Abstract

By a language we shall mean in this paper a propositional language with an innite of variables and an arbitrary set of connectives including the binary connective !, the implication. To make notations more readable we adopt the convention of associating to the left and ignoring the implication sign !. For example we will write x instead of ) ! and xyy instead of ! y) ! ! x). By a system in a given language we mean a subset of this language which is closed under the substitution rule of this language and the detachment rule for the implication. A subset of a system is called a basis i it is not contained in any proper subsystem of this system. A system is nitely axiomatizable i it has a nite basis. In this paper we give a sucient condition of nite axiomatizability of the intersection of two nitely axiomatizable systems. A trick used in the proof of suciency of this condition can be applied for showing that the intersection of the implicational fragments of the intuitionistic propositional logic and the innite-valued logic of Lukasiewicz can be axiomatized by the following axioms