The effect of premise order in conditional reasoning: a test of the mental model theory

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1 Cognition 63 (1997) 1 28 The effect of premise order in conditional reasoning: a test of the mental model theory Vittorio Girotto *, Alberto Mazzocco, Alessandra Tasso a, b b a CREPCO CNRS and University of Provence 29 Av. R.Schuman, 13100, Aix-en-Provence, France b Dipartimento di Psicologia, University of Padua, Via Venezia 8, 35100, Padua, Italy Received 22 January 1996, final version 26 November 1996 Abstract The difference in difficulty between modus ponens (if p then q; p; therefore q) and modus tollens (if p then q; not-q; therefore not-p) arguments has been traditionally explained by assuming that the mind contains a rule for modus ponens, but not for modus tollens. According to the mental model theory, modus tollens is a more difficult deduction than modus ponens because people do not represent the case not-q in their initial model of the conditional. On the basis of this theory, we predicted that conditions in which reasoners are forced to represent the not-q case should improve correct performance on modus tollens. In particular, we predicted that the presentation of the minor premise (not-q) as the initial premise should produce facilitation. Experiment 1 showed that this is the case: whereas the inversion of the premise order did not affect modus ponens, it produced a significant increase of valid conclusions for modus tollens. Experiment 2 showed that this facilitation does not depend on the negative form (contrary vs. contradictory) of the minor premise. Experiments 3 and 4 (and/ or some of their replications) demonstrated that facilitation also occurs when participants are asked to find the cases compatible with not-q or to evaluate a p conclusion. No premise order effect was found for sentences which make explicit the not-q case right from the start, i.e. p only if q conditionals and biconditionals (Experiments 5 and 6). Finally, Experiments 7 and 8 showed that the conditional fallacies are not significantly affected by the premise order. 1. Introduction As shown in recent surveys (e.g. Evans, 1991; Evans et al., 1993), the psychology of deductive reasoning is a fragmented domain of research, in which * Corresponding author. leggiro@romarin.univ-aix.fr / 97/ $ Published by Elsevier Science B.V. All rights reserved PII S (96)

2 2 V. Girotto et al. / Cognition 63 (1997) 1 28 several different theories are available, but none of them are able to take into account all the main issues concerning human deduction (i.e. the logical competence of individuals untrained in logic, their systematic errors on some relatively simple problems and the effects of both the content of the premises and the context in which these problems are presented). One of the main reasons of this state of affairs is the difficulty of comparing the available theories. Some of the theories are actually devoted to the analysis of just few experimental paradigms. More generally, they tend to cover different reasoning domains. In the last few years, however, some promising development has occurred. As far as propositional (particularly conditional) reasoning is concerned, two different interpretations, deriving from two general purpose theories of reasoning, have been proposed. Both of them take into account the same range of phenomena, so that it seems possible to assess their relative merits, by means of comparatives studies. The traditional account of propositional reasoning is based on the notion of inferential rules. According to this view, people solve reasoning problems by applying content-free inferential rules on the interpreted premises (cf. Braine, 1978; Braine and O Brien, 1991; Rips, 1994; Smith et al., 1992). For example, given the conditional, major premise. If there is a circle on the left side of the card, then there is a square on the right side of the card, and the categorical, minor premise there is a circle on the left side of the card, most of the people who are asked to draw an inference (or to evaluate it), endorse the conclusion there is a square on the right side of the card, by applying a rule corresponding to modus ponens: if p then q, p therefore q. Accordingly, difficulties in solving propositional reasoning problems are attributed to the absence of a given inferential rule or to the length of the derivation that a rule requires. For instance, given the same conditional premise as in the previous example: If there is a circle on the left side of the card, then there is a square on the right side of the card, and the categorical premise there is not a square on the right side of the card, a considerably large number of individuals do not endorse the valid conclusion there is not a circle on the left side of the card, that should follow from the application of the modus tollens schema, i.e. if p then q not-q therefore not-p. The difficulty of modus tollens, relative to modus ponens (cf. Evans et al., 1993), has been attributed to different factors. Evans (1982) stressed the charge imposed by the negation (or inconsistency) introduced by the categorical premise, or by the backward direction (from the consequent to the antecedent) of the modus tollens inference. According to other authors, individuals have difficulties in solving modus tollens problems because they lack the sophisticated reductio ad absurdum argument: If p were true, then q would have to be true, but q is false, so p must be false. This argument is considered as a specific acquisition of the higher levels of scholarship (Braine and Rumain, 1983; Rumain et al., 1983). In general, there is a consensus on the absence of an inferential rule corresponding to modus tollens from the repertoire of inferential rules with which the mind is assumed to be equipped (cf. Braine and O Brien, 1991; Rips, 1983; O Brien et al., 1994; Smith et al., 1992).

