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Logics containing K4. Part II1

Published online by Cambridge University Press:  12 March 2014

Kit Fine
Kit Fine*
Affiliation:
Department of Philosophy, University of Michigan, Ann Arbor, Michigan 48109 School of Epistemics, University of Edinburgh, Edinburgh EH8 9LW, Scotland
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Extract

This paper establishes another very general completeness result for the logics within the field of K4. With each finite transitive frame ℭ we may associate a formula — Bℭ which validates just those frames ℑ in which ℭ is not in a certain sense embeddable (to be exact, ℭ is not the p-morphic image of any subframe of ℑ. By a subframe logic we mean the result of adding such formulas as axioms to K4. The general result is that each subframe logic has the finite model property.

There are a continuum of subframe logics and they include many of the standard ones, such as T, S4, S4.3, S5 and G. It turns out that the subframe logics are exactly those complete for a condition that is closed under subframes (any subframe of a frame satisfying the condition also satisfies the condition). As a consequence, every logic complete for a condition closed under subframes has the finite model property.

It is ascertained which of the subframe logics are compact. It turns out that the compact logics are just those whose axioms express an elementary condition. Tests are given for determining whether a given axiom expresses an elementary condition and for determining what it is in case it does.

In one respect the present general completeness result differs from most of the others in the literature. The others have usually either been what one might call logic based or formula based. They have usually either been to the effect that all of the logics containing a given logic are complete or to the effect that all logics whose axioms come from a given syntactically characterized class of formulas are complete. The present result is, by contrast, what one might call frame based. The axioms of the logics to be proved complete are characterized most directly in terms of their connection with certain frames.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 50 , Issue 3 , September 1985 , pp. 619 - 651
Copyright
Copyright © Association for Symbolic Logic 1985

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Footnotes

1

The bulk of the material in this part was prepared at about the same time as the first part but, for one reason or another, was not then written up. I still have hopes of producing a third part on questions of definability and decidability.

References

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