Deductive completeness

This is an exposition of Lambek’s strengthening and generalization of the deduction theorem in categories related to intuitionistic propositional logic. Essential notions of category theory are introduced so as to yield a simple reformulation of Lambek’s Functional Completeness Theorem, from which its main consequences can be readily drawn. The connections of the theoremwith combinatory logic, and with modal and substructural logics, are briefly considered at the end. Introduction. On its own, a difficult proof is not a sufficient condition for a good theorem. Neither is it a necessary condition. As a matter of fact, it is normally better to have a simple proof, if such is available. Many a theorem started by having a difficult proof, until the notions involved were clarified enough, and the theorem perhaps reformulated, so as to obtain a simple proof. However, some important theorems always had a simple proof. Such is the theorem that p 2 is not rational, or Cantor’s theorem that the real interval [0, 1] is not denumerable, or again Cantor’s theorem on the cardinality of the power set. The deduction theorem, too, is of that kind: it always had a very simple proof. It is one of the first things one has to prove in an elementary logic course, and both the teacher and the pupil do it easily. This is nevertheless an important theorem. Its importance is perhaps the greatest for a field of logic where, except at the very beginning, it is hardly ever mentioned: classical model theory. Once we have established strong completeness and compactness, we usually need not bother any more about the consequence relation, and can work exclusively with ordinary Hilbert systems. But this is because the deduction theorem holds. However, the deduction theorem belongs rather to proof theory than to model theory. And there, it is a completeness theorem of some sort. It tells us that the system is strong enough to mirror its extensions in the same language. The completeness in question is of a syntactical, deductive, sort. For the deductions we shall be able to make in the extension we already have corresponding implications in the system. However, this completeness Received March 6, 1996. c 1996, Association for Symbolic Logic 1079-8986/96/0203-0001/$5.10