Direct analogues of the Sheffer stroke in $m$-valued logic.
Norman M. Martin · Notre Dame Journal of Formal Logic · 1976
It has been shown by Peirce 1 and, independently, by Sheffer 2 that all functions of two-valued propositional logic can be defined in terms of a single two-place primitive.This result has been extended to the m-valued case, by the explicit definition of families of such functions, which we will call Sheffer functions, by Webb, 3 Gδtlind, 4 and the present author. 5Explicit characterization of the totality of such functions has been carried out for m = 2 by Post 6 and for m = 3 by the present author. 7It is known that for ra>2, there are always a considerable number of such functions, not always describable in any simple fashion as a generalization of the two functions of the case m = 2.If we limit our considerations to functions which seem analogous to those of the case m = 2 we notice that the Sheffers for m = 2 can be partially characterized by either of two descriptions equivalent for m = 2 but not otherwise: a) There exists a#(0 < K^ 1) such that if i Φ j, f(i, j) = K. b) There exist permutations a 09 a x and b 0 , b λ of the values 0, 1 such that,Generalizing these we obtain: c) There exists a K{0 ^ K < m) such that if i Φ j, f(i, j) = K. ά) There exist permutations a 0 , . .., a m .x and b 0 , . .., b m .x of the values 0. . .., m -1 such that if i < j, f{βi, aj) = δ t .Webb has produced Sheffer functions of both types.In this paper we will prove a necessary and sufficient condition for a function of such types to be a Sheffer function.