Johnson-Laird, P.N., & Savary, F. (1996) Illusory Inferences about Probabilities. *
Acta Psychologica* **93** 69-90

Department of Psychology

Princeton University

Princeton

NJ 08544, USA

Tel: (609) 258 4432

Fax: (609) 258 1113

Email: phil@clarity.princeton.edu

Email: fabien@phoenix.princeton.edu

August 12th 1995

Problem 1. Suppose that only one of the following assertions is true about a specific hand of cards:

There is a king in the hand or there is an ace in the hand, or both.Problem 2. Suppose that only one of the following assertions is true about a specific hand of cards:

There is a queen in the hand or there is an ace in the hand, or both.Which is more likely to be in the hand: the king or the ace?

If there is a jack in the hand, then there is a queen in the hand.We will return to these two problems later; meanwhile, readers should remember to write down their answers to them.

If there is a ten in the hand, then there is a queen in the hand.Which is more likely to be in the hand: the queen or the jack?

The principal division in psychological theories of reasoning echoes a distinction in logic. On the one hand, syntactic theories propose that reasoners rely on formal rules of inference akin to those of a logical calculus; on the other hand, semantic theories propose that reasoners construct model-like entities as their interpretations of premises. In previous publications, we have defended a psychological theory based on mental models and argued that it accounts for reasoning in all the main domains of deduction (see e.g. Johnson-Laird and Byrne, 1991). In the present paper, we describe how the two main sorts of psychological theories -- formal rule theories and the mental model theory -- can be extended to deal with reasoning about probabilities. Next, we report two experiments on reasoning about relative probabilities. Their results corroborate the model theory. The decisive evidence is that, as the theory predicts, there are illusory inferences about probabilities, i.e. there are certain premises from which most people draw the same conclusions, which seem obvious, and yet which are egregious errors.

If there is a queen in the hand then there is a jack in the hand.Individuals draw this conclusion rapidly and without seeming to pause for thought. Rule theories accordingly postulate that the mind contains a formal rule of inference for modus ponens:

There is a queen in the hand.

Therefore, There is a jack in the hand.

If p then qwhich is triggered by the premises and which leads at once to the derivation of the conclusion. Such theories were first proposed twenty years ago (see e.g. Osherson, 1975; Johnson-Laird, 1975; Braine, 1978), and they continue to flourish (see e.g. Braine and O'Brien, 1991; Rips, 1994). They predict that the difficulty of an inference depends on two main factors: the length of its formal derivation (using the rules postulated by the theory) and the ease of retrieving and using each of the required rules (which can be estimated from experimental data). These theories have had some success in fitting the results of experiments (Braine, Reiser, and Rumain, 1984; Rips, 1983, 1994).

p

Therefore, q

Formal rule theories have so far been formulated to deal only with deductions, such as modus ponens, which lead to conclusions that are necessarily true. In contrast, the following problem concerns the relative probabilities of two events:

If there is a jack or queen in the hand, then there is an ace. Which is more likely to be in the hand: the jack or the ace? It has a valid answer, that is, an answer that must be true given that the premise is true: the ace is more likely to be in the hand than the jack. To the best of our knowledge, psychologists have not previously studied such problems, and it is not clear whether formal rule theories are intended to apply to them. Yet, they could be made to yield judgments of relative probability in the following way. Given the premise of the problem above:

If there is a jack or queen in the hand, then there is an ace. reasoners could proceed in the following way. They start by making a supposition, i.e. assumption for the sake or argument:

There is a jack in the handNext, according to certain formal rule theories (e.g. Braine and O'Brien, 1991), they could derive the conclusion:

There is an ace in the handusing a rule of the form:

pSimilarly, from the supposition:

if p or q then r

Therefore, r

There is a queen in the handthey can derive the same conclusion using a variant of the same rule. A system of "bookkeeping" could keep track of the respective possibilities, i.e., whenever there is a jack, there is an ace, but not vice versa, and hence conclude: an ace is more likely to be in the hand than a jack. No such theory has yet been proposed by any formal theorist, and so we will not pursue the details any further. For our purposes, it is sufficient to know that a rule theory for drawing conclusions about relative probabilities is at least feasible.