3 V. Girotto et al. / Cognition 63 (1997) According to the mental model theory (Johnson-Laird, 1983; Johnson-Laird and Byrne, 1991), people do not reason by means of inferential rules. Reasoning is a three-step semantic process in which individuals: 1) build an initial and economic representation (model) based on the meaning of the premises (and general knowledge) and containing only true contingencies; 2) derive a putative conclusion from this representation; 3) try to build further models in which that conclusion could be false. If no models of such kind are found, the initial conclusion is endorsed. Consider, for instance, the above indicated pair of premises for modus ponens. According to the model theory (cf. Johnson-Laird et al., 1992), reasoners represent the state of affairs described in the conditional premise as a disjunction of two contingencies: [O] h??? where the first line indicates the explicit model representing a card containing a circle on the left and a square on the right. The square brackets indicate that the circle is completely exhausted with respect to the square, that is, in all possible models a circle will always occur with a square. Finally, the three dots indicate the implicit models, that is the further models compatible with the major premise (see below). When the categorical premise there is a circle on the left side of the card, is presented, the implicit models can be eliminated, and the conclusion is immediately derivable from the remaining explicit model: there is a square on the right side of the card. The model theory is nondeterministic, i.e. it assumes that different individuals may built different representations of a given statement. Accordingly, in a biconditional reading of the major premise, reasoners can flesh out the following models: O h O h where the symbol represents negation. In a conditional interpretation, the same premise is represented by three explicit models: O h O h O h In both cases, the categorical premise eliminates the explicit models containing the contingency there is not a circle on the left side of the card. Thus, the same valid conclusion can be drawn, i.e. there is a square on the right side of the card. Consider now the categorical premise for modus tollens (formally not-q): there is not a square on the right side of the card. Suppose that individuals represent only one explicit model of the conditional, that is they are focused on the model representing the square on the right side of the card (cf. Legrenzi et al., 1993). Given that the categorical premise eliminates

4 4 V. Girotto et al. / Cognition 63 (1997) 1 28 that model, nothing seems to follow (as many individuals say, cf. Evans, 1982). Unless the model representing the absence of the square on the right side of the card is fleshed out, reasoners have difficulties in deriving the valid conclusion there is not a circle on the left side of the card. In other words, whereas modus ponens can be drawn with a focus on the initial models of the conditional, modus tollens requires that reasoners cease focusing on the initial models, fleshing out the alternative ones (Legrenzi et al., 1993). The model theory does not specify the factors (context and content of the premises, cf. Johnson-Laird and Byrne, 1991) that produce the fleshing out of the implicit models. It assumes only that reasoners tend to represent the meaning of the premises in the most economic way, and that the following process of representing the other models can overload working memory, so that the inferential activities will be negatively affected. In sum, according to the model theory, modus tollens difficulties are due to people s tendency to neglect the not-q contingency in their initial model of the conditional premise. If this interpretation is correct, then any manipulation that forces reasoners to represent that contingency right from the start should facilitate modus tollens performance. If so, a specific manipulation which could elicit an initial representation of the not-q case, should concern the order of presentation of premises. When the categorical premise (e.g. there is not a square on the right ) is presented in the traditional order (i.e. as second premise of the conditional syllogism), reasoners have already received the conditional premise (e.g. If there is a circle on the left, then there is a square on the right ), that is, their working memory is pre-occupied with the models of the conditional. Hence, it is difficult for them to flesh out all the models of the conditional, and they are likely merely to eliminate the explicit model of the conditional ( circle and square ). Therefore, the model representing the categorical premise ( no square on the right ) will replace the implicit one, so that it seems that no conclusion follows. By contrast, when reasoners have to treat the categorical premise before the conditional one, they can easily represent the negated consequent ( There is not a square on the right ) from the start. When they then begin to consider the conditional ( If there is a circle on the left, then there is a square on the right ), they can eliminate the model representing the antecedent and the consequent ( a circle and a square ), and thus free up the working memory capacity. It is now easier for them to represent the model containing the negated antecedent and the negated consequent ( no circle and no square ), which yields the valid conclusion ( There is not a circle on the left ). In other words, the model theory predicts that the inversion of the premise order should improve modus tollens performance. Following a revised version of the model theory proposed by Evans (1993), it is possible to make the same prediction. According to the heuristic analytic theory (Evans, 1989; Evans and Over, 1996), preconscious heuristics (related to the linguistic form of the premises or to their content) determine reasoners selective attention to the relevant features of the premises. Deductive reasoning is then achieved by the application of

5 V. Girotto et al. / Cognition 63 (1997) analytic processes. Evans (1993) considered the selective representation of features as equivalent to the notion of focusing on initial models (as defined by model theorists), and the process of analytic reasoning as equivalent to the manipulation of models. Consequently, possible effects of premise order presentation can be interpreted as the results of relevance judgements (Evans, personal communication). When the conditional premise of a modus tollens argument is presented first, the focus of attention is on the conditional and it is in this context that the relevance of the categorical premise is judged. As not-q premise is negative and does not confirm the antecedent, it has a low chance of being relevant (i.e. added to the initial model of the conditional). By contrast, the initial presentation of the categorical premise cues its relevance. Despite being presented second, the conditional rule is relevant, because reasoners have to say whether anything follows and can only do this with reference to a rule. Therefore, in this condition reasoners will tend to conclude non-p. In sum, two different versions of the model theory predict the same premise order effect in modus tollens performance. For this reason, in the following pages, reference will be made to the model theory, considering the differences between the two versions not pertinent to the present purpose. On the other hand, if the source of modus tollens difficulties depends on the unavailability of the reductio ad absurdum line of reasoning, as inferential rules supporters claim, no premise order effect should be produced. For, it is unlikely that switching round the order of the premises per se activates a reductio ad absurdum. Again, if the critical factor is the presence of negation (or inconsistency), or the logical form of the problem, the order manipulation should not affect performance, given that the order in which premises are treated does not affect these features of the problem. In the present paper, we report the results of a series of experiments concerning the possible effect of premises order on conditional reasoning. In order to avoid the residual effects that could occur by using within participants designs, the reported experiments were conducted in sequence. The samples were drawn from the same population of participants. This procedure permitted us to make comparisons across experiments. 2. Experiment 1 (draw a conclusion) The aim of this experiment was to investigate the effect of the inversion of the premises order on modus tollens and modus ponens performances. The conflicting predictions deriving from mental model and inference rule theories were tested in a simple experimental manipulation. The premises of each argument were presented either in their traditional order (TO) with the conditional premise first and the categorical premise second, or else in the opposite, inverted order (IO) with the categorical premise first.

