Classical Indeterminacy , Many - Valued Logic

The problem of representing 3-valued supervaluational languages by 2-dimensional product languages is pursued in [3], [8], and [4]. A solution is presented here that attempts to incorporate into such a representation familiar and plausible intuitive principles from traditional many-valued theories. In particular, it is argued in Sections II and III that the concept of classical indeterminacy, which is the key motivational idea behind supervaluation, also underlies the matrices of Jan ~ukasiewicz and of S.C. Kleene's strong connectives and that indeterminacy is of quite general importance to the understanding of one tradition in many-valued logic. In Section IV the kinship between supervaluations, on the one hand, and the 3-valued matrices of ~ukasiewicz and Kleene, on the other, is developed both in intuitive discussion and in a formal characterization of the former in terms of the latter. In Section V the four-valued semantics of Hans Herzberger [3] is interpreted in terms of classical indeterminacy and is used to express in two dimensions and four values the ideas of the Kleene and ~ukasiewicz theories. Then by the characterization of Section IV supervaluations are represented in two dimensions. Finally, in Section VI this representation is shown to be co-extensive to that of Herzberger in [4] and to provide, in effect, a defense of his theory in terms of traditional conceptual foundations.