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Logics containing K4. Part I

Published online by Cambridge University Press:  12 March 2014

Kit Fine*
Affiliation:
St. John's College, Oxford

Extract

There are two main lacunae in recent work on modal logic: a lack of general results and a lack of negative results. This or that logic is shown to have such and such a desirable property, but very little is known about the scope or bounds of the property. Thus there are numerous particular results on completeness, decidability, finite model property, compactness, etc., but very few general or negative results.

In these papers I hope to help fill these lacunae. This first part contains a very general completeness result. Let In be the axiom that says there are at most n incomparable points related to a given point. Then the result is that any logic containing K4 and In is complete.

The first three sections provide background material for the rest of the papers. The fourth section shows that certain models contain no infinite ascending chains, and the fifth section shows how certain elements can be dropped from the canonical model. The sixth section brings the previous results together to establish completeness, and the seventh and last section establishes compactness, though of a weak kind. All of the results apply to the corresponding intermediate logics.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1]Fraissé, R., Sur quelques classifications des relations basées sur des isomorphismes restraintes, Publications Scientifiques de l'Université d'Alger. Série A (mathématiques), vol. 2 (1955), pp. 15–60, 273295.Google Scholar
[2]Hintikka, J., Distributive normal forms in first-order logic, Formal systems and recursive functions (Crossley, and Dummett, , Editors), North-Holland, Amsterdam, 1965.Google Scholar
[3]Hughes, G. E. and Cresswell, M. J., An introduction to modal logic, Methuen, London, 1968.Google Scholar