Combinatory logic

Combinatory logic comprises a battery of formalisms for expressing and studying properties of operations constitutive to contemporary logic and its applications. The sole syntactic category in combinatory logic is that of the applicative term. Closed terms are called ‘combinators’; there is no binding of variables. Systems containing the basic combinators S and K exhibit the crucial property of combinatorial completeness: every routine expressible in the system can be captured by a term composed of these two combinators alone. Combinatory logic is a close relative of Church’s lambda calculus. M. Schönfinkel first introduced and defined basic combinators in 1920 in assaying foundations for mathematics that avoid bound variables and take operations, rather than sets, as fundamental. H. Curry later rediscovered the combinators (and coined the term ‘combinatory logic’) independently of Schönfinkel. Curry constructed various formal systems for combinatory logic and, throughout most of the subject’s history, was the central figure in the research. In 1969, D. Scott succeeded in constructing set-theoretic, functional models for the lambda calculus and combinatory logic. Since then semantic studies of combinatory systems, together with research on their applications to computer science and further development as foundational systems, have dominated the field.