CHAPTER IV - THE SECOND-ORDER PREDICATE LOGIC. THEORY OF DEFINITION

Abstract

This chapter discusses the second-order predicate logic. The distinctive feature of the second-order predicate logic or the functional calculus of second order is that there are not only individual variables but also predicate variables, with both individual and predicate variables being quantifiable, within that logic. Because of the presence of quantified predicate variables, the second-order predicate logic has, in a certain sense, more expressive power than the first-order predicate logic. Within the limits of the first-order logic, there is no way to say that for every two properties of integers, there is a property of integers that applies just to those integers to which the first of these two properties, but not the second, applies. Second-order theories contain only finitely many axioms of a specifically mathematical nature, while the corresponding first-order theories require an infinite number of such axioms. Second-order theories possess the very important property of being categorical, in the sense that all of their principal models are alike in structure; the corresponding first-order theories are not categorical.