Strategies during complex conditional inferences

DURING INFERENCES THINKINGSTRATEGIES AND REASONING, 2000,COMPLEX 6 (2), 125–160

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Strategies during complex conditional inferences Kristien Dieussaert, Walter Schaeken, Walter Schroyens, and Géry d’Ydewalle University of Leuven, Belgium

In certain contexts reasoners reject instances of the valid Modus Ponens and Modus Tollens inference form in conditional arguments. Byrne (1989) observed this suppression effect when a conditional premise is accompanied by a conditional containing an additional requirement. In an earlier study, Rumain, Connell, and Braine (1983) observed suppression of the invalid inferences “the denial of the antecedent” and “the affirmation of the consequent” when a conditional premise is accompanied by a conditional containing an alternative requirement. Here we present three experiments showing that the results of Byrne (1989) and Rumain et al. (1983) are influenced by the answer procedure. When reasoners have to evaluate answer alternatives that only deal with the inferences that can be made with respect to the first conditional, then suppression is observed (Experiment 1). However, when reasoners are also given answer alternatives about the second conditional (Experiment 2) no suppression is observed. Moreover, contrary to the hypothesis of Byrne (1989), at least some of the reasoners do not combine the information of the two conditionals and do not give a conclusion based on the combined premise. Instead, we hypothesise that some of the reasoners have reasoned in two stages. In the first stage, they form a putative conclusion on the basis of the first conditional and the categorical premise, and in the second stage, they amend the putative conclusion in the light of the information in the second premise. This hypothesis was confirmed in Experiment 3. Finally, the results are discussed with respect to the mental model theory and reasoning research in general.

People can easily make some inferences. The following inference, known as Modus Ponens, is an example of such an easy inference:

Correspondence should be addressed to Kristien Dieussaert, Laboratory of Experimental Psychology, Department of Psychology, University of Leuven, Tiensestraat 102, B/3000 Leuven, Belgium. Email: [email protected] This research was carried out with support from the IUAP/PAI P4 (Kristien Dieussaert and Géry d’Ydewalle) and from the Fund for Scientific Research Flanders (Walter Schaeken and Walter Schroyens). We wish to thank the referees, Rosemary Stevenson and Aidan Feeney, for their helpful comments. © 2000 Psychology Press Ltd http://www.tandf.co.uk/journals/pp/13546783.html

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If she has an essay to write, then she will study late in the library. She has an essay to write. Therefore, she will study late in the library.

Many psychological theories of reasoning postulate that people are equipped with formal rules of inference akin to those of a logical calculus (see e.g., Braine, 1978; Braine, Reiser, & Rumain, 1984; Johnson-Laird, 1975; Oshershon, 1975; Rips, 1983, 1994). The rule theories are syntactic theories: They claim that deductive reasoning consists of the application of inference rules to the form of the premises and conclusion of an argument. Consider again the problem just given. According to rule theories, our mind contains a rule for Modus Ponens: If p then q, p/q

This rule or reasoning schema matches the form of the problem. Therefore, the inference can be made promptly: She will study late in the library.

Consider, however, the following problem: If she has an essay to write, then she will study late in the library. She will not study late in the library.

The correct solution to this Modus Tollens problem is: She has not an essay to write.

According to rule theories, there is no rule in our mind that corresponds to the Modus Tollens problem. Therefore, it is only indirectly that we can come up with the correct solution. The latter problem requires more reasoning steps. This claim is used by the rule theories to explain the data of many studies which show that Modus Ponens inferences are easier than Modus Tollens inferences. Consider the following problem: If she has an essay to write, then she will study late in the library. She does not have an essay to write.

Many participants conclude based on these premises: She will not study late in the library.

This conclusion, which is called the fallacy of denying the antecedent, is wrong. If the conditional is interpreted as a true conditional, then it is possible that there

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are other alternatives that imply the consequent “she will study late in the library”. Of course, if the conditional is interpreted as a biconditional (if and only if), then the conclusion “she will not study late in the library” is correct. A rule theorist could propose that reasoners are also equipped with invalid inference rules (see e.g., Von Domarus, 1944). However, most of them explain reasoning errors as comprehension errors (see e.g., Braine & Rumain, 1983; Marcus & Rips, 1979; Markovits, 1985; Rumain et al., 1983). The major premise of the denial of the antecedent problem just described would invite its obverse (but for an alternative invited inference, see Evans, Clibbens, & Rood, 1996; Rips, 1994): If she does not have an essay to write, then she will not study late in the library.

Reasoners can come up with the conclusion “she will not study late in the library” if they apply the Modus Ponens rule to the obverse of the conditional and the categorical premise. Rumain et al. (1983) tested this hypothesis by giving participants a possible alternative conditional. Consider the problem again, but now accompanied with a so-called alternative conditional: If she has an essay to write, then she will study late in the library. If she has some textbooks to read, then she will study late in the library. She does not have an essay to write.

If reasoners are given such problems, they will not conclude: She will not study late in the library.

The fact that the fallacy of denying the antecedent (and the similar fallacy of affirming the consequent) can be suppressed is taken as evidence that we do not possess invalid inference rules, but that reasoners make denial of the antecedent conclusions because they translate the original premises. According to Byrne (1989), this argument has an interesting consequence: If valid inferences could be suppressed, then neither would reasoners possess valid inference rules. This would imply that people do not possess mental rules at all, which challenges the mental rule account. In the first experiment, she presented some participants with the major premise, accompanied by a so-called additional conditional . For example: If she has an essay to write, then she will study late in the library. If the library remains open, then she will study late in the library. She has an essay to write.

For such a problem, reasoners do not come up with the Modus Ponens conclusion:

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She will study late in the library.

In other words, Byrne showed that Modus Ponens can be suppressed. Similarly, Modus Tollens can be suppressed. She concluded that this suppression effect challenges the assumption of rule theories that formal rules of inference such as Modus Ponens are part of our mental logic. Byrne (1991) argues that an alternative theory, that is, the mental model theory (e.g., Johnson-Laird, 1983) can explain the suppression of the valid inferences. According to the model theory, reasoning consists of three main stages. First, the premises are understood: A mental model of the situation they describe is constructed on the basis of their meaning and of any relevant general knowledge triggered during the process of interpretation. Second, reasoners formulate a conclusion based on the model. People will only draw conclusions that convey some information that was not explicitly asserted by the premises. Third, a search is made for alternative models of the premises in which the putative conclusion is false. If there is no such model, then the conclusion is valid; that is, it must be true given that the premises are true. If there is such a model, then it is necessary to return to the second stage to determine whether there is any conclusion that holds over all the models so far constructed. The theory’s essential processing assumption is that the more models have to be constructed, the harder the inferential task will be. This is consistent with studies of syllogistic reasoning (Johnson-Laird & Bara, 1984), spatial and temporal reasoning (Byrne & Johnson-Laird, 1989; Schaeken, Johnson-Laird, & d’Ydewalle, 1996a,b; Vandierendonck & De Vooght, 1996), propositional reasoning (Johnson-Laird, Byrne, & Schaeken, 1992, 1994), and reasoning with multiple quantifiers (Johnson-Laird, Byrne, & Tabossi, 1991). Byrne (1991) argues that the model theory can explain the suppression. According to her, reasoners will integrate the two premises, dependent on the meaning of that premise and the general knowledge. If the antecedent of the second premise is understood as an additional requirement, then it will be conjoined with the antecedent of the first premise (Byrne, 1991): p …

