Complete Connectives for the 3‐Valued Propositional Calculus

1. Introduction The problem of finding complete connectives for the ra-valued proposi-tional calculus may be given a purely algebraic formulation as the problem of determining complete generators of the composition algebra on m marks. The elements of such an algebra are functions f{x x ,x 2 , ...,x n), whose n variables z 1} ..., x n range over a fixed finite set M consisting of m marks and whose values belong to the same set; that is, functions which map M x M x ... x M into M. Note that, throughout this paper, m will be used exclusively for the number of marks and n for the number of arguments in a furiction. The fundamental algebraic operation on the elements is composition. Let £ l5 £ 2 , ..., £ " be variables which range over the same set of marks; v may be greater than, equal to, or less than n. Suppose that/ i (^ 1 , f 2 , > £») (i = 1, 2, ...,n) are n given functions and that each x i is restricted by being fi($ v £ 2 , > £.) • (Of course any—or all—of the ^ may be absent from any f t .) Then f(x v x 2 ,...,x n) becomes a function/'(&, £ 2 > > D> sav > of the variables f t , ..., £ " , and we may write .