One advantage of the mental model theory is that it provides a unified account, so far lacking in formal rule theories, of reasoning that leads to necessary conclusions, probable conclusions, and possible conclusions. A conclusion is necessary -- it must be true -- if it holds in all the models of the premises; a conclusion is probable -- it is likely to be true -- if it holds in most of the models of the premises; and a conclusion is possible -- it may be true -- if it holds in at least some model of the premises (Johnson-Laird, 1994).

The fundamental representational assumption of the mental model theory is that individuals seek to minimize the load on working memory by representing explicitly only those cases that are true. Thus, a simple conjunction:

There is a king in the hand and there is a ace in it toocalls for a single model, which we represent in the following diagram where "k" denotes a king and "a" denotes an ace:

k aThere is no need to represent explicitly cases where the conjunction is false. Likewise, the exclusive disjunction:

There is a king or there is an ace, but not bothcalls for two alternative models (one for each possibility), which we represent in the following diagram:

k awhere each line represents a separate model. In this case, even those components of the assertion that would be false in these models are not explicitly represented, that is, the models do not explicitly represent that an ace does not occur in the first model and that a king does not occur in the second model. Reasoners thus need to make a mental "footnote" that the first model exhausts the hands in which a king occurs and the second model exhausts the hands in which an ace occurs. (Johnson-Laird and Byrne, 1991, used square brackets to represent such a footnote, but we will forego that notation here.) The footnote, provided it is remembered, can be used to make the models wholly explicit if necessary:

k -a -k awhere "-" denotes negation. It is these negative elements, -a and -k, that people do not ordinarily represent explicitly.

The same general principles underlie the initial representation of a conditional:

If there is a king then there is an ace.Individuals grasp that the conditional means that both cards may be in the hand, which they represent in an explicit model, but they defer a detailed representation of the case where the antecedent is false, i.e. where there is not a king in the hand, which they represent in a wholly implicit model denoted here by an ellipsis:

k a . . .Reasoners need to make a mental footnote that hands in which a king occurs are exhaustively represented in the explicit model, and so a king cannot occur in the hands represented by the implicit model. But, since hands containing an ace are not exhausted in the explicit model, they may, or may not, occur in the hands represented by the implicit model. The representation of a biconditional:

There is a king if, and only if, there is an acehas exactly the same initial models, but reasoners need to make a mental footnote that both the king and the ace are exhaustively represented in the explicit model.

Principle 1: if A occurs in each model in which B occurs, but B does not occur in each model in which A occurs, then A is more likely than B. In other words, if the models in which A occurs contain the models in which B occurs as a proper subset, then A is more likely than B.Consider again the example:

If there is a jack or a queen in the hand, then there is an ace.Which is more likely to be in the hand: the jack or the ace?

The antecedent of the conditional:

There is a jack or a queen in the handcalls for the models:

j q j qgiven an inclusive interpretation of the disjunction. These models are embedded in the interpretation of the conditional:

j a q a j q a . . .The models in which the ace occurs contain as a proper subset the models in which the jack occurs, and so the ace is more likely to be in the hand than the jack. Principle 1 is, of course, valid provided that one takes into account all of the possible models of the premise. The second principle is simpler and more general, and it includes principle 1 as a special case:-

Principle 2: if A occurs in more models than B, then A is more probable than B.This principle is risky. It is valid only if each model is equally probable, i.e., each model corresponds to a set of situations, and each of these sets is equally probable.

The computer program implementing the model theory has shown that there are premises which yield mental models supporting grossly erroneous conclusions. If the theory is right, these premises should give rise to illusory inferences, i.e. nearly everyone should draw the same conclusion, it should seem obvious, and yet it would be completely wrong. Readers have already encountered two illusory inferences (Problems 1 and 2 in the Introduction). Readers may have responded to Problem 1 that the ace is more likely to be in the hand than the king. Likewise, they may have responded to Problem 2 that the queen is more likely to be in the hand than the jack. In either case, they have succumbed to an illusion. It is impossible for an ace to be in the hand in the first problem, and it is impossible for a queen to be in the hand in the second problem. We will first outline the model theoryUs predictions about these two inferences, and then explain the correct conclusions.

