Partiality and Adjointness in Modal Logic

In Rajeev Goré, Barteld Kooi & Agi Kurucz (eds.), Advances in Modal Logic, Vol. 10. College Publications. pp. 313-332 (2014)
  Copy   BIBTEX

Abstract

Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula phi a formula f(phi) such that for every formula psi, phi -> BOX psi is provable in L if and only if f(phi) -> psi is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 90,616

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

Some Connections between Topological and Modal Logic.Kurt Engesser - 1995 - Mathematical Logic Quarterly 41 (1):49-64.
Kripke semantics for modal substructural logics.Norihiro Kamide - 2002 - Journal of Logic, Language and Information 11 (4):453-470.
A Modal Logic of Information.Krystyna Misiuna - 2012 - Logic and Logical Philosophy 21 (1):33-51.
Finite models constructed from canonical formulas.Lawrence S. Moss - 2007 - Journal of Philosophical Logic 36 (6):605 - 640.
Modal logic over finite structures.Eric Rosen - 1997 - Journal of Logic, Language and Information 6 (4):427-439.
Interpretations of intuitionist logic in non-normal modal logics.Colin Oakes - 1999 - Journal of Philosophical Logic 28 (1):47-60.
Modes of Adjointness.M. Menni & C. Smith - 2013 - Journal of Philosophical Logic (2-3):1-27.
Completeness of S4 for the Lebesgue Measure Algebra.Tamar Lando - 2012 - Journal of Philosophical Logic 41 (2):287-316.
The Dynamification of Modal Dependence Logic.Pietro Galliani - 2013 - Journal of Logic, Language and Information 22 (3):269-295.
Quasi-modal equivalence of canonical structures.Robert Goldblatt - 2001 - Journal of Symbolic Logic 66 (2):497-508.
Inverse Images of Box Formulas in Modal Logic.Lloyd Humberstone - 2013 - Studia Logica 101 (5):1031-1060.

Analytics

Added to PP
2014-06-12

Downloads
81 (#188,993)

6 months
16 (#109,309)

Historical graph of downloads
How can I increase my downloads?

Author's Profile

Wesley H. Holliday
University of California, Berkeley

Citations of this work

Supervenience, Dependence, Disjunction.Lloyd Humberstone - forthcoming - Logic and Logical Philosophy:1.
Substructural epistemic logics.Igor Sedlár - 2015 - Journal of Applied Non-Classical Logics 25 (3):256-285.
What Can You Say? Measuring the Expressive Power of Languages.Alexander Kocurek - 2018 - Dissertation, University of California, Berkeley
Locales, Nuclei, and Dragalin Frames.Guram Bezhanishvili & Wesley Holliday - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. London: College Publications. pp. 177-196.

View all 6 citations / Add more citations

References found in this work

The Connectives.Lloyd Humberstone - 2011 - MIT Press. Edited by Lloyd Humberstone.
Modal Logic.Patrick Blackburn, Maarten de Rijke & Yde Venema - 2001 - Studia Logica 76 (1):142-148.
Models for entailment.Kit Fine - 1974 - Journal of Philosophical Logic 3 (4):347 - 372.

View all 15 references / Add more references