- (logic) A fundamental notion of modern logic. Intuitively, suppose we have a class of objects about which we might be interested (a domain), and we start with a simple sentence ‘Jane is hungry’. We then strike out mention of Jane, leaving a gap that we mark with the letter x : ‘ x is hungry’. This represents something short of a sentence (it is called an open sentence, or predicate). We can ‘point’ the letter x at members of the domain in turn, giving successive sentences like the one with which we started. In such a process the letter x is said to function as a variable taking as values each member of the domain successively. We might conclude the procedure with information like this: somewhere in this process one of the sentences is true, or everywhere such a sentence is true. Such information does not tell us who is hungry, but tells us the quantity of times the predicate is satisfied. The information that somewhere the predicate F applies to the value is represented as (∃x )Fx ; the information that it always applies as (∀x )Fx . The expressions (∃…) and (∀…) are the existential and universal quantifiers . The power of the idea only becomes apparent when we consider multiple quantifications. If we start with a relational sentence, ‘Fred loves Jane’ and strike out both names, marking the spaces with different variables, we obtain ‘ x loves y ’. We can now build very different kinds of information: everyone loves someone: (∀x )(∃y ) x loves y ; someone loves everyone: (∃x )(∀y ) x loves y, and so on. The study of these forms and the relations between them is quantification theory. The basic calculus that formalizes their logic is the predicate calculus.
Philosophy dictionary. Academic. 2011.