Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T03:55:18.707Z Has data issue: false hasContentIssue false
  • English
  • Français

INSTANTIAL NEIGHBOURHOOD LOGIC

Published online by Cambridge University Press:  19 December 2016

JOHAN VAN BENTHEM ,
NICK BEZHANISHVILI ,
SEBASTIAN ENQVIST  and
JUNHUA YU
JOHAN VAN BENTHEM*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam; Department of Philosophy, Stanford University; and Changjiang Scholar Program, Tsinghua University
NICK BEZHANISHVILI*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam
SEBASTIAN ENQVIST*
Affiliation:
Institute for Logic, Language and Computation, University of Amsterdam
JUNHUA YU*
Affiliation:
Department of Philosophy, Tsinghua University
*
*INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS DEPARTMENT OF PHILOSOPHY STANFORD UNIVERSITY, CA USA and CHANGJIANG SCHOLAR PROGRAM TSINGHUA UNIVERSITY BEIJING 100084 CHINA E-mail: J.vanBenthem@uva.nl
†INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: N.Bezhanishvili@uva.nl
‡INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM, P.O. BOX 94242 1090 GE AMSTERDAM, THE NETHERLANDS E-mail: sebastian.enqvist@fil.lu.se
§DEPARTMENT OF PHILOSOPHY TSINGHUA UNIVERSITY BEIJING 100084 CHINA E-mail: junhua.yu.5036@outlook.com
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, and determine its precise SAT complexity. Finally, we discuss proof-theoretic fine-structure of INL in terms of semantic tableaux and some expressive fine-structure in terms of fragments, while discussing concrete illustrations of the instantial neighborhood language in topological spaces, in games with powers for players construed in a new way, as well as in dynamic logics of acquiring or deleting evidence. We conclude with some coalgebraic perspectives on what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

Keywords

03B45modal logicneigbourhood semanticsbisimulationsaxiomatizationcompletenesstranslationstopological semanticsgame logiccoalgebraic logic
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 1 , March 2017 , pp. 116 - 144
Copyright
Copyright © Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Aiello, M., van Benthem, J., & Bezhanishvili, G. (2003). Reasoning about space: The modal way. Journal of Logic and Computation, 13(6), 889–920.Google Scholar
Barwise, J. (1975). Admissible Sets and Structures. Berlin-New York: Springer-Verlag.Google Scholar
van Benthem, J. (1996). Exploring Logical Dynamics. Studies in Logic, Language and Information. Stanford, CA: CSLI Publications & Cambridge University Press.Google Scholar
van Benthem, J. (1998). Dynamic Odds & Ends. ILLC Prepublication Series Report ML-1998-08, Institute for Logic Language and Computation, University of Amsterdam.Google Scholar
van Benthem, J. (2003). Logic games are complete for game logics. Studia Logica, 75(2), 183–203.Google Scholar
van Benthem, J. (2010). Modal Logic for Open Minds. Stanford, CA: CSLI Publications.Google Scholar
van Benthem, J. (2014). Logic in Games. Cambridge, MA: The MIT Press.Google Scholar
van Benthem, J., Bezhanishvili, N., & Enqvist, E. (2016). A game logic for instantiated powers. To appear.Google Scholar
van Benthem, J. & Pacuit, E. (2011). Dynamic logics of evidence-based beliefs. Studia Logica, 99(1–3), 61–92.CrossRefGoogle Scholar
Berge, C. (1989). Hypergraphs: Combinatorics of Finite Sets. Amsterdam, New York, Oxford, Tokyo: North-Holland.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Chellas, B. (1980). Modal Logic. Cambridge-New York: Cambridge University Press. An introduction.CrossRefGoogle Scholar
Conradie, W., Ghilardi, S., & Palmigiano, A. (2014). Unified correspondence. In Baltag, A. and Smets, S., editors. Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, Vol. 5. Cham: Springer, pp. 933–975.Google Scholar
Enqvist, S., Seifan, F., & Venema, Y. (2015). Expressiveness of the modal mu-calculus on monotone neighborhood structures. CoRR. Available at https://arxiv.org/abs/1502.07889.Google Scholar
Fine, K. (1975). Normal forms in modal logic. Notre Dame Journal of Formal Logic, 16, 229–237.Google Scholar
Fitting, M. (1983). Proof Methods for Modal and Intuitionistic Logics. Dordrecht: Reidel.Google Scholar
Gabbay, D. M., Kurucz, A., Wolter, F., & Zakharyaschev, M. (2003). Many-dimensional Modal Logics: Theory and Applications. Studies in Logic and the Foundations of Mathematics, Vol. 148. Amsterdam: North-Holland Publishing Co.Google Scholar
Girard, P. (2008). Modal Logic for Belief and Preference Change. For ordering. Ph.D. Thesis, Stanford University.Google Scholar
Goré, R. (1999). Tableau methods for modal and temporal logics. In D’Agostino, M., Gabbay, D., Haehnle, R. and Posegga, J., editors. Handbook of Tableau Methods. Dordrecht: Kluwer Academic Publishers, pp. 297–396.CrossRefGoogle Scholar
Hansen, H. H. (2003). Monotonic Modal Logics. Master’s Thesis, ILLC, University of Amsterdam.Google Scholar
Hansen, H. H., Kupke, C., & Pacuit, E. (2009). Neighbourhood structures: Bisimilarity and basic model theory. Logical Methods in Computer Science, 5(2), 1–38.Google Scholar
Keisler, J. (1971). Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers. Studies in Logic and the Foundations of Mathematics, Vol. 62. Amsterdam-London: North-Holland Publishing Co.Google Scholar
Kracht, M. & Wolter, F. (1999). Normal monomodal logics can simulate all others. Journal of Symbolic Logic, 64(1), 99–138.Google Scholar
Leitgeb, H. (2016). New foundations of hyperintensional semantics? To appear.Google Scholar
Montague, R. (1970). Universal grammar. Theoria, 36, 373–398.Google Scholar
Moss, L. (2007). Finite models constructed from canonical formulas. Journal of Philosophical Logic, 36(6), 605–640.CrossRefGoogle Scholar
Parikh, R. (1985). The logic of games and its applications. In Karpinski, M. and van Leeuwen, J., editors. Topics in the Theory of Computation (Borgholm, 1983). North-Holland Mathematics Studies, Vol. 102. Amsterdam: North-Holland, pp. 111–139.Google Scholar
Pauly, M. (2001). Logic for Social Software. Ph.D. Thesis, ILLC, University of Amsterdam.Google Scholar
Scott, D. (1970). Advice on modal logic. In Lambert, K., editor. Philosophical Problems in Logic. Some Recent Developments. Dordrecht: Reidel, pp. 143–173.Google Scholar
Venema, Y. (2007). Algebras and coalgebras. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Studies in Logic and Practical Reasoning, Amsterdam: Vol. 3. Elsevier, pp. 331–426.Google Scholar
Voloshin, V. I. (2009). Introduction to Graph and Hypergraph Theory. New York: Nova Science Publishers, Inc.Google Scholar