferred to probabilistic ones, and ideal explanations enable prediction of the phenomenon being explained. Explanations are preferred if they are testable and fruitful. A good test requires a prediction that is (1) deducible, (2) improbable, and (3) verifiable. Science provides the antidote to superstition. There are criteria that can be used to detect pseudoscience. A reasonable scientific explanation is coherent with the paradigm for that science. Only by knowing the science's paradigm can a scientist design a controlled experiment that does not contaminate the controls and that eliminates effects unintentionally caused by the experimenter.
Chapter 9: Analogical Reasoning This chapter is based on Fundamental Methods of Logic by Matthew Knachel. I. Introduction Analogical reasoning is ubiquitous in everyday life. We rely on analogies—similarities between present circumstances and those we’ve already experienced—to guide our actions. We use comparisons to familiar people, places, and things to guide our evaluations of novel ones. We criticize people’s arguments based on their resemblance to obviously absurd lines of reasoning. In this chapter, we will look at the various uses of analogical reasoning. Along the way, we will identify a general pattern that all arguments from analogy follow and learn how to show that particular arguments fit the pattern. We will then turn to the evaluation of analogical arguments: we will identify six criteria that govern our judgments about the relative strength of these arguments. Finally, we will look at the use of analogies to refute other arguments. II. The Form of Analogical Arguments Perhaps the most common use of analogical reasoning is to predict how the future will unfold based on similarities to past experiences. Consider this simple example. When I first learned that the movie The Wolf of Wall Street was coming out, I predicted that I would like it. My reasoning went something like this: The Wolf of Wall Street is directed by Martin Scorsese, and it stars Leonardo DiCaprio. Those two have collaborated several times in the past, on Gangs of New York, The Aviator, The Departed, and Shutter Island. I liked each of those movies, so I predict that I will like The Wolf of Wall Street. Notice, first, that this is an inductive argument. The conclusion, that I will like The Wolf of Wall Street, is not guaranteed by the premises; as a matter of fact, my prediction was wrong and I really didn’t care for the film. But our real focus here is on the fact that the prediction was made on the basis of an analogy. Actually, several analogies, between The Wolf of Wall Street, on the one hand, and all the other Scorsese/DiCaprio collaborations on the other. The new film is similar in important respects to the older ones; I liked all of those; so, I’ll probably like the new one. We can use this pattern of reasoning for more overtly persuasive purposes. Consider the following: Eating pork is immoral. Pigs are just as smart, cute, and playful as dogs and dolphins. Nobody would consider eating those animals. So why are pigs any different? That passage is trying to convince people not to eat pork, and it does so on the basis of analogy: pigs are just like other animals we would never eat—dogs and dolphins. Analogical arguments all share the same basic structure. We can lay out this form schematically as follows: a1, a2, …, an, and c all have properties F1, F2, …, Fk a1, a2, …, an all have property G Therefore, c has property G. This is an abstract schema, and it’s going to take some getting used to, but it represents the form of analogical reasoning succinctly and clearly. Arguments from analogy have two premises and a conclusion. The first premise establishes an analogy. The analogy is between some thing, marked ‘c’ in the schema, and some number of other things, marked ‘a1’, ‘a2’, and so on in the schema. We can refer to these as the “analogues.” They’re the things that are similar, analogous to c. This schema is meant to cover every possible argument from analogy, so we do not specify a particular number of analogues; the last one on the list is marked ‘an’, where ‘n’ is a variable standing for any number whatsoever. There may be only one analogue; there may be a hundred. What’s important is that the analogues are similar to the thing designated by ‘c’. What makes different things similar? They have stuff in common; they share properties. Those properties—the similarities between the analogues and c—are marked ‘F1’, ‘F2’, and so on in the diagram. Again, we don’t specify a particular number of properties shared: the last is marked ‘Fk’, where ‘k’ is just another variable (we don’t use ‘n’ again, because the number of analogues and the number of properties can of course be different). This is because our schema is generic: every argument from analogy fits into the framework; there may be any number of properties involved in any particular argument. Anyway, the first premise establishes the analogy: c and the analogues are similar because they have various things in common—F1, F2, F3, …, Fk. Notice that ‘c’ is missing from the second premise. The second premise only concerns the analogues: it says that they have some property in common, designated ‘G’ to highlight the fact that it’s not among the properties listed in the first premise. It’s a separate property. It’s the very property we’re trying to establish, in the conclusion, that c has (‘c’ is for conclusion). The thinking is something like this: c and the analogues are similar in so many ways (first premise); the analogues have this additional thing in common (G in the second premise); so, c is probably like that, too (conclusion: c has G). It will be helpful to apply these abstract considerations to concrete examples. We have two in hand. The first argument, predicting that I would like The Wolf of Wall Street, fits the pattern. Here’s the argument again, for reference: The Wolf of Wall Street is directed by Martin Scorsese, and it stars Leonardo DiCaprio. Those two have collaborated several times in the past, on Gangs of New York, The Aviator, The Departed, and Shutter Island. I liked each of those movies, so I predict that I will like The Wolf of Wall Street. The conclusion is something like ‘I will like The Wolf of Wall Street’. Putting it that way, and looking at the general form of the conclusion of analogical arguments (c has G), it’s tempting to say that ‘c’ designates me, while the property G is something like ‘liking The Wolf of Wall Street.’ But that’s not right. The thing that ‘c’ designates has to be involved in the analogy in the first premise; it has to be the thing that’s similar to the analogues. The analogy that this argument hinges on is between the various movies. It’s not I that ‘c’ corresponds to; it’s the movie we’re making the prediction about. The Wolf of Wall Street is what ‘c’ picks out. What property are we predicting it will have? Something like ‘liked by me.’ The analogues, the a’s in the schema, are the other movies: Gangs of New York, The Aviator, The Departed, and Shutter Island. In this example, n is 4; the movies are a1, a2, a3, and a4. These we know have the property G (liked by me): I had already seen and liked these movies. That’s the second premise: that the analogues have G. Finally, the first premise, which establishes the analogy among all the movies. What do they have in common? They were all directed by Martin Scorsese, and they all starred Leonardo DiCaprio. Those are the F’s—the properties they all share. F1 is ‘directed by Scorsese,’ and F2 is ‘stars DiCaprio’. The second argument we considered, about eating pork, also fits the pattern. Here it is again, for reference: Eating pork is immoral. Pigs are just as smart, cute, and playful as dogs and dolphins. Nobody would consider eating those animals. So why are pigs any different? Again, looking at the conclusion—‘Eating pork is immoral’—and looking at the general form of conclusions for analogical arguments—‘c has G’—it’s tempting to just read off from the syntax of the sentence that ‘c’ stands for ‘eating pork’ and G for ‘is immoral.’ But that’s not right. Focus on the analogy: what things are being compared to one another? It’s the animals: pigs, dogs, and dolphins; those are our a’s and c. To determine which one is picked out by ‘c,’ we ask, which animal is involved in the conclusion? It’s pigs; they are picked out by ‘c.’ So we have to paraphrase our conclusion so that it fits the form ‘c has G’, where ‘c’ stands for pigs. Something like ‘Pigs shouldn’t be eaten’ would work. So G is the property ‘shouldn’t be eaten’. The analogues are dogs and dolphins. They clearly have the property: as the argument notes, (most) everybody agrees they shouldn’t be eaten. This is the second premise. And the first establishes the analogy. What do pigs have in common with dogs and dolphins? They’re smart, cute, and playful. F1 = ‘is smart,’ F2 = ‘is cute,’ and F3 = ‘is playful.’ III. The Evaluation of Analogical Arguments Unlike in the case of deduction, we will not have to learn special techniques to use when evaluating these sorts of arguments. It’s something we already know how to do, something we typically do automatically and unreflectively. The purpose of this section, then, is not to learn a new skill, but rather subject a practice we already know how to engage in to critical scrutiny. We evaluate analogical arguments all the time without thinking about how we do it. We want to achieve a metacognitive perspective on the practice of evaluating arguments from analogy; we want to think about a type of thinking that we typically engage in without much conscious deliberation. We want to identify the criteria that we rely on to evaluate analogical reasoning—criteria that we apply without necessarily realizing that we’re applying them. Achieving such metacognitive awareness is useful insofar as it makes us more self-aware, critical, and therefore effective reasoners. Analogical arguments are inductive arguments. They give us reasons that are supposed to make their conclusions more probable. How probable, exactly? That’s very hard to say. How probable was it that I would like The Wolf of Wall Street given that I had liked the other four Scorsese/DiCaprio collaborations? I don’t know. How probable is it that it’s wrong to eat pork given that it’s wrong to eat dogs and dolphins? I really don’t know. It’s hard to imagine how you would even begin to answer that question. As we mentioned, while it’s often impossible to evaluate inductive arguments by giving a precise probability of its conclusion, it is possible to make relative judgments about strength and weakness. Recall, new information can change the probability of the conclusion of an inductive argument. We can make relative judgments like this: if we add this new information as a premise, the new argument is stronger/weaker than the old argument; that is, the new information makes the conclusion more/less likely. It is these types of relative judgments that we make when we evaluate analogical reasoning. We compare different arguments—with the difference being new information in the form of an added premise, or a different conclusion supported by the same premises—and judge one to be stronger or weaker than the other. Subjecting this practice to critical scrutiny, we can identify six criteria that we use to make such judgments. We’re going to be making relative judgments, so we need a baseline argument against which to compare others. Here is such an argument: Alice has taken four philosophy courses during her time in college. She got an A in all four. She has signed up to take another philosophy course this semester. I predict she will get an A in that course, too. This is a simple argument from analogy, in which the future is predicted based on past experience. It fits the schema for analogical arguments: the new course she has signed up for is designated by ‘c’; the property we’re predicting it has (G) is that it is a course Alice will get an A in; the analogues are the four previous courses she’s taken; what they have in common with the new course (F1) is that they are also Philosophy classes; and they all have the property G—Sally got an A in each. How strong is the baseline argument? How probable is its conclusion in light of its premises? I have no idea. It doesn’t matter. We’re now going to consider tweaks to the argument, and the effect that those will have on the probability of the conclusion. That is, we’re going to consider slightly different arguments, with new information added to the original premises or changes to the prediction based on them, and ask whether these altered new arguments are stronger or weaker than the baseline argument. This will reveal the six criteria that we use to make such judgments. We’ll consider one criterion at a time. 1. Number of Relevant Similarities In the baseline argument, the only thing the four previous courses and the new course have in common is that they’re Philosophy classes. Suppose we change that. Our newly tweaked argument predicts that Alice will get an A in the new course, which, like the four she succeeded in before, is cross-listed in the Department of Religious Studies and covers topics in the Philosophy of Religion. Given this new information—that the new course and the four older courses were similar in ways we weren’t aware of—are we more or less confident in the prediction that Alice will get another A? Is the argument stronger or weaker than the baseline argument? Again, it is stronger. Unlike the last example, this tweak gives us new information both about the four previous courses and the new one. The upshot of that information is that they’re more similar than we knew; that is, they have more properties in common. To F1 = ‘is a Philosophy course’ we can add F2 = ‘is cross-listed with Religious Studies’ and F3 = ‘covers topics in Philosophy of Religion.’ The more properties things have in common, the stronger the analogy between them. The stronger the analogy, the stronger the argument based on that analogy. We now know not just that Alice did well in not just in Philosophy classes—but specifically in classes covering the Philosophy of Religion; and we know that the new class she’s taking is also a Philosophy of Religion class. I’m much more confident predicting she’ll do well again than I was when all I knew was that all the classes were Philosophy; the new one could’ve been in a different topic that she wouldn’t have liked. Similarities only affect the argument, though, if they are relevant to the conclusion we are trying to draw. This means they must have some direct connection to the property G we are trying to transfer from the analogues to object C. Irrelevant similarities have no effect on the argument. The similarities above strengthen the argument because the specific subject of the course is relevant to how well a student does. Suppose I try to add similarities like all these classes are in the same classroom, or the textbooks were all published by the same publisher. These don’t strengthen the argument, because they are very unlikely to affect how Alice does in the class. General principle: other things being equal, the more properties involved in the analogy—the more relevant similarities between the item in the conclusion and the analogues—the stronger the argument (and conversely, the fewer properties, the weaker). 2. Number of Relevant Differences An argument from analogy is built on the foundation of the similarities between the analogues and the item in the conclusion—the analogy. Anything that weakens that foundation weakens the argument. So, to the extent that there are differences among those items, the argument is weaker. Suppose we add new information to our baseline argument: the four Philosophy courses Alice did well in before were all courses in the Philosophy of Mind; the new course is about the history of Ancient Greek Philosophy. Given this new information, are we more or less confident that she will succeed in the new course? Is the argument stronger or weaker than the baseline argument? Clearly, the argument is weaker. The new course is on a completely different topic than the other ones. She did well in four straight Philosophy of Mind courses, but Ancient Greek Philosophy is quite different. I’m less confident that she’ll get an A than I was before. If I add more differences, the argument gets even weaker. Supposing the four Philosophy of Mind courses were all taught by the same professor (the person in the department whose expertise is in that area), but the Ancient Greek Philosophy course is taught by someone different (the department’s specialist in that topic). Different subject matter, different teachers: I’m even less optimistic about Alice’s continued success. Just like with similarities, differences only affect the strength of the argument if the differences are relevant to that property G we’re trying to transfer from the analogues to our object C. Generally speaking, other things being equal, the more relevant differences there are between the analogues and the item in the conclusion, the weaker the argument from analogy. There is a special case, though! With every relevant difference between the analogues and the item in the conclusion, someone benefits from the difference. Every relevant difference weakens the analogy. However, if the difference makes it easier for the item in the conclusion to obtain their property G, that difference weakens the analogy but strengthens the argument. Let’s add this information to Alice’s argument about her philosophy classes: Suppose the other four philosophy classes were taught by the same teacher, but the new one is taught by a TA—who just happens to be her boyfriend. That’s a difference, but one that makes the conclusion—that Alice will do well—more probable. This difference weakens the analogy, but it strengthens the argument, because it gives Alice a benefit that might help her get an A in the new class. 3. Number of Analogues Suppose we alter the original argument by changing the number of prior Philosophy courses Alice had taken. Instead of Alice having taken four philosophy courses before, we’ll now suppose she has taken fourteen. We’ll keep everything else about the argument the same: she got an A in all of them, and we’re predicting she’ll get an A in the new one. Are we more or less confident in the conclusion—the prediction of an A—with the altered premise? Is this new argument stronger or weaker than the baseline argument? It’s stronger! We’ve got Alice getting an A fourteen times in a row instead of only four. That clearly makes the conclusion more probable. (How much more? Again, it doesn’t matter.) What we did in this case is add more analogues. Remember: an analogue is anything we can use in our comparison that already has property G. By adding more philosophy classes she has also gotten an A in, we’ve shown she has a longer track record of success in these classes. This reveals a general rule: other things being equal, the more analogues in an analogical argument, the stronger the argument (and conversely, the fewer analogues, the weaker). The number of analogues is one of the criteria we use to evaluate arguments from analogy. 4. Number of Counterexamples A counterexample is like an analogue, with one important difference: it does not have property G. It’s something that compares well to our object C, having the relevant similarities, but it didn’t get the result we were looking for. An example will help: returning to the original argument, which had four philosophy classes where Alice got an A, suppose we find another philosophy class that she failed. It could have been an analogue, but we did not get the result we were hoping for. This breaks her track record of success in such classes, and weakens the argument. Of course, the more counterexamples you find, the weaker the argument. If we find four philosophy classes she failed, added to the four where she got an A, our argument now commits the fallacy of weak analogy. 5. Variety of Analogues You’ll notice that the original argument doesn’t give us much information about the four courses Alice succeeded in previously and the new course she’s about to take. All we know is that they’re all Philosophy courses. Suppose we tweak things. We’re still in the dark about the new course Alice is about to take, but we know a bit more about the other four: one was a course in Ancient Greek Philosophy; one was a course on Contemporary Ethical Theories; one was a course in Formal Logic; and the last one was a course in the Philosophy of Mind. Given this new information, are we more or less confident that she will succeed in the new course, whose topic is unknown to us? Is the argument stronger or weaker than the baseline argument? It is stronger. We don’t know what kind of Philosophy course Alice is about to take, but this new information gives us an indication that it doesn’t really matter. She was able to succeed in a wide variety of courses, from Mind to Logic, from Ancient Greek to Contemporary Ethics. This is evidence that Alice is good at Philosophy generally, so that no matter what kind of course she’s about to take, she’ll probably do well in it. Again, this points to a general principle about how we evaluate analogical arguments: other things being equal, the more variety there is among the analogues, the stronger the argument (and conversely, the less variety, the weaker). 6. Modesty/Ambition of the Conclusion Suppose we leave everything about the premises in the original baseline argument the same: four Philosophy classes, an A in each, new Philosophy class. Instead of adding to that part of the argument, we’ll tweak the conclusion. Instead of predicting that Alice will get an A in the class, we’ll predict that she’ll pass the course. Are we more or less confident that this prediction will come true? Is the new, tweaked argument stronger or weaker than the baseline argument? It’s stronger. We are more confident in the prediction that Alice will pass than we are in the prediction that she will get another A, for the simple reason that it’s much easier to pass than it is to get an A. That is, the prediction of passing is a much more modest prediction than the prediction of an A. Suppose we tweak the conclusion in the opposite direction—not more modest, but more ambitious. Alice has gotten an A in four straight Philosophy classes, she’s about to take another one, and I predict that she will do so well that her professor will suggest that she publish her term paper in one of the most prestigious philosophical journals and that she will be offered a three-year research fellowship at the Institute for Advanced Study at Princeton University. That’s a bold prediction! Meaning, of course, that it’s very unlikely to happen. Getting an A is one thing; getting an invitation to be a visiting scholar at one of the most prestigious academic institutions in the world is quite another. The argument with this ambitious conclusion is weaker than the baseline argument. The general principle here: the more modest the argument’s conclusion, the stronger the argument; the more ambitious, the weaker. III. Refutation by Analogy We can use arguments from analogy for a specific logical task: refuting someone else’s argument, showing that it’s bad. Recall the case of deductive arguments. To refute those—to show that they are bad, i.e., invalid—we can produce a counterexample—a new argument with the same logical form as the original that was obviously invalid, in that its premises were in fact true and its conclusion in fact false. We can use a similar procedure to refute inductive arguments. Of course, the standard of evaluation is different for induction: we don’t judge them according to the black and white standard of validity. And as a result, our judgments have less to do with form than with content. Nevertheless, refutation along similar lines is possible, and analogies are the key to the technique. To refute an inductive argument, we produce a new argument that’s obviously bad—just as we did in the case of deduction. We don’t have a precise notion of logical form for inductive arguments, so we can’t demand that the refuting argument have the same form as the original; rather, we want the new argument to have an analogous form to the original. The stronger the analogy between the refuting and refuted arguments, the more decisive the refutation. We cannot produce the kind of knock-down refutations that were possible in the case of deductive arguments, where