variable in logic and mathematics, a symbol interpreted so as to be associated with a range of values, a set of entities any one of which may be temporarily assigned as a value of the variable. An occurrence of a variable in a mathematical or logical expression is a free occurrence if assigning a value is necessary in order for the containing expression to acquire a semantic value – a denotation, truth-value, or other meaning. Suppose a semantic value is assigned to a variable and the same value is attached to a constant as meaning of the same kind; if an expression contains free occurrences of just that variable, the value of the expression for that assignment of value to the variable is standardly taken to be the same as the value of the expression obtained by substituting the constant for all the free occurrences of the variable. A bound occurrence of a variable is one that is not free. See also FORMAL LOGIC, LOGICAL SYNTAX , QUANTIFICATION , WELL — FORMED FORMUL. C.A.A.