(PDF) Human reasoning with imprecise probabilities: Modus ponens and Denying the antecedent

5th International Symposium on Imprecise Probability: Theories and Applications, Prague, Czech Republic, 2007 Human reasoning with imprecise probabilities: Modus ponens and denying the antecedent Niki Pfeifer Gernot D. Kleiter Department of Psychology, Department of Psychology, University of Salzburg, University of Salzburg, Austria Austria

Abstract mp cmp da cda The modus ponens (A → B, A ∴ B) is, along with P1 : A→B A→B A→B A→B modus tollens and the two logically not valid coun- P2 : A A ¬A ¬A terparts denying the antecedent (A → B, ¬A ∴ Concl.: B ¬B ¬B B ¬B) and affirming the consequent, the argument L-valid: yes no no no form that was most often investigated in the psychol- Vi (C) t f ? ? ogy of human reasoning. The present contribution reports the results of three experiments on the proba- Table 1: Non-probabilistic version of the modus po- bilistic versions of modus ponens and denying the an- nens (mp), denying the antecedent (da), and their tecedent. In probability logic these arguments lead respective complementary versions (cmp, and cda). to conclusions with imprecise probabilities. A and B denote propositions. → and ¬ denote the material implication and negation, respectively, and In the modus ponens tasks the participants inferred are defined as usual. L-valid denotes logical validity, probabilities that agreed much better with the coher- and Vi denotes the logical valuation-function V of the ent normative values than in the denying the an- conclusion C under the interpretation i that assigns tecedent tasks, a result that mirrors results found t (“true”), to all premises (P1 and P2 ). If the an- with the classical argument versions. For modus po- tecedents, A, of the conditional premise is false, then nens a surprisingly high number of lower and up- the truth value of the conclusion is not determined per probabilities agreed perfectly with the conjugacy (denoted by the question mark). property (upper probabilities equal one complements of the lower probabilities). When the probabilities of the premises are imprecise the participants do not ig- nore irrelevant (“silent”) boundary probabilities. The of reference and used as a criterion for the rationality results show that human mental probability logic is of human inferences. See Table 1 for often investi- close to predictions derived from probability logic for gated argument forms in psychology. the most elementary argument form, but has consid- Traditional psychological research on human reason- erable difficulties with the more complex forms involv- ing designates human inference as rational/not ra- ing negations. tional if it corresponds to logically valid/not valid argument forms. Everyday life situations, though, Keywords. Mental probability logic, modus ponens, are inherently uncertain. The uncertainty cannot be coherence, imprecise probabilities captured by classical logic. Reasoning about uncer- tainty and uncertain knowledge is a fundamental hu- 1 Introduction man competence. Thus, classical logic cannot be an adequate normative standard of reference for the psy- chology of reasoning. While there is a long tradition of probabilistic ap- proaches in human judgment and decision mak- We proposed a psychological theory of human reason- ing [10], only recently probabilistic approaches are ing, called “mental probability logic” [20, 22, 21, 23, adopted in the psychology of reasoning [16, 17, 7, 24, 25], which evaluates the rationality of human rea- 19, 12, 14, 18, 27, 26, 29]. Traditionally, classical soning not by means of logical validity but by means logic dominated the psychology of human reasoning of coherence [9, 8, 4]. Mental probability logic is a [6, 28, 2]. Classical logic was the normative standard psychological competence theory about how humans interpret common sense conditionals, represent the memory nor be stored in the long term memory. premises of everyday life arguments and draw infer- Mental probability logic suggests that humans try ences by coherent manipulations of mental represen- to keep the memory load as small as possible and tations. process only relevant informations (see also [30]). Why a competence theory? Many investigations on • Conditional probability, P (B|A), is a primitive cognitive processes report errors, fallacies, or biases. notion. The probability values are assigned di- Well known are perceptual illusions, biases in judg- rectly. Conditional probability is not “defined”— ment under uncertainty, or errors in deductive rea- as in probability textbooks—via