The initial representation in reasoning towards an ...

THE QUARTERLY JOURNAL OF EXPERIMENTAL PSYCHOLOGY 2011, 64 (2), 339 –362

The initial representation in reasoning towards an interpretation of conditional sentences Walter Schroyens1 and Senne Braem2 1

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Laboratory of Experimental Psychology, University of Leuven, Leuven, Belgium Department of Psychology, University of Gent, Gent, Belgium

All accounts of human reasoning (whether presented at the symbolic or subsymbolic level) have to reckon with the temporal organization of the human processing systems and the ephemeral nature of the representations it uses. We present three new empirical tests for the hypothesis that people commence the interpretational process by constructing a minimal initial representation. In the case of if A then C the initial representation captures the occurrence of the consequent, C, within the context of the antecedent, A. Conditional inference problems are created by a categorical premise that affirms or denies A or C. The initial representation allows an inference when the explicitly represented information matches (e.g., the categorical premise A affirms the antecedent “A”) but not when it mismatches (e.g., “not-A” denies A). Experiments 1 and 2 confirmed that people tend to accept the conclusion that “nothing follows” for the denial problems, as indeed they do not have a determinate initial-model conclusion. Experiment 3 demonstrated the other way round that the effect of problem type (affirmation versus denial) is reduced when we impede the possibility of inferring a determinate conclusion on the basis of the initial representation of both the affirmation and the denial problems. Keywords: Conditionals; Mental-models theory; Satisficing; Representation; Interpretation; If; Initial representation; Reasoning; Deduction; Inference.

We make inferences from what we believe and what we perceive. This enables us to comprehend and to predict the contingent nature of our environment. Reasoning from conditional assertions (e.g., of the form “if A then C”) is a prime example because they express a contingency between the antecedent, A, and the consequent, C. Without conditional relations we would have no way to plan, to make decisions, or to understand our world. As Evans, Handley, and Over (2003, p. 324) write: “One of the most influential theories of

human reasoning is that of mental models, and this theory has been applied in detail to conditional reasoning (Johnson-Laird & Byrne, 1991, 2002; Johnson-Laird, Byrne, & Schaeken, 1992).” Mental-models theory (Johnson-Laird & Byrne, 2002) postulates that individuals understand sentences by representing possibilities. In accordance with Gricean principles of conversation (Grice, 1975; also see Levinson, 2000), people assume that the information they are confronted with is true: Initially they will not consider incompatible

Correspondence should be addressed to Walter Schroyens, Independent Senior Research Fellow, University of Leuven, Laboratory of Experimental Psychology, Tiensestraat 102, 3000 Leuven, Belgium. E-mail: [email protected] # 2010 The Experimental Psychology Society http://www.psypress.com/qjep

339 DOI:10.1080/17470218.2010.513734


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possibilities. This is the so-called truth principle (Johnson-Laird, 1999, p. 116). Moreover, since people have limited processing resources, they cannot and hence will not even consider all true possibilities from the outset. This is the implicitmodel principle. Content and context aid the construction of alternative possibilities that are not represented ab initio. This is the idea behind the socalled principles of semantic and pragmatic modulation. The basic representational principles (the principles of truth, implicit models, and semantic and pragmatic modulation) constrain the reasoning process, which is conceptualized through a threestage process. People first form a minimal initial representation of the premises. “Basic conditionals have mental models representing the possibilities in which their antecedents are satisfied, but only implicit mental models for the possibilities in which their antecedents are not satisfied” (Johnson-Laird & Byrne, 2002, p. 655). This “implicit-model principle”, which we also refer to as the initial-model assumption, means that, as an absolute minimum, a conditional of the form if A then C is initially represented by an explicit model of the possibility in which both antecedent, A, and the consequent, C, are satisfied.1 A C ...

alternative possibilities. Content and context will also aid the process of constructing alternatives to the explicit possibility/model considered first. It helps giving body (also known as “fleshing out”) to the empty place holder—that is, the implicit model , . . ... The validation stage is not compulsory. Mental-models theory assumes that people are often cognitive misers who satisfice and will not engage in critical search for counterexamples to test their possibly mistaken believes (i.e., inferred conclusions). The present study set out to provide new empirical tests for the initial-representation principle. For the purpose of investigating the implicit-model assumption, we could suffice with considering that it is a postulate of JohnsonLaird and Byrne (2002). It has a rational basis, however (see, “The principle of parsimony”, Johnson-Laird, 2005; also see Byrne & JohnsonLaird, 2009; Frazier, 1999, chap. 2). Humans are limited in their processing capacity to represent and maintain information in working memory, and not representing all possibilities ab initio reduces the load on the representational system. Consider, for instance, the so-called modus ponens problem (henceforth MP). If A then C A

Other true possibilities (i.e., cases that are possible under the assumption that the conditional is true) are initially not represented explicitly. They are alluded to by the elliptical model, which has no explicit content. In the conclusion – formulation stage, people formulate a putative conclusion that holds in the initial models. The model theory also assumes that people can be critical thinkers: In a final processing stage they can engage in a validation stage during which they consider

(If something is a bird, then it can fly)

(Tweety is a bird)

In the context of this problem it is redundant to consider the contingencies where [not-A] might hold (e.g., Tweety is the name of an aeroplane or my dog). In other situations it is not redundant to consider such negative cases. Imagine we are given the information [not-C]. This creates the context of a so-called modus tollens problem (henceforth MT): If A then C (If something is a bird, then it can fly) Not-C (Tweety can not fly)

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The exact relational richness of the initial model need not concern us here. For present purposes it is not an issue whether the initial model includes a directional link between the antecedent and consequent. Inspired by Evans’s (1993; also see, Markovits, 1993) analyses of Johnson-Laird and Byrne (1991), Barrouillet and Lecas (1998) already introduced directionality in the model theory, which they denoted as [A!C] and which was taken over by Evans and Over (2004). Given the left-to-right order between A and C, the “arrow” is in fact redundant. Indeed, it is just a notation: Nobody supposes people have an “arrow” in their head. The notation one prefers does not change the theory, which states that “the antecedent refers to a possibility, and the consequent is interpreted in that context” (Johnson-Laird & Byrne, 2002, p. 647 italics added).

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THE INITIAL REPRESENTATION OF CONDITIONALS

In this context alternative possibilities that consider [not-C] are obviously relevant. The net gain of not considering all possibilities all the time (i.e., only considering them when there is a need to do so) is a gain that will not be lost and is a function of the divergence in the frequency with which adaptively rational people encounter MP versus MT problems. The initial-model assumption is crucial for explaining the robust finding that affirmation arguments (e.g., inferring an affirmation of the consequent, AC, e.g., “Tweety can fly” from MP) are more readily endorsed than the standard denial arguments (e.g., inferring a denial of the antecedent, DA, “Tweety is not a bird”, from MT). The initial model supports the standard affirmation inferences, but does not support the standard denial inferences. Initial models do not represent the categorical premise of the denial problems, like the MT problem. In the process of integrating the multiple sources of information that form the problem, people follow the fundamental principle that something cannot be true and false at the same time. In the context of MT, reasoners are informed that the consequent does not hold: [not-C]. This categorical premise eliminates the explicit model in the conditional’s initial representation to leave only the implicit model. This implicit model [. . .] has no explicit content. Hence, from an MT problem initially nothing follows: One draws a temporary blank. No explicit event is specified to occur in combination with the categorical fact [not-C]. Aided by pragmatic context and semantic content individuals will alleviate the temporary blank. They will need to consider models of alternative possibilities before they can make a denial inference. In the context of the affirmation problems, reasoners do not need to consider alternative models: They can satisfice with the conclusion that follows from integrating the initial representation of the two problem premises: Individuals should be inferential satisficers; That is, if they reach a credible (or desirable) conclusion, or succeed in constructing a model in which such a conclusion is true, they are likely to accept it, and to overlook models that are counterexamples. Conversely, if they reach an incredible (or

undesirable) conclusion, they are likely to search harder for a model of the premises in which it is false. (Johnson-Laird, 1995b, p. 186)

