The Existence of Combinatory Algebras: Combinatory Logic

Abstract

If the reader has tried to manufacture a useful example of a combinatory algebra for himself, at some stage he will blame me for doing away, along with type differences with a great deal of mathematical intuition. Indeed in this connection the step from the possible to the contradictory has been made by various mathematicians. From the point of view of the founders (Schönfinkel, Curry and Church) the general aim was not just the axiomatization of the concept of application for general functions, but rather a functional foundation of all logic and mathematics. In particular Curry and Church originally proposed systems, which together with reasonable appearing ingredients for combinatory algebra, also included logical rules and parts of mathematics. These expanded systems then proved to be inconsistent (Kleene & Rosser 1935). Curry considered that this lay in the nature of the subject — he compared the axioms with Frege’s system and gave the opinion that “here we are dealing with notions of such great generality, that intuition fails us. We are researching in a no-man’s land between what is certain and what is known to be contradictory”.