(PDF) Contemporary syllogistics

Niki Pfeifer Contemporary syllogistics Comparative and quantitative syllogisms Abstract: Traditionally, syllogisms are arguments with two premises and one conclu- sion which are constructed by propositions of the form “All… are…” and “At least one… is…” and their respective negated versions. Unfortunately, the practical use of traditional syllogisms is quite restricted. On the one hand, the “All…” propositions are too strict, since a single counterexample suffices for falsification. On the other hand, the “At least one …” propositions are too weak, since a single example suffices for verification. The present contribution studies algebraic interpretations of syllogisms with comparative quantifiers (e.g., “Most… are…”) and quantitative quantifiers (e.g., “n/m… are…”, “all, except n… are…”). This modern version of syllogistics is intended to be a more adequate framework for argumentation theory than traditional syllogistics. Keywords: Syllogisms, comparative and quantitative quantifiers, exceptions, rational argumentation. Author: Pfeifer, Niki; Universität Salzburg, Fachbereich Psychologie, Hellbrunner- strasse 34, A-5020 Salzburg,

. 1. Introduction The oldest and most famous reasoning system is Aristotelian syllogistics. During the last 50 years many new approaches to old logical problems were developed in philosophy, artificial intelligence, and linguistics. These generalizations or extensions of traditional syllogistics are of spe- cial interest to argumentation theory as they try to do more justice to practical reasoning than classical syllogistics did. The present paper focuses on two not very well known recent devel- opments in syllogistic inference, namely syllogisms containing compara- tive and quantitative quantifiers. The classical universal (“All …”) and the existential quantifiers (“At least one …”)1 in syllogisms are either too strict or too weak, respectively. On the one hand, the universal quantifier is too strict because it does not allow for exceptions. One simple counter- example falsifies an all-assertion. In everyday contexts exceptions are the ________________ 1 Traditionally, the existential quantifier is read as “Some …”. We prefer the unambiguous reading “At least one ...”. 58 Niki Pfeifer rule, we reason with defaults or rules of thumb that hold normally or most of the time. On the other hand, the existential quantifier is too weak be- cause a single example that fits verifies an existential statement. Such quantifiers hardly ever occur in everyday life reasoning. Comparative and quantitative quantifiers that—at least implicitly—actually occur in every- day life reasoning like “most …”, “allmost all …”, “m/n …”, or “All, except at most x …” are not expressible in classical syllogistics.2 In this paper, I propose comparative and quantitative quantifiers as promising candidates for rational argumentation and for the rational re- construction of syllogism-like arguments. After a short introduction to classical syllogistics, I sketch two formal systems of modern syllogistics, the first developed by Peterson (1979, 1985, 1991, 2000), the second by Murphree (1991, 1993, 1997, 1998). The paper concludes with first steps towards a unification of both systems within the framework of linear al- gebra. 2. Classical Syllogisms In this section I present an introduction to classical syllogistics, a com- prehensive exposition of which can be found in Bird (1964). The classical syllogisms are argument forms that have two premises and one conclu- sion. A concrete instance of a syllogism is the following one: (1) All birds have wings. (2) All birds have feathers. (3) Therefore: At least one thing has feathers and wings. Sentences (1) and (2) are the two premises, and the “Therefore:” in (3) indicates the conclusion of the syllogism. In this concrete instance of a syllogism, “birds” is the middle term, M, “wings” is the predicate term, P, ________________ 2 Such quantifiers that lie “in-between” the existential and the universal quantifier are sometimes called “intermediate quantifiers” or “generalized quantifiers”. Historically, the first system was proposed by Sir William Hamilton, his dispute with De Morgan is reprinted in De Morgan 1847. Generalized quantifiers have been developed in mathe- matics and logics (Mostowski 1957, Lindström 1966, Väänänen 2004), artificial intel- ligence and computer science (Schwartz 1997, Liu & Kerre 1998, Novák 2001), and linguistics (Barwise & Cooper 1981, van Bentham & ter Meulen 1985, Gärdenfors 