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An incomplete nonnormal extension of S3

Published online by Cambridge University Press:  12 March 2014

George F. Schumm*
Affiliation:
Ohio State University, Columbus, Ohio 43210

Extract

Fine [1] and Thomason [4] have recently shown that the familiar relational semantics of Kripke [2] is inadequate for certain normal extensions of T and S4. It is here shown that the more general semantics developed by Kripke in [3] to handle nonnormal modal logics is likewise inadequate for certain of those logics.

The interest of incompleteness results, such as those of Fine and Thomason, is of course a function of one's expectations. Define a “normal” logic too broadly and it is not surprising that a given semantics is not adequate for all normal logics. In the case of relational semantics, for example, one would want to require at least that a normal logic contain T, the logic determined by the class of all normal frames, and that it be closed under certain (though perhaps not all) rules of inference which are validity preserving in such frames. The adequacy of that semantics will otherwise be ruled out at the outset.

For Kripke a logic is normal if it contains all tautologies, □pp and □ (pq)→(□p → □q), and is closed under the rules of substitution, modus ponens and necessitation (from A infer □A). T is the smallest normal logic, and this fact, together with the “naturalness” of the definition and the enormous number of normal logics which have been shown to be complete, made it plausible to suppose that Kripke's original semantics was adequate for all normal logics. That it is not is indeed surprising and would seem to reveal a genuine shortcoming.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Fine, Kit, An incomplete logic containing S4, Theoria (Lund), vol 40 (1974), pp. 2329.CrossRefGoogle Scholar
[2]Kripke, Saul A., Semantical analysis of modal logic I, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 9 (1963), pp. 6796.CrossRefGoogle Scholar
[3]Kripke, Saul A., Semantical analysis of modal logic II, The theory of models, North-Holland, Amsterdam, 1965, pp. 206220.Google Scholar
[4]Thomason, S. K., An incompleteness theorem in modal logic, Theoria (Lund), vol. 40 (1974), pp. 3034.CrossRefGoogle Scholar