the standard of evaluation—validity—does not admit of degrees of goodness or badness, but the technique can be quite effective. Consider the following, from a 2016 article in a student-run newspaper in Indiana: “Duck Dynasty” star and Duck Commander CEO Willie Robertson said he supports Trump because both of them have been successful businessmen and stars of reality TV shows. By that logic, does that mean Hugh Hefner’s success with “Playboy” and his occasional appearances on “Bad Girls Club” warrant him as a worthy president? Actually, I’d still be more likely to vote for Hefner than Trump. The author is Austin Faulds, from the following article (link opens in a new window): “Weird celebrity endorsements fit for weird election.” Indiana Daily Student, 10/12/16. The author is refuting the argument of Willie Robertson, the “Duck Dynasty” star. Robertson’s argument is something like this: Trump is a successful businessman and reality TV star; therefore, he would be a good president. To refute this, the author produces an analogous argument—Hugh Hefner is a successful businessman and reality TV star; therefore, Hugh Hefner would make a good president—that he regards as obviously bad. What makes it obviously bad is that it has a conclusion that nobody would agree with: Hugh Hefner would make a good president. That’s how these refutations work. They attempt to demonstrate that the original argument is lousy by showing that you can use the same or very similar reasoning to arrive at an absurd conclusion. Here’s another example, from a group called “Iowans for Public Education”. Next to a picture of an apparently well-to-do lady is the following text: “My husband and I have decided the local parks just aren’t good enough for our kids. We’d rather use the country club, and we are hoping state tax dollars will pay for it. We are advocating for Park Savings Accounts, or PSAs. We promise to no longer use the local parks. To hell with anyone else or the community as a whole. We want our tax dollars to be used to make the best choice for our family.” Sound ridiculous? Tell your legislator to vote NO on Education Savings Accounts (ESAs), aka school vouchers. The argument that Iowans for Public Education put in the mouth of the lady on the poster is meant to refute reasoning used by advocates for “school choice”, who say that they ought to have the right to opt out of public education and keep the tax dollars they would otherwise pay for public schools and use it to pay to send their kids to private schools. A similar line of reasoning sounds pretty ridiculous when you replace public schools with public parks and private schools with country clubs. Since these sorts of refutations rely on analogies, they are only as strong as the analogy between the refuting and refuted arguments. There is room for dispute on that question. Advocates for school vouchers might point out that schools and parks are completely different things, that schools are much more important to the future prospects of children, and that given the importance of education, families should have to right choose what they think is best. Or something like that. The point is, the kinds of knock-down refutations that were possible for deductive arguments are not possible for inductive arguments. There is always room for further debate.
The Logic Book an open educational resource collected, edited, remixed, written, and re-written by: Benjamin Buckley Sanjay Lal Todd Janke Alex Hall Preface This textbook is a labor of love, and a true collaborative effort. The four of us have put much effort into creating this book. Most chapters are based on someone else’s open educational resource logic book; some are lightly edited, mainly for formatting and accessibility, but most have been more heavily edited, rewritten, and expanded, to be more in line with how we wanted to teach the material. (Chapter 10 is entirely original to us). For logic teachers considering adopting or adapting this book: For each chapter, I have tried to eliminate any reference to other chapters in the book (“As we learned in Chapter 3 …” for example), as well as internal references by chapter number (so sections are numbered I, II, III, etc., instead of 3.1, 3.2, 3.3). This is so other adopters and adapters of some or all of the material can easily teach the material in any order they wish, or combine it with chapters from other OER textbooks – all you should need to change is where it declares the chapter number at the beginning. One exception is the Introduction, below, which provides an overview of the chapters in this book. There should be enough flexibility here to choose what kind of course you want to teach; do you want your course to be more deductive-heavy? More inductive? Do you want to teach Venn diagrams, natural deduction, both, neither? The four of us are teaching different portions of the book, and in different orders. You’ll notice there is no chapter on truth tables. This is for accessibility reasons. Screen readers fail entirely when trying to read symbolic logic formulae, so with translations and natural deduction, I’ve turned the formulae into images and included alt-text that tells the screen reader how to read it. I couldn’t for the life of me figure out how to make truth tables accessible to the visually impaired, so I dropped it. I’ve found that moving from translations and truth conditions directly to natural deduction worked in the classroom much better than I thought it would. Of course, there are plenty of OER books that have chapters on truth tables if you’d like to insert that material. The book is copyrighted under a creative commons license, CC BY-NC-SA, which means you are free to use, adapt, remix, build upon the material in any way you wish, so long as you include attribution to the creators, it will only be used for non-commercial purposes, and any adaptation you produce is licensed the same way. Gratitude: We wish to heartily thank the authors of the OER books on which we’ve based these chapters; a full list is below. Many thanks go to the Clayton State University critical thinking students of Spring Semester, 2025 (all 323 of you!), for helping us pilot this textbook. We appreciate your patience, and your attention to detail pointing out inconsistencies, typos, and falsehoods. And we extend our deepest gratitude to Affordable Learning Georgia for the generous grant that made this work possible. Source material: For All X: The Lorain County Remix, remixed by J. Robert Loftis (2005-2017). This book’s webpage is here (link opens in a new window): For All X. (Chapters 1, 2, 3, 4) Fundamental Methods of Logic, by Matthew Knachel (2017). The book can be accessed here (link opens in a new window): Fundamental Methods (Chapters 5, 7, 9, 11) Introduction to Logic and Critical Thinking, by Matthew J. Van Cleave (2016). This book can be accessed here (link opens in a new window): Logic and Critical Thinking. (Chapter 6) Logical Reasoning, by Bradley H. Dowden (2011-2017).This book can be accessed here (link opens in a new window): Logical Reasoning. (Chapter 8) Introduction Congratulations, you are enrolled in a course that has been taught for nearly twenty-five hundred years! We study how to reason correctly: when our evidence justifies our conclusions and how to detect liars and cheats. The curriculum dates to philosophers such as Aristotle, who worked in the Greek city of Athens. Traditionally, Critical Thinking is thought of as the foundation of a well-rounded education. Critical Thinking used to be called ‘logic,’ but owing to twentieth-century developments, the field grew increasingly technical, and it now makes more sense to teach classical logic as Critical Thinking. Critical Thinking is described as the ‘Art of Arts’ because no matter what you do with your life, Critical Thinking helps you to do it better. Physics, law, customer service, mixed martial arts, theatre, parenting, take your pick – knowing how to think through issues, support conclusions with evidence, and call out bad reasoning is vital, especially in our toxic, social-media landscape that is littered with lies and bad arguments. Chapters one and two offer an overview of Critical Thinking with attention to when conclusions are guaranteed or merely probable. Chapters three, four, five, and six show how to clarify and evaluate what is really going on when someone asks you to believe something based on the reasons that they give. In chapter seven, we study common tricks (fallacies) designed to make bad arguments look good. Chapters eight, nine, ten, and eleven pursue these themes as they play out in scientific and ordinary (day-to-day) decision making. Some closing notes: Aristotle is often credited with creating logic. At best, this is only partially true, and only as regards the particular tradition that we’ll study. Aristotle worked out of the Academy, an ancient thinktank of sorts, founded in the fourth-century BCE by the philosopher Plato (and named after the Greek Olympic victor Academus). We owe to this school developments in fields such as geometry, astronomy, physics and politics. Aristotle did not work alone. In particular, Academicians (those working at the Academy) focused on mathematics. It is likely that the Greek mathematical tradition borrowed from ancient Egypt and Mesopotamia. Around the sixth-century BCE, Greek thinkers grow increasingly interested in mathematics as an instance of ‘best-case’ reasoning, applying it to fields such as physics (they gave us the term ‘Atom’ for the smallest, indestructible particle of matter). By way of example, consider the Pythagorean theorem that many of us had to memorize in high school (fun fact, Pythagoras, who lived in the sixth-century BCE, didn’t create the theorem; we find it in Babylonian and Indian works that predate him by centuries). The point is, mathematics is generally presented as a way to arrive at certainty. As Greek thinkers worked to employ mathematical-type reasoning better to understand the world, it is unsurprising that Aristotle and others sought to incorporate mathematical precision into argumentation, i.e., the process of discovering things by thinking carefully about our evidence. Consider this example developed from Aristotle’s work: Socrates is human. Humans are mortal. Therefore, Socrates is mortal. If the premises are true, the conclusion is guaranteed. Critical Thinking discovered lots of methods to arrive at necessary and probable conclusions. Add in ways to figure out when an argument is bad and how to call out people trying to fool you (or to discover when honestly intended reasoning is simply mistaken) and you have a powerful set of tools called the ‘Art of Arts’ for good reason. Finally, we should not think that logic is a Western invention. First of all, we have lost lots of writings that predate the Academy, and it would be silly to think that no one except the Greeks ever thought about this stuff. Secondly, and more to the point, we find writings on logic in the ancient Chinese Mohist canon (dating to the fifth-century BCE) and Indian, fourth-century BCE Nyaya, Vaisheshika and Nagarjuna thought. So, the Greeks didn’t create logic. Their contributions and those of thinkers in the Greek tradition are numerous and important, but they weren’t the only civilization reasoning along these lines. Chapter 1: What Is Logic? This chapter is based on For All X, The Lorain County Remix, remixed by J. Robert Loftis. I. Arguments, Premises, Conclusion Logic is a part of the study of human reason, the ability we have to think abstractly, solve problems, explain the things that we know, and infer new knowledge on the basis of evidence. Traditionally, logic has focused on the last of these items, the ability to make inferences on the basis of evidence. This is an activity you engage in every day. In logic, we don’t use the word “argument” just to refer to two people disagreeing. We use “argument” to refer to the attempt to show that certain evidence supports a conclusion. A logical argument is structured to give someone a reason to believe some conclusion. Here is an argument about a game of Clue written out in a way that shows its structure. In a game of Clue, the possible murder weapons are the knife, the candlestick, the revolver, the