Abstract
We present a general approach to quantified modal logics that can simulate most other approaches. The language is based on operators indexed by terms which allow to express de re modalities and to control the interaction of modalities with the first-order machinery and with non-rigid designators. The semantics is based on a primitive counterpart relation holding between n-tuples of objects inhabiting possible worlds. This allows an object to be represented by one, many, or no object in an accessible world. Moreover by taking as primitive a relation between n-tuples we avoid some shortcoming of standard individual counterparts. Finally, we use cut-free labelled sequent calculi to give a proof-theoretic characterisation of the quantified extensions of each first-order definable propositional modal logic. In this way we show how to complete many axiomatically incomplete quantified modal logics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bauer, S., & Wansing, H. (2002). Consequence, counterparts and substitution. The Monist, 85, 483–497.
Belardinelli, F. (2007). Counterpart semantics for quantified modal logics. In The Logica Yearbook 2006. Filosofia, Prague
Belardinelli, F. (2022). Counterpart semantics at work: Independence and incompleteness in quantified modal logics (p. 2022). Springer, Cham: In Thinking and calculating.
van Benthem, J. (2010). Frame correspondences in modal predicate logic. In Proofs, Categories and Computations: Essays in Honor of Grigori Mints. College Publications, London.
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. Cambridge University Press.
Braüner, T., & Ghilardi, S. (2006). First-order modal logic. Elsevier, New York: In Handbook of Modal Logic.
Corsi, G. (2009). Necessary for. In Logic, Methodology and Philosophy of Science, Proceedings of the\(13^{th}\)International Congress. King’s College Publications, London
Corsi, G., & Orlandelli, E. (2013). Free Quantified epistemic logics. Studia Logica, 101, 1159–1183.
Corsi, G., & Orlandelli, E. (2016). Sequent calculi for indexed epistemic logics. In: Proceedings of the 2nd International Workshop on Automated Reasoning in Quantified Non-Classical Logics (ARQNL 2016). CEUR-ws
Cresswell, M. (1995). Incompleteness and the Barcan formula. Journal of Philosophical Logic, 24, 379–403.
Cresswell, M. (2000). How to complete some modal predicate logics. In Advances in Modal Logic, vol. 2. CSLI Publications, Stanford
Dyckhoff, R., & Negri, S. (2015). Geometrisation of first-order logic. Bulletin of Symbolic Logic, 21, 123–163.
Fellin, G., Negri, S., & Orlandelli, E. (2023). Glivenko sequent classes and constructive cut elimination in geometric logics. Archive for Mathematical Logic, 62, 657–688
Fitting, M. (1991). Modal logic should say more than it does. In Computational Logic: Essays in Honor of Alan Robinson. MIT Press.
Fitting, M. (2004). First-order intensional logic. Annals of Pure and Applied Logic, 127, 171–193.
Fitting, M. (2020). De re, de dicto, and binding modalitities. In Knowledge, Proof and Dynamics: MIT Press.
Fitting, M., & Mendelsohn, R. L. (1998). First-Order Modal Logic. Springer.
Gabbay, D. M., Shehtman, V., & Skvortsov, D. (2009). Quantification in Nonclassical Logic, vol. 1. Elsevier
Garson, J. W. (2005). Unifying quantified modal logic. Journal of Philosophical Logic, 34, 621–649.
Gibbard, A. (1975). Contingent identity. Journal of Philosophical Logic, 4, 187–221.
Ghilardi, S. (1991). Incompleteness results in Kripke semantics. Journal of Symbolic Logic, 56, 517–538.
Ghilardi, S. (1992). Quantified extensions of canonical propositional intermediate logics. Studia Logica, 51, 195–214.
Goldblatt, R. (2011). Quantifiers, Propositions and Identity. CUP, Cambridge.
Hazen, A. P. (1979). Counterpart-theoretic semantics for modal logic. Journal of Philosophy, 76, 319–338.
Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. Routledge.
Kaplan, D. (1986). Opacity. Open Court, Chicago: In The Philosophy of W.V.O. Quine.
Kracht, M., & Kutz, O. (2002). The semantics of modal predicate logic. Part 1: completeness. In Advances in Modal Logic, vol. 3. CSLI Publications, Stanford
Kracht, M., & Kutz, O. (2005). The semantics of modal predicate logic. Part 2: modal individuals revisited. In Intensionality. A K Peters, Los Angeles
Kupfer, M. (2014). Weak logic of modal metaframes. In The Logica Yearbook 2013. College Publications, London
Lewis, D. (1968). Counterpart theory for quantified modal logic. Journal of Philosophy, 65, 113–126.
Montagna, F. (1984). The predicate modal logic of provability. Notre Dame Journal of Formal Logic, 25, 179–189.
Negri, S. (2003). Contraction-free sequent calculi for geometric theories with an application to Barr’s theorem. Archive for Mathemathical Logic, 42, 389–401.
Negri, S. (2005). Proof analysis in modal logic. Journal of Philosophical Logic, 34, 507–544.
Negri, S. (2009). Kriple completeness revisited. In Acts of knowledge: History, Philosophy, and Logic. College Publications, London.
Negri, S., & Orlandelli, E. (2019). Proof theory for quantified monotone modal logics. Logic Journal of the IGPL, 27, 478–506.
Negri, S., & von Plato, J. (2011). Proof Analysis. Cambridge: CUP.
Orlandelli, E. (2021). Labelled calculi for quantified modal logics with definite descriptions. Journal of Logic and Computation, 31, 923–946.
Rybakov, M, & Shkatov, D. (2018). A recursively enumerable Kripke complete first-order logic not complete with respect to a first-order definable class of frames. In Advances in Modal Logic, vol. 12. College Publications, London
Shehtman, V., & Skvortsov, D. (1993). Maximal Kripke-type semantics for modal and superintuitionistic predicate logics. Annals of Pure and Applied Logic, 63, 69–101.
Acknowledgements
Special thanks are due to Melvin Fitting, Silvio Ghilardi, Pino Rosolini, and Frank Wolter for precious suggestions. We are also grateful to two anonymous referee for their helpful comments, and to to the audience at Bonn, Bochum, Bologna, Genoa, FoMoLo, and UConn, where preliminary versions of this paper were presented.
Funding
Open access funding provided by Alma Mater Studiorum - Università di Bologna within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Orlandelli, E. Quantified Modal Logics: One Approach to Rule (Almost) them All!. J Philos Logic 53, 959–996 (2024). https://doi.org/10.1007/s10992-024-09754-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-024-09754-7