A system of formal logic without an analogue to the curry W operator

Abstract

The system of formal logic to be presented in this paper has the following main properties:(1). The Mengenlehre paradoxes do not arise in it.(2). It seems to be free from other contradictions.(3). It does not resort to a theory of types.(4). It uses negation and material implication and does not weaken the principle of excluded middle.(5). It contains no analogue to the Curry W operator; consequently, given a propositional function ϕ, a propositional function Ψ cannot always be found such that, for all x, ϕ(x, x) and Ψ(x) are equivalent propositions.(6). It employs a universal quantifier, G, such that (x)f(x), (x)f(x, x), (x)f(x, x, x), …, would be respectively expressed thus, G(f, f), G(f, f, f), G(f, f, f, f), ….(7). It employs Schönfinkel's functional notation, so that f(x1, x2, …, xn) will be written as (( … ((fx1)x2) …)xn) and abbreviated to (fx1x2… xn) and, when no ambiguity results, to fx1x2…xn.(8). A theory of integers and real numbers cannot, apparently, be deduced from it.(9). A form of the axiom of choice is provable in it.