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Gentzenizations of relevant logics without distribution. II

  • Ross T. Brady (a1)

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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.

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[1]Giambrone, S., TW+ and RW+ are decidable, Journal of Philosophical Logic, vol. 14 (1985), pp. 235254.
[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. 316329. (Russian).
[3]Kripke, S. A., The problem of entailment (abstract), this Journal, vol. 24 (1959), p. 324.
[4]McRobbie, M. A., A proof-theoretic investigation of relevant and modal logics, Ph.D. Thesis, Australian National University, Canberra, 1979.
[5]Meyer, R. K., Topics in modal and many-valued logic, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania, 1966.
[6]Meyer, R. K., Metacompleteness, Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501517.
[7]Slaney, J. K., A meta-completeness theorem for contraction-free relevant logics, Studia Logica, vol. 43 (1984), pp. 159168.
[8]Urquhart, A., Failure of interpolation in relevant logics, Journal of Philosophical Logic, vol. 22 (1993), pp. 449479.

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