6 6 V. Girotto et al. / Cognition 63 (1997) Method Participants We tested ninety-two 18 to 19 year-old students of a high school (in which the major courses concerned scientific matters) of a North Italian town Material Participants were randomly assigned to one of four conditions. There were 17 participants in the modus ponens traditional order condition (TO), 18 participants in the modus ponens inverted order condition (IO), 28 participants in the modus tollens TO condition, and 29 participants in the modus tollens IO condition. Participants were run in two large groups for each condition. The experimental materials had a neutral content designed to insulate them from participants knowledge, and were given to the participants in a five-page booklet. General task instructions were given on the first page. Participants were told to read all the pages carefully and to take whatever time they required in order to give their answer to the final question (which was defined as a reasoning problem, not a test of intelligence ). The second page described a pack of cards, each carrying two geometrical shapes: one at the top, and another at the bottom. As examples, a triangle, a square, a circle, a diamond, a trapezium and a hexagon were presented. An example of a card (with a trapezium at the top and a diamond at the bottom) was also given. The following pages presented the problem. The modus tollens version of the problem in the TO condition was as follows (in a translation from the original Italian):[3rd page] Alberto put some of the cards of the initial pack in a box, on the basis of the following rule: If there is a circle at the top of the card, then there is a square at the bottom.[4th page] Vittorio, who does not know what Alberto has done, has taken one of the cards from the box, but he can see only the bottom part, where there is a triangle.[5th page] Is it possible to draw a conclusion about the upper part (the concealed part) of the card taken from the box by Vittorio? If so, what is the conclusion? In the IO version, the problem was the same except that the information about Vittorio s card came first:[3rd page] Vittorio has taken one of the cards from the box, but he can see only the bottom part, where there is a triangle.[4th page] Vittorio does not know that Alberto put some of the cards of the initial pack in the box, on the basis of the following rule: If there is a circle at the top of the card, then there is a square at the bottom.[5th page] Is it possible to draw a conclusion about the upper part (the concealed part) of the card taken from the box by Vittorio? If so, what is the conclusion? The two conditions presenting the modus ponens problem were similar. In the

7 V. Girotto et al. / Cognition 63 (1997) TO version, the conditional was stated before the description of Vittorio s card (of which he can see only the top part, presenting a circle), while in the IO version, it was given after the description of Vittorio s card Results and discussion The results are summarised in Table 1. These results clearly support the predictions deriving from the model theory. While only 39% of the participants draw the valid conclusion in the TO version of the modus tollens problem, a significantly higher percentage of participants (69%) produced the correct solution in the IO version of the same problem (z52.1, p,0.025; rank-sum analyses for frequency tables, specific test, cf. Meddis, 1984). By contrast, no effect of premise order was found in the modus ponens problem (88% correct performance in the TO condition vs. 89% in the IO condition). As Table 1 shows, in the TO condition of modus tollens the nothing follows conclusion (produced by 11% of the participants) accounted for only 18% of errors, while, according to the model theory, it should be produced by the majority of participants who fail to solve modus tollens. A similar pattern was also found in the IO condition, where the nothing follows conclusion accounted for 22% of the errors. In both conditions, the remainder accounted for some idiosyncratic errors, and by a tendency to give a conjunctive interpretation of the conditional (see below). Most of incorrect answers tend to be based on the examples given in the second page of the booklet. For example, some participants concluded that: the top part of the card presenting a triangle at the bottom must contain a hexagon, given that the other cards present a circle with a square [given the rule], and a trapezium with a diamond [given the example]. In other words, some participants considered the features of the examples as relevant for the process of deriving a conclusion. It seems likely that this kind of error could be reduced in some less realistic versions of the problem. However, some of these errors reveal a conjunction-like interpretation, which does not seem to depend on the specific material used. Following this interpretation, participants answered as if the box contained only cards showing a circle and a square ( p and q cards). As a consequence, a card presenting a triangle at the bottom (not-q) could be in the box only by mistake. Therefore, no valid conclusion about its top part could be drawn. The model theory predicts that reasoners at the most rudimental level of performance, treat conditionals as conjunctions: they simply ignore or forget the implicit models 1 concerning not-p (Johnson-Laird et al., 1994). Contrary to Evans (1993) claim (see also Bonatti, 1994), there is some evidence that, as well as children (cf. O Brien, 1987; Taplin et al., 1974), in some cases adults do interpret conditionals as conjunction-like. Johnson-Laird and Barres (1994) found that adults, required to build the cases in which an assertion would be true, gave the same answers for 1 A similar interpretation of conditional statements comprehension processes has been proposed by Sperber et al. (1995).

