Completeness of the propositional calculus

Abstract

The completeness of the prepositional calculus was first proved by Post. His somewhat condensed proof has been succeeded by more detailed presentations of substantially the same argument, and also by several proofs of radically different forms. The present paper contains still another proof, offered because of its relative simplicity. In part this proof follows a plan which was sketched by Wajsberg for another purpose, viz. for proving the completeness of the sub-calculus involving only the material conditional.For the present proof a systematization of the propositional calculus will be used which is due to Tarski, Bernays, and Wajsberg. It involves the material conditional “⊃” and the falsehood “F” as primitive; thus the formulae are recursively describable as comprising “F”, the variables “p”, “q”, “r”, …, and all results of putting formulae for “p” and “q” in “(p⊃q)”. The denial “˜p” is definable as “(p⊃F)”, and all other truth functions are then definable in familiar fashion. (Conversely, in a system admitting “˜” instead of “F” as primitive, “F” might be explained as an abbreviation of “˜(P⊃P)”.) The postulates are four:(1) ((p⊃q)⊃((q⊃r)⊃ (p⊃ r)))(2) (((p⊃(q⊃p)⊃p)(3) (p⊃(q⊃p))(4) (F⊃p).