Generation of any N-Valued Logic by One Binary Operation.

A two-valued logic was shown in 1913 by Sheffer2 to be obtainable by the iteration of a single binary operation. It was proved in 1925 by Zylinski3 that Sheffer's function and its "dual," also introduced by Sheffer, are the only binary operations such that the iteration of either one will generate the two-valued logic of functions of two propositions. Zylinski's proof was by means of a truth table of 4 columns and 16 rows, corresponding to the two possible values (a or b) which an arbitrary function 4(x, y) can assume in a two-valued system. While theoretically applicable to an n-valued system, the method of direct inspection of the truth table is impracticable. Following another method we prove that: Any n-valued logic, where n > 2, can be generated by the iteration of one binary operation. Designate the n truth values which an "elementary proposition" may take in an n-valued logic by the marks ao, al, ... ., at, 1. For convenience, drop the a and retain only the subscript, so that our marks are now 0, 1, . . .. n-1. It is to be observed that these numbers denote merely n distinct marks without any arithemetical significance. Let p and q be any elementary propositions. Construct a truth table for two elementary propositions, p, q, of two columns and n2 rows with the n marks, 0, 1, . . ., n-1 by assigning in the ith row of the table to p the value [i-1-(i-1)']/n, and to q the value (i-1)', where j' = jmod n, j' > 0 and i = 1, 2. n. Denote the statement, p has the value i, by p = i; let p3q denote any function of p, q whose values are in the range 0, 1, ..., n-1, when p = i, q = j and i, j are in the same range; let i#j = k denote that if p = i and q = j then p(q = k, where k is in the range. Define the stroke, "j," function, plq, by ilj=OifiFj; ili= (i+1)' (i,j=O,1,...,n-1).