Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T17:50:41.105Z Has data issue: false hasContentIssue false
  • English
  • Français

Gentzenizations of relevant logics with distribution

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

We establish cut-free left-handed Gentzenizations for a range of major relevant logics from B through to R, all with distribution. B is the basic system of the Routley-Meyer semantics (see [15], pp. 287–300) and R is the logic of relevant implication (see [1], p. 341). Previously, the contractionless logics DW, TW, EW, RW and RWK were Gentzenized in [3], [4] and [5], and also the distributionless logics LBQ, LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ in [6] and [7]. This paper provides Gentzenizations for the logics DJ, TJ, T and R, with various levels of contraction, and for the contractionless logic B, which could not be included in [4] using the technique developed there. We also include the Gentzenization of TW in order to compare it with that in [4]. The Gentzenizations that we obtain here for DW and RW are inferior to those already obtained in [4], but they are included for reference when constructing other systems. The logics EW and E present a difficulty for our method and are omitted. For background to the Gentzenization of relevant logics, see [6], and for motivation behind the logics involved, see [6], [1] and [15]. Because of the number of properties that are brought to bear in obtaining these systems, we prefer to consider Gentzenizations for particular logics rather than for arbitrary bunches of axioms.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 402 - 420
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]Anderson, A. R. and Belnap, N. D. Jr., Entailment: The Logic of Relevance and Necessity. Vol. 1, Princeton University Press, Princeton, New Jersey, 1975.Google Scholar
[2]Belnap, N. D. Jr., Display logic, Journal of Philosophical Logic, vol. 11 (1982), pp. 375–417.Google Scholar
[3]Brady, R. T., The Gentzenization and decidability of RW, Journal of Philosophical Logic, vol. 19 (1990), pp. 35–73.Google Scholar
[4]Brady, R. T., Gentzenization and decidability of some contractionless relevant logics, Journal of Philosophical Logic, vol. 20 (1991), pp. 97–117.Google Scholar
[5]Brady, R. T., Simplified Gentzenizations for contractionless logics, Logique et Analyse, forthcoming.Google Scholar
[6]Brady, R. T., Gentzenizations of relevant logics without distribution. I, this Journal, later.Google Scholar
[7]Brady, R. T., Gentzenizations of relevant logics without distribution. II, this Journal, later.Google Scholar
[8]Fine, K., Semantics for quantified relevance logic, Journal of Philosophical Logic, vol. 17 (1988), pp. 27–59.Google Scholar
[9]Giambrone, G. and Meyer, R. K., Completeness and conservative extension results for some Boolean relevant logics, Studia Logica, vol. 48 (1989), pp. 1–14.Google Scholar
[10]Meyer, R. K., Meta-completeness, Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501–517.Google Scholar
[11]Meyer, R. K. and Routley, R., Classical relevant logics. I, Studia Logica, vol. 32 (1973), pp. 51–66.Google Scholar
[12]Brady, R. T., Classical relevant logics. II, Studia Logica, vol. 33 (1974), pp. 183–194.Google Scholar
[13]Priest, G. and Sylvan, R., Simplified semantics for basic relevant logics, Journal of Philosophical Logic, vol. 21 (1992), pp. 217–232.Google Scholar
[14]Restall, G., Simplified semantics for relevant logics (and some of their rivals), Journal of Philosophical Logic, vol. 22 (1993), pp. 481–511.Google Scholar
[15]Routley, R.et al., Relevant logics and their rivals, Vol. 1, Ridgeview, Atascadero, California, 1982.Google Scholar
[16]Slaney, J. K., A metacompleteness theorem for contraction-free relevant logics, Studia Logica, vol. 43 (1984), pp. 159–168.CrossRefGoogle Scholar
[17]Slaney, J. K., Reduced models for relevant logics without WI, Notre Dame Journal of Formal Logic, vol. 28 (1987), pp. 395–407.Google Scholar