Systems of illative combinatory logic complete for first order propositional and predicate calculus

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of A-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).