Akademik

(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.