Abstract
A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence.
Similar content being viewed by others
References
Boolos, G., 1979,The Unprovability of Consistencey, Cambridge, Cambridge University Press.
Corsi, G. and Ghilardi, S., 1989, Directed frames.Archive for Mathematical Logic, Vol. 29, pp. 53–67.
Corsi, G. and Ghilardi, S., 1992, Semantical aspects of quantified modal logic.Knowledge, Belief and Strategic Action, Ed. C. Bicchieri and M. L. Dalla Chiara, Cambridge University Press, pp. 167–195.
Cresswell, M. J., 1987, Magari's theorem via the recession frame.Journal of Philosophical Logic, Vol. 16, pp. 13–15.
Ghilardi, S., 1991, Incompleteness results in Kripke semantics.The Journal of Symbolic Logic, Vol. 56, pp. 517–538.
Hughes, G. E. and Cresswell, M. J., 1968,An Introduction to Modal Logic, London, Methuen (IML).
Hughes, G. E. and Cresswell, M. J., 1984,A Companion to Modal Logic, London, Methuen (CML).
Kripke, S. A., 1963, Semantical considerations on modal logics.Acta Philosophica Fennica (1963)Modal and Many-Valued Logics, pp. 83–94.
Kripke, S. A., 1967, Review of E. J. Lemmon, Algebraic semantics for modal logics II.Mathematical Reviews, Vol. 34, p. 1022. (Review no. 5662.)
Montagna, F., 1984, The predicate modal logic of provability.Notre Dame Journal of Formal Logic, Vol. 25, pp. 179–189.
Shehtman, V. B. and Skvorcov, D. P., 1991, Semantics of non-classical first-order predicate logics.Mathematical Logic, Proceedings of Heyting 88 at Chajka (Bulgaria) 1988, New York, Plenum Press.
Zeman, J. J., 1973,Modal Logic: The Lewis Systems, Oxford, Clarendon Press.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cresswell, M.J. Incompleteness and the Barcan formula. J Philos Logic 24, 379–403 (1995). https://doi.org/10.1007/BF01048353
Issue Date:
DOI: https://doi.org/10.1007/BF01048353