8 Table 1 Frequencies and percentages () of the responses produced to the different conditions used in the eight experiments. Experiment/ Order Response Correct Nothing follows Fallacy Based on ex. p and q Other n 1. Modus Ponens Traditional 15(88) 2(12) 17 Inverted 16(89) 2(11) 18 Modus Tollens Traditional 11(39) 3(11) 7(25) 4(14) 3(11) 28 Inverted 20(69) 2(6) 4(14) 2(7) 1(3) Modus Tollens (Explicit negation) Traditional 11(36) 8(27) 9(30) 2(7) 30 Inverted 16(57) 6(21) 1(3) 4(14) 1(3) 28 Replication (simple context) Modus Tollens Traditional 7(33) 7(33) 2(10) 1(5) 4(19) 21 Inverted 16(67) 7(29) 1(4) Modus Tollens (Implicit negation of p) Traditional 11(58) 4(20) 3(16) 1(5) 19 Inverted 19(66) 2(7) 8(27) 29 Replication 1 Modus Tollens Traditional 1(14) 2(29) 1(14) 3(43) 7 Inverted 7(87) 1(13) 8 Replication 2 Modus Tollens Traditional 3(27) 4(37) 1(9) 3(27) 11 Inverted 7(64) 2(18) 1(9) 1(9) Modus Tollens (Evaluation) Traditional 2(17) 5(42) 4(33) 1(8) 12 Inverted 7(58) 3(25) 1(8) 1(8) 12 8 V. Girotto et al. / Cognition 63 (1997) 1 28

9 Replication 1 Modus Tollens Traditional 9(50) 6(34) 1(5) 2(11) 18 Inverted 11(61) 4(22) 3(17) 18 Replication 2 Modus Tollens Traditional 6(50) 4(33) 2(17) 12 Inverted 12(100) 12 Piloting Traditional 8(43) 4(21) 7(37) 19 Inverted 17(85) 3(15) Modus Tollens (Only if ) Traditional 16(80) 3(15) 1(5) 20 Inverted 14(70) 2(10) 3(15) 1(5) Modus Tollens (Biconditional) Traditional 19(95) 1(5) 20 Inverted 18(95) 1(5) Fallacies Denial of the Antecedent Traditional 11(50) 5(23) 2(9) 4(18) 22 Inverted 13(59) 6(27) 2(9) 1(5) 22 Affirmation of the Consequent Traditional 13(54) 8(34) 1(4) 2(8) 24 Inverted 11(52) 10(48) Denial of the Antecedent (Explicit negation) Traditional 9(50) 5(28) 3(17) 1(5) 18 Inverted 14(70) 6(30) 20 Note. Based on ex(amples) : response in which subjects derive a conclusion on the basis of the given examples of cards; p and q : response based on a conjunctive interpretation of the conditional. V. Girotto et al. / Cognition 63 (1997)

10 10 V. Girotto et al. / Cognition 63 (1997) 1 28 compound assertions containing conditionals ( If A then 2 or if B then 3 ) and conjunctions ( A and 2 or B and 3 ). Moreover, in truth table construction tasks (for a review, see Evans et al., 1993), adults appear to consider the combinations in which the antecedent is false as counterexamples to a conditional rule, in addition to the combination true antecedent false consequent. In sum, although the syntactic and the semantic features of the modus tollens problem were not manipulated, the inversion of the premise order produced a significant improvement of performance. The model theory can explain this finding: the inversion of premise order, modifying the order of construction of the corresponding models, forces the participants to represent the crucial case not-q right from the start. The available alternative interpretations of the failures to solve modus tollens cannot easily account for the present results. 3. Experiment 2 (explicit negation) In the previous experiment, the not-q premise contained an implicit negation: on the bottom of the card, there is a triangle, where triangle is inconsistent with the consequent of the conditional rule (..., then in the bottom there is a square ). In the present experiment, the effect of premise order was tested in a modus tollens problem containing a minor premise with an explicit negation (i.e. there is not a square ). According to the model theory, the use of a contradiction should not induce an explicit representation of the case not-q from the start. Therefore, the positive effect of inverting the premise order should be obtained also with this negative form Method Participants Fifty-eight students from a high school of the same North-Italian town as in Experiment 1 were randomly assigned to one of two groups: TO (n530) and IO (n528) modus tollens conditions. The procedure ( draw a conclusion ) was the same as in Experiment Material The problems were the same as those in Experiment 1, except that the minor premise was explicitly negative: at the bottom part of the card taken by Vittorio, there is not a square Results and discussion The results (see Table 1) confirmed those obtained in Experiment 1. The percentage of correct solutions for modus tollens produced in the IO version (57%) was significantly higher than that elicited in the TO version (36%; z51.81, p,0.05). Analysis of errors revealed that in the TO version, the conjunctive