q

r

Consequently, reasoners will suppress Modus Ponens conclusions: Affirmation of the first antecedent by itself is not sufficient for drawing the Modus Ponens conclusion. If the antecedent of the second premise is understood as an alternative requirement, then the premises will be interpreted such that they are individually sufficient: p …

q

r r


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As a result, reasoners will suppress the denial of the antecedent in problems with an alternative antecedent: Denial of the first antecedent only is not sufficient for deciding that “not-r” is the case, because the model indicates that “r” can still hold given “q”. Experiment 1 and Experiment 2 focus on the suppression effect as described by Byrne (1989). Experiment 1 is a replication of Byrne’s Experiment 1 (1989). In Experiment 2, we manipulate the kind of evaluation the participants had to make. Instead of the traditional three answer alternatives, we presented a set of more than ten answer alternatives to the participants. The results of Experiment 2 force us to change our view on the nature of suppression and especially on the nature of the underlying reasoning processes: They suggest that reasoners are using one of (at least) two processing strategies when they are solving these kinds of problems. Therefore, we will focus on the underlying reasoning processes of suppression in Experiment 3. In this experiment, we changed the original evaluation task into a production task. The experiment consists of three parts. While the presentation in the first part does not differ from the one in Experiment 1 and 2, the two other parts each encourage the use of a specific strategy.

EXPERIMENT 1

In the first experiment of Byrne (1989) there were three groups of eight participants. Each group had to solve 12 problems. The first group (the control group) had to solve three problems with a different content for each of the four conditional syllogisms, that is, Modus Ponens, Modus Tollens, denial of the antecedent, and affirmation of the consequent. These problems will be called the simple problems. The participants had to evaluate three alternatives: One of them was the correct answer, the other was the negation of the correct answer, and the last answer indicated that you could not know for sure what followed. The second group received problems with two major conditionals. The second conditional contained an alternative antecedent, for example: If she has an essay to finish, then she will study late in the library. If she has some textbooks to read, then she will study late in the library.

Each of these alternative antecedents is a sufficient condition for the consequent. The third group also received two major premises, but now the second conditional contained an additional antecedent, for example: If she has an essay to write, then she will study late in the library. If the library stays open, then she will study late in the library.

We decided to repeat Experiment 1 of Byrne (1989) to see if the same materials would elicit the same pattern of answers in a different language (Dutch). Furthermore, we used about three times as many participants in order to

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increase the statistical power. Finally, there was an unexpected but unmentioned aspect of Byrne’s data, which is difficult to explain by either the model theory or the rule theory: In the condition with simple problems, participants did not give more Modus Ponens answers than Modus Tollens answers; they even gave more Modus Tollens answers than affirmation of the consequent answers.

Method

Design. There were three groups of participants. The control group received simple conditional arguments, whereas a second group received conditional arguments accompanied by a conditional containing an alternative antecedent and the third group received arguments accompanied by a conditional containing an additional antecedent. All participants received four sorts of conditional problems: Modus Ponens, Modus Tollens, denial of the antecedent, and affirmation of the consequent. Each sort of problem was presented with three different contents. Hence, each participant solved 12 problems, presented in a randomised order. Materials. We used almost the same lexical materials as Byrne (1989). The problems were tested in two separate pilot studies (with 32 and 16 participants, respectively). The translation was rather literal (see Appendix A), except for the sentence “If she meets her friend, then she goes to a play”, which was changed to “If she meets her friend, then she goes to a pub”. All participants could reason easily with the sentences. When asked to indicate if the problems sounded natural, all participants responded that the sentences were natural, with an average rating of 3.8 on a rating ranging from one (very unnatural) to five (very natural). Procedure. The experiment was completed in a single session. The instructions were written on the first page of a booklet given to each participant. The participants’ task was to answer a series of questions based on the information in the preceding assertions, and they were asked to choose the conclusion they thought followed from the sentences. In the instructions, a simple problem and the accompanying answer alternatives were given as an example. No answer was given. Each problem, together with its three answer alternatives, was printed on a separate page in the booklet. The experiment had no time limit. Participants were asked not to go back to a problem once they had answered it. Participants. A total of 70 students from the last year of a secondary school participated in the experiment. The participants were randomly assigned to one of the three groups: 23 participants received the simple problems with a single conditional, 25 participants received the problems with the second conditional


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containing an alternative antecedent (referred to as alternative problem), and 22 participants were faced with problems in which the second conditional had an additional antecedent (referred to as additional problem). Two participants did not solve all the problems and were excluded from the analysis of the problems they did not solve.

Results and discussion

All statistics in this and following experiments are done with the Mann-WhitneyU test for a between-subjects design and with the Wilcoxon signed ranks test for a within-subject design. As each participant solved three problems of each inference type, we could score the conditional answer on a specific inference type of each participant on a scale from 0 to 3. Comparisons between and within participants were done by ranking the scores as described in Siegel and Castellan (1988). Table 1 presents the percentages of conditional inferences made from the three types of problems. The pattern of the data parallels the one reported by Byrne (1989). First, an additional antecedent affected the Modus Ponens and Modus Tollens arguments, but not the denial of the antecedent and affirmation of the consequent arguments. The participants confronted with an additional antecedent made fewer Modus Ponens inferences than either the participants confronted with an alternative antecedent or the participants who received the simple problems (respectively: 60.6% vs. 93.3%; U = 121.5, p < .0005; and 60.6% vs. 88.3%, U = 142, p < .01). An additional antecedent also lowered performance on the Modus Tollens problems as compared to the single antecedent condition (43.9% vs. 69.6%; U = 159, p < .05). Modus Tollens inference was made less often on problems with an additional antecedent than on those with an alternative antecedent (43.9% vs. 69.3%; U = 173.5, p < .05). Second, an alternative antecedent affected the fallacies, but not the valid Modus Ponens and Modus Tollens arguments. On problems with an alternative TABLE 1 The percentages of conditional inferences made as a function of the type of contextual information given in Experiment 1 p

Inference type not-r not-p

r

If p then r.

88.3

69.6

49.3

55.1

If p then r. If q alt, then r.

93.3

69.3

22.0

16.0

If p then r. If q add , then r.

60.6

43.9

49.2

53.0

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antecedent, fewer denial of the antecedent inferences were made, as compared to the simple problems (22.0% vs. 49.3%; U = 162.5, p < .01). Analogously, the denial of the antecedent problems with an alternative antecedent yielded fewer conditional responses than the problems with an additional antecedent (22.0% vs. 49.2%; U = 121.5, p < .005). Likewise, in the case of an alternative antecedent, fewer affirmation of the consequent inferences were made, as compared to the performance on the simple problems (16.0% vs. 55.1%; U = 129, p < .0005). The same suppression effect is again observed when comparing the affirmation of the consequent problems with an alternative antecedent to those with an additional antecedent (16.0% vs. 53.0%; U = 96, p < .0001). Finally, contrary to the unexpected effect observed by Byrne (1989), participants made more simple Modus Ponens arguments than simple Modus Tollens arguments (88.3% vs. 69.6%; T = 50.5, n = 10, p < .0005). In sum, the results are similar to the results of Byrne (1989), except that the problematical aspect of her data was absent in our experiment: We did observe that simple Modus Ponens inferences were made more often than simple Modus Tollens inferences. Therefore, it appears that not only can invalid inferences be suppressed (by means of an alternative antecedent), but also valid inferences (by means of an additional antecedent).

