On two properties of structurally complete logics

Abstract

Let S = (S, f1,..., fn) sentential language C, possibly with index, denotes a standard consequence. Sb is the consequence operation determined by the rule of substitution. By L(C) we denote the set of all Lindenbaum's extensions of the consequence C (L(C) = {X C S : C(X) = S and C(X U {a}) = S for every a e X}). End(S) is the set of all endomorphisms of S. Let U be the consequence operation induced by a matrix U . The symbol E(U ) stands for the set of all formulas which are valid in U . We also write U C- U, iff U is a submatrix of U1. In this paper we assume that in every functionally complete matrix U(A, D) the set D is proper non-empty subset of the domain A of A. By a rule of inference we mean a non-empty subset of 2S x S. A rule r is finitary iff for every X C S and every a e S : if (X, a) e r, then X is finite. A rule r is elementary iff r = {(h(X), h(a)) : h e End(S)} for some X and for some a. In turn, CA stands for the consequence operation determined by A (A C S) and by the set of the rules R. For simplicity the symbol CR will be used instead of CR. Two particular sets of rules will be used: MP - the set which contains only the modus ponens and the Godel's rule. CL stands for the set of all classical tautologies. I is the set of all theses of the intuitionistic logic and J denotes the set of all theses of the Johansson's minimal logic (Fragment tekstu).