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Four-valued modal logic: Kripke semantics and duality

5 pagesPublished: 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.

BibTeX entry
@inproceedings{TACL2013:Four_valued_modal_logic,
  author    = {Achim Jung and Umberto Rivieccio},
  title     = {Four-valued modal logic: Kripke semantics and duality},
  booktitle = {TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic},
  editor    = {Nikolaos Galatos and Alexander Kurz and Constantine Tsinakis},
  series    = {EPiC Series in Computing},
  volume    = {25},
  publisher = {EasyChair},
  bibsource = {EasyChair, https://easychair.org},
  issn      = {2398-7340},
  url       = {/publications/paper/5bMt},
  doi       = {10.29007/12bb},
  pages     = {120-124},
  year      = {2014}}
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