EXPERIMENT 2

The results of Experiment 1 can be explained by the mechanisms proposed by Byrne (1991). Dependent on the specific content, reasoners combine the two conditionals into a conjunction or a disjunction and use mental models to come up with a conclusion. There is, however, an important problem with Experiment 1. The answer alternatives presented in the conditions with an alternative or additional antecedent preclude the subjects giving a conclusive answer: For example, they are presented with: If it is raining, then she will get wet. If it is snowing, then she will get wet. She will get wet. What follows? (a) It is raining. (b) It isn’t raining. (c) It may or may not be raining.

First, it is clear from the data of Experiment 1 that the presence and the precise semantic content of the second conditional influences the answers. If reasoners combine the two premises here into a disjunction, as Byrne (1989) supposes, and reasoners want to give a true conditional (hence, not correct) answer that is based on all the information in the premises, then reasoners would answer:


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It is raining or it is snowing.

However, reasoners cannot express anything about the relevance of this second conditional: They can only select a conclusion that conveys information based on the first conditional. In this case, answer alternative (c) might be chosen, but they might also opt to choose answer alternative (a). That is, if we want to have a better picture about the inferences that are made about both conditionals, then we must give the participants the opportunity to express what they actually want to answer. Therefore, instead of presenting 3 answer alternatives, we now gave the participants an almost exhaustive set of 15 answer alternatives. In Experiment 2, we only used the problems with either an additional or an alternative antecedent. Comparing these problems with the simple problems is always a bit tricky: The simple problems contain just one conditional, whereas the more complex problems contain a second conditional. This means that a fair comparison is not possible. However, if one compares the two complex conditions, one can make fair comparisons—as illustrated by the analyses conducted on the data of Experiment 1. That is, the suppression hypothesis can be specified as follows. First, participants will make fewer Modus Ponens and Modus Tollens inferences in the case of an additional antecedent, when compared to these types of arguments with an alternative antecedent. Second, they will make fewer denial of the antecedent and affirmation of the consequent inferences if the problems contain an alternative antecedent, as compared to when these problems include an additional antecedent.

Method

Design. There were two groups of participants: One group received alternative problems and the other group received additional problems. All participants received four sorts of conditional problems: Modus Ponens, Modus Tollens, denial of the antecedent, and affirmation of the consequent. Each sort of problem was presented with three different contents. Hence, each participant solved 12 problems, which were presented in a randomised order. Instead of presenting three answer alternatives, we increased that number to 15 alternatives, for the reasons mentioned earlier. In the Materials section, the 15 different answer alternatives for the problems are listed. Materials and procedure. We used the same kind of lexical materials and the same procedure as in Experiment 1 (see Appendix A). We presented 15 answer alternatives differing along the content and inference type of the problem. For Modus Ponens and Denial of the Antecedent of the “pub” content, for example, we presented following alternatives:

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She goes to a pub for a drink. She doesn’t go to a pub for a drink. She meets her family. She doesn’t meet her family. She goes to a pub for a drink OR she meets her family. She doesn’t go to a pub for a drink OR she meets her family. She goes to a pub for a drink OR she doesn’t meet her family. She doesn’t go to a pub for a drink OR she doesn’t meet her family. She goes to a pub for a drink AND she meets her family. She doesn’t go to a pub for a drink AND she meets her family. She goes to a pub for a drink AND she doesn’t meet her family. She doesn’t go to a pub for a drink AND she doesn’t meet her family. She goes to a pub for a drink OR she doesn’t go to a pub for a drink. She meets her family OR she doesn’t meet her family. One can’t formulate a conclusion. None of the answer alternatives is the correct one.

The choice of these alternatives was inspired by the integrated mental model proposed by Byrne (see earlier). We presented all possible combinations in conjunctive and disjunctive form as well as in positive and negative form. Furthermore, we extended the alternative set with a simple answer on the second conditional premise, also in positive and negative form. Finally, we included three alternatives that express that one is undecided about the choice. What answer choices should we predict following the results in the former experiment? If the conditional answers really are suppressed with the denial of the antecedent problems and the affirmation of the consequent problems, then the answers will equal the answers of the former experiment: Many “no valid conclusion” answers will be given for the invalid arguments in the alternative group. If the correct answers really are suppressed with the Modus Ponens and Modus Tollens problems, then the answers will also equal the answers of the former experiment: Many “no valid conclusion” answers will be given for the valid arguments in the additional group, which means that many participants will opt for the last answer alternative. Participants. We tested 70 participants. Of these, 36 participants were assigned to the group who received the problems with an alternative antecedent and 34 participants received the problems with an additional antecedent. They were all first year psychology students, who were fulfilling a course requirement. None of them had yet received a formal training in logic or had participated in previous experiments on deductive reasoning.

Results

In Table 2, we represent the answers the participants gave for each of the four problem types. We will discuss the data for each of the four problem types separately. All statistics in this experiment are done with the non-parametric

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Mann-Whitney-U test (between-subjects design). To make our discussion of the different answers more transparent, we used the following notation. We will refer to additional problems in the following way: If p then r. If q add then r.

The alternative problems will be referred to in the following way: If p then r. If q alt then r. TABLE 2 Percentages of different types of answer chosen as a function of the type of contextual information given in Experiment 2

p If p, then r If q alt, then r If q add , then r

not-r If p, then r If q alt, then r If q add , then r

Not-p If p, then r If q alt, then r If q add , then r

r If p, then r. If q alt, then r If q add , then r

r

r and q

95.3 56.9

0.0 11.8

Not-p

Solutions r or not-q

null

other

4.7 17.7

0.0 6.9

0.0 6.7

not-p and not-q

not-p or not-q

null

other

96.3 35.3

3.7 56.9

0.0 1.9

0.0 1.0

Not-r

not-r or q

null

other

9.4 37.3

67.3 22.6

12.2 28.4

11.1 11.7

p

p or q

p and q

null

other

0.0 2.9

90.7 65.7

0.9 30.4

7.4 0.0

1.0 1.0

0.0 4.9

The null group contains the percentage of “one can’t tell” answers cumulated with the percentage of “r or not r” answers for Modus Ponens and denial of the antecedent. The “other” group only contains answers given in less than 2% of the cases, except for: r or not q (denial of the antecent, alternative condition; 4.67%) q or not q (denial of the antecent, alternative condition; 2.80%) r or not q (denial of the antecent, additional condition; 3.92%).

