Incompleteness and the Barcan formula

Journal of Philosophical Logic 24 (4):379 - 403 (1995)
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Abstract

A (normal) system of propositional 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 systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without 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

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Max Cresswell
Victoria University of Wellington

Citations of this work

Modal Logic.James W. Garson - 2009 - Stanford Encyclopedia of Philosophy.
Unifying Quantified Modal Logic.James W. Garson - 2005 - Journal of Philosophical Logic 34 (5-6):621-649.
Idealist Origins: 1920s and Before.Martin Davies & Stein Helgeby - 2014 - In Graham Oppy & Nick Trakakis (eds.), History of Philosophy in Australia and New Zealand. Dordrecht, Netherlands: Springer. pp. 15-54.
Partly Free Semantics for Some Anderson-Like Ontological Proofs.Mirosław Szatkowski - 2011 - Journal of Logic, Language and Information 20 (4):475-512.

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References found in this work

Semantical Considerations on Modal Logic.Saul Kripke - 1963 - Acta Philosophica Fennica 16:83-94.
An introduction to modal logic.G. E. Hughes - 1968 - London,: Methuen. Edited by M. J. Cresswell.
An Introduction to Modal Logic.George Edward Hughes & M. J. Cresswell - 1968 - London, England: Methuen. Edited by M. J. Cresswell.
A companion to modal logic.G. E. Hughes - 1984 - New York: Methuen. Edited by M. J. Cresswell.

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