Given that people will satisfice when they can, it is thus easier to endorse the affirmation arguments than the denial arguments. The robustness of this main effect of problem type (affirmation versus denial) has been established by meta-analyses (Schroyens & Schaeken, 2003; Schroyens, Schaeken, & d’Ydewalle, 2001a). Not a single study has failed to produce the higher endorsement rates of the standard affirmation arguments. The initial-representation principle is also fundamental to mental-model theory’s account of other bench-mark phenomena. Consider the task in which people consider contingencies in relation to “if the letter is an A then the number is a 2”. For instance: The letter is not an A, and the number is a not a 2

In standard truth-table evaluation tasks people judge whether this false-antecedent – false-consequent contingency (henceforth FF) makes the rule true, makes the rule false, or is irrelevant to the rule. These studies show that many people judge the false-antecedent cases irrelevant (for meta-analyses see Schroyens, 2010a). Mentalmodels theory invokes the implicit-model principle, amongst others, to explain irrelevant judgements of false-antecedent cases: The false-antecedent cases are not part of the initial representation and might thus seem irrelevant upon a first and possibly only reading of the rule. An initial-representation principle seems intuitively plausible and is arguably impossible to circumvent. The initial-model principle of the mental-models theory has nonetheless become the subject of controversy, and critics repeatedly raise questions about its explanatory adequacy. Consider, for instance, the initial-representationbased account of irrelevant judgements. Evans, Over, and Handley (2005, p. 1049) “find this account highly implausible”. Or, as stated in Handley, Evans, and Thompson (2006): “Although Johnson-Laird and Byrne propose that people do sometimes forget these implicit models, this cannot be a strong tendency given other


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proposals in the theory” (p. 560). The literature contains other instances where negative arguments are presented against mental-models theory by minimizing or even ignoring the principle (see, e.g., the analyses in Schroyens & Schaeken, 2004, of the negative arguments in Evans et al., 2003; also see, Schroyens, 2009a, 2010b, 2010c). Though the negation of a negative argument results in a positive contribution, we do not engage here in counterargumentation against the critiques aimed at mental-model theory’s initial model principle. Instead, we use the idea of satisficing with initial-model representations to generate novel predictions and, thus, to provide potentially positive arguments in favour of mental-models theory. As it turns out, the new predictions are corroborated. That is, the experiments provide a clear and perhaps compelling corroboration of the model theory’s assumption about the initial representation of conditionals.

EXPERIMENT 1 When using the inference-evaluation format of the conditional-inference task, it is standard practice to have participants evaluate the determinate or standard conclusions shown in Table 1. Looked upon from a model-theoretic perspective,

however, one is comparing “apples and pears” in comparing the standard affirmation and denial inferences. Table 1 also shows the inferences that individuals can make from their initial model of the premises. One will see that there is a conflict between the standard conclusions and conclusions derived from the initial models in the case of the arguments based on denial. The affirmation inferences are initial-model conclusions whereas the denial inferences are not. In the standard inference-evaluation task people are not given the opportunity to satisfice with their initial conclusion that nothing seems to follows from a denial problem, whereas they can satisfice with the initial-model conclusion for the affirmation problem. To test the thesis that people tend to satisfice with the initial-model conclusion, participants evaluated both the standard arguments and the nothing-follows arguments. That is, they reasoned about the problems once with the standard conclusion and once with the nothing-follows conclusions. Since “nothing follows” is the initial-model conclusion for denial problems, people will be more likely to accept and satisfice with “nothing follows”. The likelihood of accepting the standard denial inferences should therefore be larger than the likelihood of rejecting “nothing follows” if this response is presented for evaluation with these denial problems. Let me explain.

Table 1. Formal representation of the four basic conditional inference problems and the different conclusions derived from these problems Problem type Affirmation

Conditional: If A then C Categorical premise Conclusion Initial-model conclusion Standard conclusion

Denial

Valid MP

Invalid AC

Valid MT

Invalid DA

A

C

not-C

not-A

C C

A A

null not A

null not C

Note: MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; MT ¼ denial of the consequent, i.e., “modus tollens”; DA ¼ denial of the antecedent. The “null” inference refers to the conclusion that “nothing follows”. The distinction between valid and invalid arguments refers to the logical validity of the standard determinate inferences.

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Let us assume that people accept the standard denial arguments in 70% of the cases. This means that the standard denial arguments are rejected in 30% of the cases. Everything else being equal, those who reject the standard argument would accept that nothing follows. That is, in 30% of the cases the nothing-follows argument would be accepted, and, hence, in 70% of the cases it would be rejected. In short, everything else being equal, one would expect the acceptance rates of the standard denial arguments (70%) to equal the rejection rates of the nothing-follows arguments (70%): Acceptance rates of the standard and nothing-follows arguments should sum to unity. The initial-model principle implies that not everything else is equal. According to mental-models theory the initial-model conclusions are by definition the first conclusions one considers. If people tend to satisfice with their initial-model conclusion, it follows that the acceptance rates of the nothing-follows conclusions would be higher than the 30% one needs to predict whether satisficing has no explanatory import. Hence, the rejection rates of the nothing-follows conclusions would be lower than the 70% acceptance rates of the standard denial arguments. Of course, the idea is not that people satisfice with just any given conclusion. People will tend to satisfice with initial-model conclusions. That is, they will not tend to accept a nothing-follows conclusion for the affirmation problems. Something specific can be derived on the basis of the initial-model representation. Something is not nothing. Those people who have a tendency to satisfice will therefore not accept the nothingfollows conclusion about affirmation problems, and the likelihood of accepting the initial-model conclusion (i.e., the standard affirmation argument) will tend to match the likelihood of rejecting that nothing-follows conclusion. For instance, meta-analyses show that standard MP arguments are accepted in almost 100% of the cases. We thus expect that the nothing-follows arguments are rejected in about 100% of cases. Comparing the affirmation and denial problems, it follows that there should be an interaction between problem type and the sort of conclusion

to be evaluated (standard versus “nothing follows”). The difference between the acceptance of standard conclusions and the rejection of nothing-follows conclusions should be greater for denial problems than for affirmation problems. Participants not only evaluated both the standard arguments and the nothing-follows arguments. They also evaluated opposite-conclusion arguments. Indeed, we argued that “everything else being equal, those who reject the standard argument would accept the nothing-follows argument”. This assumption is not unquestionable (Oaksford, Chater, & Larkin, 2000). It is not necessarily the case that people who reject the standard conclusion [C] would conclude that nothing follows, even if the ceteris paribus clause holds. Though intuitively absurd, they might conclude [not-C] (Bonnefon & Villejoubert, 2007).