rope, the lead pipe, and the wrench. The murder weapon was not the knife. The murder weapon was also not the revolver, the rope, the lead pipe, or the wrench. Therefore, the murder weapon was the candlestick. In the argument above, the first three statements are the evidence. We call these the premises. The word “therefore” indicates that the final statement (written below the line) is the conclusion of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion. We can define logic then more precisely as the part of the study of reasoning that focuses on argument. In more casual situations, we will follow ordinary practice and use the word “logic” to either refer to the business of studying human reason or the thing being studied, that is, human reasoning itself. While logic focuses on argument, other disciplines, like decision theory and cognitive science, deal with other aspects of human reasoning, like abstract thinking and problem solving more generally. Logic, as the study of argument, has been pursued for thousands of years by people from civilizations all over the globe. The initial motivation for studying logic is generally practical. Given that we use arguments and make inferences all the time, it only makes sense that we would want to learn to do these things better. Once people begin to study logic, however, they quickly realize that it is a fascinating topic in its own right. Thus the study of logic quickly moves from being a practical business to a theoretical endeavor people pursue for its own sake. In order to study reasoning, we have to apply our ability to reason to our reason itself. This reasoning about reasoning is called metareasoning. It is part of a more general set of processes called metacognition, which is just any kind of thinking about thinking. When we are pursuing logic as a practical discipline, one important part of metacognition will be awareness of your own thinking, especially its weakness and biases, as it is occurring. More theoretical metacognition will be about attempting to understand the structure of thought itself. Whether we are pursuing logic for practical or theoretical reasons, our focus is on argument. The key to studying argument is to set aside the subject being argued about and to focus on the way it is argued for. The above example was about a game of Clue. However, the kind of reasoning used in that example was just the process of elimination. The process of elimination can be applied to any subject. Suppose a group of friends is deciding which restaurant to eat at, and there are six restaurants in town. If you could rule out five of the possibilities, you would use an argument just like the one above to decide where to eat. Because logic sets aside what an argument is about, and just looks at how it works rationally, logic is said to have content neutrality. If we say an argument is good, then the same kind of argument applied to a different topic will also be good. If we say an argument is good for solving murders, we will also say that the same kind of argument is good for deciding where to eat, what kind of disease is destroying your crops, or who to vote for. When logic is studied for theoretical reasons, it typically is pursued as formal logic. In formal logic we get content neutrality by replacing parts of the argument we are studying with abstract symbols. For instance, we could turn the argument above into a formal argument like this: There are six possibilities: A, B, C, D, E, and F. A is false. B, D, E, and F are also false. Therefore, the correct answer is C. Here we have replaced the concrete possibilities in the first argument with abstract letters that could stand for anything. This lets us see the formal structure of the argument, which is why it works in any domain you can think of. In fact, we can think of formal logic as the method for studying argument that uses abstract notation to identify the formal structure of argument. Formal logic is closely allied with mathematics, and studying formal logic often has the sort of puzzle-solving character one associates with mathematics. You will see this when we get to the chapters on formal logic. When logic is studied for practical reasons, it is typically called critical thinking. We will define critical thinking narrowly as the use of metareasoning to improve our reasoning in practical situations. Sometimes we will use the term “critical thinking” more broadly to refer to the results of this effort at self-improvement. You are “thinking critically” when you reason in a way that has been sharpened by reflection and metareasoning. A critical thinker is someone who has both sharpened their reasoning abilities using metareasoning, and deploys those sharpened abilities in real world situations. Critical thinking is generally pursued as informal logic, rather than formal logic. This means that we will keep arguments in ordinary language and draw extensively on your knowledge of the world to evaluate them. In contrast to the clarity and rigor of formal logic, informal logic is suffused with ambiguity and vagueness. There are problems with multiple correct answers, and problems where reasonable people can disagree with what the correct answer is. This is because you will be dealing with reasoning in the real world, which is messy. Our main goal in studying arguments is to separate the good ones from the bad ones. The argument about Clue we saw earlier is a good one, based on the process of elimination. It is good because, if I’ve got all the premises right, the conclusion will also be right. Statements So far we have defined logic as the study of argument and outlined its relationship to related fields. To go any further, we are going to need a more precise definition of what exactly an argument is. We have said that an argument is not simply two people disagreeing; it is an attempt to prove something using evidence. More specifically, an argument is composed of statements intended to support a conclusion. In logic, we define a statement as a unit of language (typically a sentence) that can be true or false. The idea that statement is a sentence that can be true or false is also captured by saying that every statement has a truth value. “All cats are dogs,” is a sentence whose truth value is false. “All cats are animals,” is a sentence whose truth value is true. All of the items below are statements. (a) Tyrannosaurus rex went extinct 65 million years ago. (b) Tyrannosaurus rex went extinct last week. (c) On this exact spot, 100 million years ago, a T. rex laid a clutch of eggs. (d) George W. Bush is the king of Jupiter. (e) Murder is wrong. (f) Abortion is murder. (g) Abortion is a woman’s right. (h) Lady Gaga is pretty. (i) Murder is the unjustified killing of a person. (j) The slithy toves did gyre and gimble in the wabe. (k) The murder of logician Richard Montague was never solved. Because a statement is something that has a truth value, statements include truths like (a) and falsehoods like (b). A statement can also be something that that must either be true or false, but we don’t know which, like (c). A statement can be something that is completely silly, like (d). Statements in logic include statements about morality, like (e), and things that in other contexts might be called “opinions,” like (f) and (g). People disagree strongly about whether (f) or (g) are true, but it is definitely possible for one of them to be true. The same is true about (h), although it is a less important issue than (f) and (g). A statement in logic can also simply give a definition, like (i). This sort of statement announces that we plan to use words a certain way, which is different from statements that describe the world, like (a), or statements about morality, like (f). Statements can include nonsense words like (j), because we don’t really need to know what the statement is about to see that it is the sort of thing that can be true or false. All of this relates back to the content neutrality of logic. The statements we study can be about dinosaurs, abortion, Lady Gaga, and even the history of logic itself, as in statement (k), which is true. We are treating statements primarily as units of language or strings of symbols, and most of the time the statements you will be working with will just be words printed on a page. However, it is important to remember that statements are also what philosophers call “speech acts.” They are actions people take when they speak (or write). If someone makes a statement, they are typically telling other people that they believe the statement to be true, and will back it up with evidence if asked to. When people make statements, they always do it in a context—they make statements at a place and a time with an audience. Often the context in which statements are made will be important for us, so when we give examples, statements, or arguments, we will sometimes include a description of the context. When we do that, we will give the context in italics. For example: The Earth is 4.5 billion years old. Susan is arguing with a young-earth creationist. The Earth is 4.5 billion years old. From a college-level textbook. The Earth is 4.5 billion years old. In each of these, “The Earth is 4.5 billion years old” is the statement. The sentences in italics are our inclusion of the context in which each statement was made. The word ‘statements’ in this text does not include questions, commands, exclamations, or sentence fragments. Someone who asks a question like “Does the grass need to be mowed?” is typically not claiming that anything is true or false. Generally, questions will not count as statements, but answers will. “What is this course about?” is not a statement. “No one knows what this course is about,” is a statement. For the same reason commands do not count as statements for us. If someone bellows “Mow the grass, now!” they are not saying whether the grass has been mowed or not. You might infer that they believe the lawn has not been mowed, but then again maybe they think the lawn is fine and just want to see you exercise. An exclamation like “Ouch!” is also neither true nor false. On its own, it is not a statement. We will treat “Ouch, I hurt my toe!” as meaning the same thing as “I hurt my toe.” The “ouch” does not add anything that could be true or false. Finally, a lot of possible strings of words will fail to qualify as statements simply because they don’t form a complete sentence. In your composition classes, these were probably referred to as sentence fragments. This includes strings of words that are parts of sentences, such as noun phrases like “The tall man with the hat” and verb phrases, like “ran down the hall.” Phrases like these are missing something they need to make a claim about the world. The class of sentence fragments also includes completely random combinations of words, like “The up if blender route,” which doesn’t even have the form of a statement about the world. When we study argument, we need to express things as statements, because arguments are composed of statements. Thus, if we encounter a rhetorical question while examining an argument, we need to convert it into a statement. “Don’t you think the lawn needs to be mowed?” will become “The lawn needs to be mowed.” Similarly, commands will become ‘should’ statements. “Mow the lawn, now!” will need to be transformed into “You should mow the lawn.” The latter kind of change will be important in critical thinking, because critical thinking often studies arguments whose goal is to an get audience to do something. These are called practical arguments. Most advertising and political speech consists of practical arguments, and these are crucial topics for critical thinking. Arguments Once we have a collection of statements, we can use them to build arguments. An argument is a connected series of statements, one or more of which is designed to provide support for another statement. Let’s start with an example of an argument given to an external audience. This passage is from an essay by Peter Singer called “Famine, Affluence, and Morality” in which he tries to convince people in rich nations that they need to do more to help people in poor nations who are experiencing famine. A contemporary philosopher writing in an academic journal. If it