1987, van der Does & van Eijck 1991, Keenan & Westerståhl 1997). Peterson’s work is an improvement on Finch (1957) and belongs to the philosophy of language tradi- tion. Contemporary syllogistics: comparative and quantitative syllogisms 59 and “feathers” is the subject term, S. The position of the three terms—S, P, and M—is regimented by the four figures that are given in Table 1. Figure I Figure II Figure III Figure IV Premise 1: M—P P—M M—P P—M Premise 2: S—M S—M M—S M—S Conclusion: S—P S—P S—P S—P Table 1: The four figures indicating the positions of the subject term, S, predicate term, P, and middle term, M. The middle term, M, appears only in the premises, and the conclusion has to be in the subject-predicate order, S—P. Furthermore, each state- ment in a syllogism belongs to one of four types of propositions: (A) All S are P (universal affirmative; “affirmo”) (I) At least one S is P (particular affirmative; “affirmo”) (E) No S is P (universal negative; “nego”) (O) At least one S is not P (particular negative; “nego”) Both premises of the example above are of the type A and the conclu- sion is of type I, in short: AAI. These three letters indicate the “mood” of the syllogism. Furthermore, the above example is an instance of the AAI syllogism of Figure III. Mood and figure identify the syllogism. The total number of possible syllogisms is 256. There are 43=64 ways of constituting a two-premise argument (2 premises plus 1 conclusion over the 4 types of propositions, A, I, E, O). Multiplying the 64 moods by the four figures gives 64×4=256 possible syllogisms. Of the 256 possible syllogisms 24 are valid. A syllogism is valid if, and only if, it is impossi- ble that the conclusion is false while all premises are true. Equivalently, if all premises are true, then the conclusion must—necessarily—be true. In classical syllogistics, All S are P implies At least one S is P. This holds because of the implicit assumption that the subject term is not empty, i.e., there is at least one thing x such that x is S. This assumption is called “existential import”. In predicate logic, ∀x(Sx → Px) does not en- tail ∃x(Sx ∧ Px). The reason is well known: In predicate logic, formulae like ∀x(Sx → Px) can be “vacuously true”. This is the case when there is no x such that x has the property S. Then, clearly ∃x(Sx ∧ Px) is false (since ¬∃xSx is assumed). However, if the existential assumption is made explicit, ∃x(Sx ∧ Px) is a predicate-logically valid conclusion of ∀x(Sx → Px) ∧ ∃xSx. This is important for the formalization of the premises of 60 Niki Pfeifer syllogisms by means of predicate logical formulae: All S are P should be formalized predicate-logically as ∀x(Sx → Px) ∧ ∃xSx. Of the 24 valid syllogisms, 9 require the existential import assumption and 15 do not.3 The valid syllogisms have got names like “Barbara” for mnemotech- nic reasons.4 The vowels in these names indicate the types of propositions of the first and second premise, and the type of the conclusion of the re- spective syllogism (in the order just stated). “Darapti”, for example, de- notes the form of our introductory example (1)-(3). We note that this is a valid syllogism that requires an implicit existence assumption (there is at least one bird). A list of valid syllogisms with traditional names is given in Table 2. Explicit existence assumptions Implicit existence assumptions Figure I AAA Barbara AAI Barbari AII Darii EAO Celaront EAE Celarent EIO Ferio Figure II AEE Camestres AEO Camestrop AOO Baroco EAO Cesaro EAE Cesare EIO Festino Figure III AII Datisi AAI Darapti EIO Ferison EAO Felapton IAI Disamis OAO Bocardo Figure IV AEE Camenes AAI Bramantip EIO Fresison AEO Camenop IAI Dimaris EAO Fesapo Table 2: Valid traditional syllogisms and their mnemotechnic names (Hughes & Londey, 1965). The first two letters indicate the types of the premises, the third the type of the conclusion. ________________ 3 Notation: “∀” and “∃” are the universal and the existential quantifiers, respectively. “x” is the individual variable, bound by the quantifiers. “→”, “∧”, and “¬”, are read as “if ..., then …”, “and”, and “not”, respectively, and are defined as usual. 