Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-22T20:07:31.853Z Has data issue: false hasContentIssue false
  • English
  • Français

Intuitionistic tense and modal logic

Published online by Cambridge University Press:  12 March 2014

W. B. Ewald
W. B. Ewald*
Affiliation:
Queen's College, Oxford, England
*
Philosophisches Seminar der Georg-August Universität, 3400 Göttingen, West Germany
Get access Rights & Permissions [Opens in a new window]

Extract

In this article we shall construct intuitionistic analogues to the main systems of classical tense logic. Since each classical modal logic can be gotten from some tense logic by one of the definitions

  • (i) □ p ≡ p ∧ Gp ∧ Hp, ◇p ≡ p ∨ Fp ∨ Pp; or,

  • (ii) □ p ≡ p ∧ Gp, ◇p = p ∨ Fp

(see [5]), we shall find that our intuitionistic tense logics give us analogues to the classical modal logics as well.

We shall not here discuss the philosophical issues raised by our logics. Readers interested in the intuitionistic view of time and modality should see [2] for a detailed discussion.

In §2 we define the Kripke models for IKt, the intuitionistic analogue to Lemmon's system Kt. We then prove the completeness and decidability of this system (§§3–5). Finally, we extend our results to other sorts of tense logic and to modal logic.

In the language of IKt, we have: sentence-letters p, q, r, etc.; the (intuitionistic) connectives ∧, ∨, →, ¬; and unary operators P (“it was the case”), F (it will be the case”), H (“it has always been the case”) and G (“it will always be the case”). Formulas are defined inductively: all sentence-letters are formulas; if X is a formula, so are ¬X, PX, FX, HX, and GX; if X and Y are formulas, so are X ∧ Y, X ∨ Y, and X → Y. We shall see that, in contrast to classical tense logic, F and P cannot be defined in terms of G and H.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 1 , March 1986 , pp. 166 - 179
Copyright
Copyright © Association for Symbolic Logic 1986

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.)

References

REFERENCES

[1]Dummett, M., Elements of intuitionism, Clarendon Press, Oxford, 1977.Google Scholar
[2]Ewald, W., Time, modality, and intuitionism, to appear.Google Scholar
[3]Gabbay, D., Investigations in modal and tense logics with applications to problems in philosophy and linguistics, Reidel, Dordrecht, 1976.Google Scholar
[4]Lemmon, E. J. (with Scott, D.; edited by Segerberg, K.), An introduction to modal logic, American Philosophical Quarterly Monograph Series, no. 11, Basil Blackwell, Oxford (distributed by ISBS, Inc., Forest Grove, Oregon), 1977.Google Scholar
[5]Rescher, N. and Urquhart, A., Temporal logic, Springer-Verlag, Vienna, 1971.CrossRefGoogle Scholar
[6]Birkhoff, G., Lattice theory, new ed., American Mathematical Society, Providence, Rhode Island, 1967.Google Scholar
[7]Troelstra, A. S., Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin, 1973.Google Scholar
[8]Rabin, M., Decidable theories, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 595–629.CrossRefGoogle Scholar