is in our power to prevent something bad from happening, without thereby sacrificing anything of comparable moral importance, we ought, morally, to do so. Famine is something bad, and it can be prevented without sacrificing anything of comparable moral importance. So, we ought to prevent famine. Peter Singer, “Famine, Affluence, and Morality.” Philosophy and Public Affairs, Spring, 1972, Vol. 1 No. 3. Singer wants his readers to work to prevent famine. This is represented by the last statement of the passage, “we ought to prevent famine,” which is called the conclusion of the passage. The conclusion of an argument is the statement that the argument is trying to convince the audience of. The statements that do the convincing are called the premises. In this case, the argument has three premises: (1) “If it is in our power to prevent something bad from happening, without thereby sacrificing anything of comparable moral importance, we ought, morally, to do so,” (2) “Famine is something bad,” and (3) “It can be prevented without sacrificing anything of comparable moral importance.” Now let’s look at an example of internal reasoning. Jack arrives at the track, in bad weather. There is no one here. I guess the race is not happening. In the passage above, the words in italics explain the context for the reasoning, and the words in regular type represent what Jack is actually thinking to himself. This passage again has a premise and a conclusion. The premise is that no one is at the track, and the conclusion is that the race was canceled. The context gives another reason why Jack might believe the race has been canceled: the weather is bad. You could view this as another premise—it is very likely a reason Jack has come to believe that the race is canceled. In general, when you are looking at people’s internal reasoning, it is often hard to determine what is actually working as a premise and what is just working in the background of their unconscious. When people give arguments to each other, they typically use words like “therefore” and “because.” These are meant to signal to the audience that what is coming is either a premise or a conclusion in an argument. Words and phrases like “because” signal that a premise is coming, so we call these premise indicators. Similarly, words and phrases like “therefore” signal a conclusion and are called conclusion indicators. The argument from Peter Singer uses the conclusion indicator word, “so.” Here is an incomplete list of indicator words and phrases in English. Premise Indicators: because, as, for, since, given that, for the reason that, may be inferred from, owing to, is evidenced by Conclusion indicators: therefore, thus, hence, so, consequently, it follows that, in conclusion, as a result, it must be the case, accordingly, this implies that, this entails that, we may infer that The two passages we have looked at in this section so far have been simply presented as quotations. But often it is extremely useful to rewrite arguments in a way that makes their logical structure clear. One way to do this is to use something called “canonical form.” An argument written in canonical form has each premise numbered and written on a separate line. Indicator words and other unnecessary material should be removed from the premises. Although you can shorten the premises and conclusion, you need to be sure to keep them all complete sentences with the same meaning, so that they can be true or false. The argument from Peter Singer, above, looks like this in canonical form: If we can stop something bad from happening, without sacrificing anything of comparable moral importance, we ought to do so. Famine is something bad. Famine can be prevented without sacrificing anything of comparable moral importance. We ought to prevent famine. If you are reading this with a screen reader, there is a line between the last and second-to-last statements. In canonical form, we will always put the conclusion last, so you should be able to find the conclusion without knowing the line is there. Each premise has been written on its own line; the conclusion is written last below the line. The statements have been paraphrased slightly for brevity, and the indicator word “so” has been removed. Also notice that the “it” in the third premise has been replaced by the word “famine,” so that statements reads naturally on its own. Similarly, we can rewrite the argument Jack gives at the racetrack, like this: There is no one at the race track. The race is not happening. Notice that we did not include anything from the part of the passage in italics. The italics represent the context, not the argument itself. Also, notice that the “I guess” has been removed. When we write things out in canonical form, we write the content of the statements, and ignore information about the speaker’s mental state, like “I believe” or “I guess.” One of the first things you have to learn to do in logic is to identify arguments and rewrite them in canonical form. This is a foundational skill for everything else we will be doing in this text, so we are going to go over an example here, and there will be more in the exercises. The passage below is paraphrased from the ancient Greek philosopher Aristotle. An ancient philosopher, writing for his students. Again, our observations of the stars make it evident that the earth is round. For quite a small change of position to south or north causes a manifest alteration in the stars which are overhead. Aristotle On the Heavens, 298a2-10. The first thing we need to do to put this argument in canonical form is to identify the conclusion. The indicator words are frequently the best way to do this. The phrase “make it evident that” is a conclusion indicator phrase. He is saying that everything else is evidence for what follows. So we know that the conclusion is that the earth is round. “For” is a premise indicator word—it is sort of a weaker version of “because.” Thus the premise is that the stars in the sky change if you move north or south. In canonical form, Aristotle’s argument that the earth is round looks like this. There are different stars overhead in the northern and southern parts of the earth. The earth is spherical in shape. The ultimate test of whether something is an argument is simply whether some of the statements provide reason to believe another one of the statements. If some statements support others, you are looking at an argument. The speakers in these two cases use indicator phrases to let you know they are trying to give an argument. A final bit of terminology for this section. An inference is the act of coming to believe a conclusion on the basis of some set of premises. When Jack in the example above saw that no one was at the track, and came to believe that the race was not on, he was making an inference. We also use the term inference to refer to the connection between the premises and the conclusion of an argument. If your mind moves from premises to conclusion, you make an inference, and the premises and the conclusion are said to be linked by an inference. In that way inferences are like argument glue: they hold the premises and conclusion together. II. Inductive Arguments, Deductive Arguments Since the time of Aristotle, the study of logic has been divided into the study of two distinct types of reasoning—deductive and inductive. One way to think of the difference between these types of arguments is to think of deductive arguments in terms of “necessity,” and inductive arguments in terms of probability. In other words, we distinguish deductive arguments from inductive arguments in terms of the strength of the inferential link between premises and conclusion. The strongest inferential link possible would be one where the premises make the conclusion follow from the premises necessarily—i.e., where the premises force us to draw the conclusion. An argument like would be a deductive argument. By contrast, an argument whose purpose is to make its conclusion “probable” is an inductive argument. Here are examples of each type: Deductive All cats are animals. Rob Halford is a cat. Therefore, Rob Halford is an animal. Given the premises of this argument, we are forced to draw the conclusion that Rob Halford is an animal. Since the link between premises and conclusion here is one of necessity, and we are forced to draw this conclusion, this is a deductive argument. Inductive Ninety-nine percent of the people who like baseball live to be 120 years old. I like baseball, so I’ll probably live to be 120 years old. Given the premises of this argument, we aren’t forced to draw the conclusion, but the premises do make the conclusion highly probable. At this point we can formulate brief, useful definitions for deductive and inductive arguments: Deductive Argument: An argument in which the conclusion is claimed to follow from the premises with strict necessity, i.e., given the premises, it is claimed that we must draw the conclusion. Inductive argument: An argument in which the conclusion is claimed to follow from the premises with probability, i.e., given the premises, it is claimed that the conclusion is probable. Distinguishing Deductive and Inductive Arguments In trying to distinguish between deductive and inductive arguments, the best thing to focus on is the relationship between the premises and the conclusion. It may be helpful to think in terms of two different kinds of light switches. Deductive arguments operate on something like a light switch that's “off” or “on” with no in-between—there’s light, or there isn’t. For deductive arguments the conclusion either follows from the premises, or it doesn’t. There are no varying degrees of validity. Inductive arguments operate on something like a dimmer switch: just as a dimmer switch gives varying degrees of light intensity, inductive arguments have varying degrees of strength. Inductive Argument: Dimmer Switch on the left Deductive Argument: On/Off switch on the right Deductive Arguments Deductive arguments are always a matter of necessity. For example: “The day after Monday is always Friday. Today is Monday, so it follows necessarily that tomorrow is Friday.” The first premise is false, but if it were true, the conclusion follows necessarily—it’s guaranteed by the premises. It’s like “two plus two equals four.” If I tell you I have two apples in one hand and two apples in the other, then you have to conclude that I’ve got four apples. Nothing else would fit with those premises, just as nothing else would fit after the “equals” in the addition problem. Here’s another example: If I tell you that all dogs are animals and that Henri is a dog, then you have no choice but to conclude that Henri is an animal. The premises make that conclusion necessary. And if I say that all dolphins are Martians, and Henri is a dolphin, the only conclusion that you can draw is that Henri is a Martian. In all these cases the light switch is moved to the “on” position. But if I tell you that all dogs are animals, and that Henri is a cat, it DOES NOT follow that Henri is an animal. In that case the light stays off. Now, you and I both know that cats are animals, so that if Henri is a cat then he must be an animal, but the premises don’t say that, and we can’t assume it if it’s not in the premises—so this argument doesn’t guarantee/necessitate that conclusion. In other words, the premises don’t force us to draw that conclusion. The absolutely central thing to recognize about deductive arguments is that they have a “form” or structure, and it’s that form that makes it deductive, and it’s that form alone, and nothing else, that determines whether the argument is valid or invalid. Now, the handy thing about that is that deductive arguments can all be represented by a diagram, or a timeline, or a drawing, or by reducing them to their skeletal form by substituting letters for the terms/statements in the original argument. If you can diagram it / reduce it to a form, and can tell just from the form whether it’s good or bad, then you know it's a deductive argument. You can't do that with inductive arguments — you have to know the content to tell whether it's good or bad. Here are two examples: All dogs are animals. Henri is a dog. Therefore, Henri is an animal. The form of this argument