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Canonical Extensions and Kripke–Galois Semantics for Non-distributive Logics

Abstract

This article presents an approach to the semantics of non-distributive propositional logics that is based on a lattice representation (and duality) theorem that delivers a canonical extension of the lattice. Our approach supports both a plain Kripke-style semantics and, by restriction, a general frame semantics. Unlike the framework of generalized Kripke frames (RS-frames), the semantic approach presented in this article is suitable for modeling applied logics (such as temporal, or dynamic), as it respects the intended interpretation of the logical operators. This is made possible by restricting admissible interpretations.

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Correspondence to Chrysafis Hartonas.

Additional information

Presented at Unilog’18, Vichy, France, as the winner of the Aristotle Logic Prize and co-runner for the Universal Logic Prize 2018.

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Hartonas, C. Canonical Extensions and Kripke–Galois Semantics for Non-distributive Logics. Log. Univers. 12, 397–422 (2018). https://doi.org/10.1007/s11787-018-0195-6

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Keywords

  • Canonical lattice extension
  • Logic of lattice expansions
  • Kripke frames
  • Kripke–Galois frames
  • Generalized Kripke frames
  • Modal lattices
  • Implicative lattices

Mathematics Subject Classification

  • 03B45
  • 03B47
  • 03B60
  • 03G10