the fraction of soning. While these phenomena may be startling and the “joint”, P (A∧B), and the “marginal”, P (A), stimulating in the scientific process, they do not lead probabilities.2 Conditional probabilities are di- to theories that explain human performance in a sys- rectly encoded or just directly connected to the tematic way. Collecting slips of the tongue does not arguments of the if–then relation. lead to a theory of speaking. Psycholinguistics distin- guishes performance and competence. Competence • Because lack of knowledge (time, effort) it may be describes what functions a cognitive system can com- impossible for a person to assign precise probabil- pute. Human reasoning can solve complex problems ities to an event. If a person is uncertain about and perform sophisticated inferences. While develop- probabilities, then mental probability logic sup- ing a theory of reasoning one should have the explana- poses that human subjects make coherent impre- tion of these processes in mind. One should strive for cise probabilistic assessments (by interval-valued a competence theory. The distinction between com- probabilities or by second order probability dis- petence and performance was introduced by Noam tributions). Chomsky [3]. The analogy to deductive reasoning is • Coherence is in the tradition of subjective proba- obvious. The emphasis on the function a cognitive bility theory in which probabilities are conceived system should compute is due to David Marr [15]. as degrees of belief. Degrees of belief are naturally Mental probability logic claims that the common sense affine to psychology. conditionals are represented as subjective conditional probabilities. Based on the available information, the Imprecise versions of the argument forms presented in premises are evaluated and represented by coherent Table 1 are formalized by interval probabilities [23] or precise (point) probabilities, coherent imprecise prob- by second order probability distributions [25]. In the abilities, or logical information. Coherent imprecise present study we focus on interval probabilities only. probabilities can be represented by coherent interval We now present imprecise versions of the four argu- probabilities or second order probability distributions. ment forms of Table 1. While only the modus ponens Human reasoning is a mental process that forms new is logically valid, all four argument forms admit to representations from old ones by using probabilistic infer coherent probability intervals for the conclusion. versions of formal inference rules. We assume that The imprecise version of the modus ponens has the a certain core set of probabilistic inference rules are form: hard wired in the human inference engine. The normative standard of mental probability logic is P (B|A) ∈ [x′ , x′′ ] , P (A) ∈ [y ′ , y ′′ ] based on coherence. Coherence is the key concept in ∴ P (B) ∈ [x′ y ′ , 1 − y ′ + x′′ y ′ ] . (1) the tradition of subjective probability theory. It was originally developed by de Finetti [5]. More recent Since P (¬C) = 1−P (C) (conjugacy principle [31]) the work includes [31, 13, 4, 8]. A probability assessment complement of an interval [l, u] is [1 − u, 1 − l], it triv- is coherent1 if it does not admit one or more bets with ially follows that the imprecise complement modus sure loss. Coherence provides an adequate normative ponens has the form: foundation for the mental probability logic and has many psychologically plausible advantages compared P (B|A) ∈ [x′ , x′′ ] , P (A) ∈ [y ′ , y ′′ ] with classical concepts of probability: ∴ P (¬B) ∈ [y ′ − x′′ y ′ , 1 − x′ y ′ ] . (2) • In the framework of coherence a complete Boolean The imprecise denying the antecedent has the form: algebra is is not required for probabilistic infer- ence. Full algebras are psychologically unrealis- P (B|A) ∈ [x′ , x′′ ], P (¬A) ∈ [y ′ , y ′′ ] tic as they can neither be unfolded in working ∴ P (¬B) ∈ [(1−x′′ )(1−y ′′ ), 1−x′ (1−y ′′ )] (3) 1 Throughout we use “coherent” as synonymous with “to- 2 The definition P (B|A) = df. P (A ∧ B)/P (A) is problematic tally coherent” [9]. if P (A) = 0. The imprecise complement denying the antecedent tion is either a point percentage or a percentage be- has the form: tween two boundaries (from at least . . . to at most . . .). The booklet offered two response modalities where the P (B|A) ∈ [x′ , x′′ ], P (¬A) ∈ [y ′ , y ′′ ] participants had to choose one deliberately. ∴ P (B) ∈ [x′ (1 − y ′′ ), x′′ + y ′′ − x′′y ′′ ] (4) Response Modality 1: Equations (1)-(4) may be obtained by natural exten- If you think that the correct answer is a point per- sion [31], likewise, by de Finetti’s Fundamental The- centage, please fill in your answer here: orem [5] or by Lad’s generalized version [13]—or by Exactly . . . . . .