11 V. Girotto et al. / Cognition 63 (1997) interpretation of the conditional was the most frequent category of erroneous answers (30% of the entire sample). The nothing follows answers accounted for the 27%. Thus, the positive effect of inverting the order of the premises can be produced also with problems containing minor premises that are explicitly negative. 4. Replication of Experiment 2 (simple context) The context of the problems used in the previous experiment was not completely abstract. Reference was made to two individuals having different beliefs about cards and rules. This reference could somehow affect reasoning performance. Therefore, it was necessary to confirm the findings obtained using other contexts. For this reason, we conducted a replication study presenting a modus tollens problem (both in a TO and an IO version) in a neutral context, that is with no reference to the two individuals (Alberto and Vittorio) mentioned in the previous experiment. The problems were modelled on those used in Experiment 2 (with the draw a conclusion procedure and the explicit form of negation) Method Participants Forty-five students from the same high school as in Experiment 2 were randomly assigned to one of two groups: TO (n521) and IO (n524) modus tollens conditions. The procedure was the same as in Experiment Material The problems were the same as those in Experiment 2, except that sentences about Alberto and Vittorio were removed from the text. In the TO condition, the modus tollens problem was as follows:[3rd page] A box has been filled up with some of the cards of the initial pack, on the basis of the following rule: If there is a circle at the top of the card, then there is a square at the bottom.[4th page] Imagine you take one of the cards from the box, but you can see only the bottom part, where there is not a square.[5th page] Is it possible to draw a conclusion about the upper part (the concealed part) of the card taken from the box? If so, what is the conclusion? In the IO condition, the modus tollens problem reads as follows:[3rd page] Imagine finding a box containing this kind of card and taking one of them, but you can see only the bottom part, where there is not a square.[4th page] The box has been filled up with some of the cards of the initial pack, on the basis of the following rule:

12 12 V. Girotto et al. / Cognition 63 (1997) 1 28 If there is a circle at the top of the card, then there is a square at the bottom.[5th page] Is it possible to draw a conclusion about the upper part (the concealed part) of the card taken from the box? If so, what is the conclusion? 4.2. Results and discussion The results (see Table 1) confirmed those obtained in Experiment 2. The percentage of correct solutions for modus tollens produced in the IO version (67%) was significantly higher than that elicited in the TO version (33%; z52.5, p,0.01). Thus, the good performance observed in the IO version in Experiment 2 (and, by extension, also in Experiment 1) was the result of the inversion of the premise order, independently of the context. Analysis of erroneous answers revealed that in the TO version the nothing follows conclusion accounted for 33% of all responses (50% of errors), while only 10% of the participants based their inferences on the initial examples. This pattern of errors corroborates the model theory interpretation of modus tollens. We have no confident explanation for the different patterns of errors obtained in the different experiments reported. It might be argued that the less concrete context used in the replication study could have inhibited some of the participants from considering the initial examples as relevant features of the problem. However, it should be noted that a similar high percentage of nothing follows conclusions was found in Experiment 2, where the context was not completely abstract. 5. Experiment 3 (implicit negation of p) In order to have a further test of the order effect, we used it in a new task. Instead of drawing a conclusion about the hidden part of the card having a triangle at the bottom (not-q), participants had to indicate all the shapes that could be found at the top of that card, given the usual conditional ( If there is a circle at the top of the card, then there is a square at the bottom ). In other words, participants had to indicate the possible cases compatible with not-q. The correct answer was an exhaustive list of all the shapes other than the circle. As for the previous experiments, a positive effect of inverting the premise order was predicted Method Participants Forty-eight students from the same pool as in previous experiments were randomly assigned to one of two groups: TO and IO modus tollens conditions (Due to a distribution error, 29 participants were assigned to the IO group and only 19 to the TO one).

13 V. Girotto et al. / Cognition 63 (1997) Material The problems were the same as those in Experiment 1 (with an implicit negation in the minor premise There is a triangle at the bottom ), except that participants were asked to Indicate all the shapes that can be found at the top part of the card taken from the box by Vittorio (i.e. the card having the triangle at the bottom) Results and discussion Differently from previous experiments, no significant order effect was found (cf. Table 1). Although performance in the IO condition turned out to be better than that elicited in the TO condition (66% vs. 58% correct responses, respectively), this difference failed to reach a significant level. The error patterns were similar to those found in previous experiments. We have no confident explanation for this failure to replicate the order effect, apart from possible sampling errors. 6. First replication of Experiment 3 Given that we had no reasons for attributing the obtained results to the procedure used in Experiment 3, we conducted a replication of it with another sample of students (7 in the TO condition, 8 in the IO condition) from the same school. The results of the replication study are consistent with those obtained previously. Modus tollens performance was better in the IO than in the TO condition (87% vs. 14%, z52.74, p,0.01; a significant difference was also found by adding the results of the replication study with those obtained in Experiment 3; IO total correct performance: 70% vs. TO total correct performance: 46%, z51.92, p,0.05). Analysis of errors revealed the same patterns as in previous experiments. 7. Second replication of Experiment 3 A second replication was conducted, in which the problems were the same as those of Experiment 3, except that the card presented in the second page of the booklet did not depict real shapes. It contained two occurrences of the word shape, one at the top and one at the bottom. Twenty-two students (11 per condition: IO vs. TO) served as participants. Despite the slightly less concrete example used, the results were similar to those elicited in the first replication of Experiment 3. In the IO condition 64% of the participants solved the problem, whereas only 27% did the same in the TO condition (z51.69, p,0.05).

