Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T23:34:16.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Gentzenizations of relevant logics without distribution. II

Published online by Cambridge University Press:  12 March 2014

Ross T. Brady
Ross T. Brady*
Affiliation:
La Trobe University, Bundoora, Victoria 3083, Australia
Get access Rights & Permissions [Opens in a new window]

Extract

In Part I, we produced a Gentzenization L6LBQ of the distributionless relevant logic LBQ, which contained just the one structural connective ‘:’ and no structural rules. We compared it with the corresponding “right-handed” system and then proved interpolability and decidability of LBQ. Knowledge of Part I is presupposed.

In Part II of this paper, we will establish Gentzenizations, with appropriate interpolation and decidability results, for the further distributionless logics LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ, using essentially the same methods as were used for LBQ in Part I. LRWQ has been Gentzenized by Grishin [2], but the interpolation result is new and the decidability result is proved by a substantial simplification of his method. LR has been Gentzenized and shown to be decidable by Meyer in [5], by extending a method of Kripke in [3], and McRobbie has proved interpolation for it in [4], but here the Gentzenization and interpolation results are extended to quantifiers.

We axiomatize these logics as follows. The primitives and formation rules are as before, except that LTWQ and LEWQ require the extra primitive ‘o’, a 2-place connective (called ‘fusion’), and LEWQ also requires ‘t’, a sentential constant.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 379 - 401
Copyright
Copyright © Association for Symbolic Logic 1996

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]Giambrone, S., TW+ and RW+ are decidable, Journal of Philosophical Logic, vol. 14 (1985), pp. 235–254.Google Scholar
[2]Grishin, V. N., Herbrand's theorem for logics without contraction, Studies in nonclassical logics and set theory (Mikhaĭlov, A. I., editor), “Nauka”, Moscow, 1979, pp. 316–329. (Russian).Google Scholar
[3]Kripke, S. A., The problem of entailment (abstract), this Journal, vol. 24 (1959), p. 324.Google Scholar
[4]McRobbie, M. A., A proof-theoretic investigation of relevant and modal logics, Ph.D. Thesis, Australian National University, Canberra, 1979.Google Scholar
[5]Meyer, R. K., Topics in modal and many-valued logic, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania, 1966.Google Scholar
[6]Meyer, R. K., Metacompleteness, Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501–517.Google Scholar
[7]Slaney, J. K., A meta-completeness theorem for contraction-free relevant logics, Studia Logica, vol. 43 (1984), pp. 159–168.Google Scholar
[8]Urquhart, A., Failure of interpolation in relevant logics, Journal of Philosophical Logic, vol. 22 (1993), pp. 449–479.Google Scholar