Method Participants and design Forty-four University of Leuven students received credit points or were paid for their participation. They evaluated 12 different arguments, which were presented twice in a blocked order. The arguments varied as a function of logical validity (valid vs. invalid), problem type (affirmation vs. denial), and conclusion type (standard/opposite/nothing follows). Materials and procedure Each participant participated by running a custom-made computer program. Participants first read the instructions, then received two exercise problems (about “or else”), and subsequently received two blocks of 12 arguments. Within each block they were randomly given one of the four problems with one of the three conclusion types. With each of the four problem types participants were given the affirmation of the inferential clause, the denial of the inferential clause, or the nothing-follows conclusions, thus producing the 12 (3 × 4) arguments. The affirmation of the inferential clause is the standard for the affirmation problems (MP/AC); it is the opposite of the standard denial inferences (MT/DA). The


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denial of the inferential clause is the standard denial inference; it is the opposite of the standard affirmation inference. A standard DA inference was presented as: Given: If the letter is a B, then the number is a 2. Given: The letter is NOT a B. Walter concludes: Hence, the number is NOT a 2.


The conclusion of the opposite DA read “Hence, the number is a 2”. Participants had to evaluate whether the derived conclusion was “logically correct or incorrect”, by clicking the mouse on a “correct” or “incorrect” button (“juist” or “fout” in Dutch).

Results and discussion Figure 1 presents the acceptance rates of the standard arguments and rejection rates of the null inferences (see Appendix A for a replication of Experiment 1). The standard arguments replicate previous findings (see, Schroyens & Schaeken, 2003, for meta-analyses). First, there is a simple main effect of logical validity. In mental-models theory this is a straightforward implication of the presumed meaning of conditionals. If the conditional is true, then the true– false (TF; A and not-C) cases are impossible whereas the false – true (FT, not-A and C) cases are possible. These cases capture the counterexamples to the logically valid and invalid arguments, respectively. Hence, people are less likely to defeat the logically valid arguments. Second, there is a simple main effect of problem type, which in mental-models theory is a consequence of the initial-representation assumption. Third, logical validity and problem type interact: The logical validity effect on the standard argument is larger for the affirmation problems.2 This interaction between inference type and logical validity confirms the thesis that the process of a validating search for

counterexamples, as proffered by mental-models theory, is conditional upon the likelihood of accepting the provisional conclusions. One cannot engage in a test of a conclusion when there is no such conclusion. Reasoners are generally less likely to accept the state of affairs denoted by the denial inference, such that a subsequent search for a counterexample has a smaller effect on the final endorsement rate of these inferences. That is, we have a two-way interaction between the type of problem (affirmation vs. denial) and its logical validity because testing the affirmation or denial inferences is conditional upon evaluating and accepting them as provisional conclusions, which is iffier in the context of denial problems. Unity minus the acceptance rates of “nothing follows” corresponds in principle to the acceptance rates of the standard inferences. The results show that though this equivalence is true in principle, in practice there is a clear discrepancy. The overall standard acceptance rates amount to .747, whereas the overall nothing-follows rejection rates amount to .614 (Wilcoxon T ¼ 48.0, N ¼ 30, Z ¼ 3.794, p , .001). Most importantly, the difference depends on the type of problem (affirmation vs. denial: d ¼ – 0.028 vs. d ¼ 0.295; T ¼ 28.0, N ¼ 32, Z ¼ 4.413, p , .001). People are as likely to accept the standard affirmation arguments as they are likely to reject the conclusion that nothing follows from the same premises (.784 vs. .812). Phrased otherwise, the likelihood of accepting the standard affirmation inferences and accepting the nothing-follows inference nicely sums to unity. In case of denial (MT/DA), people are much more likely to accept the standard denial argument than they are likely to reject the nothing-follows inference (.710 vs. .415; T ¼ 11.0, N ¼ 27, Z ¼ 4.276, p , .001). In summary, people are more likely to accept the initial-model conclusion (i.e., “nothing follows”)

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Performance with the standard arguments deviates slightly from the general picture, as established by meta-analyses (Schroyens & Schaeken, 2003). Overall, AC endorsement rates are generally higher than DA endorsement rates. This is not observed in the present study. Numerically DA was even more likely to be endorsed than AC. This is only an apparent difference. Statistically speaking, the difference was not significant. Moreover, every experiment is in essence just a sample. That is, statistically speaking, one is bound to observe some “deviations” from the central tendency. Meta-analyses have established that overall robustness of the AC– DA difference. We therefore do not interpret the present null effect in the difference between AC and DA.

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Figure 1. Acceptance rates of standard arguments and rejection rates of nothing-follows arguments (Experiment 1). MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; MT ¼ denial of the consequent, i.e., “modus tollens”; DA ¼ denial of the antecedent.

when this conclusion is given for evaluation. Conclusion type (standard vs. nothing follows) interacts with the type of problem. As expected, only on the denial problems do the acceptance rates of standard, opposite, and nothing-follows conclusions not sum to unity. This is an important observation. It shows that people are not simply accepting the conclusion that is given for evaluation. Our argument hinges on the observed difference between accepting the standard conclusion and rejecting the nothing-follows conclusions. One might argue that the equivalence between the standard acceptance rates and the nothingfollows rejection rates would not be expected in the first place because people might endorse the opposite conclusions. Those people who do not endorse the standard inferences would accept “nothing follows” or would accept the opposite conclusion. The results show that virtually nobody accepts the latter type of conclusion

(acceptance rates on MP, AC, MT, DA are, respectively: .00, .00, .02, and .01). That is, as assumed in mental-models theory, responses fall into two classes. People infer the default standard conclusion, or infer that nothing conclusive can be inferred from the premises. Virtually nobody endorses the opposite conclusions. These are supported neither by the initial models, nor by a fully explicated representation of all possibilities (i.e., models) that would be consistent with the premises. This finding strengthens our argument and analysis of the difference between the acceptance rates of the standard arguments and the rejection rates of the nothing-follows arguments. The new predictions and test of these predictions fail to disrepute the initial-model assumption. They provide converging evidence for this assumption in mental-models theory and, thus, give counterweight to resent questioning of its explanatory adequacy.


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Another alternative explanatory hypothesis for the results might be that it is people’s uncertainty about the arguments that explains the findings. The more uncertain people are about the arguments, the more likely they will endorse the “nothing follows” conclusion. It seems this uncertainty hypothesis is begging the question. Indeed, the difference in uncertainty is in itself a phenomenon that needs explanation. Mental-model theory’s initial-model principle explains this phenomenon. Setting aside the theoretically more fundamental and interesting question regarding people’s uncertainty about the standard arguments, the data provide evidence against using uncertainty as an explanation or predictor for the size of the nothing-follows effect. Figure 1 shows then that the present subject pool seems to be less certain about AC than about DA (.57 vs. .68). Under the uncertainty hypothesis one should thus expect the nothing-follows effect to be larger (certainly not smaller) on AC than on DA. Figure 1 shows this is clearly not the case and that the nothing-follows effect is actually smaller on AC than on DA. Uncertainty about the arguments therefore seems insufficient to explain the novel nothing-follows effects observed in the present study.

