Axiomatization of propositional calculus with Sheffer functors.
Thomas W. Scharle · Notre Dame Journal of Formal Logic · 1965
The two binary functors of the (two-valued) propositional calculus known as Sheffer functors have the property that all other functors are definable by each of them.2 Hence, one is able to base a functionally complete propositional calculus on either of these functors, and it is of interest to axiomatize such systems, which is the main purpose of this paper.We will employ the parenthesis-free notation of Lukasiewicz, in which the Sheffer functors are given by Dpq = NKpq, i.e., not both p and q Spq = KNpNq, i.e., neither p nor q For such reasons, all investigations have been for D axioms, using the ruleDL 3 Following Nicod [6], the conventional rule of detachment for D is D1.Clearly, this is a stronger rule than, say, D3, for we are allowed more freedom in the first line.However, the only investigations carried out have