Skip to main content
Log in

First-Order Classical Modal Logic

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].

We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alur, R., T. A. Henzinger, and O. Kupferman, ‘Alternating-time temporal logic’, in Compositionality: The Significant Difference, LNCS 1536, Springer, 1998, pp. 23-60.

  2. Arló Costa, H., ‘Qualitative and Probabilistic Models of Full Belief’, Proceedings of Logic Colloquim'98, Lecture Notes on Logic 13, S. Buss, P. Hajek, P. Pudlak (eds.), ASL, A. K. Peters, 1999.

  3. Arló Costa, H., ‘First order extensions of classical systems of modal logic’, Studia Logica 71:87–118, 2002.

    Article  Google Scholar 

  4. Arló Costa, H., ‘Non-Adjunctive Inference and Classical Modalities’, Journal of Philosophical Logic 34 (5-6):581–605, 2005.

    Article  Google Scholar 

  5. Arló Costa, H., and E. Pacuit, ‘Free Quantified (Classical) Modal Logic: Varying Domains’, typescript CMU and ILLC-Amsterdam, 2006.

  6. Arló Costa, H., and R. Parikh, ‘Conditional Probability and Defeasible Inference’, Journal of Philosophical Logic 34 (1):97–119, 2005.

    Article  Google Scholar 

  7. Bacharach, M. O. L., L.-A. Gérard Varet, P. Mongin, and H.S. Shin (eds.), Epistemic Logic and the Theory of Games and Decisions, Theory and Decision Library, Vol. 20, Kluwer Academic Publishers, 1997.

  8. Barcan (Marcus), R. C., ‘A functional calculus of First Order based on strict implication’, Journal of Symbolic Logic XI:1–16, 1946.

    Google Scholar 

  9. Barcan (Marcus), R. C., ‘The identity of individuals in a strict functional calculus of First Order’, Journal of Symbolic Logic XII:12–15, 1947.

    Google Scholar 

  10. Barcan Marcus, R. C., ‘Modalities and intensional languages,’ Synthese XIII, 4:303–322, 1961. Reprinted in Modalities: Philosophical Essays, Oxford University Press, 1993, pp. 3–36.

  11. Barcan Marcus, R. C., ‘Possibilia and Possible Worlds,’ in Modalities: Philosophical Essays, Oxford University Press, 1993, pp. 189–213.

  12. Barwise, J., and L. Moss, Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena, C S L I Publications, February 1996.

  13. Battigalli, P., and G. Bonanno, ‘Recentresults on belief, knowledge andthe epistemic foundations of game theory,’ Proceedings of Interactive Epistemology in Dynamic Games of Incomplete Information, Venice, 1998.

  14. Benthem, J.F.A.K., van, ‘Two simple incomplete logics,’ Theoria 44:25–37, 1978.

    Article  Google Scholar 

  15. Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, 58, Cambridge University Press, 2001.

  16. Carnap, R., Meaning and Necessity; A study of semantics and modal logic, Second edition, Midway Reprint, Chicago, 1988.

  17. Chellas, B. Modal logic an introduction, Cambridge University Press, 1980.

  18. Corsi, G., ‘A Unified Completeness Theorem for Quantified Modal Logics’, J. Symb. Log. 67(4):1483–1510, 2002.

    Google Scholar 

  19. Cresswell, M. J., ‘In Defense of the Barcan Formula’, Logique et Analyse 135–6:271–282, 1991.

    Google Scholar 

  20. De Finetti, B., Theory of Probability, Vol I, Wiley Classics Library, John Wiley and Sons, New York, 1990.

    Google Scholar 

  21. Dubins, L.E., ‘Finitely additive conditional probabilities, conglomerability, and disintegrations,’ Ann. Prob. 3:89–99, 1975.

    Google Scholar 

  22. Fagin, R., J. Y. Halpern, Y. Moses, and M. Y. Vardi, Reasoning about knowledge, MIT Press, Cambridge, Massachusetts, 1995.

    Google Scholar 

  23. Fitting, M., and R. Mendelsohn, First Order Modal Logic, Kluwer, Dordrecht, 1998.

    Google Scholar 

  24. Gabbay, D., Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics, Dordrecht, Reidel, 1976.

    Google Scholar 

  25. Gaifman, H., and M. Snir, ‘Probabilities Over Rich Languages, Testing and Randomness’, J. Symb. Log. 47 (3): 495–548, 1982.

    Article  Google Scholar 

  26. Garson, J., ‘Unifying quantified modal logic’, Journal of Philosophical Logic 34 (5–6):621–649, 2005.

    Article  Google Scholar 

  27. Gerson, M., ‘The inadequacy of neighborhood semantics for modal logic’, Journal of Symbolic Logic 40 (2):141–8, 1975.

    Article  Google Scholar 

  28. Garson, J., ‘Quantification in modal logic’, Handbook of Philosophical Logic, D. Gab-bay and F. Guenthner (eds.), vol II, Kluwer Academic Publishers, Dordrecht, 2nd edition, 2002, pp. 249–307.