EXPERIMENT 2 The standard inference-evaluation task gives the initial-model conclusion for affirmation problems but not for denial problems. This is supposedly why there is a robust effect of problem type and why in Experiment 1 this problem-type effect increased in size when both the affirmation and denial problems are presented for evaluation with their initial-model conclusion. The affirmation problems were not affected by presenting them with the nothing-follows conclusion, while the denial problems were: Nothing-follows rejection rates were significantly lower than the acceptance rates of the standard denial arguments. At first glance, there seems to be another, simpler solution to the problem of the difference in the conclusions’ theoretical status (an initial-model conclusion or

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not). Indeed, not providing any conclusion for evaluation simply avoids the problem that the standard affirmation inference captures an initialmodel conclusion whereas the standard denial inference does not. However, the production format—in which people have to derive their own conclusion—has its own problems. People would have a reluctance to conclude that nothing follows, especially when they are asked to infer what follows from the premises (see, e.g., Revlis, 1975; also see Braine & O’Brien, 1998, for a similar assumption and arguments based on it). Asking “what follows” presumes that something follows. Hence, people would not tend to generate the conclusion that nothing follows. This is why the standard production task often instructs people to write down what follows, if anything. In a “guided-production task” one stresses even more clearly that it is not necessarily the case that something follows. This is accomplished by adapting the response format and highlighting the possibility that nothing follows. Participants have to choose whether something or nothing follows, and if something follows, they have to write down their conclusion (cf., Barrouillet, Grosset, & Lecas, 2000). The predictions about the production versus the guided-production task are analogous to the corroborated predictions of Experiment 1. The standard production task establishes the baseline, which should replicate previous findings (cf. Schroyens & Schaeken, 2003). First, people are more likely to produce affirmation than denial arguments. Second, people are more likely to produce logically valid than logically invalid arguments, and, third, the logical validity effect would be larger on affirmation than on denial problems. The guided-production task was expected to yield a similar pattern of results, though with some crucial shifts in the absolute production rates as compared to the standard production task. It should yield fewer denial arguments (i.e., more acceptance of nothing follows), and this nothing-follows effect should be larger than that on the affirmation problems. The guidedproduction task gives people the possibility to select a nothing-follows conclusion, which is the



initial-model conclusion for the denial problems. Since the nothing-follows effect should be larger on denial problems than on affirmation problems, it follows that the problem-type effect should be larger in the guided-production task.


Method Participants and design All 102 participants were 12th-grade students of a Flemish high-school. They evaluated four conditional problems that varied as function of logical validity (valid: MP and MT vs. invalid: AC and DA) and problem type (affirmation: MP and AC vs. denial: DA and MT). Task format (standard vs. guided production) formed a between-groups factor (N ¼ 38 and N ¼ 64, respectively). Materials and procedure The four inference problems were presented on a single sheet of paper. The instruction introduced a conditional rule (about a set of coloured geometrical shapes), and participants were told that in the problems they are also given information (“a fact they have to assume is true”) about the colour or shape of a figure such that their task is to indicate “what follows, if anything from the rule and given fact”. The problems were presented as follows (translated from Dutch): Take the following rule: If the figure is a square, then it is coloured red. Take the following fact: The figure is not coloured red. What follows about the shape of the nonred figure?

In the standard production task there was a single line on which participants could fill in their responses: Conclusion: . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..

In the guided-production task this was extended as: O Nothing follows. O Something follows, namely (write down your conclusion): Conclusion: . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..

Results and discussion Preliminary analyses has shown that some participants indicated that “nothing follows” from MP. Meta-analyses have firmly established the nearuniversal acceptance of the MP arguments on the type of knowledge-lean problems used in the present study. Almost everybody will accept the MP argument and will be certain that the standard conclusion follows. It follows that nothing follows is an “absurd” response caused by inattentiveness and/or failure to engage in the reasoning task. To reduce the noise in the data set caused by these inattentive and/or unengaged participants, their results were excluded from the analyses, leaving, respectively, 38 and 54 participants in the standard and guided-production tasks. One reason for the use of an evaluation or conclusion-selection format is probably the ease of scoring, inserting the data, and analysing them. Indeed, in a production task one typically observes some participants who generate a modal response. For instance, instead of the conclusion “the figure is a square”, they generate the modal responses “the figure might be a square”. Such uncertainty responses are correct for the logically invalid arguments. Given “if square then red” and “red”, it is possible but not necessary that the figure is a square. All analyses presented below were conducted with nonparametric statistics on the frequency of the standard conclusions—that is, the determinate conclusions that did not make use of the modal “may be” or “could be”. Figure 2 presents the production rates of the standard arguments (Appendix B presents all three types of response). Combined over the two task formats, we replicate the standard effects. That is, first, there is a main effect of affirmation versus denial (.804 vs. .690; Wilcoxon T ¼ 81.0, nontied N ¼ 28, Z ¼ 2.778, p , .01). Second, people generated more valid than invalid arguments (.908 vs. .587: T ¼ 0.0, N ¼ 38, Z ¼ 2.778, p , .01), and the effect of logical validity was larger on the affirmation problems than on the denial problems (d ¼ 0.391 vs. d ¼ 0.25; T ¼ 36.0, N ¼ 19, Z ¼ 2.294, p , .05). Of course, these overall effects are not particularly


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Figure 2. Percentages of different conclusions produced for the standard conditional inference problems as a function of the task format (Experiment 2). MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; MT ¼ denial of the consequent, i.e., “modus tollens”; DA ¼ denial of the antecedent.

interesting, especially since we expected to see significant shifts in the conclusion production rates as a function of the task format. Comparing performance on the standard and guided-production tasks corroborates our predictions. That is, first, there was a main effect of task format: Participants generated fewer standard conclusions in the guided-production tasks: .836 vs. .685; Mann – Whitney U ¼ 681, Z ¼ 2.953, p , .005. This could merely suggest that people have a tendency to select the given conclusion option that nothing follows. Indeed, for those who are not particularly engaged in the task it would obviate the need to write down a specific conclusion. However, we explicated that the initial-model principle yields the crucial prediction that there should be an interaction with problem type. Such an interaction also falsifies the alternative hypothesis that people simply

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accept the given conclusion option. People do not just satisfice with any given conclusion; they tend to satisfice with conclusions that seem to follow, and the first conclusions that seem to follow are the so-called initial-model conclusions. As in Experiment 1, the data corroborate the initial-representation principle: There was a reliable nothing-follows effect on the denial problems (.829 vs. .593: U ¼ 676.0, Z ¼ 3.083, p , .005) but not on the affirmation problems (.842 vs. .779: U ¼ 894.0, Z ¼ 1.238, p . .20), and the interaction was significant (U ¼ 844, Z ¼ 1.776, p , .05, one-tailed). This larger nothing-follows effect on denial than on affirmation problems is exactly what one would expect given that nothing follows from the initial presentation of the denial problems, whereas it is the standard conclusion that follows from the initial representation of the affirmation problems.




Figure 2 shows that there is an overall increase of correct solutions to the logically invalid problems (both AC and DA): There is a reduction in the proportion of standard conclusions. This means that the sum of the two logically correct responses (the modal “maybe p” and the nothing-follows conclusion) does not match across tasks and that more is going on in the partial shift from the correct “maybe” conclusion to the correct “nothing follows” conclusion in the guided-production task. The increase of logically correct responding accordingly shows that generating a conclusion (as in the unguided task) is somewhat more difficult than evaluating a given conclusion (as in the guided-production task). This also means that people’s evaluation of a given conclusion (like the nothing-follows conclusion in the guided-production task) is not merely determined by whether it allows them to satisfice. Some people go beyond the initial representations and evaluate “nothing follows” more profoundly. But, this is trivial and in any case already presumed. Satisficing is but a tendency. The validation principle in mental-models theory (people tend to look for counterexamples) and the satisficing principle are two sides of the same proverbial coin. If 60% of people satisfice, 40% of people do not and, according to model theory, attempt to look for counterexamples. The fact that almost 38.9% of people reach a logically correct evaluation of AC does not explain why this is less than the 50% of people who reach the logically correct evaluation of “nothing follows” when this is suggested for a DA problem. The crucial interaction between the type of problem and the task did not depend on the logical validity of the arguments (Z ¼ 0.067, p ..90). The nothing-follows effect tended to be larger on DA than on AC (29.3 vs. 12.8; U ¼ 875, Z ¼ 1.515, p , .10, one- tailed) and was larger on MT than on MP (18.0 vs. 0.0; U ¼ 841, Z ¼ 2.182, p , .05). That the nothingfollows effects were present on both the logically valid and the logically invalid arguments, without a statistically significant interaction, shows that the conclusions cannot be questioned by pointing towards the fact that the task format increased

their absolute level of logically correct responding on the invalid arguments.

