Generalized Truth Values and Many-Valued Logics: Harmonious Many-Valued Logics

Abstract

In this chapter, we reconsider the notion of an $n$-valued propositional logic. In many-valued logic, sometimes a distinction is made not only between designated and undesignated (not designated) truth values, but also between designated and antidesignated truth values. Even if the set of truth values is, in fact, tripartitioned, usually only a single semantic consequence relation is defined that preserves the possession of a designated value from the premises to the conclusions of an inference. We argue that if in the set of semantical values the sets of designated and antidesignated truth values are not complements of each other, it is natural to define at least $two$ entailment relations, a “positive” one that preserves the possession of a designated value from the premises to the conclusions of an inference, and a “negative” one that preserves the possession of an antidesignated value from the conclusions to the premises. Once this distinction has been drawn, it is quite natural to reflect it in the logical object language and to contemplate many-valued logics $\Uplambda$ whose language is split into a positive and a matching negative logical vocabulary. If the positive and the negative entailment relations do not coincide, if the interpretations of matching pairs of connectives are distinct, and if the positive entailment relation restricted to the positive vocabulary is nevertheless isomorphic to the negative entailment relation restricted to the negative vocabulary, then we say that $\Uplambda$ is a $harmonious$ many-valued logic. We reconstruct some of the logical systems considered in this book as harmonious, finitely-valued logics. At the end of the chapter, we outline some possible ways of generalizing the notion of a harmonious $n$-valued propositional logic.