A short article for the Encyclopedia of Artificial Intelligence: Second Edition “Logic, Higher-order”

While first-order logic has syntactic categories for individuals, functions, and predicates, only quantification over individuals is permitted. Many concepts when translated into logic are, however, naturally expressed using quantifiers over functions and predicates. Leibniz's principle of equality, for example, states that two objects are to be taken as equal if they share the same properties; that is, a = b can be defined as ∀P [P (a) ≡ P (b)]. Of course, first-order logic is very strong and it is possible to encode such a statement into it. For example, let app be a first-order predicate symbol of arity two that is used to stand for the application of a predicate to an individual. Semantically, app(P, x) would mean P satisfies x or that the extension of the predicate P contains x. In this case, the quantified expression could be rewritten as the first-order expression ∀P [app(P, a) ≡ app(P, b)] (appropriate axioms for describing app are required). Such an encoding is often done in a multi-sorted logic setting, where one sort is for individuals and another sort is for predicates over individuals. Set-theory is another first-order language that encodes such higher-order concepts using membership ∈ as the converse of app. Higher-order logics arise from not doing this kind of encoding: instead, more immediate and natural representation of higher-order quantification are considered. Indeed naturalness of higher-order quantification is part of the reason why higher-order logics were initially considered by Frege and Russell as a foundation for mathematics. SYNTAX OF HIGHER-ORDER LOGIC A common approach to describing the syntax of a higher-order logic is to introduce some kind of typing scheme. One approach types first-order individuals with ι, sets of individuals with ι, sets of pairs of individuals with ιι, sets of sets of individuals with ι, etc. Such a typing scheme does not provide types for function symbols. Since in some treatments of higher-order logic, functions can be represented by their graphs, i.e. certain kinds of sets of ordered pairs, this lack is not a serious restriction. Identifying functions up to their graphs does, of course, treat functions extensionally, something that might be too strong in some applications. (A logic is extensional if whenever two predicates or two functions are equal on all their arguments, they themselves are equal.) A more general approach to typing is that used in the Simple Theory of Types (Church, 1940). Here again, the type …