EXPERIMENT 3 Both Experiment 1 and Experiment 2 provided evidence in favour of the initial-representation principle by setting up conditions in which people can satisfice with the hypothesized initialmodel conclusion for the denial arguments. “Nothing follows” from the initial representation of denial problems, and providing this conclusion yields the nothing-follows effects observed in Experiments 1 and 2 (i.e., acceptance rates of nothing follows conclusions are higher than the rejection rates of the standard conclusions). The present experiment was set up to provide converging evidence by working the other way round. Instead of allowing people to satisfice with their initial-model conclusion for the denial arguments, we decided to set up conditions such that the determinate initial-model conclusion no longer follows from the affirmation problems. Experiments 1 and 2 used an explicit type of referencing between the antecedent/consequent and the categorical premise that affirms or denies this so-called referred clause of the conditional. The denial problems (e.g., MT) invariably used an explicit negation: If the letter is an A, then the number is a 2 The number is not a 2.

Such a denial can be conveyed implicitly: If the letter is an A, then the number is a 2 The number is a 3.

The number 3 is not the number 2. Hence it is denied that the number is a 2. By introducing a negation in the referred clause, it is similarly possible to construct implicit versus explicit affirmation problems. Take the following conditional: If the letter is not an G, then the number is 5.

The antecedent is affirmed explicitly by the categorical premise stating “the letter is not a G”, whereas it is affirmed implicitly by a categorical premise stating, for example, “The letter is an


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H”. The letter H is an element of the contrastclass delineated by denying “the letter is a G”. Implicitness effects have already been demonstrated in the conditional-inference task (cf. Barrouilllet & Lecas, 1998, Evans & Handley, 1999; Schroyens, Schaeken, Verschueren, & d’Ydewalle, 1999; Schroyens, Verschueren, Schaeken, & d’Ydewalle, 2000). These studies all showed a main implicitness effect: People are less likely to endorse the standard conclusion when referencing between the major and minor premise is implicit. These implicitness effects have been observed in other experimental paradigms and are often referred to as matching bias (Evans, 1998). Indeed, all contemporary theories presume that a fast-and-frugal heuristic processing of the implicit problems yields an initial irrelevancy judgement. The two sources of information do not match, and some people will tend to satisfice with this initial irrelevancy judgement and conclude that nothing follows. The initial-representation principle yields the additional prediction that the implicitness effect should be larger on the affirmation problems than on the denial problems. To illustrate that a single processing hurdle of coping with implicit references implies a larger implicitness effect on affirmation versus denial problems, consider the average explicit MP and MT rates: p(MP) ¼ .973, p(MT) ¼ .768 (Schroyens & Schaeken, 2003). There seems to be a consensus that the implicitness problem (also known as matching bias) is a problem in the process of relating the two premises and integrating their respective representations. This means that the processing hurdle poses itself from the outset. Mentalmodels theory presumes the primacy of this processing hurdle (also see, Evans & Handley, 1999), even though this is not necessarily the case. The likelihood of accepting the implicit arguments is accordingly a proportion “a” of accepting the explicit arguments. In a parameterized model (Schroyens & Schaeken, 2003; Schroyens et al., 2001) this “a” captures the difficulty of dealing with the implicitness problem. Let us take an arbitrary value for a—for example, .50 (changing this parameter only changes the absolute size of the

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effects). This means that 50% of participants are able to cope with the implicit type of referencing. But, once these people have dealt with the implicitness problem they obviously have not solved the problem yet. They have only established the relation between the categorical premise and the conditional premise and, thus, still have to go through the same processes associated with processing the explicit problems. That is, acceptance rates of the implicit MP arguments would amount to .50 × .973 ¼ .486, and the acceptance rates of the implicit MT arguments would amount to .50 × .768 ¼ .384. The implicitness effect is computed as the difference between the explicit and implicit problems. This implies that the effect on MP would amount to .973 – .486 ¼ .487 whereas the implicitness effect on MT would amount to .768 – .384 ¼ .384. This illustrates the prediction that the implicitness effect should be larger on affirmation than on denial problems. It is similarly easy to show that the problem-type effect should become smaller when referencing is implicit than when it is explicit. The problem-type effect (in our example computed only on the valid arguments) would amount to .973 – .768 ¼ .205 for the explicit problems and would be half the size on the implicit problems when a ¼ .50 (.486 – .384 ¼ .102). When framed implicitly the affirmation arguments lose their head start as compared to the denial arguments: Satisficing with a determinate initial-model conclusion is no longer possible under implicit conditions. There is no determinate initial-model conclusion for both the affirmation problems and the denial problems. Hence, the presumed effect of being able to infer and satisfice with a determinate initial-model conclusion (i.e., the problem-type effect) should be reduced, as illustrated by our numerical example.

Method Design Participants served as their own control as regards the 32 arguments conveyed as a function of the conditional-inference problem (modus ponens, affirmation of the consequent, modus tollens,




and denial of the antecedent), conclusion type (the standard determinate inference or its opposite), and the presence of a negation in the antecedent and/or consequent of the conditional premise (“if A then C”, “if not-A then C”, “if A then not-C”, and “if not-A then not-C”). A betweengroups factor was formed by the referential status of the categorical premise (implicit versus explicit: see Table 2). Participants Eighty participants were assigned to the implicitor explicit-negations task (N ¼ 40 in both groups). Participants were students at the University of Plymouth, between 18 and 45 years of age. They were paid for their participation or received credit points towards a course requirement. Materials and procedure Each participant participated by running a custom-made program on an individual PC. The experiment was run in groups of 1 – 5 participants and lasted between 10– 20 min. Participants first read the instructions, then received 1 exercise

problem (about “or else”), and subsequently received the 32 experimental problems. They were randomly confronted with an MP/DA/ AC/MT problem with its standard determinate or opposite conclusion. All problems concerned a proposition describing a conditional relation between letter – number combinations written on cards. The arguments were presented in the following format: Given: If the letter is a B, then the number is a 2. Given: The letter is NOT a B. Conclusion: Hence, the number is NOT a 2. Is this conclusion TRUE or FALSE? True Impossible to tell False

The instructions informed the participants that their task was to evaluate the conclusions, by selecting one of three possible evaluations. They were told that “when you think that the conclusion MUST BE TRUE, you select ‘TRUE’”, that “when you think that the conclusion MUST BE FALSE, you select ‘FALSE’”, and that “when you think that it is impossible to tell whether the conclusion is true or false, you select ‘IMPOSSIBLE TO TELL’”.

Table 2. Exemplary representation of the conditional-inference problems (and their standard determinate conclusion) with an explicit or implicit type of referencing Problem type Affirmation Conditional premise If A then 2 Categorical premise Determinate inference If A then not-2 Categorical premise Determinate inference If not-A then 2 Categorical premise Determinate inference If not-A then not-2 Categorical premise Determinate inference

Denial

MP

AC

DA

MT

A 2

2 A

not-A (G) not 2

not-2 (3) not A

A not 2

not-2 (3) A

not-A (G) 2

2 not A

not-A (G) 2

2 not A

A not 2

not-2 (3) A

not-A (G) not 2

not-2 (3) not A

A 2

2 A

Note: MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; DA ¼ denial of the antecedent; MT ¼ denial of the consequent, i.e., “modus tollens”. Explicit or implicit type of referencing in parentheses.