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Modus Ponens. In the case of an alternative antecedent, 95.3% of the participants made the Modus Ponens argument by giving “r” as their conclusion, whereas with the additional problems “r” was selected in only 56.9%. That is, there is a clear suppression effect (U = 225, p < .000005). More interesting, however, are the other types of answers that were given to the problems with an additional antecedent. Given a multitude of answer alternatives, these participants did not pick the “you can’t know” answer as their response (only 6.9 %). In the majority of the remaining answers “r” was included, for example: “r or not-qadd” (1 7.7%) and “r and q add” (11.8%). Modus Tollens. With alternative problems, none of the participants gave “not-p” or “not-qalt” as a conclusion. The participants confronted with an additional problem did select this answer alternative (4.9%; U = 522, p < .05). This observation seems to go against the hypothesis of Byrne (1991) and is certainly in contrast with the frequency by which this alternative was chosen in Experiment 1 (43.9%). However, the alternative problems resulted in the conclusion “not-p and not- qalt” in 96.3%, whereas this answer was selected in 35.3% of the additional problems (U = 98, p < .000005). This pattern can be interpreted as a suppression effect. However, this conclusion should be put into perspective as well. The additional problems resulted in the conclusion “not-p or not-qadd” in 56.9%, whereas only 3.7% of the participants confronted with an alternative antecedent gave this conclusion (U = 138, p < .00001). Denial of the antecedent. Problems with an additional antecedent yielded conclusions of the form “not-r” in 37.3% of the cases, whereas only 9.4% of the conclusions had this form when the problems had an alternative antecedent (U = 321, p < .0005). This supports the suppression hypothesis. However, when given an additional antecedent, 22.6% of the conclusions were of the form “notr or qadd”, whereas participants confronted with an alternative antecedent selected this conclusion more often (67.3%; U = 251, p < .00001). The answer “not-r or qadd” can also be considered as a conditional answer: Reasoners would infer that “not-r” is the case, unless “q” is the case. When combining these two sorts of conditional answers (“not-r” and “not-r or q”), it is even so that participants confronted with an alternative antecedent more frequently gave a conditional answer than participants confronted with an additional antecedent (76.7% vs. 59.9%; U = 406.5, p < .05). This unexpected pattern for Byrne (1991) and Rumain et al. (1983) is also present in the percentages of the correct “you can’t know” answers. For the problems with an additional antecedent, more correct solutions were given than for the problems with an alternative antecedent (28.4% vs. 12.2%; U = 366.5, p < .005). Affirmation of the consequent. In the case of an additional problem only in a few cases was the conclusion “p” or “q add” selected (2.9%), but in the case of an alternative problem neither one of these answer alternatives was selected


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(U = 558, p < .05). This result could be interpreted as support for the suppression hypothesis. However, with an additional problem 65.7% of the conclusions were of the form “p or qadd”, whereas the alternative problems yielded 90.7% conclusions of this form (U = 361, p < .0005). In contrast, with additional problems 30.4% of the conclusions were of the form “p and qadd ”, whereas with an alternative one only 0.9% of the conclusions were of this form (U = 318.5, p < .00001). The “p and qadd” conclusion can also be interpreted as a conditional response to additional problems. Reasoners infer that “p” is the case (by affirmation of the consequent on the first conditional) but also infer “qadd ” (by making the affirmation of the consequent argument with respect to the second conditional).

Discussion

Given a broad set of answer alternatives, the absolute level of participants coming up with the standard conditional responses to the four arguments is much lower than expected. Most answers (especially in the suppressed conditions) contain more than one item. In addition and most importantly, these complex answers seem to reflect a conditional line of reasoning, as we will argue in the following. Consider an important problem that emerges when analysing the answers that were given to, for instance, the denial of the antecedent problems with an alternative antecedent: If p then r. If qalt then r. Not-p.

In 67.3% of the cases, the participants gave a conclusion “not-r or qalt”. Byrne (1991) argued that reasoners would combine the first conditional and the second conditional into a disjunctive antecedent. This would lead to the following implicit models: p qalt …

r

r

Consequently, “r” can still hold even if “not-p” is the case. However, the conclusion “not-r or qalt” requires that at least the following models be fleshed out (the fully explicit model is mentioned in Note 1 at the end of this paper): not-p not-p not-p

not-q alt not-q alt qalt

r not-r r

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Johnson-Laird et al. (1992) claim that considering multiple models is difficult because of the load it places on working memory: As soon as the capacity of working memory is exceeded, reasoners are unlikely to reach a conclusion that depends on considering multiple models. The answer “not-r or qalt” requires three explicit models and each of these models contains three atomic propositions, which is a very serious amount of information for working memory. Indeed, the number of models as well as the size of a single model, that is, the number of entities it contains, taxes working memory. Schaeken and Johnson-Laird (2000) show that the number of events matters in temporal reasoning. As soon as a single model contains more than six events, reasoning is more difficult. Therefore, reasoning with the complex multiple models that are a result of the integration of the two premises is likely to be difficult. Thus, the explanation of these results in terms of the model theory might not be as straightforward as Byrne (1989) suggested. For the other problems the same pattern is found: In the suppressed conditions, participants gave conclusions that can only be made if they considered multiple models that contain many tokens. Other observations are also problematic for Byrne’s thesis. For denial of the antecedent problems, more than twice as many participants chose “impossible to tell” in the additional condition than in the alternative condition. For Modus Tollens, the difference between the rate at which “not-p and not-q” and “not-p or not-q” conclusions were chosen was not as great as might be expected if participants were interpreting the additional antecedents as being conjointly necessary. Nevertheless, the answers given in Experiment 2 still seem to reflect a conditional line of reasoning. Consider for instance the additional Modus Ponens problems. Many participants conclude “r or not-q” or “r and q”. Selecting this conclusion can be interpreted as the result of the following line of reasoning. Reasoners first infer “r”, but notice that this conclusion also depends on the content of the second conditional with “r” in its consequent clause. That is, it appears as if reasoners make an amendment to their putative conclusion “r” by considering that this conclusion would be falsified if “not-q” were the case. The same argument can be made concerning the answers given to the alternative denial of the antecedent problems. The conclusion “not-r or q” is consistent with the hypothesis that reasoners make the denial of the antecedent argument that results in the conclusion “not-r”, which they subsequently validate by inferring that “not-r” would not be the case if “q” were the case. The same line of argumentation can be used for the Modus Tollens and the affirmation of the consequent problems. That is, it appears that, when reasoners have the opportunity to give a conclusion about all parts of the problems, they engage in elaborate chains of conditional reasoning. Johnson-Laird and Byrne (1991, p. 84) wrote that with an additional antecedent “the subjects’ knowledge leads them to construct one model in which both antecedents occur and an implicit alternative model.” This has an important consequence for the affirmation of the consequent problems. One would expect


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that reasoners come up more often with a conclusion “p and qadd” than with a conclusion “p or qadd ”, if they were interpreted as conjointly necessary. However, we found the opposite: 65.7% of the conclusions were of the form “p or qadd” and 30.4% were of the form “p and qadd”. This pattern, however, might reflect a response bias. Out of caution some participants might prefer the less stringent conclusion “p or qadd” over the semantically more informative conclusion “p and qadd”. In sum, the results agree with the findings of Byrne (1989), by showing that not only invalid inferences, but also valid inferences can be suppressed. However, if one takes into account the more complex answers, then no suppression remains (for denial of the antecedent, we even found the opposite result: more conditional inferences in the case of an alternative antecedent). Moreover, the nature of the more complex answers implies that the explanation of Byrne (1989) of the suppression effect in terms of combining the premises and in terms of the model theory is more complicated than she suggested. At least some reasoners were not combining the information of the two premises in the way she proposed.