14 14 V. Girotto et al. / Cognition 63 (1997) Experiment 4 (evaluate p conclusion) The aim of this experiment was to test whether the positive effect produced by the inversion of the premises in the modus tollens performance could be obtained by using the traditional evaluation task. Evaluating a given conclusion may limit participants reasoning (cf. Mosconi, 1970). However, this task has been utilised in virtually all previous research on conditional reasoning (exceptions are Byrne, 1989; Evans et al., 1995/Exp. 2; Johnson-Laird et al., 1992). For this reason, we decided to investigate the order effect in modus tollens problems in which participants had to evaluate the validity of the p conclusion Method Participants Twenty-four students from the same pool as in previous experiments were randomly assigned to one of two equally-sized groups: TO and IO modus tollens condition (n512) Material The problems were the same as those in Experiment 1, except that the last page read as follows:[5th page] Is it possible to conclude that there is a circle on the upper part (the concealed part) of the card taken from the box by Vittorio? Choose one of the following answers: YES, NO, I CANNOT CONCLUDE Results and discussion The results (see Table 1) confirmed those obtained in Experiment 1. In the IO condition, 58% of the participants correctly evaluated that a p conclusion does not follow from the modus tollens argument, whereas only 17% of the participants in the TO condition gave the correct answer (z52.07, p,0.025). In the latter condition, the nothing follows conclusion accounted for 42% of all responses (50% of errors). The frequency of this answer (predicted by the model theory) was higher in this experiment than in the previous ones, possibly because it was one of the given options. 9. First replication of Experiment 4 The aim of this replication was to test the order effect in modus tollens evaluation problem with a not-q premise containing an explicit negation. Thirty-six students (eighteen in each condition TO vs. IO) had to solve the same problem as in Experiment 4, except that the not-q premise read as follows: At the bottom part of the card taken by Vittorio, there is not a square.

15 V. Girotto et al. / Cognition 63 (1997) The performance elicited in these two conditions did not turn out to be significantly different, although a tendency in the predicted direction was observed (61% correct performance in the IO condition vs. 50% in the TO condition). As in the previous experiment, nothing follows conclusions were the commonest erroneous responses in both conditions (57% of errors in the IO condition vs. 66% in the TO condition). 10. Second replication of Experiment 4 A second replication of Experiment 4 was conducted. Twenty-four students from another school in the same town as in the previous experiments served as participants. The results were consistent with those obtained in Experiment 4. Whereas only 50% of the participants in the TO condition correctly solved the problem, the entire group of participants who had to solve it in the IO condition produced the correct answer (z52.77, p,0.01). 11. Pilot studies for Experiment 4 Besides the second replication, two pilot studies elicited the same result as in the Experiment 4, i.e. a significantly better performance in the IO condition. The pilot studies differed from each other only for the card presented in the second page of the booklet. In the first study, the card-example presented a circle at the top and a square at the bottom (i.e. a p and q combination). In the second study, it was replaced with the example used in all the other experiments: a card with a trapezium at the top and a diamond at the bottom. In both studies, the not-q premise indicated that the box had no card containing a triangle (i.e. a q case). It read as follows (the sentence in the square brackets refers to the TO condition):[4th page in the TO condition, 3rd page in the IO condition] Vittorio [who did not know what Alberto had done] eliminated from the box all the cards containing a square at the bottom. Thus, on the remaining cards there are no squares Given that performance did not differ in the two pilot studies, a general analysis was carried out on the results collapsed across studies. These results confirm those obtained in Experiment 4. The large majority of participants (85%) in the IO condition (n520) correctly concluded that a card containing the p case (the circle) at the top could not be found in a box in which all the cards containing the q case (the square) had been eliminated. By contrast, only 43% (n519) made the same evaluation in the TO condition (z52.9, p,0.01). In the latter version, a high rate (37%) of unusual errors was observed, but the nothing follows conclusion accounted for 21% of all responses (36% of errors).