is: All D are A H is D Therefore, H is A. “H is A” is guaranteed by the premises here. Nothing else will go in its place. This argument, then, or rather its form, is what we will call “valid.” Any argument in which the terms are arranged that way will be a valid argument. (The concept of validity will be discussed in more detail below). Here’s another deductive argument: Steve is taller than Cheryl, and Cheryl is taller than Mary. It follows that Steve is taller than Mary. You can write this out using different sized letters to represent the people, and use the letters S, C and M. You’d get something like this: Since you can represent the argument like this, it’s deductive. Inductive Arguments Inductive arguments are always a matter of probability. The premises of an inductive argument can only make a conclusion more or less probable, but never guaranteed or necessary. Consider the following argument: People who get flu shots have a 93% chance of not getting the flu. I just got a flu shot, therefore I probably won't get the flu. The conclusion here is not guaranteed, of course. Some people who get a flu shot obviously DO get the flu (7% to be precise). But 93% is a high degree of probability (anything over 50% is considered probable), so the premises of this argument make the conclusion highly probable. An inductive argument where the premises make the conclusion probable is called “strong.” (Like validity, the concept of strength will be discussed in more detail below). An essential point here, and one that bears frequent repeating: It will strike you as completely counter-intuitive at first, but in trying to determine whether deductive and inductive arguments support their conclusions, it doesn’t matter whether any of the statements in the arguments are true. Taking the flu shot example—I have no actual idea what the numbers are on flu shots, and I didn’t just get one. But logic doesn’t care about that. Logic is only concerned with whether, given what the premises say, the premises support the conclusion. You are going to have to continually tell yourself not to answer questions about what kinds of arguments you are looking at, and whether their premises support their conclusions, based on your knowledge of whether their conclusions are true. Here’s an inductive argument example to illustrate: NASA astronauts have gone to the Moon 647,832 times so far this year. Every single time they’ve landed on the moon a little green man comes out of a hole in the ground and serves them hamburgers and beer. Therefore, the next time astronauts make a trip to the moon, that little green man will probably come out and serve them hamburgers and beer. The conclusion of this argument follows with an exceptionally high degree of probability. And none of it is true. But in logic we don’t care about that when we are trying to determine whether the premises of an argument support its conclusion. Another helpful way of recognizing deductive and inductive arguments is to familiarize ourselves with a handful of common types of arguments that fall under each heading. If you recognize that something is an argument based on mathematics, for example, you know that that argument is deductive. If you recognize that something is a causal inference, by contrast, you know that that argument is inductive. Common Types of Deductive Arguments In deductive arguments like the ones in this section, the conclusion is supposed to contain only information that is already in the premises. You simply shuffle information around. (The word “deductive” comes from the Latin words de and ducere, and means “to take out”). An argument based on mathematics is one in which the arguer draws a conclusion by doing some mathematical computation: addition, subtraction, division, multiplication, etc. Imagine a person who goes to the hardware and puts four hammers and six screwdrivers in their cart, and that each item costs two dollars with tax. They conclude that their purchase will amount to twenty dollars. Arguments that depend on calculations like this are always best seen as deductive. (Be advised, though, that most statistical reasoning, though it involves math, is almost always inductive). An argument from definition is one in which the conclusion of an argument involves stating the definition of a word in the premise. For example, you could argue that since Humbertimus is riding a tricycle, it follows that he is riding something that has three wheels. Or someone could argue that someone is duplicitous because they are always being deceitful. Arguments from definition are always deductive. There are three types of syllogism (a syllogism is an argument with two premises and one conclusion): Categorical Syllogism: A syllogism in which each statement begins with the word “All,” “No,” or “Some.” For example: All Mustangs are Fords Some Mustangs are red sports cars. Therefore, some Fords are red sports cars. Hypothetical Syllogism: A syllogism containing at least one premise which is a conditional (if…then…) statement. For example: If Atlanta is the capital of Georgia, then Atlanta is a southern city. Atlanta is the capital of Georgia. Therefore, Atlanta is a southern city. Disjunctive Syllogism: A syllogism in which one of the premises is an either/or statement: Either George Bush was a president, or Beyoncé was a president. George Bush was not a president. Therefore, Beyoncé was a president. Common Types of Inductive Arguments In inductive arguments, as in the examples below, the conclusion goes beyond the premises and adds something to the information given. You try to give as much evidence as you can for the conclusion. (“Inductive” also comes from Latin: in plus ducere means “to put in”). A prediction is an argument that uses information about the past to make a claim about what will happen in the future. For example, someone might argue that since every squirrel people have ever seen has eaten nuts, that probably the next squirrel someone sees will also be a nut eater. Or someone might argue that, given all the data we have linking smoking with lung disease, that a person who smokes three packs of cigarettes ever day will likely develop lung disease. Predictions never deal with absolutes, but only with probabilities, so these types of arguments are almost always best read as inductive. A causal inference is an argument that proceeds from knowledge of a cause and makes a claim about a subsequent effect, or proceeds from knowledge of an effect and makes a claim about a previous cause. For example, you might argue that since Kat hit their hand with a hammer, they developed a bruise. Or you might argue that other way, that since Kat is sniffling and sneezing and coughing, they must have caught a cold. (*Note: it’s important to distinguish between predictions and causal inferences; predictions always gather data from the past and try to establish what will happen in the future. Causal inferences where you know the effect and are reasoning out what caused it always draw a conclusion about something that has already happened. Causal inferences where you know the cause and are reasoning out the effect could have a conclusion about the past, present, or future, whenever we think the effect would most likely happen). An argument from authority is one in which the arguer claims that something must be the case because someone taken by the arguer to be an expert has said so. Arguments from authority then typically look like this. “X said Y must be the case. Since X is an expert, Y must be the case.” For example: “The cable repair person said there must be something wrong my cable box. Since they’re the expert, it’s probably true that there’s something wrong with my cable box.” Experts can be wrong, though, and they sometimes fail to tell the truth, so these types or arguments are always only a matter of probability. An argument based on signs is one in which the arguer claims that some conclusion follows simply because of something stated on a sign. Here we have to interpret the word “sign” broadly, to include literal signs like stop-signs and theater marquees, but also any visual messages produced by an intelligent being: things like clocks, pop-up messages, product labels, price-tags, a row of orange cones in the road. Any visual clue someone left as a form of communication. Signs can be mistaken, though, or misplaced, or out of date, so these arguments never make their conclusions strictly necessary (they don’t force us to draw their conclusions), but only probable. A generalization is an argument that proceeds from knowledge of specific cases to a general claim about an entire group, or all members of a certain class of things. For example, you might argue that since the people from Wisconsin you met on your last vacation were really nice, that all people from Wisconsin are really nice. Or you might claim that all Waffle Houses make great hashbrowns, since every Waffle House you’ve eaten at has always had great hashbrowns. An argument from analogy is an inductive argument in which the arguer compares things and draws a conclusion based on the similarities between those things. We make these kinds of arguments whenever we ask people we trust about their experiences with things we are interested in pursuing. Say, for example, you are thinking of buying the latest iPhone. You might ask your friend who has the latest model whether they are satisfied with the phone. If they’re very satisfied, you might conclude that you will also be very satisfied if you get the same (exactly similar) phone. These types of arguments at best make their conclusions probable. Summing Up The key to distinguishing Inductive and Deductive arguments is to focus on the support relationship between the premises and conclusion. An argument is best interpreted as deductive if: The support relationship between premises is one of necessity. You can isolate the form of the argument by replacing terms with letters, drawing a diagram or circles or a timeline, etc., and you can evaluate the argument as good or bad based on the form. It has a recognizable deductive form. An argument is best interpreted as inductive if: The support relationship between premises and conclusion is one of probability. Trying to construct a diagram or abstract representation of the argument does not help you evaluate it. It shares recognizable features with common inductive argument types. III. Evaluating Arguments Two Ways an Argument Can Go Wrong Arguments are supposed to lead us to the truth, but they don’t always succeed. There are two ways they can fail in their mission. First, they can simply start out wrong, using false premises. Consider the following argument: It is raining heavily. If you do not take an umbrella, you will get soaked. You should take an umbrella. If premise (1) is false—if it is sunny outside—then the argument gives you no reason to carry an umbrella. The argument has failed its job. Premise (2) could also be false: Even if it is raining outside, you might not need an umbrella. You might wear a rain poncho or keep to covered walkways and still avoid getting soaked. Again, the argument fails because a premise is false. Even if an argument has all true premises, there is still a second way it can fail. Suppose for a moment that both the premises in the argument above are true. It is actually raining heavily. You do not own a rain poncho. You need to go places where there are no covered walkways. Now does the argument show you that you should take an umbrella? Not necessarily. Perhaps you enjoy walking in the rain, and you would like to get soaked. In that case, even though the premises were true, the conclusion would be false. The premises, although true, do not support the conclusion. Back when we defined an inference, we said it was like argument glue: it holds the premises and conclusion together. When an argument goes wrong because the premises do not support the conclusion, we say there is something wrong with the inference. Consider another example: You are reading this book. This is a logic book. You are a logic student. This is not a terrible argument. Most people who read this book are logic students. Yet, it is possible for someone besides a logic student to read this book. If your roommate picked up the book