A new strategy: Amendment

The results can be explained if one hypothesises that some reasoners reasoned with each premise separately. Consider the following alternative denial of the antecedent problem: If she has an essay to write, then she will study late in the library. If she has some textbooks to read, then she will study late in the library. She does not have an essay to write.

First, reasoners combine the categorical premise with the first premise, and come up with the conditional answer: She will not study late in the library.

Next, they consider whether this conclusion holds with respect to the second premise. This can explain the answer “not-r or qalt” that was given by 67.3% of the participants: She will not study late in the library or she has some textbooks to read.

Indeed, different answers will occur according the applied strategy. From now on, we will name the strategy proposed by Byrne (1991) the integration strategy, and the alternative strategy that we propose the amendment strategy. We chose the latter because after a first conclusion is drawn the second part of the conclusion may be “amended” in a later stage. We can make theoretical predictions of the conditional answers that should occur for the four inference types when the

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amendment strategy is applied. The working of the amendment strategy is the following: Stage 1: The first conditional premise and categorical premise are taken into consideration. One forms a putative conclusion based on this information. Stage 2: The putative conclusion is analysed and validated in the light of the information in the second conditional premise. This leads to an eventual amendment of the putative conclusion. Given the premises “if p, then r”, “if q, then r”, and “not p”, one takes into consideration the first premise “p may occur together with r” in Stage 1. When the categorical premise is brought to attention, one may infer a conditional conclusion. In Stage 2, this putative conclusion is reconsidered in the light of the (new) information given (see Note 2 at the end of this paper). One could either opt to leave the putative conclusion as it was, which would lead to the conditional answer “not r”—37% of the participants in Experiment 2 opted for this solution. On the other hand, one may also think that the new information changes the putative conclusion considerably—23% of the participants in Experiment 2 opted for amending “or q” to the putative conclusion “not r”. We expect different answers for the integration and amendment strategies. With the amendment strategy, one can explain a broader set of answers. Indeed, when the relation between the two premises is fixed before the reasoning process starts, one is bound to that relation during the inference process. However, when one first makes an inference based on one conditional premise and the categorical premise, and then takes into consideration the second conditional premise only after one has formulated this putative conclusion, there is no bound relation that one has to obey. For that reason, answers in which the second conditional premise is considered useful (or not useful) can be explained by the use of the amendment strategy. We will describe the theoretically predicted answers for each of the four conditional problems. Modus Ponens. We expect an “r” answer with an alternative problem, no matter which strategy is followed. Indeed, only the first of the conditional premises contains useful semantic information for the Modus Ponens problem. When a participant uses the amendment strategy, we might additionally expect that he/she amends the putative conclusion (“r”) with “if qalt”. An answer of this kind is unexplainable from the point of view of the integration strategy. Following the integration strategy, we would expect a disjunctive model as follows: p

qalt

r r

This means that (first line) “p may occur together with r, whatever the value of qalt is” or (second line) “q alt may occur together with r whatever the value of p is”.


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Therefore, when “p” is affirmed, “r” can be affirmed, whatever the value of “qalt” is. On the contrary, for the amendment strategy we have concluded that when “p” is affirmed, “r” may be affirmed in Stage 1. This conditional and correct conclusion may be amended in Stage 2 when stating that the affirmed “r” occurs with the affirmation of “qalt”. With an additional problem, we expect an “r and qadd ” answer when the premises are integrated. A predicted answer for the amendment strategy is “r if also qadd ”. Eventually, one could explain this latter answer by means of the integration strategy: Stating “r if also qadd ” is equal to “r and qadd” in a weaker sense. Another possible amendment answer that is not explainable by the integration hypothesis is simply stating “r”, with which one confirms that the second conditional does not matter, or does not have any influence on the putatively inferred conclusion. Modus Tollens. The predictions for the integration strategy are very clear in this case: “Not p and not q alt” with an alternative problem and “not p or not qadd ” with an additional problem. For the amendment strategy, the predictions are less defined. In Stage 1, the conditional answer “not p” is very likely to be inferred, but how precisely the information in the second premise will be included stays unclear for the moment; “not p and/or not q” is our prediction. Furthermore, we can expect simple answers; reasoners might not want to change their conclusion in the light of the information in the second premise. We do not expect simple answers when participants follow the integration strategy. Denial of the antecedent. With an alternative problem, the answer predicted from the integration strategy is something like “not r and not q alt”, or less firmly stated: “not r if not qalt”. The earlier mentioned answers might also count as an amendment answer. Another possibility according to the amendment strategy is that people consider the second conditional premise to be of no further use once they have inferred the initial putative conclusion “not r”. With an additional problem, we may expect a “not-r” answer for both strategies, because participants may think that only the first conditional premise contains useful semantic information (amendment) or may think that the negation of one of the conjunctively related antecedents is a sufficient condition to negate the conclusion (integration). Another possible answer according to the amendment strategy is that the putative conclusion “not r” is amended with “unless qadd”. This kind of answer is impossible when a participant uses the integration strategy. Affirmation of the consequent. Similar to Modus Tollens, we predict “p and q” as a conditional answer with an additional problem, and “p or q” with an alternative problem, when reasoners use the integration strategy. Again, for the amendment strategy, the predictions are less defined. As a putative conclusion

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we can expect “p”, but how precisely the information in the second premise will be included stays unclear for the moment: “p and/or q” is what we predict. Furthermore, we can expect simple answers, that is, reasoners might not want to change their conclusion in the light of the information in the second premise. We do not expect simple answers when participants follow the integration strategy. In conclusion, the integration strategy specifies that people integrate the two conditionals in a single conditional with a disjunctive or conjunctive antecedent, depending on whether or not the two antecedents determine alternative or additional conditions with respect to the mutual consequent. That is, before people start making inferences, they integrate the two conditionals in a single conditional upon which they base their inferences. The amendment strategy stipulates that people will first draw an inference with respect to one of the conditionals, and the resulting putative conclusion is amended, if necessary, on the basis of the second conditional. The results from Experiment 2 indicate that participants use both strategies to infer a conclusion from a complex conditional problem. Therefore, rather than opposing one strategy to the other, we want to contrast the answers resulting from them.

EXPERIMENT 3

We conducted a third experiment in order to contrast the answers resulting from both strategies. In the first part we presented the same problems as we did in Experiment 2, but the task was a production task. In the second part, the problems were presented in such a way that the amendment strategy was induced, whereas in the third part, the problems were presented in an integrated manner. First, we will discuss the general idea behind the three parts of the experiment and their respective methods. A problem with Experiment 2 was that the participants could not produce their own answers, but had to choose their answer(s) from several answer alternatives. In order to test more directly the hypothesis that some participants reasoned with each premise separately, we repeated the experiment as a production task (Experiment 3.1), which gives participants the opportunity to formulate answers as: She will study late in the library, unless the library is closed.

Experiment 3.2 induced a line of reasoning that is based on the amendment strategy. In order to achieve this, we conducted this experiment on a computer, whereby we presented the two conditionals in two stages. First, the participants saw one conditional and one categorical premise: If it is raining, then she will get wet. It is raining.