% of the cars on this parking lot are elementary probability theory (for a demonstration two-door-cars. see [23]). Point percentage: In the psychological literature, the non-probabilistic versions of the modus ponens and the denying the an- |——————————————| tecedent were studied extensively. Meta-analytical 0 25 50 75 100 % results show that the modus ponens is endorsed by 89-100% of human subjects [6]. The denying the antecedent is endorsed by 17-73% of the subjects Response Modality 2: [6]. We do not judge the 17-73% of subjects as ir- If you think that the correct answer lies within two rational. Rather, we propose to reinterpret the data boundaries (from at least . . . to at most . . .), please in the light of mental probability logic. The question mark the two values here: is not whether the human subjects endorse the non- At least . . . . . .% and at most . . . . . .% of the cars probabilistic denying the antecedent, but whether on this parking lot are two-door-cars. they infer coherent probabilities from the premises of Within the bounds of: the imprecise denying the antecedent. In the next sections we present empirical data on the four impre- |——————————————| cise argument forms (1)-(4). 0 25 50 75 100 % 2 Experiment 1 The subsequent two tasks were formulated accord- ingly. In the second task the numerical values in two 2.1 Method and Procedure premises were 20 and 40%, and in the third task 60 and 90%, respectively. Each task was on a separate Thirty students of the University of Salzburg partici- page. After the third task the participants answered pated in Experiment 1. No students with special log- eight analogous tasks presented on one page in tab- ical or mathematical education were included. ular form. Again, the cover story was kept constant, Each participant received a booklet containing a gen- only the numerical values contained in the premises eral introduction, one example explaining the re- varied (see Table 2). sponse modality with point percentages, and one ex- The thirty participants were divided into two groups, ample explaining the response modality with interval fifteen participants received the modus ponens tasks, percentages. Three target tasks were presented on as just described, and fifteen participants received separate pages. Eight additional target tasks were denying the antecedent tasks. The denying the an- presented in tabular form. The first three modus po- tecedent tasks were formulated exactly as the modus nens target tasks had the following form: ponens tasks, with two differences. First, a negation Please imagine the following situation. Several cars was added in the second premise: “Exactly 90% of are parked on a parking lot. About these cars we the cars on this parking lot are not red cars”. Sec- know the following: ond, a negation was added to the question: “How many of these cars are not two-door-cars?”. Both re- Exactly 80% of the red cars on this parking sponse modalities were adopted accordingly. Adding lot are two-door-cars. the negations to the second premise and to the conclu- Exactly 90% of the cars on this parking lot sion as just described clearly reflects the form of the are red cars. corresponding imprecise denying the antecedent. In both conditions, we presented the same percentage Imagine all the cars that are on this parking lot. numbers to the participants. How many of these cars are two-door-cars? The booklets were mixed and assigned arbitrarily to Then, the participants were informed that the solu- P1 P2 clb cub lbr ubr P1 P2 clb cub lbr ubr 80 90 72 82 75.80 (5.16) 82.60 (8.33) 80 90 2 92 27.13 (33.58) 64.13 (37.94) 20 40 8 68 14.60 (11.15) 55.60 (30.24) 20 40 48 88 36.07 (27.71) 62.80 (31.97) 60 90 54 64 51.93 (10.13) 63.87 (14.21) 60 90 4 94 22.47 (22.17) 73.00 (29.30) 40 40 16 76 25.13 (12.65) 64.07 (35.43) 40 40 36 76 33.87 (20.87) 62.67 (23.25) 80 70 56 86 53.00 (21.91) 71.20 (27.26) 80 70 6 76 22.93 (23.20) 55.27 (34.74) 20 60 12 52 18.60 (13.37) 46.27 (23.59) 20 60 32 92 29.07 (17.90) 66.53 (24.65) 100 100 100 100 100.00 (0.00) 100.00 (0.00) 100 100 0.00 100 10.00 (28.03) 36.67 (48.06) 60 70 42 72 48.00 (15.08) 67.60 (22.04) 60 70 12 82 28.40 (26.26) 60.93 (30.39) 40 60 24 64 33.27 (13.88) 59.47 (23.55) 40 60 24 84 22.53 (17.01) 59.07 (23.71) 70 80 56 76 61.33 (13.80) 76.73 (12.40) 70 80 6 86 19.93 (23.91) 54.93 (31.25) 30 50 15 65 20.67 (12.80) 51.33 (30.32) 30 50 35 85 27.87 (20.76) 62.67 (25.13) Table 2: Mean