Notre Dame Journal of Formal Logic

New Consecution Calculi for Rt

Katalin Bimbó and J. Michael Dunn

Abstract

The implicational fragment of the logic of relevant implication, R is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on LR, a Gentzen-style calculus. In this paper, we add the truth constant t to LR, but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve t. This calculus, LT, extends the consecution calculus LTt formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of Rt. For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of Rt. These results serve as a basis for our positive solution to the long open problem of the decidability of T, which we present in another paper.

Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 491-509.

Dates
First available in Project Euclid: 8 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1352383228

Digital Object Identifier
doi:10.1215/00294527-1722719

Mathematical Reviews number (MathSciNet)
MR2995416

Zentralblatt MATH identifier
1345.03046

Subjects
Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}
Secondary: 03F05: Cut-elimination and normal-form theorems 03B25: Decidability of theories and sets of sentences [See also 11U05, 12L05, 20F10] 03F52: Linear logic and other substructural logics [See also 03B47]

Keywords
relevance logics Ackermann constants sequent calculi admissibility of cut ticket entailment

Citation

Bimbó, Katalin; Dunn, J. Michael. New Consecution Calculi for $R^{t}_{\to}$. Notre Dame J. Formal Logic 53 (2012), no. 4, 491--509. doi:10.1215/00294527-1722719. https://projecteuclid.org/euclid.ndjfl/1352383228


