ABSTRACT
We know that it is possible to extend the truth-functional semantics to predicate logic. However, this is not at all trivial. The trouble is that a quantified sentence need not contain any (closed) sentences as its parts – hence, its truth value cannot be generally seen as determined by those of its sub-sentences. What we need is for the truth values of sentences to not necessarily be determined by the truth values of their parts but rather, more generally, by their denotations. And as we consider denotations, instead of merely truth values, we can do what Tarski proposed to do: we can take the truth values of at least some sentences to be determined not only by the denotations of their parts, and not even only by the denotations of any other expressions of our language, but directly by the objects which are eligible as (though in fact they need not be) denotations of the expressions. This leads us to the usual extensional semantics for the languages of predicate logic as mapping subjects (individual terms) on elements of a universe and predicates on relations over the universe. And though this may look as though it is going from the merely truth-functional apparatus to one that engages entities of the real world, the chapter argues that this is unwarranted. Extensional semantics for the first-order predicate logic can also be naturally generalized to higher-order logics up to the lambda-categorial grammar.