lower (lbr) and mean upper bound Table 3: Mean lower lbr and mean upper bound re- responses (ubr) of the modus ponens tasks of Exper- sponses ubr of the denying the antecedent tasks of iment 1 (n1 = 15). The standard deviations are in Experiment 1 (n2 = 15). The standard deviations parenthesis. P1 and P2 denote the percentages in the are in parenthesis. P1 and P2 denote the percent- premises. clb and cub denote the normative/coherent ages in the premises. clb and cub denote the norma- lower and upper bounds, respectively. tive/coherent lower and upper bounds, respectively. the participants. All participants were tested individ- This may be a consequence of the explicit presentation ually in a quiet test room in the department. They of the point value response modality. The participants were told to take as much time as they wanted. In could actually have some imprecise value in mind, but case of questions, the they were asked to reread the nevertheless respond with a representative point value instructions carefully. just to reduce the complexity of the task. Such point value responses bias the mean lower and upper bound 2.2 Results and Discussion responses. To avoid constantly pointing explicitly to the possibility to give an interval value response we Table 2 lists the probabilities presented in the dropped the response point response modality in Ex- premises, the normative lower and upper bounds, and periment 2. Dropping the point response modality the participants’ mean lower and upper bound re- forces the participants to respond by intervals while sponses for the modus ponens tasks. In the task with still allowing point value responses by equating the certain premises (i.e., “100%” in both premises) all fif- lower and the upper bound responses. teen participants responded with point value of 100%, Table 4 reports the frequencies of interval response which is normatively correct. categories of the modus ponens condition in 3 × 3 Table 3 lists the respondents’ mean lower and up- tables. Each table contains the six possible inter- per bound responses for the denying the antecedent val responses together with the according empirical tasks. In the task with certain premises four of the frequencies of the interval responses. The columns fifteen participants responded with the unit interval designate whether the participants’ lower bounds are 0-100%. These four participants clearly understood below (LB), within (LW ), or above (LA) the norma- that the denying the antecedent is probabilistically tive intervals. The rows designate whether the upper not informative if all premises have probabilities equal bounds are above (U A), within (U W ) or below (U B) to 1. One participant responded with the point value the normative intervals. 100 and one with the point value 50%. The majority Table 5 reports the frequencies of interval response (nine out of the fifteen participants) responded with categories of the denying the antecedent condition the point value 0%. in 3×3 tables. Figure 1 presents the averaged interval In the modus ponens tasks on the average 30% of response frequencies in the modus ponens tasks. The the responses were point values (the task with the data of Task 2 and of Task 7 were not averaged. The certain premises not included). In the denying the normative lower bound of Task 2 is ≤ 10%. Both antecedent tasks on the average 23% of the responses normative bounds of task 7 are equal to 100%. Figure were point values (the task with the certain premises 1 shows that in the modus ponens tasks the majority was not averaged). of the participants gave coherent interval responses. In both conditions the standard deviations are high. Figure 2 presents the averaged interval response fre- quencies in the denying the antecedent tasks. The Schema Task 1 Task 2 data of Task 1, 3, 5, 6, 7, and 10 were not aver- UA a b c 0 4 1 0 4 0 aged. The normative upper bounds of Task 1, 3, UW d e - 0 10 - 0 11 - 6, and 7 are ≥ 90%. The normative lower bounds UB f - - 0 - - 0 - - of Task 2, 5, 7 and 10 are ≤ 10%. Figure 2 shows LB LW LA LB LW LA LB LW LA that in the denying the antecedent tasks the par- Task 3 Task 4 Task 5 ticipants responded with more incoherent intervals UA 0 5 0 0 5 0 0 2 0 than in the modus ponens tasks. More coherent in- UW 2 7 - 0 9 - 1 10 - terval responses were observed in the modus ponens UB 1 - - 1 - - 2 - - tasks (62.93% of the participants) compared with the LB LW LA LB LW LA LB LW LA denying the antecedent tasks (41.33% of the partic- Task 6 Task 7 Task 8 ipants). UA 0 4 1 - - - 0 4 1 In the modus ponens condition, the mean responses UW 1 9 - 0 15 - 0 9 - agree very well with the normative lower (rLBR,CLB = UB 0 - - 0 - - 1 - - .99) and upper (r(UBR,CUB) = .92) probabili- LB LW LA LB LW LA LB LW LA ties. The good agreement remains when the lower Task 9 Task 10 Task 11 (r(LBR,CLB).P 1 = .91) and upper (r(UBR,CUB).P 2 = UA 0 5 0 0 3 1 0 4 0 .95) percentages in the premises are partialled out. UW 0 10 - 1 10 - 0 11 - Partialling out the values contained in the tasks UB 0 - - 0 - - 0 - - reduces the possible influence of anchoring and/or LB LW LA LB LW LA LB LW LA matching effects. Table 4: Frequencies of the interval responses in the In the denying the antecedent condition we ob- modus ponens condition of Experiment 1 (n1 = 15). served a different pattern. While the mean re- U A: the participants’ upper bound response is above sponses still agree well with the normative lower the normative upper bound, U W : upper bound re- (r(LBR,CLB) = .76) probabilities, the correlation is sponse is within the normative interval, U B: upper slightly negative for the upper (r(UBR,CUB) = −.20) bound response is below the normative lower bound; probabilities. Partialling out Premise 1 and Premise LA, LW , and LB: same for the participants’ lower 2 reduces the correlations to r(LBR,CLB).P1 = .25 and bound responses. a: too wide interval responses, b: r(UBR,CUB).P2 = .03. The results may be explained lower bound responses coherent, c: both bound re- by assuming that the participants just respond with sponses above, d : upper bound responses coherent, e: values close to those contained in the description of both bound responses coherently within ±5% (bold), the tasks (known as “matching heuristic”). f : both bound responses below the normative lower It is well known that logical tasks involving nega- bounds. tions are difficult. In probabilistic inference tasks we consider the correlations between the probabil- ities of the premises and the normative lower and dition are very close to the normative values while in upper probabilities of the conclusions. For the set the denying the antecedent condition the responses of our modus ponens tasks the four correlations are might be explained by matching based guessing. all positive, r(P1 ,CLB) = .97 and r(P2 ,CLB) = .92 for the lower probabilities, and r(P1 ,CUB) = .86 and The presence of the negations in the denying the an- r(P2 ,CUB) = .51 for the upper ones. tecedent is a possible explanation, why there were less coherent interval responses compared with the For the denying the antecedent tasks (with the modus ponens tasks. It is easier to cognitively rep- identical numerical probabilities of the premises!) resent an affirmed than a negated proposition. An the lower bound correlations are highly negative, affirmed proposition can be visualized, for example, r(P1 ,CLB) = −.92 and r(P2 ,CLB) = −.93, and for more directly than a negated one. Classical modus the upper bounds positive, r(P1 ,CUB) = .24 and ponens was proposed as a basic and “hard wired” in- r(P2 ,CUB) = .62. The weighting and integration ference rule [28, 2]. Probabilistic modus ponens is a of affirmative and non-affirmative information makes similar candidate. tasks like the denying the antecedent especially dif- ficult. We note that linear regression predicts lower In addition to the modus ponens and the denying the probabilities better than upper ones. antecedent, the respective complementary versions are investigated in Experiment 2. By investigating The overall conclusion of Experiment 1 is that the re- the complementary versions as well, the presence of sponses of the participants in the modus ponens con- the negation is more balanced. 3 Experiment 2 Schema Task 1 Task 2 UA a b c - 2 0 0 1 0 3.1 Method and Procedure UW d e - - 13 - 3 6 - UB f - - - - - 5 - - Method and procedure of Experiment 2 are analog LB LW LA LB LW LA LB LW LA to Experiment 1. Sixty students of the University of Task 3 Task 4 Task 5 Salzburg participated in Experiment 2. No students UA 0 1 0 0 3 0 0 4 0 with special logical or mathematical education were UW 0 14 - 5 6 - 2 9 - included. Thirty participants were assigned to the UB 0 - - 1 - - 0 - - modus ponens condition and thirty participants were LB LW LA LB LW LA LB LW LA assigned to the denying the antecedent condition. Task 6 Task 7 Task 8 In the modus ponens condition, each participant UA 0 2 0 - - - 0 4 0 worked out three modus ponens tasks and three com- UW 6 6 - - 15 - 4 7 - plement modus ponens tasks. To counterbalance UB 1 - - - - - 8 - - position position effects, fifteen participants got the LB LW LA LB LW LA LB LW LA modus ponens tasks at the beginning, and fifteen