Non-finitely axiomatisable modal product logics with infinite canonical axiomatisations

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Abstract

Our concern is the axiomatisation problem for modal and algebraic logics that correspond to various fragments of two-variable first-order logic with counting quantifiers. In particular, we consider modal products with Diff, the propositional unimodal logic of the difference operator. We show that the two-dimensional product logic Diff×Diff is non-finitely axiomatisable, but can be axiomatised by infinitely many Sahlqvist axioms. We also show that its ‘square’ version (the modal counterpart of the substitution and equality free fragment of two-variable first-order logic with counting to two) is non-finitely axiomatisable over Diff×Diff, but can be axiomatised by adding infinitely many Sahlqvist axioms. These are the first examples of products of finitely axiomatisable modal logics that are not finitely axiomatisable, but axiomatisable by explicit infinite sets of canonical axioms.

Original languageEnglish
Article number102786
Pages (from-to)1-46
Number of pages46
JournalAnnals of Pure and Applied Logic
Volume171
Issue number5
Early online date31 Jan 2020
DOIs
Publication statusPublished - 1 May 2020

Keywords

  • Algebraic logic
  • Canonical and Sahlqvist axiomatisations
  • Elsewhere quantifiers
  • Non-finite axiomatisability
  • Products of modal logics

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