Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T06:03:21.162Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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

REFERENCES

Boolos, G. [79], The unprovability of consistency, Cambridge University Press, Cambridge.Google Scholar
Bull, R. A. [66], That all normal extensions of S4.3 have the finite model property, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12, pp. 341–344.CrossRefGoogle Scholar
Cresswell, M. J. and Hughes, G. E. [68], An introduction to modal logic, Methuen, London.Google Scholar
Fine, K. [71], Logics containing S4.3, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 17, pp. 371–376.CrossRefGoogle Scholar
Fine, K. [74a], Logics containing K4. Part I, this Journal, vol. 39, pp. 31–42.Google Scholar
Fine, K. [74b], An ascending chain of S4 logics, Theoria, vol. 40, pp. 110–116.CrossRefGoogle Scholar
Fine, K. [74c], An incomplete logic containing S4, Theoria, vol. 40, pp. 23–29.CrossRefGoogle Scholar
Fine, K. [75], On some connections between elementary and modal logic, Proceedings of the Third Scandinavian Logic Symposium (Kanger, Stig, editor), North-Holland, Amsterdam, pp. 15–51.Google Scholar
Gabbay, D. [76], Investigations in modal and tense logics with applications to problems in philosophy and linguistics, Reidel, Dordrecht.CrossRefGoogle Scholar
Goldblatt, R. I. and Thomason, S. K. [75], Axiomatic classes in prepositional modal logic, Algebraic logic (Crossley, J. N., editor), Lecture Notes in Mathematics, vol. 450, Springer-Verlag, Berlin, pp. 163–173.CrossRefGoogle Scholar
Segerberg, K. [71], An essay in classical modal logic, Ph.D. Thesis, Stanford University, Stanford, California; mimeographed by Department of Philosophy, University of Uppsala, Uppsala.Google Scholar
Tarski, A. [54], Contributions to the theory of models, I, II, Indagationes Mathematicae, vol. 16, pp. 572–588.Google Scholar
Urquhart, A. [81], Decidability and the finite model property, Journal of Philosophical Logic, vol. 10, pp. 367–370.CrossRefGoogle Scholar