The Sheffer functions of 3-valued logic

Abstract

In previous papers, Post, Webb, Götlind and the present author have described some Sheffer functions (in Swift's terminology, “independent binary generators”) in m -valued logic. Professor J. Dean Swift has recently isolated the symmetric Sheffer functions of 3-valued logic. In the present paper, we will prove some properties of Sheffer functions in m -valued logic and isolate all of the Sheffer functions of 3-valued logic. Before we proceed we will define some terms which we will find convenient. A set of functions in m -valued logic is functionally complete , if the set of the functions which can be defined explicitly from the functions of the set is exactly the set of all functions of m -valued logic. A function is functionally complete , if its unit set is functionally complete. A Sheffer function is a two-place functionally complete function. If i and j are truth values (1 i , j ≤ m ), we will say i ~ j ( D ), if D is a decomposition of the truth values 1, …, m into 2 or more disjoint non-empty classes and i and j are elements of the same class. A binary function f ( p, q ) satisfies the substitution law for a decomposition D , if for any truth values h, i, j, k , whenever h ~ j ( D ) and i~k ( D ), then f ( h, i ) ~ f ( j, k ) ( D ). The function f ( p,q ) satisfies the co-substitution law for D , if for any truth values h, i, j, k , whenever f ( h, i ) ~ f ( j, k ) ( D ), then h ~ j ( D ) or i ~ k ( D ). We will say f ( p, q ) has the proper substitution property , if there is a decomposition of the truth values into less than m classes for which it satisfies the substitution law.