Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T10:59:34.187Z Has data issue: false hasContentIssue false

On the decidability of implicational ticket entailment

Published online by Cambridge University Press:  12 March 2014

Katalin Bimbó
Affiliation:
Department of Philosophy, University of Alberta, Edmonton, AB, T6G 2E7, Canada, E-mail: [email protected], URL: www.ualberta.ca/~bimbo
J. Michael Dunn
Affiliation:
School of Informatics and Computing, and Department of Philosophy, Indiana University, Bloomington, IN 47408-3912, USA, E-mail: [email protected]

Abstract

The implicational fragment of the logic of relevant implication, R is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, T is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of T to the decidability problem of R. The decidability of T is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base {B, B′, I, W}.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2013

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

Anderson, Alan R. [1960], Entailment shorn of modality, (abstract), this Journal, vol. 25, p. 388.Google Scholar
Anderson, Alan R. [1963], Some open problems concerning the system E of entailment, Acta Philosophica Fennica, vol. 16, pp. 918.Google Scholar
Anderson, Alan R. and Belnap, Nuel D. [1975], Entailment. The logic of relevance and necessity, vol. I, Princeton University Press, Princeton, NJ.Google Scholar
Anderson, Alan R., Belnap, Nuel D., and Dunn, J. Michael [1992], Entailment. The logic of relevance and necessity, vol. II, Princeton University Press, Princeton, NJ.Google Scholar
Belnap, Nuel D. and Wallace, John R. [1961], A decision procedure for the system of entailment with negation, Technical Report 11, Contract No. SAR/609 (16), Office of Naval Research, New Haven, (published in Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 11, (1965), pp. 277289).CrossRefGoogle Scholar
Bimbó, Katalin [2005a], Admissibility of cut in LC with fixed point combinator, Studia Logica, vol. 81, pp. 399423.CrossRefGoogle Scholar
Bimbó, Katalin [2005b], Types of ∣-free hereditary right maximal terms, Journal of Philosophical Logic, vol. 34, pp. 607620.CrossRefGoogle Scholar
Bimbó, Katalin [2007a], LK and cutfree proofs, Journal of Philosophical Logic, vol. 36, pp. 557570.CrossRefGoogle Scholar
Bimbó, Katalin [2007b], Relevance logics, Philosophy of logic (Jacquette, D., editor), Handbook of the Philosophy of Science (Gabbay, D., Thagard, P. and Woods, J., editors), vol. 5, Elsevier, pp. 723789.Google Scholar
Bimbó, Katalin [2012], Combinatory logic: Pure, applied and typed, Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL.Google Scholar
Bimbó, Katalin and Dunn, J. Michael [2008], Generalized Galois logics. Relational semantics of non-classical logical calculi, CSLI Lecture Notes, vol. 188, CSLI Publications, Stanford, CA.Google Scholar
Bimbó, Katalin and Dunn, J. Michael [2012a], From relevant implication to ticket entailment, (abstract), The Bulletin of Symbolic Logic, vol. 18, p. 288.Google Scholar
Bimbó, Katalin and Dunn, J. Michael [2012b], New consecution calculi for , Notre Dame Journal of Formal Logic, vol. 53, pp. 491509.CrossRefGoogle Scholar
Bimbó, Katalin and Dunn, J. Michael [to appear], The decision problem of T , (abstract), The Bulletin of Symbolic Logic.Google Scholar
Curry, Haskell B. [1963], Foundations of mathematical logic, McGraw-Hill Book Company, New York, NY, (reprint, Dover, New York, NY, 1977).Google Scholar
Dunn, J. Michael [1973], A ‘Gentzen system’ for positive relevant implication, (abstract), this Journal, vol. 38, pp. 356357.Google Scholar
Dunn, J. Michael [1986], Relevance logic and entailment, Handbook of philosophical logic (Gabbay, D. and Guenthner, F., editors), vol. 3, D. Reidel, Dordrecht, 1st ed., pp. 117224, (updated as Relevance Logic (with G. Restall), Handbook of Philosophical Logic, 2nd ed., (D. Gabbay and F. Guenthner, editors), vol. 6, Kluwer Academic Publishers, 2002, pp. 1–128).CrossRefGoogle Scholar
Dunn, J. Michael and Meyer, Robert K. [1997], Combinators and structurally free logic, Logic Journal of IGPL, vol. 5, pp. 505537.CrossRefGoogle Scholar
Giambrone, Steve [1985], TW+ and RW+ are decidable, Journal of Philosophical Logic, vol. 14, pp. 235254.CrossRefGoogle Scholar
Kleene, Stephen C. [1952], Introduction to metamathematics, P. Van Nostrand Company, Inc., Princeton, NJ.Google Scholar
Kripke, Saul A. [1959], The problem of entailment, (abstract), this Journal, vol. 24, p. 324.Google Scholar
Mares, Edwin D. and Meyer, Robert K. [2001], Relevant logics, The Blackwell guide to philosophical logic (Goble, L., editor), Blackwell Philosophy Guides, Blackwell Publishers, Oxford, UK, pp. 280308.Google Scholar
Meyer, Robert K. [2001], Improved decision procedures for pure relevant logic, Logic, meaning and computation. Essays in memory of Alonzo Church (Anderson, C. A. and Zelëny, M., editors), Kluwer Academic Publishers, Dordrecht, pp. 191217.CrossRefGoogle Scholar
Meyer, Robert K. and McRobbie, Michael A. [1982], Multisets and relevant implication I, Australasian Journal of Philosophy, vol. 60, pp. 107139.CrossRefGoogle Scholar
Meyer, Robert K. and Routley, Richard [1972], Algebraic analysis of entailment I, Logique et Analyse, vol. 15, pp. 407428.Google Scholar
Padovani, Vincent [201x], Ticket Entailment is decidable, Mathematical Structures in Computer Science, to appear.Google Scholar
Riche, Jacques and Meyer, Robert K. [1999], Kripke, Belnap, Urquhart and relevant decidability & complexity, Computer science logic (Brno, 1998) (Gottlob, Georg, Grandjean, Etienne, and Seyr, Katrin, editors), Lecture Notes in Computer Science, no. 1584, Springer, pp. 224240.CrossRefGoogle Scholar
Routley, Richard, Meyer, Robert K., Plumwood, Val, and Brady, Ross T. [1982], Relevant logics and their rivals, vol. I, Ridgeview Publishing Company, Atascadero, CA.Google Scholar
Schönfinkel, Moses [1924], On the building blocks of mathematical logic, From Frege to Gödel. A source book in mathematical logic (van Heijenoort, J., editor), Harvard University Press, Cambridge, MA, 1967, pp. 355366.Google Scholar