On Some Method of Axiomatization of Some Propositional Calculi

Let Fm be the set of formulas formed in the usual manner by means of the infinite set {p0, p1, . . .} of propositional variables and the finite set Con of connectives. For C ∈ Con let ar(C) be the arity of C. Formulas are denoted by a, b, c, . . .. By Fm we mean the restriction of Fm into the variables po, . . . , pk−1. We assume that in terms of the connectives of Con, two peculiar (but not necessarily different) binary connectives “⊃” and “→”, which we call implications, are defined. By Sb,MP, Cn we denote the consequence operators based on substitution, modus ponens for “⊃” as well as substitution and modus ponens for “ ” respectively. Then for every X ⊆ Fm,Cn(X) = MP (Sb(X)) (cf. eg. [1]). The letters M,N denote logical matrices of the type of Con, and by |M | we mean the number of truth-values of M . The set of all tautologies of M is denote by E(M) and E(M) = E(M)∩Fm. It is well-known that the set E(M) is closed under Sb.