4 The first mnemotechnic verses of valid syllogisms appeared in William of Sher- wood's Introductiones Logicam (Summulae), 13th Century (cf. Kneale & Kneale 1984, p. 231-232). Contemporary syllogistics: comparative and quantitative syllogisms 61 3. Syllogisms with comparative quantifiers As stated in the introduction, the universal quantifier is too strict and the existential quantifier is too weak, hence hardly applicable on a priori grounds. I therefore propose comparative and quantitative quantifiers as a useful tool for rational argumentation theory. We denote by “comparative quantifiers” quantifiers that do not ex- plicitly involve numbers, e.g., “Many …”, “Most ...”, “Almost-all …”, etc. Statements of the form “Most S are P” inform us about the relations be- tween the cardinalities of the set of things that are S and that are P. A common view of “Most S are P”, which will be adopted in the present paper, is that there are more things that are both S and P than things that are both, S and non-P, formally: |{S ∧ P}| > |{S ∧ ¬P}| and |{S ∧ P}| > 0 , where “and |{S ∧ P}| > 0” makes the existential import explicit. 5 Here, only an implicit reference to the number of things is made. This inequality can actually hold generally and independently of how many things there are in the domain of reference. I.e., if the domain contains only 10 objects, the inequality holds when there are at least 6 S that are P. However, when there are 100 objects in the domain, at least 51 S that are P must exist to assert truly that Most S are P. In this sense, implicit refer- ence to the number of things is made, but not an explicit one. To formalize Most S are P in the language of predicate logic, one needs to know the cardinality of the set of things that are both S and P, and the cardinality of the set of things that are both S and non-P. Then predicate logic is able to express statements of the form “n S are P” (where n is an integer). The idea is, that “n S are P” is equivalent to the conjunction “at least n and at most n of the S are P”: to state that at least 51 and at most 51 S are P means just the same as to state that exactly 51 S are P. Generally, to speak about n objects, one needs n existential quanti- fiers and one universal quantifier. The greater n, the more complex the predicate logical formula grows. However, if we are ignorant about these cardinalities, then we cannot express comparative quantifiers in predicate logic. ________________ 5 Notation: “{A}” means the same as “set A”. “|{A}|” denotes the cardinality of {A}. “>” is the usual arithmetic “greater than” sign. By the way, “and |{S ∧ P}| > 0” is redundant since there cannot be negative quantities (frequencies are always positive or zero!). 62 Niki Pfeifer Quantifiers that contain explicit references to numbers as in “At least n/m …”, or “All, except exactly 12 …” will be called “quantitative quan- tifiers”. We use “comparative syllogism” as shorthand for “a syllogism that contains a comparative quantifier”. The same holds for quantitative syllogisms. Syllogisms that contain both a comparative and a quantitative quantifier will be called “quantitative syllogisms”. We will come back to quantitative syllogisms in the next section. In this section we are con- cerned with comparative quantifiers, i.e. quantifiers that do not explicitly refer to numbers. Figure 1: Venn diagram representing the subject, S, predicate, P, and middle term, M. The lower-case letters, a, …, h, denote the numbers of things that are in the re- spective region. b + e, e.g., denotes the number of things that are both S and P. Peterson (1979, 1985, 2000) provides algorithms to evaluate syllo- gisms with comparative quantifiers. These algorithms are correct and complete with respect to comparative and quantitative syllogisms. For his generalization of traditional syllogistics it is useful to consider Venn dia- grams. Venn diagrams represent graphically the relations among terms. A general form of a Venn diagram that represents the three syllogistic terms is reproduced in Figure 1. Every circle represents a term, and the lower case letters denote the number of things that are in this region. “a”, e.g., denotes the number of things that are S and not P and not M. “h” denotes the number of things that are neither S, nor P, nor M. Moreover, the statement, e.g., that there are exactly 10 things that are S is equivalent to the following equation: a + b + d + e = 10 . To state that there are exactly 5 things that are S and P is equivalent to: b+e=5. Contemporary syllogistics: comparative and quantitative syllogisms 63 To state that Most S are P is to say that there are more S that are P than S that are not P, in the Venn diagram: b + e > a + d , where “+”, “=”, and “>” are the usual arithmetic symbols. It is easy to see that Venn diagrams plus a little arithmetic are useful tools to handle the traditional as well as the comparative