Procedure and Materials

The subjects were tested individually in a quiet room. The experimenter explained that the purpose of the experiment was to elicit judgments about which cards were more likely to be in a specific hand of cards, and that these estimates were to be based solely on the information presented in the verbal statement of the problems. He then showed the subjects how to use the micro-computer and the mouse (an Amiga 2000 computer running a program written in Basic to control the experiment). The subjects carried out one practise trial in which they made two simple assessments of probability. Each trial began with the verbal presentation of the premises with two horizontal scales presented below -- one scale for one card, and the other scale for the other card. The scales were labeled at equal intervals with the following terms:

The subjects made their responses by moving a mouse that in turn directed the cursor to the desired point on the scale. The subject then clicked on the mouse, and the scale was divided with a vertical line at that point. The program recorded the distance of the subject's response along the scale. After the subject had made both responses to a problem, the trial ended, and the next trial began when the subject clicked in the ready box. The particular cards for each problem were assigned at random, but no problem had the same pair of cards as any other.

In the other part of the experiment, the subjects were asked to state what followed from each of the problems, that is, they had to write down what conclusion, if any, must be true given the information in the premises. This part of the experiment was administered as a simple paper-and-pencil test, and the twelve problems were presented in a random order.

Subjects

Twenty Princeton students volunteered to participate in the experiment. None of them had any training in logic or had taken in part in any previous experiments on reasoning. They were paid $5 per hour to carry out the experiment, which lasted for approximately 40 minutes.

Results

For our first analysis of the probability judgments, we divided them into three categories: 1. jack assigned a higher probability than queen (J > Q) 2. queen assigned a higher probability than jack (Q > J) 3. jack and queen assigned equal probabilities (J =Q) where, for convenience, we have assumed that each problem concerned a jack and a queen. We then scored the number of correct responses for each of the problems. Control problem 2' has no clear correct answer because it expresses a contradiction, and so we dropped both it and its matching illusory problem 2 from this analysis. Table 3 presents the percentages of correct responses for each of the remaining ten problems. Overall, the subjects made 13% correct responses to the illusory problems, but 64% correct responses to the control problems. The difference between the two conditions was highly reliable: 17 subjects made fewer correct inferences to the illusory problems than to the control problems, 1 subject had the opposite pattern of results -- this subject made only one correct response, and there were two ties (Wilcoxon's T = 1, n = 18, p << .005). Likewise, all five pairs of materials showed the predicted difference (Sign test, p < .04). Insert Table 3 about here

The subjects' responses matched the model theory's predictions on 56% of the illusory problems and, of course, on 64% of the control problems. In the case of problems that are not illusory, the model theory predicts that the greater the number of explicit models that have to be constructed in order to respond correctly, the harder the task should be (Johnson-Laird and Byrne, 1991). This prediction is corroborated by the control problems: problems 3' and 4' require only one explicit model whereas problems 1', 5', and 6' require two explicit models (see Table 2). The former elicited 83% correct responses, whereas the latter elicited only 52% correct responses, and the difference was reliable (Wilcoxon's T = 9, n = 16, p << .005).

Each problem called for a subject to infer the relative probabilities of two cards, and 89% of these judgments aligned a card on one of the five canonical points labeled on the scale (allowing for an error of no more than two hundredths of the scale). Four out of the twenty subjects were responsible for just over two thirds of the judgements that were not on one of the five canonical points, and nine subjects made only canonical judgments. The percentages of judgments were as follows:

We have pooled the data for the illusory and the control problems because their patterns were highly similar. As the percentages show, the subjects overwhelmingly judged the cards as impossible, 50/50, or certain. This distribution of responds is to be expected from the models of the premises: cards in the models are certain, or impossible, or occur in one of two models but not the other (see Table 2).

In the other part of the experiment, the subjects were asked to state what conclusion, if any, followed from each problem. The results were relatively noisy, perhaps because the subjects were reluctant to draw conclusions corresponding to a categorical premise (even though it occurred as part of an exclusive disjunction or biconditional). They were revealing, however, about problems 2 and 2', where the model theory predicts that subjects construct only an implicit model in both cases (see Table 2). Subjects had a tendency to conclude that there was a contradiction (60% of responses were of this form for problem 2 and 50% of responses were of this form for problem 2'). In fact, the illusory problem is not self-contradictory, but the control problem is self- contradictory.

Discussion

The results showed that the phenomenon of illusory inferences occurs with a variety of connectives. The main connective can be an exclusive disjunction or a biconditional. For example, given problem 1:

1. Only one is true:
If J then Q.
If not J then Q.

few subjects grasped that the queen was impossible, and the modal response was that the queen was more probable than the jack. The experiment also showed that illusions could be created in a minimal way by assertions containing only two connectives, e.g.:

3. If one is true so is other:
J iff Q.
J.

where most subjects erroneously inferred that the two cards had the same probabilities of being in the hand, though in fact the queen is certain to be in the hand but the jack is not.

