Topological completeness of first-order modal logic

Steve Awodey, Kohei Kishida

Research output: Chapter in Book/Report/Conference proceedingConference contribution


As McKinsey and Tarski [20] showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the "necessity" operation is modeled by taking the interior of an arbitrary subset of a topological space. This topological interpretation was recently extended in a natural way to arbitrary theories of full first-order logic by Awodey and Kishida [3], using topological sheaves to interpret domains of quantification. This paper proves the system of full first-order S4 modal logic to be deductively complete with respect to such extended topological semantics. The techniques employed are related to recent work in topos theory, but are new to systems of modal logic. They are general enough to also apply to other modal systems.

Original languageEnglish (US)
Title of host publicationAdvances in Modal Logic
EditorsTorben Brauner, Lawrence Moss, Thomas Bolander, Silvio Ghilardi
PublisherCollege Publications
Number of pages17
ISBN (Electronic)9781848900684
StatePublished - 2014
Externally publishedYes
Event9th Conference on Advances in Modal Logic, AiML 2012 - Copenhagen, Denmark
Duration: Aug 22 2012Aug 25 2012

Publication series

NameAdvances in Modal Logic


Conference9th Conference on Advances in Modal Logic, AiML 2012


  • Completeness
  • First-order modal logic
  • Topological semantics

ASJC Scopus subject areas

  • Logic
  • Computational Theory and Mathematics
  • Computational Mathematics


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