THE DEGREE OF COMPLETENESS OF THE m-VALUED LUKASIEWICZ PROPOSITIONAL CALCULUS CORRECTION AND ADDENDUM

P,,...,PnVQ. If, for this formalisation, the value of n is bounded above and the greatest possible value of n is a then the formalisation is saidf to have degree of completeness a+1. Let us denote by C(m) the degree of completeness of a formalisation of the w-valued Lukasiewicz propositional calculus of the type considered in the previous paper with the above title [3]. In that paper it was stated (p. 102) that, since the number of classes is d{m— 1), C(m)^d(m-\) + \. (1) The inference is not, however, valid, since, as we shall now show, there are formulae which are not of any class. We note first that if the formulae P, Q, KPQ, Jt P take| the truth-values x, y, k(x, y), ji(x) respectively, where k(x, y) = min(x, y), Mim-OKm-l))'^! ( /=l , . . . ,m),