quantifiers. Table 3 lists the traditional types of propositions and the types with com- parative quantifiers in their affirmative and negative form. What “greatly exceeds” in the P, B, K, and G types means is context dependent. Its meaning can be fixed by numerical weights. Type Interpretation (A) All S are P a = 0 and d = 0, and (b ≠ 0 or e ≠ 0) (I) At least one S is P b ≠ 0 or e ≠ 0 (E) No S is P b = 0 and e = 0, and (a ≠ 0 or d ≠ 0) (O) At least one S is not P a ≠ 0 or d ≠ 0 (P) Almost-all S are P b + e » a + d, and (b ≠ 0 or e ≠ 0) (T) Most S are P b + e > a + d, and (b ≠ 0 or e ≠ 0) (K) Many S are P Not (a + d » b + e), and (b ≠ 0 or e ≠ 0) (B) Almost-all S are not P a + d » b + e, and (a ≠ 0 or d ≠ 0) (D) Most S are not P a + d > b + e, and (a ≠ 0 or d ≠ 0) (G) Many S are not P Not (b + e » a + d), and (a ≠ 0 or d ≠ 0) Table 3: Semantical interpretation of the four traditional types of propositions and their generalization by comparative quantifiers (Peterson, 1979, 2000). The “and…” clause makes the existential import explicit. “»” is read as “greatly ex- ceeds”. Usually, “or more” can be added to the P, T, B and D types (e.g., Most S are P is understood as Most, or more S are P). Let us now turn to the syllogisms with comparative quantifiers. Con- sider the following example: (1) Most birds have wings. (2) Most birds have feathers. (3) Therefore: At least one thing has feathers and wings. The general form of this syllogism is: (4) Most M are P. (5) Most M are S. (6) Therefore: At least one S is P. The premises (4) - (5) must satisfy the following inequalities: (7) e + f > d + g, and (e ≠ 0 or f ≠ 0) , and (8) d + e > g + f, and (d ≠ 0 or e ≠ 0) . 64 Niki Pfeifer Figure 2: Valid syllogisms with comparative quantifiers (Peterson, 1985, 2000, modi- fied). The classical syllogisms are in the boxes. Solid arrows indicate strengthening of the premises, dashed arrows indicate weakening of the conclusion. For further expla- nation see text. Contemporary syllogistics: comparative and quantitative syllogisms 65 e ≠ 0 follows algebraically from conditions (7) and (8), which implies (Peterson, 2000, p. 62): (9) b ≠ 0 or e ≠ 0 , and (9) means just At least one S is P, which is the conclusion (6). Figure 2 lists the affirmative valid syllogisms with the comparative quantifiers Almost-all S are P (P), Most S are P (T), and Many S are P (K) and their negative counterparts Almost-all S are not-P (B), Most S are not-P (D), and Many S are not-P (G), respectively. Syllogism ATK of Figure I, e.g., is, All M are P (major premise), Most S are M (minor premise), therefore Many S are P (conclusion). The validity of the syllo- gisms of figures I, II, and IV can be directly inspected, since the com- parative quantifiers strengthen the premises or weaken the conclusion (solid or dashed arrows, respectively). The valid syllogisms of Figure III in the shaded boxes are not derived trivially by strengthening or weak- ening. The above example (4)-(6) is an instance of the syllogism TTI of Figure III. It can be found in the middle of the left shaded box of. 4. Syllogisms with quantitative quantifiers As stated above, comparative quantifiers cannot be expressed in predicate logic if the total number of things in the domain is not known. Another restriction of predicate logic and hence of classical syllogistics is that it cannot handle quantitative quantifiers, like Half of the … are …. The fol- lowing subsection deals with fractionated quantifiers (e.g., m/n of the … are …; Peterson, 1991, 2000), as one type of quantitative quantifiers. Another type of quantifiers, the numerically exceptive quantifiers (Murphree, 1991, 1993, 1997, 1998) and their relation to syllogistics is presented in Section 4.2. 4.1. Fractionated quantifiers Fractionated quantifiers are important for the formalization and evaluation of arguments that involve fractions. As a simple example, consider the following quantitative syllogism with a fractionated quantifier in the second premise (Peterson, 2000): (1) All M are P. (2) 1/2, or more, S are M. (3) Therefore: Many S are P. 66 Niki Pfeifer The syllogism (1)-(3) is of the form AFK of Figure I, where “F” denotes the fractionated quantifier 1/2, or more …. The premises (1)-(2) must satisfy the following inequalities: (4) d = 0 and g = 0, and (e ≠ 0 or f ≠ 0) , and (5) d + e ≥ a + b, and (d ≠ 0 or e ≠ 0) . The proof by reductio ad absurdum runs as follows (we omit mentioning arithmetical and logical justifications): (6) ¬¬(a + d » b + e) (denial of the conclusion) (7) a ≥ b + e (from (4) and (6)) (8) a + b » b + e (from (4), (6), and (7)) (9) a + b » e (from (8)) (10) ¬(a + b > d + e) (from (5)) (11) ¬(a + b > e) (from (10)) (12) a + b > e (from (9)) Since (11) and (12) contradict each other, it follows that (13) ¬(a + d » b + e) , and (13) means just Many S are P, which is the conclusion (3). This com- pletes the proof. Peterson (1991, 2000) provides algorithms to evaluate syllogisms with comparative and fractionated quantifiers. These algorithms are cor- rect and complete with respect to arbitrarily many syllogisms. I.e., by the use of fractionated quantifiers we get as many steps between the universal and the existential quantifier as we like. Since n/m S are P means n/m, or more, S are P, Peterson (2000, p. 208) defines Exactly m/n of the S are P as follows: Exactly m/n of the S are P =df. (m/n of the S are P) ∧ ((n – m)/n S are not P) , i.e., m(b+e) = (n-m)(a+d) if, and only if [(m(b + e) ≥ (n – m)(a + d)) ∧ ((n – m)(a + d) ≥ m(b + e))] . Finally, we note that Peterson's logic of intermediate quantifiers can easily be related to a probability interpretation based on relative frequen- cies. 4.2. Numerically exceptive quantifiers In everyday life reasoning and rational discussions we are often con- fronted with law-like rules that hold normally but not strictly generally. That is, there are exceptions to the law-like rules. E.g., all birds can fly, except penguins, chickens, emus, etc. For the case where the number of exceptions is given or can at least be estimated, Murphree (1991, 1993, 1997, 1998) developed a formalism that generalizes traditional syllogis- tics in a way that is different from Peterson’s. Murphree’s approach is Contemporary syllogistics: comparative and quantitative syllogisms 67 called numerically exceptive, because it reasons explicitly with the num- bers of exceptions. As an example, consider the following (valid) numeri- cally exceptive syllogism: (1) At least all but 3 M are P. (2) At least all but 3 S are M. (3) Therefore: At least all but 6 S are P. According to Murphree, At least all but x S are P, where x is an integer ≥ 0 means that there are at most x exceptions to All S are P. In other words: All, except at most x, S are P. Consequently, the traditional syllo- gistic A-type, (A) All S are P, is read as (A, 0) At least all but 0 S are P. Murphree’s generalization of the traditional types involves the intro- duction of “… but 0 …”, and the qualification of the expression by “At least …”. Appending “… but 0 …” to “All” means that all S are P with no (=0) exceptions. In general, appending “… but x …” means that there are at most x exceptions to: All S are P. Equivalently, there are at most x S that are not P. The “but x” clause makes the number of the exceptions to the expression regimented by the preceding quantifier(s) explicit. More- over, prefixing “At least …” makes the ambiguous expression “… all but x S are P” precise. “… all but x S are P” is ambiguous, since it could mean (Murphree, 1993, p. 107): (4) At least all but x S are P, (5) At most all but x S are P, or (6) Exactly all but x S are P. (6) is, of course, the conjunction of (4) and (5). The four traditional types and their reading in the numerical exceptive logic and their generalizations are given in Table 4. We can now show the validity of syllogism (1)-(3), which is an instance of the numerically ex- ceptive Modus Barbara, (AAA of Figure I). Consider Figure 3, which intuitively shows why the following syllogism (7)-(8) is valid: (7) At least all but x M are P. (8) At least all but y S are M. (9) Therefore: At least all but x + y S are P. 68 Niki Pfeifer Traditional type Murphree’s generalization Algebraic interpretation (A) All S are P (A, 0) At least all but 0 S are P a+d=0 (A, x) At least all but x S are P a+d≤x (E) No S is P (E, 0) At most 0 S are P b+e=0 (E, x) At most x S are P b+e≤x (I) At least one S is P (I, 1) At least 1 S is P b+e≥1 (I, x) At least x S are P b+e≥x (O) At least one S is ¬P (O, 1) At least 1 S is ¬P a+d≥1 (O, x) At least x S is ¬P a+d≥x Table 4: The four syllogistic types of propositions in their traditional version and in terms of Murphree’s numerically exceptive logic. The respective generalizations are given in a separate line each. x is an integer ≥ 0. For the algebraic interpretation given in the right column consult Figure 1. Figure 3: Venn diagram (see legend of Figure 1) validating the numerically excep- tive Modus Barbara. The dashed lines indicate the regions where the exceptions could be. The x and the y value