Consider problem 1, which we abbreviate as:

Only one is true:The models of the first premise are:

King or ace, or both.

Queen or ace, or both.Which is more likely: king or ace?

k a k aand the models of the second premise are:

q a q aThe assertion that only one of the two premises is true means that either one assertion or the other assertion is true, but not both of them. That is, it calls for an exclusive disjunction of them, and the models for an exclusive disjunction, X or else Y, are:

X Yand so the disjunction calls for a list of all the models in the two alternatives. Hence, the problem as a whole calls for the following models:

k a k a q a q aIf subjects estimate probabilities using the second riskier principle of the two described above, then they judge the probability of an event on the basis of the proportion of models in which it holds. They will therefore respond that the ace is more probable than the king. If, however, subjects assume that the models may not be equiprobable, they will conclude that the problem is indeterminate, e.g. the probability of the king alone could be greater than the probabilities of all the other models summed together. Both of these responses are wrong, however.

What has gone wrong? If only one of the two assertions is true, then the other assertion is false: the two premises are in an exclusive disjunction, and so when one is true, the other is false. The models, however, represent only the true cases. When the false cases are taken into account, the correct answer emerges. When the first disjunction is false there is neither a king nor an ace, and when the second disjunction is false there is neither a queen nor an ace. Either way, there is no ace -- it cannot occur in the hand. Hence, the king, which can occur in the hand, is more probable than the ace, which cannot occur in the hand.

Now, consider problem 2, which we abbreviate:

Only one is true:Once again, if readers answered that the queen is more likely than the jack, then they succumbed to an illusion. It arises because an exclusive disjunction calls for listing the two sets of models:

If jack then queen.

If ten then queen.Which is more likely: queen or jack.

j q 10 q . . .and now even the sound first principle for estimating probabilities dictates that the queen is more probable than the jack. But, as with the first problem, the correct answer depends on bearing in mind the false contingencies, that is, when one conditional is true the other is false. The first conditional is false when there is a jack but not a queen, and the second conditional is false when there is a ten but not a queen. Either way, there is not a queen: it is impossible, but the jack is not impossible, and so the correct answer is that the jack is more probable than the queen.

The subjects acted as their own controls and carried out two illusory inferences of the form:

- Only one assertion is true about a specific hand of cards:

King or ace, or both.

Queen or ace, or both.

Which is more likely: king or ace? - Only one assertion is true about a specific hand of cards:

If king then ace.

If queen then ace.

Which is more likely: ace or king?In fact, each problem concerned different cards, but for convenience we have stated the problems as though they were always about the same particular cards. The subjects also carried out two control problems:

- If king then ace.

Which is more likely: king or ace?

- If king or queen then ace.

Which is more likely: ace or king?

k a . . .with a footnote to the effect that kings are exhausted in the explicit model, i.e. kings cannot occur in any other model, whereas aces can occur in the model signified by the ellipsis. Subjects should therefore infer that the ace is more likely to be in the hand than the king. If subjects make a biconditional interpretation, however, then they will treat the initial model as exhaustively representing both kings and aces, and so they will judge the two cards to be equiprobable.

Control problem 4 calls for the initial models:

k a q a k q a . . .The third of these models will be omitted by those individuals who interpret "or" as an exclusive disjunction. But, in either case, the models containing kings are a proper subset of the models containing aces, and so the subjects should judge that the ace is more likely to be in the hand than the king. These conclusions to the two control problems are correct, that is, even when the models are made completely explicit they still support the same conclusions.

Each subject carried out the four inferences in a different order, i.e. one of the 24 possible orders. The materials for each problem concerned a specific hand of cards, and four distinct hands of cards were assigned to each problem at random.