According to the model theory, the illusions arise because reasoners represent true cases, but not false cases. The difference between the illusory and the control problems is simply that this tendency has no effect on correctness in the case of the controls. The control problems were selected to be as similar as possible to the illusory problems in the choice of connectives and negatives. One consequence was that some of the control problems called for two distinct models to be constructed in order to reach the correct answer, whereas others called for only one model. Previous studies of reasoning with sentential connectives have shown that two model problems can be difficult for subjects (see Johnson-Laird, Byrne, and Schaeken, 1992). Our results with the control problems corroborated this phenomenon: the two-model control problems were reliably harder than the one-model control problems.

General Discussion

The model theory was originally developed as an account of how people draw logically-necessary conclusions. However, it also gives a simple explanation of how reasoners reach conclusions about what is probable (Johnson-Laird, 1994): a situation is probable if it holds in most models of the premises, and, assuming a principle of "indifference" according to which models are equiprobable, one event is more probable than another if it occurs in more models than the other. The twist in these predictions is that reasoners are likely to construct explicit models only of true cases and to forget about the false cases, especially if the premises are complex. Where these models yield a different conclusion from the one supported by fully explicit models of both true and false components, subjects should draw illusory inferences: their conclusions should be totally wrong. It is worth emphasizing that the prediction is based on the models needed for deduction, that is, we did not develop a new theory of models to account for reasoning about probabilities.

Experiment 1 confirmed the existence of illusory inferences. It also showed that subjects do tacitly assume a principle of "indifference" and infer that whichever event occurs in more models is the one that is more likely to occur. Thus, given problem 1:

Only one assertion is true:
king or ace, or both.
queen or ace, or both.

Which is more likely: king or ace?

most subjects inferred that the ace is more likely. This illusory inference rests on the construction of the following set of models:

in which there are more models containing the ace than models containing the king. The fully explicit models for this problem, however, are as follows:

The ace is impossible and so less likely to occur in the hand than the king.

Experiment 2 provided further corroboration of the model theory's predictions by showing that illusory inferences occur with a variety of different sorts of connectives. It also established that illusions can be constructed with just two sentential connectives. The superficial similarity between these illusory problems and their matched control problems is quite striking. For example, a cursory examination of the following pair of problems:

is unlikely to suggest that reasoners will perform in a radically different way with them. Yet, only one subject made the correct judgments about problem 4, i.e. the queen has a higher probability of being in the hand than the jack, whereas 14 subjects made the correct judgments for problem 4', i.e. the jack has a higher probability of being in the hand than the queen.

Is there any obvious alternative explanation for subjects' susceptibility to the illusions? Colleagues who have succumbed to the illusions -- and they include a number of distinguished cognitive psychologists -- have suggested several alternative explanations for them. We will consider three possibilities.

First, subjects may misinterpret an assertion of the form:
Only one of the following assertions is true

to mean:

One of the assertions is true and the other is of an unknown truth value

and then reason in a wholly correct way. Likewise, they may misinterpret an assertion of the form:

If one of the following assertions is true about a specific hand of cards, then so is the other assertion

to mean:

Either both of the assertions are true or else they have unknown truth values.

Given a disjunction of two assertions, X and Y, the first of these hypotheses implies that individuals consider one case in which X is true and Y is either true or false, and another case in which Y is true and X is either true or false. When these cases are spelt out explicitly, they are as follows:

In other words, the interpretation is equivalent to an inclusive disjunction of the two assertions, X and Y. However, an inclusive disjunction of the two conditionals in problem 1 of Experiment 2:

If J then Q. If not J then Q.

yields a tautology: there is either a jack or not a jack, and there is either a queen or not a queen. Hence, there is no reason to infer that the queen is more likely than the jack -- yet subjects made exactly that inference, and so it follows they are not treating the two conditionals as being in an inclusive disjunction.

Given two assumptions, X and Y, the second of these two hypothesis implies that subjects consider one case in which they are both true, and other cases in which they are each either true or false. This treatment yields a tautology based on X and Y:

Once again, however, this interpretation provides no basis for inferring that one card is more probable than another, and so it is refuted by the results of Experiment 2 (see problem 6').

Second, subjects may have a formal rule of inference that converts an exclusive disjunction of two conditionals:

If there is a king then there is an ace.
If there isn't a king then there is an ace.

into a single conditional with a disjunctive antecedent:

If there is a king or isn't a king then there is an ace.

For problems of this sort, the hypothesis makes the same prediction as the model theory, because it postulates that subjects construct the models:

The difficulty for the formal rule, however, is that it cannot explain the other sorts of illusion, such as the ones where the main connective is a biconditional.

Third, subjects may have interpreted the two premises in an illusory inference as though they were in a conjunction, and then reasoned in a wholly correct way. In our view, it is again unlikely that intelligent students would take the assertion equivalent to an exclusive disjunction:

Only one of the following assertions is true

to mean:

Both of the following assertions are true.