par- Task 9 Task 10 Task 11 ticipants got the modus ponens tasks at the end of the UA 0 2 0 0 2 0 0 2 0 session. The modus ponens tasks had the following UW 5 7 - 3 10 - 6 5 - form: UB 1 - - 0 - - 2 - - Please imagine the following situation. Around LB LW LA LB LW LA LB LW LA Christmas time a certain ski-resort is very busy. This region is very popular among sportsmen, like Table 5: Frequencies of the interval responses in the skiers, snow-boarders, and sledge-rider. Every hour denying the antecedent condition (n2 = 15). For a cable-car brings the sportsmen to the top. About explanation of the schema see Table 4. this cable-car we know: Exactly 100% of the skiers wear red caps. Exactly 100% of the sportsmen are skiers. 0.00% 26.67% 3.73% Imagine all the sportsmen in this cable car. How many of these sportsmen wear a red cap? 3.73% 2.93% 62.93% Speaking about a closed room (cable-car) instead of an unspecified parking lot (Experiment 1) should help 0% 100% to represent and visualize the problems. As in Ex- coherent interval periment 1, participants were free to respond either in terms of point percentages or in terms of interval percentages. In Experiment 2, however, the response Figure 1: Averaged interval response frequencies over modality 1 (point response) was dropped. The partic- nine selected modus ponens tasks (see text, n1 = 15). ipants were informed by two examples at the begin- ning that point values can be given by equating the lower and the upper bounds. 0.00% All three modus ponens tasks had the same structure. 30.67% 16.00% The percentages of the two premises in the first task were 100 and 100%, in the second task were 70 and 12.00% 0.00% 90%, and in the third task the percentages were 70 41.33% and 50%, respectively. The three complement modus 0% 100% ponens tasks contained the same percentages and dif- coherent interval fered from the modus ponens task only in one respect: a negation was added to the conclusion (“How many of these sportsman do not wear a red cap?”). Figure 2: Averaged interval response frequencies over The denying the antecedent condition was ana- five selected denying the antecedent tasks (see text, logue. Fifteen participants first received the three n2 = 15). denying the antecedent tasks and then the three complementary versions of the denying the an- P1 P2 clb cub lbr ubr modus ponens), between 50 and 60% of the responses modus ponens were point value responses, which is about double 100 100 100 100 100.00 (.00) 100.00 (.00) compared with Experiment 1. 70 90 63 73 62.43 (11.77) 69.17 (9.71) This result is surprising, since dropping the explicit 70 50 35 85 42.5 (15.13) 54.83 (21.19) point value response modality should decrease and not complement modus ponens increase the number of point value responses. A pos- 100 100 .00 .00 .00 (.00) .00 (.00) sible explanation is that, as for all participants the 70 90 27 37 35.40 (16.73) 42.03 (17.45) first task contained certain premises and as all par- 70 50 15 65 41.00 (18.82) 53.67 (17.71) ticipants responded by point values in the first task, denying the antecedent they simply continued to give point value responses 100 100 .00 100 37.37 (47.53) 85.00 (35.11) later on. 70 20 20 44 18.63 (15.25) 41.63 (15.97) 70 50 15 65 25.4 (21.12) 59.23 (20.73) In the denying the antecedent tasks with certain compl. denying the antecedent premises (i.e., “100%” in both premises) fourteen 100 100 .00 100 0.83 (4.56) 53.33 (49.01) of the thirty participants inferred a unit interval, 70 20 56 80 51.9 (19.12) 75.87 (20.19) [≤ 1, 100]%. Four participants inferred a point value 70 50 35 85 32.70 (12.92) 65.17 (27.43) equal to zero, and ten inferred a point value equal to 100%. One participant inferred a point value of 50% and one an interval between 70 and 100%. In the ac- Table 6: Mean lower lbr and mean upper bound re- cording complement denying the antecedent tasks sponses ubr of the modus ponens condition (n1 = 30) fifteen of the thirty participants inferred a unit inter- and of the denying the antecedent condition (n2 = val, [≤ 1, 100]%. Thirteen responded a point value 30) of Experiment 2. The standard deviations are in equal to zero. One participant inferred a [25, 50]% in- parenthesis. P1 and P2 denote the percentages in the terval and one inferred a [0, 50]% interval. Practically premises. clb and cub denote the normative/coherent half of the participants understood that only a non- bounds. informative interval can be inferred if each premise is certain. tecedent tasks. The order was reversed for the other In the two denying the antecedent tasks with 70 fifteen participants. The premises and the conclu- and 20%, and 70 and 50%, in the premises, 0 and sions were adopted accordingly. The only difference 16.67% point value responses were observed, respec- in the percentages between the modus ponens condi- tively. This amount of point value responses is smaller tion and the denying the antecedent condition was, than in Experiment 1. In both according complement that “90%” was replaced by “20%”. The reason for denying the antecedent tasks 26.67% point value this was to avoid non-informative assessments. responses were observed, which is comparable to the results of Experiment 1. 