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Some results in modal model theory

Published online by Cambridge University Press:  12 March 2014

Michael Mortimer
Michael Mortimer*
Affiliation:
Bedford College, London, England
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Extract

This paper is concerned with extending some basic results from classical model theory to modal logic.

In §1, we define the majority of terms used in the paper, and explain our notation. A full catalogue would be excessive, and we cite [3] and [7] as general references.

Many papers on modal logic that have appeared are concerned with (i) introducing a new modal logic, and (ii) proving a weak completeness theorem for it. Theorem 1, in §2, in many cases allows us to conclude immediately that a strong completeness theorem holds for such a logic in languages of arbitrary cardinality. In particular, this is true of S4 with the Barcan formula.

In §3 we strengthen Theorem 1 for a number of modal logics to deal with the satisfaction of several sets of sentences, and so obtain a realizing types theorem. Finally, an omitting types theorem, generalizing the result for classical logic (see [5]) is proved in §4.

Several consequences of Theorem 1 are already to be found in the literature. [2] gives a proof of strong completeness in languages of arbitrary cardinality of various logics without the Barcan formula, and [8] for some logics in countable languages with it. In the latter case, the result for uncountable languages is cited, without proof, in [1], and there credited to Montague. Our proof was found independently.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 3 , September 1974 , pp. 496 - 508
Copyright
Copyright © Association for Symbolic Logic 1974

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References

REFERENCES

[1]Cocciarella, Nino, Ph.D. Dissertation, University of California, Berkeley, 1965.Google Scholar
[2]Gabbay, Dov M., Craig's interpolation theorem for modal logics, Conference in Mathematical Logic—1970 (Hodges, W., Editor), Springer Lecture Notes, no. 255.Google Scholar
[3]Hughes, C. E. and Cresswell, M. J., An introduction to modal logic, Methueo, London, 1968.Google Scholar
[4]Lambert, K. (Editor), Philosophical problems in logic, Reidel, Dordrecht, 1970.CrossRefGoogle Scholar
[5]Sacks, G., Saturated model theory, Benjamin, New York, 1972.Google Scholar
[6]Scott, Dana, Advice on modal logic, in[4].Google Scholar
[7]Shoenfield, J. R., Mathematical logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar
[8]Thomason, R. H., Some completeness results for modal predicate calculi, in[4].Google Scholar