But, it is feasible that the subjects took the biconditional assertion:

If one of the following assertions is true about a specific hand of cards then so is the other assertion

to mean that both assertions are, in fact, true. Experiment 2 refutes the first of these hypotheses. The control problem 1':

Only one of the following assertions is true:
If jack then queen.
If jack then not queen.

would yield the inference that the jack is impossible, because its presence would yield a contradiction. Only 10% of subjects judged that the queen had a higher probability than the jack, whereas 55% of subjects made the response predicted by the model theory: the jack is more probable than the queen (see Table 3). The treatment of a biconditional as though it were a conjunction is similar to the model theory's account, which postulates that reasoners represent explicitly the case where the two assertions are true, and that they represent the case where they are both false with only an implicit model (signified by the ellipsis in our notation). Indeed, if subjects omit the implicit model, or forget it, then the two accounts are one and the same. The implicit model, however, does seem to be necessary in order to explain how subjects make the following sort of inference based on the more conventional expression of a biconditional:

There is a king if, and only if, there is a queen.
There isn't a queen.

Therefore, There isn't a king.

Many reasoners are able to make this inference, presumably by fleshing out the content of the implicit model explicitly before they take into account the information in the second premise (see Johnson-Laird et al, 1991).

In general, the alternative explanations can account for only some of our data, whereas the model theory predicts the existence of the illusory inferences in general. The illusions refute the extension of formal rule theories to deal with probabilities -- at least, the extension that we described earlier, because it yields only valid inferences. Of course, one could invoke a different (invalid) formal rule to deal with each of the different sorts of illusory inference, but such an account would be entirely post hoc. It might well lead to invalid inferences that reasoners do not, in fact, make. Moreover, there is a general problem in invoking invalid rules to explain the illusions. If human reasoners were guided by such a system, they would be intrinsically irrational and their capacity for rational thinking as manifest in mathematics and logic would be wholly inexplicable. In contrast, the model theory assumes that human reasoners are rational in principle because they grasp that an argument is valid if, and only if, there are no counterexamples to it, i.e. no models of the premises in which the conclusion is false. They err in practise, however, because their working memory is limited and they tend to represent explicitly only true cases.

Errors in reasoning in previous studies of deduction can be explained in terms of failures to use appropriate rules of inference (e.g. Braine and O'Brien, 1991; Rips, 1994), or in terms of failures to consider all possible models of the premises (e.g. Johnson-Laird and Byrne, 1991). However, illusory inferences of the sort established in our experiments are not a result of such oversights. What is novel about them is that a conclusion that nearly everyone draws is totally wrong, e.g. what is judged more probable of two alternatives is impossible. In a recent unpublished experiment, we have shown that illusions can also occur in deductive reasoning. Given the following premises, for example:

If there is a king in the hand then there is an ace in the hand; or else
if there isn't a king in the hand then there is an ace in the hand.
There is a king in the hand.

100% of subjects drew the conclusion:

Therefore, There is an ace in the hand.

Yet, it is impossible for there to be an ace in the hand. Such deductions appear to refute current theories based on rules of inference (e.g. Braine and O'Brien, 1991; Rips, 1994), just as our present results refute an extension of formal rules to deal with probable conclusions. Current theories use only rules that yield valid conclusions, and so they have no way to explain the systematically invalid conclusions that individuals draw to illusory inferences. Rule theorists could well follow Jackendoff (1988) and invoke unsound rules that deliver invalid conclusions. Rips (1994) clearly countenances the possibility: "If people possess ... normatively inappropriate rules for reasoning with uncertainty, it seems a short step to assuming that they have similarly inappropriate rules for reasoning deductively" (p. 383). It remains to be seen whether anyone will succeed in formulating a rule theory that falls into deductive illusions but copes satisfactorily with the control problems.

We have just begun to explore the space of possible premises in search of illusory inferences. They are relatively rare, but there are many sorts of putative illusion. If the model theory is on the right lines, they arise because reasoners overlook cases in which a state of affairs is false. To rely on as little explicit information as possible is a sensible solution to the all-pervasive problem of limited processing capacity. Just occasionally, however, it leads us into a profound illusion about what is probable.

Acknowledgements

We thank Ruth Byrne, Jack Gelfand, Vittorio Girotto, Sam Glucksberg, Danny Kahneman, Geoffrey Keene, Joel Lachter, Paolo Legrenzi, Rick Lewis, Bonnie Meyer, Eldar Shafir, Patrizia Tabossi, and Amos Tersky, for their helpful suggestions. The research was carried out with support from the James S. McDonnell Foundation and from Fonds pour la Formation de Chercheurs et l'aide a la Recherche (Quebec). A version of this paper was presented to a meeting of SPUDM-15 (Subjective Probability, Utility, and Decision Making) in Jerusalem, August, 1995, and we are grateful to the participants for their comments. Experiment 1 was presented to the 17th Annual Conference of the Cognitive Science Society, Pittsburgh, July, 1995.

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Tables

Table 1: The percentages of responses to illusory and control inferences in Experiment 1. The correct responses are shown in bold print.