A revision of the fundamental rules of combinatory logic

The purpose of this paper is to put on record some theorems relating to improvements in the primitive frame of combinatory logic. These improvements were, for the most part, suggested by the work of Rosser, who formulated a weakened system of combinatory logic in which the rules had a simple character not possessed by those of the original system. In the latter the rules B, C, W, K were in reality axiom-schemes, and their postulation amounted to assuming infinitely many axioms. Rosser had rules of procedure such that no propositions were deducible from them except in combination with axioms (or previously proved propositions); moreover, the conclusion of each rule was uniquely determined by the premises. He also eliminated equality as a primitive term, defining it (essentially) according to the traditional method. This paper shows that these and related advantages apply to certain formulations of the full system of combinatory logic, so far as it concerns the theory of combinators.The method of procedure is as follows. Instead of setting up a primitive frame at the start and then deriving its properties, I begin (after some preliminary explanations in §2) by stating in §3 the properties which it is desired to establish. The next few sections are devoted to the formulation and proof of certain general theorems concerning possible bases for the system of §3. These theorems are, perhaps, more general than is necessary for the immediate purpose, but they are of some interest on their own account. A formulation in terms of the primitives of original system (i.e., B, C, W, and K), which is of the same general type as Rosser's formulation, is obtained at the end of §6. In §7 are discussed the changes in this formulation which are sufficient in order to base it on the primitive combinators S and K of Schönfinkel.