Four-valued modal logic: Kripke semantics and duality

5 pages•Published: July 28, 2014

Abstract

Along the lines of recent investigations combining many-valued and modal systems, we address the problem of defining and axiomatizing the least modal logic over the four-element Belnap lattice. By this we mean the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued. Our main result is the introduction of two Hilbert-style calculi that provide complete axiomatizations for, respectively, the local and the global consequence relations associated to the class of all four-valued Kripke models. Our completeness proofs make an extensive and profitable use of algebraic and topological techniques; in fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach.

Keyphrases: Belnap logic, Jónsson-Tarski duality, many-valued modal logic, N4-lattices, paraconsistent Nelson logic, twist-structures

In: Nikolaos Galatos, Alexander Kurz and Constantine Tsinakis (editors). TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic, vol 25, pages 120--124

Links:	https://easychair.org/publications/paper/5bMt
	https://doi.org/10.29007/12bb