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They had to write down a conclusion based on these two premises. Next, we presented the second conditional: If it is snowing, then she will get wet.

We asked the participants if they thought they had to change their previous conclusion in the light of this new premise. If they thought so, they could write down the new conclusion or adapt the first conclusion. With this procedure, we predicted mainly answers that are in accordance with the amendment strategy. Moreover, we expected more amendment answers in this experiment than in Experiment 3.1. Experiment 3.3 induced a line of reasoning based on an integrated conditional. Byrne (1989, 1991) proposed that the alternative antecedent be integrated as a disjunctive antecedent to the consequent. Consider for instance the following conditional with an alternative antecedent from the previous experiment: If she has an essay to write, then she will study late in the library. If she has some textbooks to read, then she will study late in the library.

These premises would be integrated in a conditional like: If she has an essay to write or has some textbooks to read, then she will study late in the library.

Similarly, the integration strategy proposes that additional antecedents be incorporated in a conjunctive antecedent. Hence, the following premises: If she has an essay to write, then she will study late in the library. If the library remains open, then she will study late in the library.

would be integrated in a conditional like: If she has an essay to write and the library remains open, then she will study late in the library.

In Experiment 3.3, we presented our participants with such integrated conditionals. When they received such integrated conditionals, we predicted mainly answers that are in accordance with the integration strategy. Moreover, we expected more integration answers in this experiment than in Experiment 3.1. In Table 3, we indicate the predicted integration and amendment answers. Categories without an index are answers that could follow from both strategies.

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3.1 3.2 3.3

Additional

3.1 3.2 3.3

3.1 3.2 3.3

Alternative

Additional

Modus Tollens

3.1 3.2 3.3

Alternative

Modus Ponens

r if (also) q

am 0 19.0 0

int/am 25.3 57.1 68.4

not-p and not-q

int/am 71.3 31.0 83.7

am 29.7 14.3 8.7

r

int/am 93.8 76.2 96.7

am 61.6 28.6 8.7

not-p/ not-q

am 7.7 35.7 1.1

am 36.2 31.0 10.9

int/am 29.7 31.0 69.6

am 8.5 14.3 9.8

not-p or not-q

2.2 9.5 1.1

nr if nq

0 11.9 0

5.4 4.8 1.1

maybe p

2.9 0 7.6

both r if q & nr if nq

0 0 7.6

nothing follows

0 0 13.1

nothing follows

4.3 11.9 3.3

7.0 14.3 4.3

other

int 6.5 0 0

r and q

1.4 4.8 1.1

6.2 4.8 3.3

other

TABLE 3 Percentages of different types of answers chosen as a function of the type of contextual information given in Experiment 3

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3.1 3.2 3.3

Additional

3.1 3.2 3.3

Additional

p and q

am 11.6 11.9 0

int/am 55.0 45.2 78.3

am 10.0 28.6 0

am 31.1 27.0 6.5

am 7.9 21.9 0

int/am 64.6 50.0 64.1

p/q

3.9 28.6 20.7

r if q

am 42.6 33.3 18.4

not-r

am 7.9 7.1 0

int/am 66.0 38.1 82.6

p or q

2.9 0 0

int/am 3.1 4.8 8.7

not-r if not-q

0 18.2 9.8

5.4 16.7 13.0

nothing follows

2.9 0 0

14.7 0 16.3

both, r if q & nr if nq

5.8 2.4 5.4

7.0 4.8 4.3

other

am 2.9 6.3 0

18.6 9.5 10.9

not-r unless q

int 7.2 0 21.7

not-r (even) if q

“Other” answers contain answers given in less than 3% of the cases or answers that were not directed to the problem.

3.1 3.2 3.3

Alternative

Affirmation of the consequent

3.1 3.2 3.3

Alternative

Denial of the antecedent

0 18.7 12.0

3.9 23.8 18.5

nothing follows

11.5 0 2.2

13.2 0 6.5

other


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Method

Design. The design of Experiment 3.1 was the same as in Experiment 1. The only difference was that the participants were asked to produce their own conclusions, instead of selecting the conclusion that resulted from the premises. Answers in the baseline condition will not be discussed further, because they did not significantly differ from the answers in the baseline condition (simple answers) in Experiment 1. In Experiments 3.2 and 3.3, the participants served as their own control with respect to the presented conditional problems accompanied by either a conditional containing an alternative antecedent or a conditional containing an additional antecedent. In both conditions, the participants received two versions of the four sorts of conditional problems: Modus Ponens, Modus Tollens, denial of the antecedent, and affirmation of the consequent. Materials. The materials used in Experiment 3.1 were the same as the ones used in Experiment 1 (see Appendix A). We decided to use more diverse contents in Experiment 3.2 than in the previous experiments, in order to minimise the influence of previous sentences and inferences on the subsequent ones. In a pilot study with 18 participants, we tested seven different problem contents on the strength of their alternative and additional premises. In one of the seven different contents, the additional premise was not rated as very additional and the alternative premise was not rated as very alternative. Although we presented all 14 problems in the experiment, we did not include this problematical content in the statistical analysis (see Appendix B; un-analysed contents are not mentioned). Appendix C represents the different contents of the conditionals that were used to construct the four types of conditional problems for Experiment 3.2. Procedure. The procedure of Experiment 3.1 was the same as the procedure of Experiment 1. Experiment 3.2 differed from the previous experiments, as the participants were tested on a computer. At the beginning of each trial, the screen signalled “press space bar for the next problem”. When participants pressed the space bar, the first premise appeared together with the categorical premise and the question “what follows?”. At this point, the participants typed their first answer. When they had entered their answer, we presented the following question under the first two premises: Do you have to change your conclusion: [and then the conclusion of the participant was repeated] in the light of the following information: [and then the second conditional premise appeared on the screen]. If you think so, how would you change your conclusion (if not, press “enter”)?

The computer recorded their first answer and the second answer (if they changed their first answer). The instructions were presented on the screen. They


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explained the participants’ task. None of the participants had any difficulty with the procedure. Experiment 3.3 was a paper and pencil test. The participants received two booklets: The first booklet contained the instructions and the second one presented the 16 problems, each on a separate page. As in Experiments 3.1 and 3.2, the participants had to produce their own conclusion to the problems. Participants. A total of 178 participants took part in the three parts of the experiment. In Experiment 3.1, we tested 111 participants, of whom 22 solved simple problems that are not discussed further. In the alternative condition, we tested 43 participants; in the additional condition, we tested 46 participants. All participants were students of a secondary school, between 16 and 19 years old. In Experiment 3.2, 21 students at the University of Leuven, who had not taken part in previous reasoning experiments, participated. They had not yet received a formal introduction to logic. In Experiment 3.3, 46 first-year psychology students at the University of Leuven who had not taken part in previous reasoning experiments, participated. They received credit points towards a course requirement and had not yet received a formal introduction to logic.