    Google Scholar 

  29. Ghilardi, S., ‘Incompleteness results in Kripke semantics’, Journal of Symbolic Logic 56 (2):517–538, 1991.

    Article  Google Scholar 

  30. Gilio, A., ‘Probabilistic reasoning under coherence in System P’, Annals of Mathematics and Artificial Intelligence 34:5–34, 2002.

    Article  Google Scholar 

  31. Goldblatt, R., and E. Mares, ‘A General Semantics for Quantified Modal Logic’, Advances in Modal Logic, Volume 6, King's College Publisher, 2006.

  32. Halpern, J., ‘Intransitivity and vagueness,’ Ninth International Conference on Principles of Knowledge Representation and Reasoning (KR 2004), 121–129, 2004.

  33. Hansen, H. H., Monotonic modal logics, Master's thesis, ILLC, 2003.

  34. Harsanyi, J., ‘Games of incomplete information played by Bayesian players, Parts I, II, III’, Management Science 14:159–182, 320–334, 486–502, 1967–68.

    Article  Google Scholar 

  35. Hughes, G. E., and M. J. Cresswell, A new introduction to modal logic, Routledge, 2001.

  36. Kyburg, H. E., Jr., ‘Probabilistic inference and non-monotonic inference,’ in R. D. Schachter, T. S. Evitt, L. N. Kanal, and J. F. Lemmer (eds.), Uncertaintyin Artificial Intelligence, Elsevier Science (North Holland), 1990.

  37. Kyburg, H. E., Jr., and C. M. Teng, ‘The Logic of Risky Knowledge’, Proceedings of WoLLIC, Brazil, 2002.

  38. Kracht, M., and F. Wolter, ‘Normal Monomodal Logics can simulate all others’, Journal of Symbolic Logic 64:99–138, 1999.

    Article  Google Scholar 

  39. Kratzer, A. ‘Modality’, in A. von Stechow and D. Wunderlich (eds.), Semantik. Ein internationales Handbuch der zeitgenossischen Forschung, Walter de Gruyter, Berlin, 1991, pp. 639-650.

    Google Scholar 

  40. Kripke, S., ‘Semantical considerations on modal logics’, Acta Philosophica Fennica 16:83–94, 1963.

    Google Scholar 

  41. Lehmann, D., and M. Magidor, ‘What does a conditional base entails?’, Artificial Intelligence 55:1–60, 1992.

    Article  Google Scholar 

  42. Levi, I., For the sake of the argument: Ramsey test conditionals, Inductive Inference, and Nonmonotonic reasoning, Cambridge University Press, Cambridge, 1996.

    Google Scholar 

  43. Linsky, B., and E. Zalta, ‘In Defense of the Simplest Quantified Modal Logic’, Philosophical Perspectives, 8 (Logic and Language), 431–458, 1994.

    Article  Google Scholar 

  44. Montague, R., ‘Universal Grammar’, Theoria 36:373–98, 1970.

    Article  Google Scholar 

  45. Parikh, R., ‘The logic of games and its applications’, in M. Karpinski and J. van Leeuwen (eds.), Topics in the Theory of Computation, Annals of Discrete Mathematics, 24, Elsevier, 1985.

  46. Pauly, M., ‘A modal logic for coalitional power in games’, Journal of Logic and Computation 12 (1):149–166, 2002.

    Article  Google Scholar 

  47. Scott, D., ‘Advice in modal logic’, in K. Lambert (ed.), Philosophical Problems in Logic, Dordrecht, Netherlands: Reidel, 1970, pp. 143–73.

    Google Scholar 

  48. Scott, D., and P. Krauss, ‘Assigning probability to logical formulas,’ in Hintikka and Suppes (eds.), Aspects of Inductive Logic, North-Holland, Amsterdam, 1966, pp. 219–264.

    Google Scholar 

  49. Segerberg, K., An Essay in Classical Modal Logic, Number 13 in Filosofisska Studier. Uppsala Universitet, 1971.

  50. van Fraassen, B. C., ‘Fine-grained opinion, probability, and the logic of full belief’, Journal of Philosophical Logic, XXIV:349–77, 1995.

    Article  Google Scholar 

  51. Vardi M., ‘On Epistemic Logic and Logical Omniscience,’ in Y. Halpern (ed.), Theoretical Aspects of Reasoning about Knowledge. Proceedings of the 1986 Conference, Morgan Kaufmann, Los Altos, 1986.

    Google Scholar 

  52. Williamson, T., ‘Bare Possibilia,’ Erkenntnis 48:257–273, 1998.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Horacio Arló-Costa.

Additional information

Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arló-Costa, H., Pacuit, E. First-Order Classical Modal Logic. Stud Logica 84, 171–210 (2006). https://doi.org/10.1007/s11225-006-9010-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11225-006-9010-0

Keywords

Navigation