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Axiom Systems for Trilattice Logics

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

Abstract

This chapter is devoted to introducing and discussing the axiom systems for trilattice logics obtained by Odintsov (Studia Logica 93:407–428, 2009). First-degree proof systems for logics related to SIXTEEN3 in the language \({\cal L}_{tf}\) are introduced. Moreover, the language \({\cal L}_{tf}\) is extended by a truth implication → t , a falsity implication → f , or both. Adding implications to the truth and falsity vocabulary of \({\cal L}_{tf}\) is of independent interest. We start with considering two ways of defining a many-valued logic: the method of valuation systems (matrices) and the lattice approach as exemplified by the bilattice FOUR2 and the trilattice SIXTEEN3. We conclude that the lattice approach has the advantage of admitting a general and uniform way of defining implications. An implication connective is required for Odintsov’s Hilbert-style proof systems with modus ponens as the only rule of inference. These Hilbert-style systems axiomatize truth entailment and falsity entailment in the languages comprising at least one implication. The axiom systems are instructive insofar as they reveal that falsity conjunction and falsity disjunction have an indeterministic interpretation with respect to truth entailment in the trilattice SIXTEEN3, whereas truth conjunction and truth disjunction have an indeterministic interpretation with respect to falsity entailment in SIXTEEN3. Moreover, the matrix presentation of the algebraic operations of SIXTEEN3 will be used in  Chap. 6 to obtain cut-free sequent calculi for truth entailment and falsity entailment in the languages with or without truth or falsity implication.

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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