Results and discussion

The three parts of Experiment 3 have a different design. This manipulation was necessary to induce the use of the amendment (3.2) and the integration strategy (3.3). It is as well to note, however, that the difference in design implies less power in the statistical analysis. In Table 3, we present all the answers and the corresponding percentages of the three parts of the experiment. We did not restrict the different answers by putting them into categories. The diversity of different answers given by the participants is in itself already an important finding. As is immediately observed, almost all answers given in 3.1 are given in 3.2 or 3.3, or in both 3.2 and 3.3. The theoretically predicted integration and amendment answers are indicated. We distinguish four kinds of answers: theoretically predicted pure amendment answers, theoretically predicted pure integration answers, answers that are expected from both theories, and finally answers that are not specifically predicted (not indicated). Very few answers are pure integration answers. The amendment hypothesis, being more general, can account for a larger set of possible answers than the integration hypothesis. Of all given answers, 72.6% were theoretically predicted. Overall, the purely predicted integration answers account for 1.7%, 0%, and 2.7% of all answers, respectively. Also in line with expectations, the purely predicted amendment answers account for 32.2%, 34.9% and 8.0% of all answers, respectively. The statistical comparisons of the pure amendment answers in Experiment 3 mentioned in Table 4. These comparisons are consistent with the idea that

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amendment is encouraged in Experiment 3.2. Finally, the answers predicted from both hypotheses account for 42.9%, 36.9%, and 58.7% of all answers, respectively. Table 5 shows the statistical comparisons of the mixed amendment and integration responses in Experiment 3. These findings together with the comparisons of Table 4 are consistent with the idea that Experiment 3.3 encourages integration. As shown in Table 3, most answers given in 3.2 are in line with the amendment strategy (71.8%), and most answers given in 3.3 are in line with the integration strategy (61.4%). Our hypothesis for the two existing strategies is largely confirmed. There are, nevertheless, a minority of answers that we did not predict. We want to focus on these answers now, showing that even some of these answers fit in the proposed strategies. One example in this category is the “maybe p” (alt: 4.8%; add: 11.9%) answer for the Modus Tollens problem (alternative and additional condition). It is explainable when one uses the amendment strategy. The putative “not p” conclusion is put into question when the second premise is brought to the attention of the participant. One can express his/her doubt about the putative conclusion in a way that is more (not p … or not q, not p … and not q) or less (not p … or maybe still p) specific. This latter answer is difficult to explain with the integration strategy, because one would expect an answer containing the two (disjunctive or conjunctive) related propositions of the antecedent in that case. Indeed, this answer is not given in 3.3 (alt: 1.1%; add: 0%). Another possible explanation of the “maybe p” answer is that participants start doubting the original premise when an extra premise is added (David Over, personal communication, May 1999). We can agree with this point of view. It might be the case that this answer and some and of the other, are given because the participants consider a stronger answer (e.g., not-r) too strong or improbable bearing in mind the doubt they have about the major premise. However, this viewpoint cannot explain all observed answers. Indeed, the “maybe p” answers represent only 7.7% of the Modus Tollens answers of Experiment 3, but most answers for these problems as well as some others are clearly predicted by the amendment or integration strategies. Also and most importantly, some of these answers (e.g., not-r unless q) cannot be explained by means of doubt about the premises. Another group of unpredicted answers that we observed are “both, r if qadd and not r if not qadd” for the Modus Ponens problem and “both, r if q alt, and not r if not qalt” for the denial of the antecedent problem. It is easy to see why these answers only appear in Experiment 3.1 and 3.3. Because the latter answer is a specification of the predicted “r if qadd ” (Modus Ponens) and “not r if not qalt” (Modus Tollens), which stresses the “if and only if” meaning of the expected answers, it fits perfectly the line of thinking of the integration strategy. However, it does not fit with the amendment strategy. Indeed, why should participants, when they


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TABLE 4 Pure amendment answers in Experiment 3 Modus Ponens r if (also) q alt radd : Modus Tollens not p alt: not p add : Denial of the antecedent not ralt: r if q add

Affirmation of the consequent p alt p and q alt p or q add p add :

3.1 (0%) 3.1 > 3.2: 3.2 > 3.3

3.2 (19.0%) 3.3 (0%) 61.6% vs. 28.6%; U = 275.5, p 3. 1: 3.1 > 3.3: 3.1 > 3.3: 3.2 > 3.3:

35.7% vs. 7.7%; U = 309, p < .005 7.7% vs. 1.1, U = 851, p < .05 36.2% vs. 10.9, U = 670, p < .0005 31.0% vs. 10.9; U = 183, p < .0001

3.1 > 3.3: 3.2 > 3.3: 3.1 (7.9%)

42.6% vs. 18.4%, U = 641.5, p < .00 1 33.3% vs. 18.4%, U = 380, p < .05 3.2 (21.9%) 3.3 (0%)

3.1 (10.0%) 3.1 (11.6%) 3.1 (7.9%) 3.1 > 3.3: 3.3 > 3.2:

3.2(28.6%) 3.3 (0%) 3.2 (11.9%) 3.3 (0%) 3.2 (7.1%) 3.3 (0%) 31.1 % vs. 6.5%, U = 586, p < .000 1 27.0% vs. 6.5%, U = 309.5, p < .0005

TABLE 5 Mixed amendment and integration answers in Experiment 3 Modus Ponens ralt: Modus Tollens not p and not q alt: not p or not q add : Denial of the antecedent not ralt +: Affirmation of the consequent p or q alt: p and q add :

3.1 > 3.2: 3.3 > 3.2:

93.8% vs. 76.2%; U = 348, p < .05 96.7% vs. 76.2%; U = 349, p < .005

3.3 > 3.1: 3.1 > 3.2: 3.3 > 3.2: 3.3 > 3. 1:

83.7% vs. 71.3%, U = 749.5, p < .05 71.3% vs. 31.0; U = 309, p < .005 69.6% vs. 31.0, U = 264, p < .001 69.6% vs. 29.7%, U = 483, p < .0001

3.1 > 3.2: 3.3 > 3.2:

36.4% vs. 14.3%, U = 312, p < .05 35.9% vs. 14.3%, U = 337.5, p 3.2: 3.1 > 3.2: 3.3 > 3. 1: 3.3 > 3.2:

82.6% vs. 38.1%, U = 250.5, p < .0001 66.0% vs. 38.1%, U = 301, p < .05 78.3% vs. 55.0%, U = 681, p < .001 78.3% vs. 45.2%, U = 287, p < .005

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want to express a revision of the putative conclusion in Stage 2, work out the relation between the propositions that far? A related category of observed answers are “not r if not qadd ” for the Modus Ponens problem and “r if qalt”/“not r unless qalt” for the denial of the antecedent problem. These answers appear in all parts of Experiment 3, and should therefore be explainable from both strategies. If one reasons in two stages, one may conditionally conclude from the first premise “if p, then r” and the categorical premise “not p” that “not r” is the case. This putative conclusion may be reconsidered in Stage 2 when one takes into account the premise “if q, then r”, to “(not r, but still) r if qalt”/“not r, unless (when) qalt (then r)”. When one follows the integration strategy, on the contrary, the following models may be made explicit: not-p not-q

qalt not qalt

r r

This leads to the same “r if qalt” answer. An intriguing finding is the “r” answer that appears in 8.7% of the cases in Experiment 3.3 for the Modus Ponens problem. We presented the problems in the additional condition of Experiment 3.3 as follows: If p and qadd , then r. / p