The model theory rests on the assumption that individuals focus on true states of affairs, which they represent explicitly, and rapidly forget, if they represent them at all, false states of affairs. This assumption led to the prediction of the existence of illusory inferences, but perhaps our results are merely a happy coincidence, and the true cause of the illusions is quite different from model theory's account. We will discuss the possibility of such alternative explanations later. However, we designed Experiment 2 in part to examine their plausibility.

- Exclusive disjunction: "Only one of the following assertions is true about a specific hand of cards."
- Biconditional: "If one of the following assertions is true about a specific hand of cards, then so is the other assertion."

The illusory problem in the first pair of problems based on three connectives had an exclusive disjunction as its main connective:

1. Only one of the following assertions is true about a specific hand of cards:The subjects' task was to estimate the probability that the jack was in the hand and to estimate separately the probability that the queen was in the hand. The subjects made both responses by indicating the relevant position on two separate computer-presented scales running from "impossible" to "certain". According to the model theory, this problem should elicit models of the following form:

If there is a jack in the hand then there is a queen in the hand.

If there isn't a jack in the hand then there is a queen in the hand.

j q -j q . . .and so subjects should assign a higher probability to the queen than to the jack. It is by no means certain whether individuals will represent the implicit model signified here by the ellipsis. The implicit model may be forgotten, or it may be omitted because the two antecedents of the conditionals exhaust the possibilities. In either case, the models support the same inference: the queen should be assigned a higher probability than the jack. This conclusion, of course, is an illusion: one of the premises must be false, and so there cannot be an queen. The correct models for the premise, as shown by a program that represents all possible contingencies in a fully explicit way, are as follows:

-j -q j -qThe correct response is accordingly to assign the jack a higher probability than the queen, whose presence in the hand is impossible.

The matched control problem took the form:

1'. Only one of the following assertions is true about a specific hand of cards:We abbreviate the statement of this problem as follows:

If there is a jack in the hand then then there is a queen in the hand.

if there is a jack in the hand then there is not a queen in the hand.

Only one is true:using the conventions that "J" denotes "there is a jack", and "not Q" denotes "there is not a queen." This problem should elicit the following models:

If J then Q.

If J then not Q.

j q j -q . . .and so subjects should assign a higher probability to the jack than to the queen. The fully explicit models are in this case:

j q j -qHence, the response is correct.

The second pair of problems with three connectives were based on biconditionals. According to the model theory, both problems yield only implicit models, that is, models with no explicit content (see problems 2 and 2' in Table 2 below), and so they were included in the experiment primarily to see how subjects would respond when a problem seemed not to have any explicit model. In the illusory case, the correct response is that the queen is more probable than the jack; in the control case, the correct response is unclear because the premises are self-contradictory.

We also used four illusory problems and four matched
controls based on only two sentential connectives. For two of these
pairs, the main connective was an exclusive disjunction, and for the
other two of these pairs, it was a biconditional. One of the illusory
problems based on a biconditional was:
3. If one of the following assertions is true about a specific hand of
cards then so is the other assertion:

There is a jack if and only if there is a queen.

There is a jack.
We can again abbreviate this problem:

If one is true so is other:It should elicit the models:

J iff Q.

J.

j q . . .and so subjects should estimate that the two cards have equal probabilities of occurring in the hand. But, the fully explicit models of the problem are:

j q -j qand so the correct answer is that the queen is certain to be in the hand whereas the jack is not.

The matching control problem was of the form:which should elicit the single model:

3'. If one is true so is other:

If J then Q.

J.

j qand so subjects should again infer that the two cards have equal probabilities of occurring in the hand. In this case, the models are correct, and so the conclusion is, too. The full set of six illusory problems and their matched controls are shown in Table 2. The problems were presented in a different random order to each subject.

impossible unlikely 50/50 likely certainThe 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.

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:

Impossible: 16 unlikely: 3 50/50: 33 likely: 3 certain: 34We 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.

