Models and illusory inferences about probabilities

The model theory predicts that conclusions which hold in most of the models of the premises will be judged to be probable. It also makes predictions about judgments that one event is more probable than another. There are two potentially relevant principles.

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 hand

calls for the models:

given an inclusive interpretation of the disjunction. These models are embedded in the interpretation of the conditional:

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:
King or ace, or both.
Queen or ace, or both.

Which is more likely: king or ace?

The models of the first premise are:

and the models of the second premise are:

The 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:

and 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:

If 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:
If jack then queen.
If ten then queen.

Which is more likely: queen or jack.

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:

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.

Experiment 1

No previous experiments, as far as we know, have examined inferences about relative probabilities based on information conveyed to subjects in terms of sentential relations, such as "if", "and", and "or". The first aim of Experiment 1 was accordingly to determine whether logically-untrained individuals could make such inferences. The second aim was to test the model theory's prediction that certain of these inferences are illusory and that the subjects would therefore be prone to err on them. The experiment compared the two illusions described above with two simpler control problems, which should not be illusory according to the model theory. The third aim was to assess whether subjects based their judgments of relative probability on principle 1 concerning proper subsets or on the more general, but riskier, principle 2 based on the assumption of equiprobability.

Method