3.2 Results and Discussion Table 7 reports the frequencies of interval response categories of the modus ponens condition and of the The results of t-tests indicate that there were no po- denying the antecedent condition in 3 × 3 tables. sition effects. We therefore pooled the data in the 80.22% of the participants inferred coherent inter- modus ponens condition (n1 = 30) and in the deny- vals in the modus ponens condition on the average ing the antecedent condition (n2 = 30). (the tasks with certain premises were not averaged). Table 6 lists the probabilities presented in the 55.00% of the participants inferred coherent intervals premises, the normative lower and upper bounds, and in the denying the antecedent condition on the av- the participants’ mean lower and upper bound re- erage (the tasks with certain premises were not aver- sponses for the modus ponens tasks and the comple- aged). ment modus ponens tasks of Experiment 2. In the modus ponens tasks 85.00% (Experiment 1: In the modus ponens tasks with certain premises 62.93%) of the interval responses were coherent on (“100%” in both premises) all thirty participants re- the average. In the denying the antecedent tasks sponded with that point value 100%, which is norma- 56.66% (Experiment 1: 41.33%) of the interval re- tively correct. In the according complement modus sponses were coherent on the average. The improved ponens tasks all thirty participants responded cor- cover-story explains why more coherent interval re- rectly with the point value 0.00%. Thus, in Task 1 sponses in Experiment 2 than in Experiment 1 were all participants inferred (correctly) point values. In observed. the other tasks (both modus ponens and complement modus ponens tasks reasonable that humans are better in argument forms Task 2 Task 3 Coh. Bounds that guarantee high probabilities of the conclusion if UA 0 4 1 0 0 0 Task 2: each premise is highly probable. If the premises of UW 2 22 - 1 29 63–73 the modus ponens are certain, then the conclusion is UB 1 - - 0 - - Task 3: certain. However, if the premises of the denying the LB LW LA LB LW LA 35–85 antecedent are certain, then the probability of the complement modus ponens tasks conclusion is in the unit interval [0, 1]. Task 2 Task 3 Coh. Bounds UA 0 0 6 0 2 1 Task 2: 4 Experiment 3 U W 5 19 - 0 27 27–37 UB 0 - - 0 - - Task 3: This section reports data of an experiment with im- LB LW LA LB LW LA 15–65 precise probabilities in the premises conducted by Flo- denying the antecedent tasks rian Bauerecker [1]. Specifically, we focus on human Task 2 Task 3 Coh. Bounds understanding of what we call “silent bounds”. We UA 3 4 0 0 5 0 Task 2: call a probability bound b of a premise silent if, and U W 6 16 - 7 18 20–44 only if, b is irrelevant for the probability propagation UB 1 - - 0 - - Task 3: from the premise(s) to the conclusion. E.g., in the LB LW LA LB LW LA 15–65 probabilistic modus ponens y ′′ is silent (y ′′ doesn’t complement denying the antecedent tasks occur in the lower or upper probabilities of the con- Task 2 Task 3 Coh. Bounds clusion, see (1)). Experiment 3 introduces an espe- UA 1 8 0 1 6 0 Task 2: cially critical test of the claim that human subjects U W 6 14 - 4 18 56–80 are capable to make coherent probabilistic inferences. UB 1 - - 1 - - Task 3: LB LW LA LB LW LA 35–85 Method and procedure of Experiment 3 are analog to Experiment 1. Eighty participants were recruited Table 7: Frequencies of the interval responses in the for investigating questions going beyond the scope of modus ponens (n1 = 30) and the denying the an- the present study. Therefore, we report selected data tecedent condition (n2 = 30) of Experiment 2. For on the imprecise modus ponens only (n = 40). The explanation see Table 4. modus ponens tasks were formulated as follows: Please imagine the following situation. Claudia works at blood donation services. She investigates All participants inferred a probability(interval) of a to which blood group the donated blood belongs and conclusion C, P (C) ∈ [zC′ , zC′′ ], and the probability of ′ ′′ whether the donated blood is Rhesus-positive. the negated conclusion, P (¬C) ∈ [z¬C , z¬C ]. To test the conjugacy principle of the interval responses, we Claudia is 60% certain: If the donated checked for each participant whether (i) zC′ + z¬C ′′ = blood belongs to the blood group 0, then ′ ′′ 100%, and whether (ii) z¬C + zC = 100%. the donated blood is Rhesus-positive. In the modus ponens tasks with certain premises, all Claudia knows as well that donated blood participants satisfied both equalities, (i) and (ii). In belongs with more than 75% the tasks with 70% and 90% in the premises sixteen certainty to the blood group 0. of the thirty participants satisfied both, (i) and (ii). How certain should Claudia be that a recent donated In the tasks with 70% and 50% in the premises fif- blood is Rhesus-positive? teen of the thirty participants satisfied both, (i) and (ii). It is surprising that in the modus ponens tasks Contrary to Experiment 1 and Experiment 2, the con- more than half of the participants gave intervals that ditional premise is here formulated in a if–then form. with lower/upper probabilities that exactly add up to The cover-story remained constant, only the numbers 1. In the denying the antecedent tasks with cer- in the premises varied. Table 8 lists the probabilities tain premises, twenty of the thirty participants satis- presented in the premises, the normative lower and fied both, (i) and (ii). In the tasks with 70 and 20% upper bounds, and the participants’ mean lower and premises nobody satisfied both, (i) and (ii), eleven upper bound responses for the modus ponens tasks satisfied (i), and one satisfied (ii). In the tasks with with and without silent bounds. 70 and 50% ten satisfied both, (i) and (ii). The participants inferred higher upper bounds in the In sum, more additive responses and more coherent modus ponens task containing silent bounds (M = interval responses were observed in modus ponens 71.79) compared with the according task not contain- tasks than in the denying the antecedent tasks. It is ing silent bounds (M = 60.20; t(39)=3.53, p=.001). P1 /P2 ci lbr ubr Task 1 Task 2 Coh. Bounds 60/[75,100∗] [45,70] 44.50 (21.57) 71.78 (20.07) (60/[75,100∗]) (60/75) 60/75 [45,70] 46.83 (23.76) 60.20 (16.86) UA 1 14 0 0 2 4 Task 1: [75,100]/60 [45,100] 43.42 (22.00) 72.38 (22.98) UW 8 16 - 7 25 45–70 75/60 [45,85] 46.27 (21.73) 59.90 (17.19) UB 1 - - 2 - - Task 2: LB LW LA LB LW LA 45–70 Table 8: Mean lower lbr and mean upper bound re- Task 3 Task 4 Coh. Bounds sponses ubr of the modus ponens tasks (n = 40) in [1]. ([75,100]/60) (75/60) ∗ denotes the silent bound. The standard deviations UA - - - 0 2 0 Task 3: are in parenthesis. P1 and P2 denote the percentages UW 7 32 - 8 29 45–100 in the premises. ci denotes the normative/coherent UB 1 - - 1 - - Task 4: interval. LB LW LA LB LW LA 45–85 Table 9: Frequencies of the interval responses in the modus ponens (n = 40) tasks of Experiment 3. Co- herent interval responses are bold (±5% tolerance in- terval). Further explanation in Table 4. Thus the participants were sensitive to the silent bounds. They did not understand the irrelevance of 5 Concluding Remarks the silent bound for the probability propagation from the premises to the conclusion. We reported three psychological experiments on the [21] report data on a conjunction problem where in probabilistic versions of two prominent argument one condition interval-values in the premises were pre- forms in the framework of mental probability logic. sented. All upper bounds were equal to 100%. In the Clearly more coherent responses were observed in other condition only corresponding point values were modus ponens than in denying the antecedent presented. The point values were equal to the lower tasks. Human subjects employ inference rules that bounds of the interval condition. Higher mean lower guarantee high probability conclusions if the premises bounds were observed in the interval condition than are highly probable. Practically no participant in- in the point condition. An explanation for this finding ferred “too wide” intervals such that the coherent in- is, that the participants reduced the processing load tervals are subintervals. While most participants did of the interval valued premises by representing only not understand the irrelevance of the silent bounds in the means of the lower and upper bounds. Then, of the modus ponens task of Experiment 3 they are not course, the coherent lower bound must be higher. completely “blind” for them. 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