The axiomatic system of the factorial implication

Abstract

In the present paper I submit my results concerning the systems Q and Qf of S. Ja?kowski [3], [4] and I quote the set of axioms for the system Qf with the factorial implication as a primitive notion. The considerations are divided into two parts, one of them concerning non-axiomatic, the second concerning axiomatic systems. In ? 1 I prove the necessary and sufficient condition for a formula to be a theorem of the system Qf. The ? 2 contains the proof of the mutual adequate interpretability1 of the systems Q and Qf. In ? 3 I construct the axiomatic system Q? of factorial implication. Next to it I demonstrate that in this system the axiomatic calculus of dependent sentential variables (?4) and the modal system S5 of Lewis2 (? 5) are interpretable. In ? 6 I show that the system Q? forms an abstract cylindric co-dimensional algebra without diagonal elements3.