1. Only one is true: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.:

If J then Q.

If not J then Q.

3. If one is true so is other: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.

J iff Q.

J.

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.

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

king or ace, or both.

queen or ace, or both.Which is more likely: king or ace?

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

-k q -a k -q -aThe 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:

4. If one is true so is other: 4'. If one is true so is other: J or else not Q. J or else Q. J. Not Q.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:to mean:

Only one of the following assertions is true

One of the assertions is true and the other is of an unknown truth valueand 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 assertionto 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:

X Y X -Y -X YIn 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:

X Y X -Y -X Y -X -YOnce 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.into a single conditional with a disjunctive antecedent:

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

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:

k a -k aThe 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 trueto 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 assertionto 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: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:

If jack then queen.

If jack then not queen.

There is a king if, and only if, there is a queen.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).

There isn't a queen.Therefore, There isn't a king.

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 else100% of subjects drew the conclusion:

if there isn't a king in the hand then there is an ace in the hand.

There is a king in the hand.

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.

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_________________________________________________________________________ Type of problem Responses and their percentages ace king equiprobable ________________________________________________________________________ Illusory inferences: 1. Only one assertion is true: king or ace, or both. queen or ace, or both. Which is more likely: king or ace? 75 21 4 2. Only one assertion is true: If king then ace. If queen then ace. Which is more likely: ace or king? 79 13 8 Control inferences: 3. If king then ace. Which is more likely: king or ace? 62 17 21 4. If king or queen then ace. Which is more likely: ace or king? 79 17 4 __________________________________________________________________________ _________________________________________________________________________ Table 2: The six illusory problems and their matched controls in Experiment 2. The models that subjects should construct are on the left, and the fully explicit correct models are on the right. "Iff" denotes "if and only if." __________________________________________________________ Illusory Control 1. Only one is true: 1'. Only one is true: If J then Q. If J then Q. If not J then Q. If J then not Q. j q -j -q j q j q -j q j -q j -q j -q . . . . . . 2. If one is true so is other: 2'. If one is true so is other: J and Q. J iff Q. J and not Q. J iff not Q. . . . -j -q . . . null (i.e. one -j q premise contradicts the other.) 3. If one is true so is other: 3'. If one is true so is other: J iff Q. J. If J then Q. J. j q j q j q j q . . . -j q 4. If one is true so is other: 4'. If one is true so is other: J or else not Q. J. J or else Q. Not Q. j j q j -q j -q j -q -j q . . . j q . . . 5. Only one is true: 5'. Only one is true: Not J or else not Q. Q. Not J or else Q. Not J. -j j -q -j j q -q j q q -j q q 6. Only one is true: 6'. Only one is true: J iff not Q. J. not J iff not Q. J. j -q -j q -j -q -j -q j j q j j -q _____________________________________________________________________________ ______________________________________________________________________________ Table 3: The correct responses and their percentages for five illusory problems and their matched controls in Experiment 2. J > Q indicates that the jack was given a higher probability than the queen, Q > J indicates that the queen was given a higher probability than the jack, J = Q indicates that the jack and the queen were given the same probabilities. Problems 2 and 2' were not included in this analysis (see text). _____________________________________________________________________________ Illusory Control 1. Only one is true: 1'. Only one is true: If J then Q. If J then Q. If not J then Q. If J then not Q. J > Q: 15 J > Q: 55 3. If one is true so is other: 3'. If one is true so is other: J iff Q. J. If J then Q. J. Q > J: 10 J = Q: 95 4. If one is true so is other: 4'. If one is true so is other: J or else not Q. J. J or else Q. Not Q. Q > J: 5 J > Q: 70 5. Only one is true: 5'. Only one is true: Not J or else not Q. Q. Not J or else Q. Not J. J > Q: 20 Q > J: 35 6. Only one is true: 6'. Only one is true: J iff not Q. J. not J iff not Q. J. Q > J: 15 J > Q: 65 _____________________________________________________________________________ Overall Percentages: 13 64 ______________________________________________________________________________