Combinatory logic is a branch of mathematical logic that analyzes certain processes, such as substitution, which are associated with variables. These processes are taken for granted in most formulations of logic, but they are complex, and since a fundamental part of the resulting theory is recursively undecidable the analysis is not trivial. Combinatory logic contributes to simplifying the ultimate foundations of mathematical logic and to explaining the paradoxes; it contains an arithmetic in which exactly those numerical functions that are partial recursive are representable; and it has potential applications to the deeper study of such areas as logical calculuses of higher order, computer programming, and linguistics.

Before one can define combinatory logic precisely, it is necessary to explain some notions concerning formal systems. This will be done in the next section. In the following section the definition will be given and a plan presented according to...