a lot more work must be carried out. First, the implicit aspects of the initial models must be fleshed out with the explicit models, for example:

and then these models must be combined with the model of the second premise (i.e., -A), to get the following model:

which can be described by the conclusion: "There is not a circle". Thus, modus tollens is more difficult than modus ponens because people have to flesh out the models and keep multiple alternatives in mind.

Invalid inferences in the model theory

Invalid inferences arise for two reasons: people can have different interpretations of the conditional premise and they may fail to flesh out the explicit models for their interpretation. First, people can interpret the original premise—"If there is a circle, there is a triangle"—as a biconditional; "if and only if there is a circle, there is a triangle".

In the minimal set of models adopted by people, this interpretation of the premise would be represented implicitly as:

where both components are exhaustively represented; indicating that no other models will contain circles or triangles. Invalid inferences arise from these initial models. When subjects are given the extra information that "There is a triangle" (A), the second model will be eliminated, leaving the affirmation of the consequent inference: "There is a circle".

When subjects are given the information "There is not a circle", they can get the right answer for the wrong reason; that is, they may say that "nothing follows", simply because they cannot combine the model (-O) with the initial set of models, regardless of whether those models are for the conditional or biconditional interpretation. However, if subjects flesh out the explicit models for the biconditional (see Table 16.2):

then when they combine these models with the model of the second premise (-O), the first model will be ruled out and they will draw the denial of the antecedent inference: "There is not a triangle". In this way the theory explains why the denial of the antecedent inference is made less often than the affirmation of the consequent inference, as people can get the right answer for the wrong reason (although see Evans et al, 1993, and Evans, 1993a).

Context effects

Regarding context effects, Byrne (1989a) has proposed that extra information leads to a different interpretation of the premises. Premises that contain alternatives as opposed to additionals result in different sets of models being constructed. The validation procedures that revise the models find that alternative antecedents act as counterexamples to the invalid inferences, whereas additional antecedents act as counterexamples to the valid inferences. Indeed, Byrne et al. (1999) have shown that combined additional and antecedent premises can suppress all of four inferences. Byrne et al. (1999; Byrne, 1991) propose that the effects of causality and saliency are due to the immediate availability of counterexamples based on background knowledge. So, for example, Chan and Chua's (1994) saliency effects are attributed to the differential effectiveness of the counterexamples called to mind for the weak and strong saliency premises. Similarly, Stevenson and Over (1995) discuss extensions to the model theory that would account for their uncertainty effects. Although such accounts are not fully worked out, it is clear that several feasible proposals are on the table.

The selection task

The model theory has also been applied to account for the results of the selection task (Johnson-Laird, 1995; Johnson-Laird & Byrne, 1991). The explanation is based on three key points:

• People consider only those cards that are represented explicitly in their models of the rule.

• Different contexts (e.g., deontic ones) can affect what comes to be represented explicitly in the model.

• People then select those cards for which the hidden value could have a bearing on the truth or falsity of the rule.

It may be remembered that people typically choose the P and Q cards on this task, when the P and not-Q cards are the optimal choices. The model representation of the conditional "if there is a vowel on one side of the card, then there is an even number on the other side" (instantiated with E and 4, see Figure 16.3) would be represented by the following models:

which given the aforementioned proposals suggests that P (E-card) and Q (4-card) should be chosen. Furthermore, this account suggests that any manipulation that makes the not-Q card explicit in the model should facilitate performance on the task. The suggestion, therefore, is that effects due to a deontic reading of the task achieve just this sort of explicit representation. Matching bias effects are also explained in terms of how rules with negations are represented as explicit models, although there has been some debate on exactly what form these models might take (see Evans, 1993a; Johnson-Laird, 1995; Johnson-Laird & Byrne, 1991). In short, the model theory also has a story to tell about the selection task.

Further evidence for the model theory

The model theory accounts for other evidence and many novel predictions from it have been confirmed (see Evans et al., 1993, Chapter 2 for a review). Johnson-Laird, Byrne, and Schaeken (1992) have shown that the difficulty data, which abstract-rule theorists propose to be due to length of the mental derivation, can be

The percentage of subjects making modus ponens and modus tollens inferences when given two-premise conditional or biconditional problems from Johnson-Laird, Byrne, and Schaeken (1992, Experiment 2).

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