Probabilities and theories based on mental models

According to the mental model theory, reasoning is a semantic process rather than a syntactic one. Reasoners imagine the states of affairs that satisfy the premises, that is, they build mental models of the relevant situations based on their understanding of the premises and on general knowledge; they formulate a conclusion that is true in these models; and they establish its validity by ensuring that there are no models of the premises in which the conclusion is false (Johnson-Laird, 1983). By a mental model, we mean a representation that corresponds to a set of situations, and that has a structure and content that captures what is common to these situations. The contrast between formal, or syntactic, methods in logic and semantic methods is a familiar one; and logicians have shown that there exists a sharp division between the "proof theoretic" method based on formal rules and the "model theoretic" method based on semantics. In certain branches of logic, proof theory is incomplete in that one cannot formulate a consistent set of rules that captures all and only the valid inferences (see e.g. Jeffrey, 1981).

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 too

calls for a single model, which we represent in the following diagram where "k" denotes a king and "a" denotes an ace:

There 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 both

calls for two alternative models (one for each possibility), which we represent in the following diagram:

where 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:

where "-" 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:

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 ace

has 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.