The logically correct answer therefore is not “r”, but “one can’t know” (13.1%), “maybe r” (0%), or “r if q add” (68.4%). This answer is neither a simple conditional answer nor an integration answer. Given the following model, we cannot see how the “r” answer would be inferred: p …

q

r

Hence, we think that people used the amendment strategy even in Experiment 3.3. Indeed, the disjunctive or conjunctive combined answers may still be quite easy to solve for the Modus Ponens problem, but they are far more difficult for the other problems. This is an example of why we suppose that some people, even when forced to use the integration strategy, split up the problems and handled them in two stages. As the problems were randomised, participants may have developed the amendment strategy for the more complex problems and used it to solve the less complex ones. This explanation might also account for the following answer categories: “not p/not q”, “not p and not qadd”, “not p or not qalt” for the Modus Tollens problem; “not r”, “not r unless qalt” for the denial of the antecedent problem; “not p/not qadd” for the affirmation of the consequent problem. An alternative explanation that can account for the strange “not p or not qalt” and “not p and not qadd ” answers for the Modus Tollens problem is that this


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problem is difficult to solve: Including a conjunction, disjunction that has to be negated makes the problem even more complex (De Morgan rule). We end the discussion of Experiment 3 with three general conclusions. First, the existence of the two strategies is well confirmed. More than 70% of all given answers are predicted by at least one of the proposed strategies. Second, inducing the amendment like the integration strategy worked very well: Most answers given in 3.2 are predicted or (ad hoc) explained by the amendment strategy, and most answers given in 3.3 are predicted or (ad hoc) explained by the integration strategy. Third, the amendment strategy seems to be preferred over the integration strategy, which we attribute to the lesser load on working memory.

GENERAL DISCUSSION

In Experiment 1, we found that conditional inferences can be suppressed, repeating the findings of Byrne (1989). Indeed, reasoners made fewer denial of the antecedent and fewer affirmation of the consequent inferences (which are invalid inferences) when the major premise was accompanied by an alternative antecedent. Reasoners also made fewer Modus Ponens and Modus Tollens inferences (which are valid inferences) when an additional antecedent accompanied the major premise. However, we revealed a serious shortcoming in the design of both our replication Experiment 1 and Byrne’s Experiment 1 (1989): The answer alternatives only deal with the inferences that can be made with respect to the first conditional. In Experiment 2, reasoners could choose their answers among many answer alternatives. Because of this procedure, many reasoners gave a complex answer. Indeed, they did not only give information about the first conditional, they also tried to give information about both conditional premises. It even seems as though they attempted to reason conditionally. Indeed, in a somewhat broader definition, we can describe a “conditional” answer as one in which at least one part of the answer coordinates with the normal conditional pattern. This means that the single conditional answer is extended with another element mentioned in one conditional premise which enlarges the scope of that single conditional answer or makes it smaller. For example, given an affirmation of the consequent problem such as: If it rains, then Marianne gets wet. If she takes a walk, then Marianne gets wet. Marianne gets wet.

This example clearly illustrates how the conclusion “it rains” may be extended with “and she takes a walk”, which makes the scope smaller in this case. It is necessary to consider also that if we take into account such a broader definition, we did not find suppression in Experiment 2. Conditional reasoning with two

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conditionals is not the same as reasoning with one conditional. Consider the double conditionals in Experiments 1 and 2. Making the Modus Ponens argument means confirming the consequent (and doing something with the antecedent of the second conditional premise). Similarly, making the Modus Tollens argument means denying the antecedent (and doing something with the antecedent of the second conditional premise); making the denial of the antecedent argument means denying the consequent (and doing something with the antecedent of the second conditional premise); and making the affirmation of the consequent argument means confirming the antecedent (and doing something with the antecedent of the second premise). If reasoners are given a less arbitrary set of choices in a forced choice paradigm (as in Experiment 2), the suppression phenomenon disappears. Indeed, reasoners choose or provide answers that show clear signs of conditional reasoning. Our results indicate that it is not the case that the first premise of the argument was rendered false by the additional premise, so that people simply refuse to use it in reasoning, as Savion (1993) claimed. The results also show that reasoners did not reject the first premise of the argument (see Bach, 1993). Reasoners take into account the first premise and they reason with this premise. However, some of them change their conclusion in the light of the second (additional or alternative) premise, while some others may combine the two premises, as Byrne (1989) proposed. Politzer and Braine (1991) also responded to the experiments of Byrne (1989) by stating that the procedure caused the participants to doubt the truth of one of the premises due to its perceived inconsistency with other premises. However, Byrne (1991) did show data that falsified this account. Most of our data confirmed Byrne (1991), although a few of the answers (but clearly not all) observed in our experiments can be explained by means of this mechanism of doubt (see e.g., the “maybe p” answer in Experiment 3.1). With Experiment 3, we tried to solve some questions about the underlying reasoning processes by taking into account the broad palette of answers. In Experiment 3.1, we did this by changing the task of Experiment 1 into a production task. In Experiment 3.2, we presented the conditionals in two stages. The first conditional was presented together with a categorical premise and the participants were asked to give a conclusion. Next, we gave the second conditional and asked the participants if they wanted to change their putative conclusion, and if so, how. In Experiment 3.3, we presented the participants integrated conditionals, like: If she has an essay to write and the library stays open, then she will study late in the library.

We hypothesised that such a presentation would elicit reasoning from integrated conditionals. Most of the answers in Experiment 3.2 were amendment answers and most of the answers in Experiment 3.3 were integration answers, consistent


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with our predictions. Moreover, neither the answer categories of Experiment 3.2 alone nor those of Experiment 3.3 alone can account for all answer categories in Experiment 3.1 (i.e. both strategies are used in Experiment 3.1). Thus far, the results confirm our predictions. However, we must admit that some of our predictions were not confirmed: in Experiment 3.3 we observed answers that we classified as amendment answers. Only if the reasoning progresses in two stages would these answers be explainable. What is brought to light by this is that, at the moment, we do not yet know enough of the strategic and interpretive components that play a crucial role in reasoning. What we do know, however, is that it requires a lot from working memory to take into account more than two models at the same time, especially when they each contain three atomic propositions (Schaeken & Johnson-Laird, 2000). We conclude with some theoretical points of view. Some other researchers have reported suppression effects. Chan and Chua (1994) interpreted their findings in terms of the relative salience model. According to them, the critical component is the relative salience of the two antecedents, with respect to the consequent as interpreted by the reasoners. Consider the following problem: If p then r. If q add then r.

Chan and Chua predict that suppression of the valid inferences (Modus Ponens and Modus Tollens) would occur only if “qadd ” was more salient relative to “p” for the occurrence of “r”, and that the probability of suppression would increase with an increase of relative salience. The results, which are not predicted directly by the mental model theory, confirmed their hypothesis. We agree with them that this finding shows that the mental model theory fails to give a principled account of the critical interpretative component involved in reasoning (see also Fillenbaum, 1993). However, we do not agree with their proposed implementation of the principle of salience. They suggest a production system (see Anderson, 1983) comprising two fairly simple production rules: Prodl: IF qadd or not-q add is unknown, THEN scale qadd according to the assertion p (i.e., qadd /P) Prod2: IF (qadd /P)