16 16 V. Girotto et al. / Cognition 63 (1997) 1 28 In what follows, we report the results of two experiments in which no premise order effect was predicted on modus tollens. 12. Experiment 5 (only if conditional) As indicated, according to the model theory, a conditional which induces the representation of the not-q case right from the start should improve modus tollens performance. The results of the reported experiments have shown that this is the case for if p then q conditionals presented as a second premise. However, some conditionals (e.g. only if conditionals and biconditionals) elicit good modus tollens performance, even when they are presented as first premise. For these conditionals, the model theory does not predict a premise order effect. From a formal point of view, p only if q and if p then q statements are equivalent. Both are false when p is true and q is false. However, from a psychological point of view they are not equivalent. For example, obligation rules (If condition P occurs, then one must undertake action Q) cannot be paraphrased by only if statements (cf. Cheng and Holyoak, 1985). In general, only if statements are more easily paraphrased as If not-q then not-p, than as If p then q (cf. McCawley, 1981). Compare, for instance, I ll give you the present only if you come home with If I give you the present, then you come home, and with If you don t come home, then I won t give you the present. These phenomena have been interpreted in different ways: p only if q statements can be considered as equivalent to p in no event other that one in which q (Braine, 1978; Geis, 1973; McCawley, 1981) or, at least some of them, as equivalent to p on the one condition: that q (Sanford, 1989). In any case, following both interpretations, p only if q and if p then q statements behave differently because the only if form stresses the necessity of q for p. Empirical evidence exists that reasoning performance differs in arguments containing the two forms of conditionals. In particular, modus tollens is easier when the conditional premise is expressed with an only if statement (about 60% of correct answers vs. 40% in the if p then q version, cf. Evans, 1977). According to the model theory (Johnson-Laird et al., 1992), only if statements yield a representation containing two explicit models right from the start. For example, the premise. There is a circle only if there is a square is represented with two models: [O] h O [ h]??? From this representation and the categorical premise: there is not a square, it is easy to deduce the valid conclusion of the modus tollens argument: there is not a circle. Following this interpretation (for a related explanation of the use of only as a quantifier, see Johnson-Laird and Byrne, 1989), no specific premise order effect

17 V. Girotto et al. / Cognition 63 (1997) should occur for modus tollens problems presenting an only if premise. If the initial representation of only if conditionals contains the explicit model corresponding to the not-q case, modus tollens problems should be easy to solve in both 2 conditions (TO and IO) Method Participants Forty students from the same pool as in previous experiments were randomly assigned to one of two equally-sized groups: TO and IO modus tollens conditions (20 per group) Material The problems were the same as those in Experiment 1, except that the conditional rule had the form: There is a circle at the top of the card only if there is a square at the bottom Results and discussion As in previous research, the modus tollens performance obtained in the TO only if condition was good (80% correct; see Table 1). But, there was no reliable difference between this condition and the IO condition (70% correct). Thus, as predicted by the model theory, the inversion of the premise order does not affect modus tollens performance for only if conditionals 13. Experiment 6 (biconditional) According to the model theory, a biconditional: If and only if there is a circle, then there is a square is represented with two explicit models: O h O h This representation allows modus tollens to be made without any further fleshing out. Indeed, Johnson-Laird et al. (1992) have actually demonstrated that modus tollens is easier with a biconditional than with a conditional premise. Therefore, no specific premise order effect was predicted for modus tollens problems with a biconditional. In both premise orders, the set of explicit models was predicted to be sufficient to make the valid conclusion. 2 Evans (1993) version of the model theory does not assume that the case not-p and not-q is modelled in the initial representation of only if conditionals. However, Evans et al. (1995) results corroborated the present interpretation (e.g. these Authors found a tendency to draw the fallacy indicating the modelling of not-p and not-q case DA, see below more often with only if than with if then affirmative conditionals, 76% vs. 65%, respectively).

18 18 V. Girotto et al. / Cognition 63 (1997) Method Participants Thirty-nine students from the same pool as in previous experiments were randomly assigned to one of two groups: TO (n520) and IO (n519) modus tollens conditions Material The problems were the same as those in Experiment 2, except that the main premise had a biconditional form: There is a circle at the top of the card if and only if there is a square at the bottom Results and discussion As predicted, modus tollens performance was good with both premise orders. In each condition, all but one participant solved the problem. Thus, as for only if conditionals, the inversion of the premise order does not affect modus tollens with a biconditional premises, given that this statement is represented with two explicit models, one of which corresponds to the not-q case. 14. Experiment 7 (the fallacies) In this experiment, we investigated the possible order effect in the two classical fallacies. Consider first the denial of the antecedent (DA). When presented with the conditional premise If there is a circle, then there is a square and with the categorical premise there is not a circle, many participants accept as a valid conclusion the negation of the consequent: there is not a square. According to the model theory, reasoners make the DA fallacy when they represent the conditional premise in a biconditional way, i.e. with two explicit models, O h O h??? In some context and given some content (e.g. conditional promises), a conditional evokes a biconditional representation (cf. Fillenbaum, 1975). However, even neutral conditionals may be interpreted as biconditionals: people are neither consistent with one another nor from one occasion to another (Johnson-Laird and Byrne, 1991, p. 46) Reasoners who build a complete representation of the conditional premise: O h O h O h

