A Single‐Axiom Impligational Calculus of Given Unsolvability

Abstract

in Bristol (England) Q 1. Introduction Erst a paragraph of terminology.A propositional calculus, as the phrase is used here, consists of the following three things :(1) A set of coianectives, adequate to express implication (a connective conaists of a ( 2 ) A particular espression of implication in terms of the given connectives;(3) A set of tautologies, to be kiaown as "atuioms", in which no connectives appearThe wff of a propositional calculus are built up in the usual way from variables (throughout this paper '. variable " means "propositional variable "), the given connectives, and, possibly (according to the convention used), brackets and coinmas. The theorems are those of the wff which can be derived from the axioms, using as rules of inferencc,We are interested in propositional calculi which are uwolvable; that is, there is no effective procedure for deciding whether an arbitrary wff is a theorem.The purpose of this paper is to show that even a, propositional calculus of the Sirrrplwt possiblc structure, may be unsolvable. More precisely, we can satisfy the following three conditions simultaneously :( I ) The calculus is of any given recursively enumerable degree of unsolvability.(2) There is just one axiom.(3) Thc only connective is one representing implication. Say this connective is 3 ; then the designated expression for " A implies U" is " ( A 3 B ) " (without this stipulation, thc designated cxpression could be, for instancc, " ( A 3 ( A 3 B ) ) " ) . Tn other wolds, we have an irnplicational calculus.The strongest result of this kind hitherto obtained, as far as I know, is as above but without the single-axiom condition; it is given in my own paper [2], also by HARKWP in (41. For an extensive survcy of the literature, the reader is referred t o H ZKKC)P')R paper 131.symbol and an assignment to that symbol of LL classical truth-function) ; other than the given. one$.15 X h h r f iiiritli Locik 13*