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Generalized Truth Values: SIXTEEN3 and Beyond

Chapter
Part of the Trends in Logic book series (TREN, volume 36)

Abstract

In this chapter we define two principal entailment relations on \(SIXTEEN_{3}\), one determined by the truth order and another determined by the falsity order. It turns out that the logics generated separately by the algebraic operations under the truth order and under the falsity order in \(SIXTEEN_{3}\) coincide with the logic of \(FOUR_{2}\), namely first-degree entailment. In the present setting, however, it becomes rather natural to consider also logical systems in the language obtained by combining the vocabulary of the logic of the truth order and the falsity order. Some important fragments of these combined logics are axiomatized. We also consider iterated powerset formation applied to 4 and introduce Belnap trilattices. These structures again give rise to relations of truth entailment and falsity entailment. It is observed that the logic of truth and the logic of falsity for every Belnap trilattice is one and the same, namely again, first-degree entailment. Finally, we consider Priest’s notion of hyper-contradiction and examine some approaches to generalizations of Priest’s logic of Paradox and Kleene’s logic of uncertainty.

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of PhilosophyState Pedagogical UniversityKryvyi RihUkraine
  2. 2.Department of Philosophy IIRuhr-University BochumBochumGermany

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