indicate the maximum numbers of exceptions. If all the y are in a and all the x are in d, then At least all S are P, except 6 (= x + y). If all the y are in b and all the x are in g, then At least all S are P, except 0. Hence, At least all S are P, except at most x+y, which is the conclusion (9). The algebraic version of the numerically exceptive Modus Barbara (7)-(9) is: (10) 0 ≤ d + g ≤ x (11) 0 ≤ a + b ≤ y (12) Therefore: 0 ≤ a + d ≤ x + y Other examples of valid numerically exceptive syllogisms can be found in Murphree (1991, 1993, 1997). For schemas to check systemati- Contemporary syllogistics: comparative and quantitative syllogisms 69 cally the validity of numerically exceptive syllogisms refer to Murphree (1991). I’m not aware of any approaches that try to combine the numerically exceptive and the comparative quantifiers.6 As an example of a possible combination consider the following generalization of the comparative syllogism TTI of Figure III of Section 3: (13) Most birds have wings and at least all but x birds have wings. (14) Most birds have feathers and at least all but y birds have feathers. (15) Therefore: More than x and more than y things have feathers and wings. As above, the x and the y denote the numbers of the exceptions. Here, x is the number of birds that don’t have wings, and y is the number of birds that don’t have feathers. Consequently, the algebraic interpretation of the premises (13) and (14) may be: (16) e + f > x ≥ d + g (Most M are P and at least all but x M are P.) (17) d + e > y ≥ f + g (Most M are S and at least all but x M are S.) “There are at most x exceptions involved in the claim that most M are P” is equivalent to (16). (16) and (17) imply that the intersection of S and P (i.e., b + e) must exceed the maximum of the numbers of the exceptions (b + e > max(x, y))7, which is the conclusion (15). The proof of the valid- ity of syllogism (13)-(14) runs as follows: (18) ¬(e > x) (indirect proof assumption) (19) ¬(e + f > x) (18) (20) e + f > x (16) (21) e > x (indirect proof (18)-(20)) (22) ¬(e > y) (indirect proof assumption) (23) ¬(d + e > d + y) (22) (24) ¬(d + e > y) (23) (25) d + e > y (17) (26) e > y (indirect proof (22)-(25)) (27) e > x and e > y (21) + (26) (28) e > max(x, y) (27) (28) implies the conclusion (15), namely, (29) b + e > max(x, y) . Here, the conclusion that “More than x and more than y things have feathers and wings” is of course more informative then the conclusion of the pure TTI of Figure III, that states that “There is at least one thing that has feathers and wings”. This is clear, because the premises of syllogism (13)-(14) are more informative than the premises of the pure TTI of Fig- ure III. ________________ 6 The unpublished work of Lorne Szabolski is an exception, as Wallace Murphree mentioned in an email to me. 7 “> max(x, y)” means “exceeds the maximum value of x and y”. 70 Niki Pfeifer 5. Concluding remarks After an introduction to traditional syllogistics I presented an algebraic interpretation of syllogisms with comparative and quantitative quantifiers. Section 4.2. concludes with an example of a syllogism that contains both comparative and numerically exceptive quantifiers. The semantics and the proofs of validity involved solving systems of inequalities. A general me- thod to solve problems with equalities and inequalities is linear program- ming (Danzig 1963, Gass 1985). Linear programming may be useful for the systematic investigation of the algebraic interpretation of syllogisms. Besides their application in argumentation theory, systems of com- parative and quantitative syllogistics are useful tools for the rational evaluation of human inferences in cognitive psychology (Pfeifer & Klei- ter 2005). 6. Acknowledgements The author would like to thank Georg J. W. Dorn, Reinhard Kleinknecht, Gernot D. Kleiter, and Wallace Murphree for helpful comments on earlier versions of the paper and for stimulating discussions. The research was partly financed by a grant of the Deanary of the Faculty of the Natural Sciences of the University of Salzburg. 7. References Barwise, J., & Cooper, R. (1981). Generalized quantifier and natural language. Lin- guistics and Philosophy, 4, 159-219. Bird, O. (1964). Syllogistic and its extensions. Englewood Cliffs, NJ: Prentice-Hall. Danzig, G. B. (1963). Linear programming and extensions. Princeton, NY: Princeton University Press. De Morgan, A. (1847). Formal logic: or, The calculus of inference, necessary and probable. 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