Export citation

References

  • [1] Ackermann, W., “Begründung einer strengen Implikation,” Journal of Symbolic Logic, vol. 21 (1956), pp. 113–28.
  • [2] Anderson, A. R., “Entailment shorn of modality” (abstract), Journal of Symbolic Logic, vol. 25 (1960), p. 388.
  • [3] Anderson, A. R., and N. D. Belnap, Entailment. The Logic of Relevance and Necessity, Vol. I, Princeton University Press, Princeton, 1975.
  • [4] Anderson, A. R., N. D. Belnap, and J. M. Dunn, Entailment. The Logic of Relevance and Necessity, Vol. II, Princeton University Press, Princeton, 1992.
  • [5] Belnap, N. D., and J. R. Wallace, “A decision procedure for the system $E_{\overline{I}}$ of entailment with negation,” Technical Report 11, Contract No. SAR/609 (16), Office of Naval Research, New Haven, 1961.
  • [6] Belnap, N. D., and J. R. Wallace, “A decision procedure for the system $E_{\overline{I}}$ of entailment with negation,” Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 11 (1965), pp. 277–89.
  • [7] Bimbó, K., “Admissibility of cut in ${LC}$ with fixed point combinator,” Studia Logica, vol. 81 (2005), pp. 399–423.
  • [8] Bimbó, K., “Types of I-free hereditary right maximal terms,” Journal of Philosophical Logic, vol. 34 (2005), pp. 607–20.
  • [9] Bimbó, K., “$LE^{\,t}_{\to}$, $LR^{^{_{\scriptstyle\circ}}}_{\overset{\scriptscriptstyle\land}{\scriptscriptstyle\sim}}$, $LK$ and cutfree proofs,” Journal of Philosophical Logic, vol. 36 (2007), pp. 557–70.
  • [10] Bimbó, K., “Relevance logics,” pp. 723–89 in Philosophy of Logic, edited by D. Jacquette, vol. 5 of Handbook of the Philosophy of Science, Elsevier (North-Holland), Amsterdam, 2007.
  • [11] Bimbó, K., and J. M. Dunn, Generalized Galois Logics: Relational Semantics of Nonclassical Logical Calculi, vol. 188 of CSLI Lecture Notes, CSLI Publications, Stanford, 2008.
  • [12] Bimbó, K., and J. M. Dunn, “From relevant implication to ticket entailment” (abstract), Bulletin of Symbolic Logic, vol. 18 (2012), p. 288.
  • [13] Bimbó, K., and J. M. Dunn, “The decision problem of $T_{\to}$” (abstract), to appear in Bulletin of Symbolic Logic.
  • [14] Bimbó, K., and J. M. Dunn, “On the decision problem of $T_{\to}$,” to appear in Journal of Symbolic Logic.
  • [15] Church, A., “The weak theory of implication,” pp. 22–37 in Kontrolliertes Denken, Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften, edited by A. Menne, A. Wilhelmy, and H. Angsil, Verlag Karl Alber, Munich, 1951.
  • [16] Ciabattoni, A., and K. Terui, “Towards a semantic characterization of cut-elimination,” Studia Logica, vol. 82 (2006), pp. 95–119.
  • [17] Curry, H. B., Foundations of Mathematical Logic, corrected reprint, Dover, New York, 1977.
  • [18] Curry, H. B., J. R. Hindley, and J. P. Seldin, Combinatory Logic, vol. II, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1972.
  • [19] Došen, K., “The first axiomatization of relevant logic,” Journal of Philosophical Logic, vol. 21 (1992), pp. 339–56.
  • [20] Dunn, J. M., “The Algebra of Intensional Logics,” Ph.D. dissertation, University of Pittsburgh, 1966.
  • [21] Dunn, J. M., “A ‘Gentzen system’ for positive relevant implication” (abstract), Journal of Symbolic Logic, vol. 38 (1973), pp. 356–57.
  • [22] Dunn, J. M., “Relevance logic and entailment,” pp. 117–224 in Handbook of Philosophical Logic, vol. III, edited by D. Gabbay and F. Guenthner, vol. 166 of Synthese Library, D. Reidel, Dordrecht, 1986.
  • [23] Dunn, J. M., and R. K. Meyer, “Combinators and structurally free logic,” Logic Journal of the IGPL, vol. 5 (1997), pp. 505–37.
  • [24] Dunn, J. M., and G. Restall, “Relevance logic,” pp. 1–128 in Handbook of Philosophical Logic, vol. 6, 2nd ed., edited by D. Gabbay and F. Guenthner, Kluwer, Dordrecht, 2002.
  • [25] Gentzen, G., “Untersuchungen über das logische Schließen, I” Mathematische Zeitschrift, vol. 39 (1935), pp. 176–210.
  • [26] Gentzen, G., “Investigations into logical deduction,” American Philosophical Quarterly, vol. 1 (1964), pp. 288–306.
  • [27] Kripke, S. A., “The problem of entailment” (abstract), Journal of Symbolic Logic, vol. 24 (1959), p. 324.
  • [28] Mares, E. D., and R. K. Meyer, “Relevant logics,” pp. 280–308 in The Blackwell Guide to Philosophical Logic, edited by L. Goble, Blackwell Philosophy Guides, Blackwell, Oxford, UK, 2001.
  • [29] Meyer, R. K., “Intuitionism, entailment, negation,” pp. 168–98 in Truth, Syntax and Modality (Philadelphia, 1970), edited by H. Leblanc, vol. 68 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1973.
  • [30] Meyer, R. K., “Sentential constants in $\mathrm{R}$ and $\mathrm{R}_{\lnot}$,” Studia Logica, vol. 45 (1986), pp. 301–27.
  • [31] Meyer, R. K., “Improved decision procedures for pure relevant logic,” pp. 191–217 in Logic, Meaning and Computation. Essays in Memory of Alonzo Church, edited by C. A. Anderson and M. Zelëny, vol. 305 of Synthese Library, Kluwer Academic Publishers, Dordrecht, 2001.
  • [32] Meyer, R. K., and M. A. McRobbie, “Multisets and relevant implication, I,” Australasian Journal of Philosophy, vol. 60 (1982), pp. 107–39.
  • [33] Meyer, R. K., and M. A. McRobbie, “Multisets and relevant implication, II,” Australasian Journal of Philosophy, vol. 60 (1982), pp. 265–81.
  • [34] Meyer, R. K., and R. Routley, “Algebraic analysis of entailment, I,” Logique et Analyse (New Series), vol. 15 (1972), pp. 407–28.
  • [35] Moh, S.-K., “The deduction theorems and two new logical systems,” Methodos, vol. 2 (1950), pp. 56–75.
  • [36] Orlov, I. E., “The calculus of compatibility of propositions” (in Russian), Matematicheskii Sbornik, vol. 35 (1928), pp. 263–86.
  • [37] Routley, R., R. K. Meyer, V. Plumwood, and R. T. Brady, Relevant Logics and Their Rivals, Part I, Ridgeview Publishing, Atascadero, Calif., 1982.
  • [38] Schönfinkel, M., “On the building blocks of mathematical logic,” pp. 355–66 in From Frege to Gödel. A Source Book in Mathematical Logic, edited by J. van Heijenoort, Harvard University Press, Cambridge, MA, 1967 (1924).
  • [39] Urquhart, A., “The undecidability of entailment and relevant implication,” Journal of Symbolic Logic, vol. 49 (1984), pp. 1059–73.