On the inconsistency of systems similar to

Abstract

This note shows that the inconsistency proof of the system of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:Also the inconsistency follows almost as easily foras it does for ⊢ H2X for arbitrary X, so we will consider the more general case.The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,andLet G = [x] Hn−1x⊃: . … H2x⊃ :Hx⊃ . x ⊃ A, where A is arbitrary. Then if Y is the paradoxical combinator and X = YG, X = GX.Now X ⊂ X, i.e.,