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Results and discussion If falls outside the scope of the present manuscript to discuss the effects of introducing a negation in the antecedent or consequent of the conditional premise in the standard conditions where the affirmation or denial is explicit. The negation effects concur with Schroyens et al.’s (2001b) meta-analyses on the processing of negations. Space limitations also do not allow elaborations on the effect of giving the opposite conclusions. Appendix C presents the detailed findings for future reference. The main focus of the present experiment was to test the novel prediction that the implicitness effects should be larger on affirmation than on denial problems. Introducing negations in the conditional premises is a requisite to construct implicit affirmation arguments. Combined over the four conditionals in the implicit-negations groups, two problems for each

of the four arguments are phrased implicitly while two problems for each of the four arguments are phrased explicitly. Of course, one cannot simply compare the two implicit and explicit problems in the implicit-negations paradigm since one would introduce a confound with negations in the major premise (e.g., the implicit affirmations concern a conditional with a negative referred clauses, while the explicit affirmations concern a conditional in which the referred clause is affirmative). The implicitness effects are therefore established by comparing performance in the explicit versus implicit groups. Figure 3 presents the mean acceptance rates of the standard conclusions for the four conditional-inference problems. The results confirm the predictions and, thus, provide further corroborating evidence for the initial-representation principle. First, there was a main effect of the type of referencing (implicit versus explicit: .784

Figure 3. Acceptance rates of standard conclusion for the conditional inference problems as a function of an explicit or implicit type of referencing between the premises. MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; MT ¼ denial of the consequent, i.e., “modus tollens”; DA ¼ denial of the antecedent.

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vs. .667, Mann – Whitney U ¼ 427.5, Z ¼ 3.584, p , .001). Second, and most importantly, the effect of problem type (affirmation vs. denial) was larger when explicit referencing was used (.928 vs. .640, T ¼ 10, N ¼ 36, Z ¼ 5.074, p , .0001) than when an implicit type of referencing was used (.750 vs. .584: T ¼ 95.5, n ¼ 34, Z ¼ 3.453, p , .005). This means that there was a reliable interaction between type of referencing (implicit vs. explicit) and the type of problem (affirmation vs. denial; U ¼ 617.0, Z ¼ 1.761, p , .05, one-tailed). Figure 2 clearly shows this: The significant implicitness effect on the affirmation problems (.928 vs. .750: U ¼ 342.5, Z ¼ 4.402, p , .001) is significantly larger than the small and in casu nonsignificant implicitness effect on the denial problems (.640 vs. .584: U ¼ 679.5, Z ¼ 1.159, p . .10). The larger implicitness effect on the affirmation versus denial problems provides further corroborating evidence that converges upon the thesis that the implicit-representation principle provides an adequate explanation for the most robust finding in the conditional reasoning literature— that is, the problem type effect. Standard affirmation arguments would be accepted more readily because they follow from the initial representation of the premises in affirmation problems, whereas the standard denial arguments do not follow from the initial representation. By conveying the problems implicitly neither the initial representation of the affirmation problems nor that of the denial problems yields a determinate conclusion. Hence, as observed, the problem type effect is smaller when using an implicit type of referencing. Looking back at the data in Evans and Handley (1999), it seems we can be confident about the reliability of the interaction between the type of referencing and the type of problem. For instance, the two “implicit-negation groups” in their first experiment showed a 33% implicitness effect on the affirmation problems, whereas these groups exhibited a 20% implicitness effect on the denial problems. The same pattern was observed in their second and third experiments (29.5% vs. 22.5% and 43% vs. 19%, respectively). Evans and Handley (1999) neither discussed these results

nor presented statistical analyses of what is a rather straightforward interaction between two factors. This also means that Evans and Handley (1999) did not present an explanation of the phenomenon that we have presently established as a real statistically reliable effect (as compared to merely a suggestive numerical difference). We have illustrated how mental-models theory predicts the observed effects on the basis of the proposed initial representation of conditionals. The interaction between problem type and implicitness corroborates the initial-representation principle. This obviously does not prove the principle. Indeed, one could generate an alternative hypothesis. Schroyens et al. (2000) suggested, for instance, that people’s perception of a negative might contribute to the larger effect of implicit referencing on affirmation versus denial problems. To negotiate the hurdle of an implicit denial it seems one needs to call upon the knowledge that “B is not an A”. In this case the linguistic function of negation seems to be that of denying the supposition that there is an A (Greene, 1970; Wason, 1959). To negotiate the hurdle of an implicit affirmation it seems more appropriate to call upon the knowledge that “B is a not-A”, and in this case the negation seems to identify a contrast class (Wason, 1965; Wason & Jones, 1963). The more difficult aspect of coping with implicit affirmation might be that of considering the presence of “B” (or the presence of “C”, or “D”, or . . . ) to signify an affirmation of the entire class. In terms of the additional inference that people would need to make, this means that people need to invoke a so-called “OR-introduction” mechanism (Rips, 1994): If it is true that B is the letter on the front of a card, then it is also true that B, or C, or D, or . . . , or Z is the letter on the front of the card. In contrast, the knowledge that a B is not an A is sufficient to know that the affirmation of the letter B denies the presence of the letter A, as in the case of implicit denial problems. The “processing-of-negations” hypothesis might sound plausible. As things stand, parsimony rules against considering the hypothesis in further detail. Consider again our example, where the implicitness weight a was kept constant. The


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minimal assumption that a is the same for affirmation and denial suffices to predict an interaction between problem type and type of referencing (implicit vs. explicit). However, this does not mean that the size of the interaction (a difference between two differences) might not increase due to other, additional processing difficulties like the one associated with processing negatives. When we estimate the implicitness weights on the basis of the observed data (implicit MP ¼ a × explicit MP, hence a ¼ implicit/explicit MP), the estimates indeed suggest that there is an additional difficulty associated with negotiating implicit affirmation (a-MP ¼ .81, a-AC ¼ .80, aMT ¼ .90, a-DA ¼ .93). That is, the present data suggest that Schroyens et al.’s (2000) processing-of-negations hypothesis might be investigated further and that the presumed processes are not redundant to provide a complete account of the data within a theory like mental-models theory (which predicts the interaction without making additional processing assumptions).

GENERAL DISCUSSION People ’s beliefs are not bestowed upon them via divine intervention. This is trivial, but it is nonetheless neglected by theorists who proffer that people reason from (vs. reason towards) an interpretation. When the human processing system is confronted with the premises of a reasoning problem, it will have to start building a representation from the bottom up. The initialmodel assumption holds that, ab initio, they only represent what is explicitly narrated by means of the sentence. The sentence “if it rains, then the streets get wet” is about it raining and the streets getting wet. People accordingly first consider only this contingency. Constructing a minimal initial representation allows a saving of limited

processing resources. When the initial representation yields a conclusion that satisfies the task demands and/or processing goals, people can satisfice without considering alternatives. Satisficing and the initial-model representation are thus intrinsically linked. Recent negative arguments against mental-models theory, however, questioned the explanatory adequacy of satisficing with initial-model conclusions. These critiques are rendered null and void by taking into account that satisficing with the initial-model conclusion depends on the specific environmentally determined processing demands (i.e., the task). Our studies provided new predictions and new evidence for the context- and/or goal-dependent nature of satisficing with putative initial-model inferences. All three experiments demonstrated that when one changes the task such that one changes (i.e., stimulates or blocks) the possibility to satisfice with an initial-model conclusion, performance is affected accordingly. Ormerod (2000) already elaborated upon the underlying idea of cognitive economy and provided an insightful explication of a “principle of minimal completion” in reasoning within mental-models-based theories of reasoning: “Individuals endeavour to flesh out or add as few models as possible under the control of their current inferential goal” (Ormerod & Richarson, 2003, p. 468). People can cash in on the possibility to save on processing resources by satisficing with conclusions that follow from the initial representation. When the initial representation yields a determinate conclusion (as is generally the case for affirmation problems), people are in the position to satisfice. They do not need to consider alternative possibilities to meet the task requirement.3 It was accordingly shown that when people are given the initial-representation conclusion for denial problems, the robust problemtype effect—a bench mark phenomenon for all

