quantifier

Informally, a quantifier is an expression that reports a quantity of times that a predicate is satisfied in some class of things (i.e. in a ‘domain’). Thus, thinking about a class of children and their diets, one might report that some eat cake, or that all eat cake, or that not all eat cake, or that none eat cake. ‘Some’ and ‘all’ are represented in modern logic by the quantifiers. The important point is that the treatment fends off thinking of ‘something’, ‘nothing’, and their kin as kinds of names.

In classical logic the two interdefinable quantifiers are the existential quantifier (∃x )… x, read as saying that something is…, and the universal quantifier (∀x )… x, read as saying that all things are…. Existential propositions, claiming that things of some kind exist, are represented by the existential quantifier. Less common quantifiers include the plurality quantifiers ‘many…’ and ‘few…’, and there are definable mathematical quantifiers such as ‘more than half…’, ‘exactly one…’.

More formally, a quantifier will bind a variable, turning an open sentence with n distinct free variables into one with n – 1 (an individual letter counts as one variable, although it may recur several times in a formula). When no variables remain free we have a closed sentence, i.e. one that can be evaluated as true or false within a domain. For example, from the open sentence F x & G x we can form (∃x )(F x & G x ), meaning that something is both F and G. The one variable x is bound on each occurrence.