19 V. Girotto et al. / Cognition 63 (1997) cannot make the DA fallacy. Similarly, reasoners who do not flesh out the initial models: [O] h??? can derive the (apparently) correct nothing follows conclusion, just as they derive the same (incorrect) conclusion for modus tollens. As for the latter problem, the inversion of the premise order in a DA problem could force reasoners to flesh out models. However, the IO version could force reasoners to built both alternative models (conditional reading) or just one more model (biconditional reading). In the latter case, they will endorse the fallacy. Given that it is not possible to specify how many models will be fleshed out in the IO version, we do not predict any premise order effect on DA problems. Consider the other fallacy: affirmation of the consequent (AC). When presented with a conditional statement (if p then q) and the categorical premise q, people often endorse the erroneous conclusion p. For example, given the premises If there is a circle, then there is a square and there is a square many reasoners draw, or accept, the conclusion: there is a circle. According to the model theory (cf. also Evans, 1993), the AC fallacy should be more frequent than the DA fallacy. Whereas the latter requires some fleshing out, AC can be made even from the initial models of the conditional, in which the q contingency is always represented. Indeed, Evans et al. (1995) have found that AC inferences occur more often than DA inferences. In particular, with a draw inference paradigm, 88% of participants made the AC fallacy, whereas 65% of participants made the DA fallacy. Therefore, no order effect should be produced for the AC fallacy, given that the contingency ( q) which invites the erroneous conclusion ( p) is represented right from the start for any conditional statement. The following experiment was aimed to test these predictions. Given that the correct answer in both DA and AC problems is nothing follows, we used an evaluation task (i.e. reasoners had to evaluate a not-q conclusion for DA arguments, and a p conclusion for AC argument). A draw your own conclusion task could have produced false negative answers (i.e. some fallacies), given the reasoners tendency (defined as horror negativi by Mosconi, 1970) to avoid nothing follows conclusions, even for arguments in which this conclusion is the correct one Method Participants Eighty-nine students from the same pool as in previous experiments were randomly assigned to one of four conditions. There were 22 participants in the two DA conditions (TO and IO), 24 in the TO version and 21 in the IO version of AC problem.

20 20 V. Girotto et al. / Cognition 63 (1997) Material The problems were similar to those in previous experiments. In the TO version of the DA problem, the minor premise read as follows:[4th page] Vittorio, who doesn t know what Alberto has done, has taken one of the cards from the box, but he can see only the top part, where there is a triangle.[5th page] Is it possible to conclude that there is a square on the bottom part (the concealed part) of the card taken from the box by Vittorio? Choose one of the following answers: YES, NO, I CANNOT CONCLUDE. In the IO version, the problem was the same except that the information about Vittorio s card came first:[3rd page] Vittorio has taken one of the cards from the box, but he can see only the top part, where there is a triangle. In the two versions (TO and IO) of the AC problem, the minor premise referred to the bottom part of Vittorio s card, where he could see a square. The question concerned the top card of this card (i.e. the possibility to find or not a circle on it) Results and discussion As predicted, no order effect was elicited for both problems. In all conditions the majority of participants (from 50% to 59%) gave the correct answer. For the DA problems, the low rate of fallacy (23% and 27% in TO and IO, respectively) is not surprising given the predicted (relatively) high rate of correct performance. The IO condition produces a (non-significant) increase of the sum of fallacy and correct performance, when compared to the same sum in the TO condition (85% vs. 73%, respectively). AC problems elicited a low rate of fallacies. The (non-significant) increase of the fallacy rate in the IO condition (48% vs. 33% in TO), could be attributed to a backward line of reasoning (see below) elicited by this premise order condition. Finally, in the TO condition, there were less fallacies in AC problems than correct answers in modus ponens problems (cf. Expt. 1). This result is similar to what was reported in the literature (Evans, 1993). 15. Experiment 8 (DA fallacy with explicit negation) Differently from Experiment 7, in the final experiment the minor premise of DA problems contained a contradiction ( at the top of the card there is not a circle ), rather than a contrary negation ( at the top of the card there is a triangle ). According to the model theory, contradictions should not induce an explicit representation of the not-p contingency, nor should they induce this representation in the IO condition.

21 V. Girotto et al. / Cognition 63 (1997) Method Participants Thirty-eight students from the same pool as in previous experiments were randomly assigned to one of two conditions. There were 18 participants in the TO condition, and 20 participants in the IO condition Material The problems were the same as the DA ones in Experiment 7, except that the categorical premise contained an explicit negation: at the top of the card [...] there is not a circle Results and discussion As in the previous experiment, the majority of participants gave the correct answer (50% in the TO condition vs. 70% in the IO condition). Similarly, the rate of fallacious performance did not differ in the two conditions (30% in the IO condition vs. 28% in the TO condition). Therefore, the inversion of premise order does not affect DA performance whatever form of negation is used in the minor premise. However, in the IO condition all the elicited answers were either correct or fallacious. Although this pattern does not differ from that obtained in the TO condition (78% of correct and fallacious answers), an increase of these two kinds of answer in the IO condition was predicted by the model theory. 16. General discussion In the reported experiments, the manipulation of an apparently irrelevant aspect of conditional syllogisms (i.e. the order of the two premises) produced a significant effect in participants performance. In particular, an inversion of the premise order improved the performances in modus tollens problems, whereas that was not the case with modus ponens problems (Experiment 1). This main finding was replicated in the subsequent experiments: The facilitation obtained by the inversion of the premise order in modus tollens turned out independent of: a) the negative form (contrary vs. contradictory) used in the minor premise (Experiment 2); b) the context of the problem (replication of Experiment 2); c) the participants task: draw your own conclusion (Experiments 1 and 2), find the cases compatible with not-q, and evaluate a p conclusion (Experiments 3, 4 and their replications). However, the evaluation task offered only mixed evidence in favour of the hypothesis tested. Experiments 5 and 6 showed that the inversion of premise order does not affect performance in modus tollens problems, when the conditional premise is either an only if or a biconditional statement. Similarly, no order effect was found for the two classical fallacies: DA and AC (Experiments 7 and 8). These findings were predicted on the basis of the model theory of propositional

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