3 The task- or goal-dependent nature of satisficing is also illustrated by the instruction effects reported in Schroyens and Schaeken (2003; also see Schroyens, Schaeken, & Dieussaert, 2008; Schroyens, Schaeken, & Handley, 2003). When instructions stress the need to draw logical inferences and explicate that such inferences are conclusions that follow necessarily and not just possibly, the task demands are stricter. People are accordingly less likely to satisfice with conclusions that follow possibly (like the initialmodels conclusion), but do not follow necessarily.

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theories of reasoning—becomes even larger (Experiments 1 and 2): People tend to accept the nothing-follows conclusion more often than they reject the standard denial argument. Experiment 3 conversely showed that when we hamper construction of a determinate initial-model conclusion for the affirmation problems, this robust difference between the affirmation and the denial problems becomes smaller. We also referred to the initial-model assumption as the “initial-representation assumption”. This is a more theoretically neutral label. It does not presume that sentential information is represented via mental-model like representations— that is, representations of states of affairs that would be possible when it is assumed to be in line with the Gricean principles of communication that the sentence is true. We suggest that any theory will need to incorporate an initial-representation assumption irrespective of whether it proffers model-like or higher level proposition-like representations. Failure to do so—in one or the other form—seems to render a theory incomplete and inapt to explain the basic findings of Experiments 1 – 3, as well as other findings reported in the literature (see, e.g., the so-called illusory inferences reported by Johnson-Laird & Savary, 1995, 1996, 1999). We illustrated that the findings are predictable on the basis of an initial-representation principle and the tendency for people to satisfice with a conclusion that follows from the presumed initial representation. The additional body of evidence in favour of an initial-representation principle should provide counterweight to recent questioning of the principle’s explanatory adequacy and the construction of strawman arguments against mental-models theory that are based on the mistaken idea that conditionals are material-implication-like expressions (also see, Schroyens, 2009a, 2010b). The initial-representation principle, which reflects the idea that everybody starts with a minimal representation (i.e., typically one model), implies that one can never make simplistic arguments that derive from assuming that all people construct a representation in which they explicitly consider all possibilities consistent with the conditional. It

implies that one cannot argue as if mental-models theory predicts that the probability of “if p then q” has to take account of false-antecedent cases (cf. Schroyens & Schaeken’s, 2004, analyses of Evans et al., 2003). At best some people will sometimes do so, as shown by Schroyens et al. (2008). It implies that one cannot argue as if mental-models theory predicts that “if p then q” and “if p then not-q” are compatible by virtue of sharing a common possibility (Handley et al., 2006). At best some people might sometimes evaluate them to be such (see Schroyens, 2009b). It implies that one cannot argue as if mental-models theory predicts that false-antecedent cases make a conditional true (e.g., Evans, 2006). At best some people might sometimes judge cases as such (see Schroyens, 2010a). The primary aim of the present paper was to investigate and test the initial-model principle of mental-models theory. Testing hypotheses about cognitive processes does not imply that theories need to make opposing predictions. If our analyses are right then all theories that make an initial-representation assumption and subscribe to the principle of minimal completion (also known as satisficing) will be able to explain the novel effects observed in Experiments 1, 2, and 3. Consider, for instance, the so-called “rulebased” or “mental-logic” theories. To explain the robust difference between affirmation and denial arguments, these theories distinguish between direct and indirect inferences rules or schemata. An example of a direct inference rule is MP, or “IF” elimination in Rips (1994). People have a production – action-like inference rule having the import that when working memory contains a conditional (e.g., If p then q) and at the same time contains an affirmation of its antecedent (p), then the consequent (q) can be added to working memory. There is no such direct inference rule to solve MT. A longer and thus more errorprone indirect line of inference can be used. People generate the hypothesis “p”, which yields the inference “q” via application of IF-elimination. Given “q”—which is hypothetical because it is inferred on the basis of a hypothesis—the processing system now contains a contradiction.


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The hypothetical “q” is contradicted by the factual “not-q” (i.e., the categorical premise) in MT. The contradiction completes the reductio ad absurdum argument. Since the hypothesis yields a contradiction, the hypothesis can not hold: Barring contradictions, it cannot be true that “p”, hence “not-p” must be the case (Q.E.D.). One will see that the process of generating the hypothesis might explain why rejection rates of nothing follows are lower than the acceptance rates of the standard denial arguments. When the standard denial argument is given for evaluation, people have “not-p” as a basis to generate the hypothesis “p”: One supposes the opposite of the conclusion one wants to prove.4 When people evaluate whether “nothing follows”, people do not have the determinate conclusion (e.g., “not-p”) to generate the hypothesis. That is, people do not have a guideline to initiate the hypothetical reductio ad absurdum strategy, and solving MT becomes more difficult. Consider, for instance, that the additional difficulty of having to generate the hypothesis completely unguided has a weight of .80, and consider that standard MT arguments (when the reductio argument is guided by the given conclusion) are accepted in about 75% of cases (see Figure 1). This means that the nothing-follows conclusion would be rejected in .75 × .80 of cases. Given this .60 nothingfollows acceptance rate, the rejection rates would equal .40. This is the situation depicted for MT in Figure 1: There is a 35% nothingfollows effect (.75 vs. .40). In contrast, the direct IF-elimination rule suffices both to accept the MP argument and to reject the nothing-follows conclusion. The conclusion follows directly from the categorical premises, which almost automatically triggers application of the IF-elimination rule. As before, given that something follows almost automatically, people will tend to reject that nothing follows.

In recent years the so-called suppositionalconditional theory (e.g., Evans & Over, 2004) has presented itself as a new theory. Its account of inference-task performance about knowledge-lean conditionals does not seem to differ, however, in any substantial way from the processes proffered by rule-based theories. It proffers the above reductio-ad-absurdum strategy to “prove” denial arguments and like rule-based theories invokes socalled pragmatic implicatures or invited inferences to account for endorsements of the logical fallacies (i.e., AC, DA). The theory would thus seem able to account for the present findings. More recently, however, the theory has proffered an alternative, heuristic strategy to generate inferences, which has been labelled “simple-equivalence reasoning” (Evans, 2007, p. 65; Evans, Handley, Nielens, & Over, 2007, p. 1782). As expressed more than 30 years ago by Braine (1978, p. 8): “Subjects who fail to work out these commitments [FT is a possible case] assume that the events or states of affairs represented by p and q are mutually contingent, that is, both present of both absent.” Addition of simple-equivalence reasoning has rendered suppositional theory unclear. It seems simple-equivalence reasoning is proposed to replace the invited inferences (also known as pragmatic implicatures) account. It would indeed eliminate the ad hoc and circular character of supposing that “if not-p then not-p” is added as an invited inference in the context of DA (i.e., “not-p therefore not-q”) while “if q then p” is added as an invited inference in the context of AC (i.e., “q therefore p”). However, if simple-equivalence reasoning is presumed to replace the invited inferences, then the theory is decidedly defective. Indeed, it cannot even account for the robust difference in MP versus AC. Both MP and AC follow from simple-equivalence reasoning. Invoking directionality (AC is a backward inference from the consequent to the antecedent) does not salvage the theory, as it

4

Experiments 1 and 3 show that the rejection of the opposite conclusions are higher than the acceptance rates of the standard denial arguments, This is consistent with the presumed difficulty of generating the opposite conclusion as the hypothetical start of a reductio argumentation. While the standard denial arguments require people to generate the opposite conclusion themselves (though with the guideline of the standard denial conclusion), it is given in the context of opposite conclusion arguments.