and thumbed through it, they would not immediately become a logic student. So the premises of this argument, even though they are true, do not guarantee the truth of the conclusion. Its inference is less than perfect. Again, for any argument, there are two ways that it could fail. First, one or more of the premises might be false. Second, the premises might fail to support the conclusion. Even if the premises were true, the form of the argument might be weak, meaning the inference is bad. In logic, we are almost exclusively concerned with evaluating the quality of inferences, not the truth of the premises. The truth of various premises will be a matter of whatever specific topic we are arguing about, and, as we have said, logic is content neutral. Remember that whether the things in the arguments are true is irrelevant to determining what kind of argument it is (deductive/inductive) and this goes as well for determining whether it’s valid/invalid or strong/weak. Logic is largely unconcerned with whether the statements in the argument are true. Instead, logic is almost exclusively concerned with whether the premises of an argument, true or false, provide support for the conclusion, true or false. Here are two examples, one inductive and one deductive, where everything in the arguments is false, and yet the premises support the conclusion: Here's the inductive example: The average temperature for December in Atlanta has been 142 degrees every year for last 113 years. Probably next year the average temperature for December in Atlanta will be 142 degrees. Given the premise here (which is false), the conclusion follows with a high degree of probability. Here’s the deductive example: All cats are Martians. Barack Obama is a cat. Therefore Barack Obama is a Martian. Given what the premises here say (and they are false as a matter of fact) we MUST draw the conclusion here. The premises make that conclusion necessary. Deductive Arguments Deductive and inductive arguments are evaluated differently, so we have different vocabulary for whether each type of logic succeeds or fails. A deductive argument with good logic is called valid; a deductive argument with bad logic is called invalid. Validity For the purposes of this textbook, we will adopt the following definitions for validity: Valid: The conclusion is supported by (follows from) the premises. So: if the premises of a valid argument are true, it is impossible for the conclusion to be false. Invalid: The conclusion is not supported by (does not follow from) the premises. So: if the premises of an invalid argument are true, it is still possible for the conclusion to be false. Another way to put this is to say that the only thing that matters for validity is whether the premises, as stated, support the conclusion. So, for example, you can have valid and invalid arguments with all true premises and a true conclusion, valid and invalid arguments with false premises and a true conclusion, valid and invalid arguments with one false and one true premise and a true conclusion, etc. The only thing you cannot have with a valid argument is all true premises and a false conclusion. A valid deductive argument is one where the premises support the conclusion. Given that, it is also not possible for a valid deductive argument to have true premises and a false conclusion. In other words, given a valid argument (the premises support the conclusion), then if the premises are true, the conclusion must also be true. But, the premises of a valid argument do not have to be true. There are valid arguments with false premises for example, in which case the argument can have either true or false premises. So, the only arrangement of truth values that is not possible is a valid argument with true premises and false conclusion. It’s important to always keep in mind that valid arguments can have false conclusions, because people naturally tend to think that any argument must be good if they agree with the conclusion. And the more passionately people believe in the conclusion, the more likely we are to think that any argument for it must be brilliant. Conversely, if the conclusion is something we don’t believe in, we naturally tend to think the argument is poor. And the more we don’t like the conclusion, the less likely we are to like the argument. But this is not the correct way to evaluate inferences at all. The quality of the inference is entirely independent of the truth of the conclusion. You can have great arguments for false conclusions and horrible arguments for true conclusions. We have trouble seeing this because of biases built deep in the way we think called “cognitive biases.” A cognitive bias is a habit of reasoning that can be dysfunctional in certain circumstances. Generally, these biases developed for a reason, so they serve us well in many or most circumstances. But cognitive biases also systematically distort our reasoning in other circumstances, so we must be on guard against them. There is a particular cognitive bias that makes it hard for us to recognize when a poor argument is being given for a conclusion we agree with. It is called “confirmation bias” and it is in many ways the mother of all cognitive biases. Confirmation bias is the tendency to discount or ignore evidence and arguments that contradict one’s current beliefs. It really pervades all of our thinking, right down to our perceptions. Because of confirmation bias, we need to train ourselves to recognize valid arguments for conclusions we think are false. Remember, an argument is valid if it is impossible for the premises to be true and the conclusion false. This means that you can have valid arguments with false conclusions, they just also have to have at least one false premise. Consider this example: Oranges are either fruits or musical instruments. Oranges are not fruits. Oranges are musical instruments. The conclusion of this argument is nonsensical. Nevertheless, it follows validly from the premises. This is a valid argument. If both premises were true, then the conclusion would necessarily be true. This shows that a valid argument does not need to have true premises or a true conclusion. Conversely, having true premises and a true conclusion is not enough to make an argument valid. Consider this example: London is in England. Beijing is in China. Paris is in France. The premises and conclusion of this argument are, as a matter of fact, all true. This is a terrible argument, however, because the premises have nothing to do with the conclusion. The argument is not valid. If an argument is not valid, it is called invalid. In general the actual truth or falsity of the premises, if known, do not tell you whether or not an inference is valid. There is one exception: when the premises are true and the conclusion is false, the inference cannot be valid, because valid reasoning can only yield a true conclusion when beginning from true premises. Here is another invalid argument: All dogs are mammals. All dogs are animals. All animals are mammals. In this case, we can see that the argument is invalid by looking at the truth of the premises and conclusion. We know the premises are true. We know that the conclusion is false. This is the one circumstance that a valid argument is supposed to make impossible. Some invalid arguments are hard to detect because they resemble valid arguments. Consider this one: An economic stimulus package will allow the U.S. to avoid a depression. There is no economic stimulus package. The U.S. will not avoid a depression. This reasoning is not valid since the premises do not definitively support the conclusion. To see this, assume that the premises are true and then ask, “Is it possible that the conclusion could be false in such a situation?” There is no inconsistency in taking the premises to be true without taking the conclusion to be true. The first premise says that the stimulus package will allow the U.S. to avoid a depression, but it does not say that a stimulus package is the only way to avoid a depression. Thus, the mere fact that there is no stimulus package does not necessarily mean that a depression will occur. When an argument resembles a good argument but is actually a bad one, we say it has a fallacy. Fallacies are similar to cognitive biases, in that they are ways our reasoning can go wrong. Fallacies, however, are always mistakes you can explicitly lay out as arguments in canonical form, as above. Soundness If an argument is not only valid, but also has true premises, we call it sound. “Sound” is the highest compliment you can pay an argument. We said earlier that there were two ways an argument could go wrong, either by having false premises or weak inferences. Sound arguments have true premises and undeniable inferences. An argument that fails, either by having invalid logic, or by having at least one false premise, is called unsound. This argument is valid, but not sound: Socrates is a person. All people are carrots. Therefore, Socrates is a carrot. This argument both valid and sound: Socrates is a person. All people are mortal. Therefore, Socrates is mortal. Both arguments have the exact same form. They say that a thing belongs to a general category and everything in that category has a certain property, so the thing has that property. Because the form is the same, it is the same valid inference each time. The difference in the arguments is not the validity of the inference, but the truth of the second premise. People are not carrots, therefore the first argument is not sound. People are mortal, so the second argument is sound. Often it is easy to tell the difference between a valid but unsound argument, and a valid and sound argument, if you are using completely silly examples. Things become more complicated with false premises that you might be tempted to believe, as in this argument: Every Irishman drinks Guiness. Smith is an Irishman. Therefore, Smith drinks Guiness. You might have a general sense that this argument is bad—you shouldn’t assume that someone drinks Guinness just because they are Irish. But the argument is completely valid (at least when it is expressed this way). The inference here is the same as it was in the previous two arguments. The problem is the first premise. Not all Irishmen drink Guinness, but if they did, and Smith was an Irishman, he would drink Guinness. The important thing to remember is that validity is not about the actual truth or falsity of the statements in the argument. Instead, it is about the way the premises and conclusion are put together. It is really about the form of the argument. A valid argument has perfect logical form. The premises and conclusion have been put together so that the truth of the premises is incompatible with the falsity of the conclusion. The following table gives examples of both valid and invalid arguments with a variety of truth values for the premises and the conclusion. Notice that there are many ways for an argument to be unsound, but only one way to be sound. Notice also that the only time truth values can tell us anything about validity is when the premises are actually true and the conclusion is actually false. If an argument is valid, it is impossible to have true premises and a false conclusion, so if these are the real-world truth values, something went wrong with the logic, and the argument has to be invalid. Table 1: Deductive Arguments Valid Invalid True Premises True Conclusion All mammals are animals. Cats are mammals. Therefore, cats are animals. (sound) All mammals are animals. Cats are animals. Therefore, cats are mammals. (unsound) True Premises False Conclusion Does Not Exist All mammals are animals. Snakes are animals. Therefore, snakes are mammals. (unsound) False Premises True Conclusion All mammals are fruits. Bananas are mammals. Therefore, bananas are fruits. (unsound) All mammals are fruits. Cats are fruits. Therefore, cats are mammals. (unsound) False Premises False Conclusion All mammals are shoes. Bananas are mammals. Therefore, bananas are shoes. (unsound) All mammals are fruits. Snakes are fruits. Therefore, snakes are mammals. (unsound) A general trick for determining whether an argument is valid is to try to come up with just one way in which the premises could be true but the conclusion