Variable
Variable
A letter occurring in a mathematical or logistic formula and serving, not as a name of a particular, but as an ambiguous name of any one of a class of things — this class being known as the range of the variable, and the members of the class as values of the variable.
Where a formula contains a variable, say x, as a free variable, the meaning of the formula is thought of as depending on the meaning of x. If the formula contains no other free variables than x, then it acquires a particular meaning when x is given a value — i.e., when a name of some one value of x is substituted for all free occurrences of x in the formula — or, what comes to the same thing for this purpose, when the free occurrences of x are taken as denoting some one value.
Frequently an (interpreted) logistic system (q.v.) is so constructed that the theorems may contain free variables. The interpretation of such a theorem is that, for any set of values, of the variables which occur as free variables, the indicated proposition is true. I.e., in the interpretation the free variables are treated as if bound by universal quantifiers (q.v.) initially placed.
A bound variable, or apparent variable, in a given formula, is distinguished from a free variable by the fact that the meaning of the formula does not depend on giving the variable a particular value. (The same variable may be allowed, if desired, to have both bound occurrences and free occurrences in the same formula, and in this case the meaning of the formula depends on giving a value to the variable only at the places where it is free.) For examples, see Abstraction, and Logic, formal, 3.
For the terminology used in connection with functions, see the article function. Cf. also the articles Constant, and Combinatory logic. — A.C.