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would then predict that MT (a backward inference) is endorsed less frequently than the DA, which is the opposite from what is actually observed (Schroyens et al., 2001). Until the suppositional-conditional theory is clarified, a further discussion seems unwarranted. The so-called conditional-probability model (Oaksford et al., 2000) is almost by default defective in explaining the present set of findings. Indeed, the model has not received any substantial algorithmic level specification—for instance, about what an initial representation of conditionals would be and how such representations (whether symbolic or subsymbolic) are transformed and manipulated in the reasoning process. One could engage in a post hoc model-fitting exercise of the computational model, but this was identified by Gigerenzer (1998) to be one of the “surrogates for theories”. (See Schroyens & Schaeken, 2003, for a detailed discussion of the problematical nature of the implicitness effects like the ones reported in Experiment 3, and see Schroyens, 2005, for a detailed discussion of Experiment 1 in relation to the conditional-probability model.) In summary, we derived new empirical tests of the initial-representation principle. This has resulted in establishing three new phenomena that are readily explained when considering an initial-representation assumption in a theoretical framework that subscribes to the idea that people abandon a chain of reasoning with a (potentially suboptimal) conclusion as soon as this conclusion satisfies the task demands. The mental-models theory explicitly proffers an initial-representation principle and also adheres to a satisficing principle. These principles are at the core of mental-model theory’s explanation of the most robust finding in the conditional reasoning literature: Affirmation arguments are easier than denial arguments (see Schroyens & Schaeken, 2003, for a meta-analysis). Experiments 1 and 2 provided the initial-model conclusion for the denial problems (“nothing follows”). This allowed people to satisfice in the context of denial problems. Experiment 3 worked the other way round. There is no longer a determinate conclusion that follows from the initial representation of implicitly related premises. This

imputes the possibility to satisfice with a conclusion, as indeed there is no longer a conclusion that follows from the initial representation of the problems. Original manuscript received 20 July 2009 Accepted revision received 30 November 2009 First published online 18 November 2010

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APPENDIX A Replication of Experiment 1


Figure A1 presents the representation format of the conditional inference problems, presented on screen with E-prime software.

Figure A1. Representation and response format of the conditional inference problems. To view a colour version of this figure, please see the online issue of the Journal.

The eight inference problems (MP, AC, DA, or MT with the standard or nothing-follows conclusion) were the first set of problems in a larger battery of conditional reasoning problems, which are not discussed here. Participants were first presented with only the conditional premises and had to press the space bar to see the complete argument, whose conclusion they had to evaluate as correct or incorrect (“juist” vs. “fout” in Dutch). A total of 39 first-year psychology undergraduates were paid for their participation (8 euros) or participated for partial completion of a course requirement. Figure A2 presents the mean acceptance rates of the standard arguments and the rejection rates of the nothing-follows arguments. The standard arguments replicate the standard main effect of problem type (Wilcoxon T ¼ 32.0, N ¼ 18, Z ¼ 2.330, p , .05), the main effect of logical validity (T ¼ 16.0, N ¼ 23, Z ¼ 3.71, p , .05), and their interaction (T ¼ 57.5, N ¼ 27, Z ¼ 3.159, p , .05). The study also replicates the findings of Experiment 1. There was a main nothingfollows effect (T ¼ 21.0, N ¼ 20, Z ¼ 3.159, p , .05), which, most importantly, interacted with problem type (T ¼ 78.5, N ¼ 23, Z ¼ 1.81, p , .05, one-tailed). That is, the null effect on the affirmation problems (T ¼ 18.0, N ¼ 40, Z ¼ 0.969, p ..30) was smaller than the effect on the denial problems (T ¼ 28.0, N ¼ 20, Z ¼ 2.876, p , .005).

Figure A2. Endorsement rates of the standard conditional arguments and rejection rates of the nothing-follows arguments.

360



APPENDIX B Detailed findings of Experiment 2 Table B1. Percentages of different conclusions produced for the standard conditional inference problems as a function of the task format


Production task format

Modus ponens: A, therefore . . . C Maybe C Nothing Affirmation of the consequent: C therefore . . . A Maybe A Nothing Modus tollens: Not-C therefore . . . Not-A Maybe not-A Nothing Denial of the antecedent: Not-A, therefore . . . Not-C Maybe not-C Nothing

Standard

Guided

100 0 0

100 0 0

68.4 23.7 7.9

55.6 5.6 38.9

92.1 2.6 5.3

74.1 0 25.9

73.7 15.8 10.5

44.4 5.6 50


361



Basic results of Experiment 3 Table C1. Acceptance rates, rejection rates, and “impossible to tell” evaluations of the standard and opposite conclusions for the conditional inference problems within the explicit or implicit negations paradigm If p then q Negations

N

Conclusions

Explicit

40

Standard

Implicit

40

if not-p then q

if p then not-q

if not-p then not-q

MP

AC

MT

DA

MP

AC

MT

DA

MP

AC

MT

DA

MP

AC

MT

DA

True False I.t.t.

1.0 .0 .0

.9 0 0

.9 .1 .0

.8 .0 .2

.90 0 0

.95 .05 0

.58 .18 .3

.75 .05 .20

.98 .03 0

.85 .05 .10

.88 .05 .08

.42 .15 .43

.98 0 .03

.88 0 .08

.48 .33 .20

.40 .33 .28

Opposite

True False I.t.t.

0 .9 0

0 .9 .05

0 .93 .05

0 .90 .10

.05 .95 0

.03 .88 .10

.18 .73 .10

.08 .85 .08

0 1 0

0 .85 .10

0 .93 .08

.20 .55 .25

.03 .90 .08

.05 .90 .05

.20 .68 .13

.18 .65 .18

Standard

True False I.t.t.

1 .00 .00

.95 .00 .05

.85 .08 .08

.6 .1 .3

.65 .18 .18

.95 .05 .00

.30 .13 .58

.75 .15 .10

.93 .05 0

.43 .13 .45

.75 .20 .05

.28 .08 .65

.53 .25 .23

.58 .20 .23

.58 .25 .18

.55 .20 .25

Opposite

True False I.t.t.

0 .98 .00

.05 .95 .00

.05 .90 .05

0 .8 .2

.23 .58 .20

.10 .83 .08

.13 .70 .18

.10 .90 .00

.05 .93 .03

.25 .43 .33

.05 .93 .03

.08 .38 .55

.20 .43 .38

.28 .28 .45

.13 .75 .13

.18 .58 .25

Note: MP ¼ affirmation of the antecedent, i.e., “modus ponens”; AC ¼ affirmation of the consequent; MT ¼ denial of the consequent, i.e., “modus tollens”; DA ¼ denial of the antecedent. I.t.t. ¼ impossible to tell.

SCHROYENS AND BRAEM

362

APPENDIX C

The initial representation in reasoning towards an ...

The initial representation in reasoning towards an ...

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