false. If you can think of one, the reasoning is invalid. This is called the counterexample method (more on that below). Inductive Arguments Instead of “valid” and “invalid,” when we evaluate the logic of inductive arguments, we use the terms “strong” for good arguments and “weak” for bad ones. Strength Inductive arguments, remember, can never guarantee their conclusion; there’s always room for error. So we can’t really apply “validity” or “invalidity” to them. I mean, you can. But given inductive arguments always leave room for error, according to the definitions of validity and invalidity, all inductive arguments are invalid. This is hardly helpful in evaluating them, however—some inductive arguments are much better than others, and we need to be able to tell the difference. Inductive arguments can, however, give evidence that the conclusion is probably. We evaluate them according to how probable the premises make the conclusion—how strongly the premises support the conclusion. For the purposes of this textbook we will adopt the following definitions of strength: Strong: Conclusion is made probable in light of the premises. So: if the premises are true, the conclusion is probably true. The premises give good support for the conclusion. Weak: Conclusion is NOT made probable in light of the premises; i.e., the premises give little or no support for the conclusion. An argument is strong if the premises would make the conclusion more probable, were they true. In a strong argument, the premises don’t guarantee the conclusion, but they do make it a good bet—we say that they make the conclusion probable. You may have noticed that the word “probable” is a little vague. How probable do the premises have to make the conclusion before we can count the argument as strong? The answer is a very unsatisfying “it depends.” As a general rule, any argument where the premises give the conclusion a probability greater than 50% is a strong argument. We don’t often have actual numbers support our inductive arguments, so we have to use common sense. Take the following argument: The highway sign says “Atlanta, 34 miles.” Therefore, it’s probably 34 miles to Atlanta. This is an argument based on signs. We don’t possess statistics on how frequently highway signs are accurate, but common sense tells us that if they weren’t accurate more than half the time there’d be no point to having them at all. The vagueness of the word “probable” brings out an interesting feature of strong arguments: some strong arguments are stronger than others. Consider the following two arguments: My friend Tessa has the iPhone Infinity and they love it. I bet if I get the iPhone Infinity, I will also love it. My friends Tessa, Lester, George, and Imogen have the iPhone Infinity and they love it. I bet if I get the iPhone Infinity, I will also love it. The argument gives only one instance of another iPhone user loving their phone. The argument is not very strong as it stands but it can become stronger if we add more users as examples. The more evidence we have, the better a bet the conclusion is. When we do induction, we try for strong inferences, where the premises, assuming they are true, would make the truth of the conclusion highly probable, though not necessary. Consider these two arguments. First, look at this one: 92% of Republicans from Texas voted for Bush in 2000. Jack is a Republican from Texas, so Jack probably voted for Bush. This is a strong argument. Now compare it to this one, which is weak: One half of all drivers are men. There’s a person driving the car that just cut me off, so the person driving that car is a probably a man. There is a big difference between how much support the premises, if true, would give to the conclusion in the first and how much they would in the second. The premises in the first, assuming they are true, would provide very strong reasons to accept the conclusion. This, however, is not the case with the second: if the premises in it were true then they would give only weak reasons for believing the conclusion. Thus, the first is strong while the second is weak. Cogency A cogent argument is the inductive equivalent of sound deductive argument: it’s a strong inductive argument with true premises. Cogent equals strong, plus all true premises. Always think of the definition in that order. First determine whether the argument is strong, and then ask whether the premises are true. It’s also important that the “true premises” condition requires that you not leave out any important evidence that would tend to weigh against the conclusion. For example, if it’s a warm, sunny day, with only gentle breezes and no big waves, you may conclude it’s a nice day for swimming in the ocean. That would typically be a strong argument. But if you’ve ignored the fact that a large ship just dumped huge quantities of toxic waste near the beach, your argument is weak. This is often referred to as the total evidence requirement. In sum, then, an argument is cogent if it is strong and has true premises. It is uncogent if it is weak, or has false premises. The following table gives examples of strong and weak inductive arguments, with a variety of truth values for the premises and conclusions. Table 2 Inductive Arguments Strong Weak True Premise Probably True Conclusion All previous Halloween’s were celebrated in October. Probably the next Halloween will be celebrated in October. (cogent) Several Halloweens were celebrated in the 19th century. Probably the next Halloween will be celebrated in October. (uncogent) True Premise Probably False Conclusion Does Not Exist Several Halloweens were celebrated in the 19th century. Probably the next Halloween will be celebrated in January. (uncogent) False Premise Probably True Conclusion All previous Halloweens were celebrated in the 21st century. Probably the next Halloween will be celebrated in the 21st century (uncogent) Several Halloweens were celebrated in January. Probably the next Halloween will be celebrated in October. (uncogent) False Premise Probably False Conclusion All previous Halloweens were celebrated in January. Probably the next Halloween will be celebrated in January. (uncogent) Several Halloweens were celebrated in January. Probably the next Halloween will be celebrated in January. (uncogent) IV Counterexample Method It has been stated repeatedly that validity is entirely a function of the form of a deductive argument, and that the actual truth values of the statements involved are irrelevant. The following argument, for example, is valid: All cats are dogs. All dogs are fish. Therefore, all cats are fish. Since validity is determined by the form of the argument, it follows that any argument with the same form will also be valid. The form of the above argument is: All C are D All D are F All C are F Similarly, the following argument is invalid because it has an invalid form: Some cats are mammals Some cats are animals. Some mammals are animals. Everything in this argument is true, so it would be tempting to think this is a valid argument. It’s not, though, because it’s possible for an argument with true premises and a false conclusion to have this same form. Any argument where the form would allow true premises and a false conclusion is an invalid argument. Here’s the form of the argument immediately above: Some C are M Some C are A Some M are A The following argument has the very same form, but has true premises and a false conclusion. Table 1 above tells us that no valid argument form would allow that. Some animals are cats Some animals are dogs Some cats are dogs Again, the question is not whether the premises and conclusion are true or false, but whether the premises support the conclusion. That’s what table 1 above illustrates. The validity of a deductive argument, to put it another way, is determined solely by the original argument’s form. If that form would allow true premises and a false conclusion, then the form is invalid. Thus, there is only one empty box in table 1—the box representing a valid argument with true premises and a false conclusion. Every other arrangement of truth values is possible for both valid and invalid arguments—true premises and true conclusion, false premises and true conclusion, false premises and false conclusion. But no valid argument form will ever allow true premises and a false conclusion. This seems counter-intuitive, but you can have a valid deductive argument in which everything is false, so long as the conclusion actually follows from the premises. And you can have an invalid deductive argument in which everything is true, so long as the conclusion does not follow from the premises. And whether the conclusion does follow from the premise will be determined by the argument’s form—the arrangement of the terms in the argument, and not what the argument is about. In order to prove that a deductive argument is invalid, we have been using the counterexample method. In the counterexample method we look for a substitution instance for the terms in the original argument so that the premises come out true and the conclusion false. Here are the steps: Write out the original argument. Isolate the form of the original argument. Find terms to substitute for those in the original argument. Plug substitute terms into the form so that you make the premises true and the conclusion false. 1. Original argument Some humans are doctors. Some humans are men. Therefore, some men are doctors. This is a categorical syllogism. All the statements in it are true, so it would be tempting to think that it is valid. This would be a mistake. This argument is invalid, but it’s hard to see that until you break it down into its form and then do substitutions for the terms. Here’s the form of the original argument: 2. Form of the original argument: Some H are D Some H are M Some M are D 3. Find substitute terms for those in the original argument H = Animals D = Cats M = Dogs 4. Plug substitute terms into the original argument form Some Animals are Cats (True) Some Animals are Dogs (True) Some Dogs are Cats (False) Note about “terms” used for substitutions. For categorical syllogisms, the terms you use for substitutions must be plural nouns (dogs, cats, animals, mammals) or plural noun phrases (dogs who like to watch football, cats who smoke cigarettes, mammals riding bicycles). Counterexample Method and non-Categorical Syllogisms The substitution method can also be used for hypothetical syllogisms, disjunctive syllogisms and more—and follows the same steps. You need to find substitutions that make the premises true and the conclusion false. For hypothetical and disjunctive syllogisms, what you need to use for substitutions are full statements, and not just terms like we used with categorical syllogisms. Remember, the steps are: Write out the original argument. Isolate the form of the original argument. Find statements to substitute for those in the original argument. Plug substitute statements into the form so that you make the premises true and the conclusion false. 1. Original argument: If Beyoncé is a cat, then Beyoncé is a feline. (True) Beyoncé is not a cat. (True) Therefore, Beyoncé is not a feline. (True) 2. Original argument form: If C then F Not C Not F This is an invalid argument, even though everything in it is true. We can prove it’s invalid by substituting the following statements for C and F (you can always use these two statements to check invalidity for hypothetical syllogisms). 3. Find substitute terms for those in the original argument C = Atlanta is the capital of Florida F = Atlanta is a southern city 4. Plug substitute terms into the original argument form If (C) Atlanta is the capital of Florida, then (F) Atlanta is a southern city. (True) (not C) Atlanta is not the capital of Florida. (True) (not F) Atlanta is not a southern city. (False) Summing Up A valid deductive argument form will not allow you to substitute terms or statements to make the premises true and have the conclusion come out false. If you can do that, find substitute terms or statements for the terms in the original argument that make the premises true and the conclusion false, then you have proved that the original argument has an invalid form, i.e., you have proved that the original argument is invalid.