Predicates and Sets

Abstract

In this chapter we show how to formalize predicates in higher-order logic and how to reason about their properties in a general way. Sets are identified with predicates, so the formalization of predicates also gives us a formalization of set theory in higher-order logic. The inference rules for predicates and sets are also special cases of the inference rules for Boolean lattices. These structures are in fact complete lattices, so the general rules for meets and joins are also available. We complete our formalization of higher-order logic by giving the final constants and inference rules that need to be postulated: an axiom of infinity and an axiom of choice.