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Gentzenizations of relevant logics without distribution. I

  • Ross T. Brady (a1)

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The history of the Gentzenization of relevant logics goes back to Kripke [17], who in 1959 Gentzenized R and went on to prove its decidability. Formulae were separated by commas on the left side of the turnstile, the commas just representing nested implications. Kripke employed just a singleton formula to the right of the turnstile. He also considered adding negation, as well as other connectives, but it was not until 1961 that Belnap and Wallace, in [5], Gentzenized and proved its decidability, though their Gentzenization employed commas on both sides of the turnstile. Subsequently, in 1966, the logic R without distribution, now called LR (for lattice R), was Gentzenized in a similar style by Meyer in [20]. He also went on to show decidability for LR by extending Kripke's argument. Later, in 1969, Dunn Gentzenized R+ (published in [1], pp. 381–391) using two structural connectives (commas and semicolons) to the left of the turnstile, and with a single formula to the right. Here, the commas represent conjunction and the semicolons represent an intensional conjunction, called “fusion”. This is all nicely set out in McRobbie [19], where he also introduces left-handed Gentzenizations and analytic tableaux for a number of fragments of relevant logics. In 1979, further work on distributionless logic was done by Grishin, in a series of papers, including [16], in which he produced a Gentzenization of quantified RW without distribution (which we will call LRWQ), and used it to prove the decidability of this quantified logic.

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[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.
[2]Avron, A., The semantics and proof theory of linear logic, Theoretical Computer Science, vol. 57 (1988), pp. 161184.
[3]Belnap, N. D. Jr., A formal analysis of entailment, Technical Report No. 7, Contract No. SAR/Nonr-609(16), Office of Naval Research, New Haven, Connecticut, 1960.
[4]Belnap, N. D., Display Logic, Journal of Philosophical Logic, vol. 11 (1982), pp. 375417.
[5]Belnap, N. D. Jr., and Wallace, J. R., A decision procedure for the system of entailment with negation, Technical Report No. 11, Contract No. SAR/609(16), Office of Naval Research, New Haven, Connecticut, 1961; reproduced in Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 11 (1965), pp. 277–289.
[6]Brady, R. T., The simple consistency of a set theory based on the logic CSQ, Notre Dame Journal of Formal Logic, vol. 24 (1983), pp. 431449.
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[9]Brady, R. T., Gentzenization and decidability of some contractionless relevant logics, Journal of Philosophical Logic, vol. 20 (1991), pp. 97117.
[10]Brady, R. T., Hierarchical semantics for relevant logics, Journal of Philosophical Logic, vol. 21 (1992), pp. 357374.
[11]Brady, R. T., Relevant implication and the case for a weaker logic, Journal of Philosophical Logic, forthcoming.
[12]Curry, H. B., Foundations of mathematical logic, McGraw-Hill, New York, 1963.
[13]Dunn, J. M., The algebra of intensional logics, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania, 1966.
[14]Giambrone, S., TW+ and RW+ are decidable, Journal of Philosophical Logic, vol. 14 (1985), pp. 235254.
[15]Girard, J.-Y., Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1102.
[16]Grishin, V. N., Herbrand's theorem for logics without contraction, Studies in nonclassical logics andset theory (Mikhaĭlov, A. I., editor), “Nauka”, Moscow, 1979, pp. 316329. (Russian)
[17]Kripke, S. A., The problem of entailment (Abstract), this Journal, vol. 24 (1959), p. 324.
[18]Maehara, S., On the interpolation theorem of Craig, Sugaku, vol. 12 (1960/1961), pp. 235237. (Japanese)
[19]McRobbie, M. A., A proof-theoretic investigation of relevant and modal logics, Ph.D. Thesis, Australian National University, Canberra, 1979.
[20]Meyer, R. K., Topics in modal and many-valued logic, Ph.D. Thesis, University of Pittsburgh, Pittsburgh, Pennsylvania, 1966.
[21]Meyer, R. K., Metacompleteness, Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501517.
[22]Ono, H. and Komori, Y., Logics without the contraction rule, this Journal, vol. 50 (1985), pp. 169201.
[23]Routley, R.et al., Relevant logics and their rivals, Vol 1, Ridgeview, Atascadero, California, 1982.
[24]Slaney, J. K., A meta-completeness theorem for contraction-free relevant logics, Studia Logica, vol. 43 (1984), pp. 159168.
[25]Slaney, J. K., A general logic, Australasian Journal of Philosophy, vol. 68 (1990), pp. 7488.

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