Even middle-school mathematics relies on the use of variables to capture the generality of theorems and proofs. Ordinary nonmathematical language also relies on similar devices – determiners, pronouns, and sometimes other parts of speech – to specify information about entities without having to describe or to name them individually. In psychology, variables are a central means of representing, retrieving, and manipulating information in memory, according to many cognitive theories (e.g., Anderson, 1983; Newell, 1990). In these theories, for example, a simple procedure for recognizing a triangle might be spelled out in terms of mental rules such as IF Closed(x) & Three-sided(x) & Two-dimensional(x) THEN Triangle(x), much as in conventional computer-programming languages.
These psychological theories assume that all people – even those who have never had math or logic training – manipulate variables mentally, but until recently there have been no general proposals about reasoning with variables. If people do represent generality in this way, then it is useful to know something about the limits of their ability to deduce information from such representations. It is possible, of course, to capture generality by other means, for example, replacing variables with combinators (e.g., Hindley & Seldin, 1986; Schönfinkel, 1924/1967). The research that I describe here, however, follows the lead of theories such as Anderson's and Newell's in assuming variable-based representations, and it examines people's deductive skills within this framework.