Abstract
We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction (fission, ‘cotensor’, par) against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we call ipsilateral: applying such a rule does not make any (sub)formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions.
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References
Allwein G., Dunn J.M. (1993). Kripke models for linear logic. Journal of Symbolic Logic 58, 514–545
Avron A. (1988). The semantics and proof theory of linear logic. Theoretical Computer Science 57, 161–187
Battilotti G., Sambin G. (1999). Basic logic and the cube of its extensions. In Cantini A., et al. (eds) Logic and foundations of Mathematics. Dordrecht, Kluwer, pp. 165–186
Bierman G.M. (1996). A note on full intuitionistic linear logic. Annals of Pure and Applied Logic 79, 281–287
Blamey S. (2002). Partial Logic. In: Gabbay D.M., Guenthner F.(eds) Handbook of philosophical logic 2nd ed. Vol. 5. Dordrecht, Kluwer, pp. 261–353
Chellas B. (1980). Modal logic: An Introduction. Cambridge, Cambridge University Press
Došen K. (1989). Sequent-systems and groupoid models. II. Studia Logica 48, 41–65
Dunn J.M. (1986). Relevance logic and entailment. In: Gabbay D., Guenthner F.(eds) Handbook of philosophical logic III: Alternatives to classical logic. Dordrecht, Reidel, pp. 117-224
Fine K. (1974). Models for entailment. Journal of Philosophical Logic 3, 347–372
Gentzen, G. (1934). Untersuchungen über das Logische Schliessen. Math. Zeitschrift, 39, 176–210, 405–431 (English translation in The collected papers of Gerhard Gentzen, ed. M. Szabo, North-Holland, Amsterdam 1969).
Girard J.-Y. (1987). Linear Logic. Theoretical Computer Science 50, 1–102
Goldblatt R. (1993). Mathematics of Modality. Stanford, California: Center for the Study of Language and Information
Hirokawa, S. (1996). Right weakening and right contraction in LK. In M. E. Houle & P. Eades (Eds.), CATS ’96 (= Australian Computer Science Communications Vol. 18, No. 3, (pp. 167–174).
Humberstone L. (1988). Operational semantics for positive R. Notre Dame Journal of Formal Logic 29, 61–80
Humberstone L. (1997). Singulary extensional connectives: A closer ook. Journal of Philosophical Logic 26, 341–356
Humberstone L. (2000). Parts and partitions. Theoria 66, 41–82
Humberstone L. (2006). Identical twins, Deduction theorems, and pattern functions: Exploring the implicative BCSK Fragment of S5. Journal of Philosophical Logic 35, 435–487
Hyland M., de Paiva V. (1993). Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic 64, 273–291
Ishihara H. (2000). A canonical model construction for substructural logics. Journal of Universal Computer Science 6, 177–168
Kashima R. (1997). Contraction-elimination for implicational logics. Annals of Pure and Applied Logic 84, 17–39
MacCaull W. (1996). Kripke semantics for logics with BCK implication. Bulletin of the Section of Logic 25, 41–51
Ono H., Komori Y. (1985). Logics without the contraction rule. Journal of Symbolic Logic 50, 169–201
Paoli F. (2002). Substructural Logics: A Primer. Dordrecht, Kluwer
Rautenberg W. (1981). 2-Element Matrices. Studia Logica 40, 315–353
Rautenberg W. (1989). A calculus for the common rules of \({\wedge}\) and \({\vee}\) . Studia Logica 48, 531–537
Restall G. (2000). An Introduction to Substructural Logics. London, Routledge
Rogerson S. (2003). Investigations into properties of structural rules on the right (Abstract). Bulletin of Symbolic Logic 9: 263
Schotch P.K., Jennings R.E. (1979). Modal logic and the theory of modal aggregation. Philosophia 9, 265–280
Shoesmith D.J., Smiley T.J. (1971). Deducibility and many-valuedness. Journal of Symbolic Logic 36, 610–622
Shoesmith D.J., Smiley T.J. (1978). Multiple-Conclusion Logic. Cambridge, Cambridge University Press
Surarso B., Ono H. (1996). Cut elimination in noncommutative substructural logics. Reports on Mathematical Logic 30, 13–29
Troelstra, A. S. (1992). Lectures on Linear Logic. CSLI Lecture Notes #29, Stanford University.
Urquhart A. (1972). Semantics for relevant logics. Journal of Symbolic Logic 37, 159–169
Wójcicki R. (1974). Note on deducibility and many-valuedness. Journal of Symbolic Logic 39, 563–566
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Humberstone, L. Investigations into a left-structural right-substructural sequent calculus. JoLLI 16, 141–171 (2007). https://doi.org/10.1007/s10849-006-9026-x
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DOI: https://doi.org/10.1007/s10849-006-9026-x