Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T21:58:46.523Z Has data issue: false hasContentIssue false
  • English
  • Français

MODAL LOGIC WITHOUT CONTRACTION IN A METATHEORY WITHOUT CONTRACTION

Published online by Cambridge University Press:  13 February 2019

PATRICK GIRARD  and
ZACH WEBER
PATRICK GIRARD*
Affiliation:
School of Humanities, University of Auckland
ZACH WEBER*
Affiliation:
Department of Philosophy, University of Otago
*
*PHILOSOPHY, SCHOOL OF HUMANITIES UNIVERSITY OF AUCKLAND 14A SYMONDS ST, AUCKLAND CENTRAL AUCKLAND 1010, NEW ZEALAND E-mail: [email protected]
†DEPARTMENT OF PHILOSOPHY UNIVERSITY OF OTAGO PO BOX 56, DUNEDIN 9054 NEW ZEALAND E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Standard reasoning about Kripke semantics for modal logic is almost always based on a background framework of classical logic. Can proofs for familiar definability theorems be carried out using a nonclassical substructural logic as the metatheory? This article presents a semantics for positive substructural modal logic and studies the connection between frame conditions and formulas, via definability theorems. The novelty is that all the proofs are carried out with a noncontractive logic in the background. This sheds light on which modal principles are invariant under changes of metalogic, and provides (further) evidence for the general viability of nonclassical mathematics.

Keywords

03B4703B45non-classical logicmodal logicsubstructural logiccontraction-free logicdefinability theoryclassical recapture
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 4 , December 2019 , pp. 685 - 701
Copyright
Copyright © Association for Symbolic Logic 2019 

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

BIBLIOGRAPHY

Allwein, G. & Dunn, J. M. (1993). Kripke semantics for linear logic. Journal of Symbolic Logic, 58 (2), 514–545.CrossRefGoogle Scholar
Badía, G. & Weber, Z. A substructural logic for inconsistent mathematics. In Reiger, A. and Young, G., editors. Dialetheism and Its Applications, to appear.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2001). Modal Logic. New York: Cambridge University Press.CrossRefGoogle Scholar
Cantini, A. (2003). The undecidability of Grisin’s set theory. Studia Logica, 74(3), 345–368.CrossRefGoogle Scholar
Chellas, B. F. (1980). Modal Logic: An Introduction. New York: Cambridge University Press.CrossRefGoogle Scholar
Cintula, P. & Paoli, F. (2016). Is multiset consequence trivial? Synthese, 1–25.Google Scholar
Feferman, S. (1984). Toward useful type-free theories, I. Journal of Symbolic Logic, 49(1), 75–111.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices. Amsterdam: Elsevier.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science, 50(1), 1–102.CrossRefGoogle Scholar
Girard, J.-Y. (1998). Light linear logic. Information and Computation, 143, 175–204.CrossRefGoogle Scholar
Girard, P. & Weber, Z. (2015). Bad worlds. Thought, 4, 93–101.Google Scholar
Gris̆in, V. N. (1982). Predicate and set theoretic calculi based on logic without contraction rules. Mathematics of the USSR-Izvestiya, 18(1), 41–59.CrossRefGoogle Scholar
Hughes, G. E. & Cresswell, M. J. (1996). A New Introduction to Modal Logic. New York: Routledge.CrossRefGoogle Scholar
Humberstone, L. (2011). The Connectives. Cambridge, MA: MIT.CrossRefGoogle Scholar
Kamide, N. (2002). Kripke semantics for modal substructural logics. Journal of Logic, Language and Information, 11, 453–470.CrossRefGoogle Scholar
MacCaull, W. (1996). Kripke semantics for logics with bck implication. Bulletin of the Section of Logic, 25, 41–51.Google Scholar
Mares, E. D. & Meyer, R. K. (1993). The semantics of R4. Journal of Philosophical Logic, 22, 95–110.CrossRefGoogle Scholar
Meadows, T. & Weber, Z. (2016). Computation in nonclassical foundations? Philosophers’ Imprint, 16, 1–17.Google Scholar
Ono, H. (1993). Semantics for substructural logics. In Schroeder-Heister, P. and Došen, K., editors. Substructural Logics. Oxford: Clarendon Press, pp. 259–291.Google Scholar
Ono, H. & Komori, Y. (1985). Logic without the contraction rule. Journal of Symbolic Logic, 50, 169–201.CrossRefGoogle Scholar
Petersen, U. (2000). Logic without contraction as based on inclusion and unrestriced abstraction. Studia Logica, 64, 365–403.CrossRefGoogle Scholar
Petersen, U. (2003). .${l^i}d_\lambda ^Z$as a basis for pra. Archive for Mathematical Logic, 42, 665–694.CrossRefGoogle Scholar
Ripley, D. (2015). Contraction and closure. Thought, 4, 131–138.Google Scholar
Terui, K. (2004). Light affine set theory: A naive set theory of polynomial time. Studia Logica, 77, 9–40.CrossRefGoogle Scholar
Troelstra, A. S. & Schwichtenberg, H. (1996). Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science, Vol. 43. Cambridge: Cambridge University Press.Google Scholar
van Benthem, J. (1984). Correspondence theory. In Gabbay, D. M. and Günthner, F., editors. Handbook of Philosophical Logic, Vol. 2. Dordrecht: Reidel, pp. 167–247.CrossRefGoogle Scholar
Wansing, H. (1993). The Logic of Information Structures. Lecture Notes in Artificial Intelligence, Vol. 681. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Weber, Z., Badia, G., & Girard, P. (2016). What is an inconsistent truth table? Australasian Journal of Philosophy, 94(3), 533–548.CrossRefGoogle Scholar
Weber, Z. & Omori, H. (2019). Observations on the trivial world. Erkenntnis, 84, 975–994.CrossRefGoogle Scholar