Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-20T09:04:34.400Z Has data issue: false hasContentIssue false

CONTRACTIONS OF NONCONTRACTIVE CONSEQUENCE RELATIONS

Published online by Cambridge University Press:  18 December 2014

ROHAN FRENCH*
Affiliation:
Department of Philosophy, Monash University
DAVID RIPLEY*
Affiliation:
Department of Philosophy, University of Connecticut
*
*DEPARTMENT OF PHILOSOPHY WELLINGTON RD MONASH UNIVERSITY CLAYTON, VICTORIA, AUSTRALIA E-mail:rohan.french@gmail.com
DEPARTMENT OF PHILOSOPHY 101 MANCHESTER HALL, 344 MANSFIELD RD UNIVERSITY OF CONNECTICUT, STORRS CT 06269 USA E-mail: davewripley@gmail.com

Abstract

Some theorists have developed formal approaches to truth that depend on counterexamples to the structural rules of contraction. Here, we study such approaches, with an eye to helping them respond to a certain kind of objection. We define a contractive relative of each noncontractive relation, for use in responding to the objection in question, and we explore one example: the contractive relative of multiplicative-additive affine logic with transparent truth, or maalt.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2014 

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

Beall, J. (2009). Spandrels of Truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
Beall, J., & Murzi, J. (2013). Two flavors of Curry’s paradox. Journal of Philosophy, 110(3), 143165.Google Scholar
Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841866.Google Scholar
Curry, H. B., Feys, R., & Craig, W. (1958). Combinatory Logic, Volume 1. Amsterdam: North-Holland Publishing Company.Google Scholar
Dyckhoff, R., & Pinto, L. (1999). Permutability of proofs in intuitionistic sequent calculi. Theoretical Computer Science, 212(1), 141155.Google Scholar
Field, H. (2008). Saving Truth from Paradox. Oxford: Oxford University Press.Google Scholar
Friedman, H., & Sheard, M. (1987). An axiomatic approach to self-referential truth. Annals of Pure and Applied Logic, 33, 121.Google Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Amsterdam: Elsevier.Google Scholar
Gentzen, G. (1969). Investigations into logical deduction. In Szabo, M. E., editor. The Collected Papers of Gerhard Gentzen. Amsterdam: North-Holland Publishing Company, pp. 68131.Google Scholar
Gris̆in, V. N. (1982). Predicate and set-theoretic calculi based on logic without contractions (English translation). Mathematics of the USSR Izvestija, 18(1), 4159.Google Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge: Cambridge University Press.Google Scholar
Kleene, S. C. (1952). Permutability of inferences in Gentzen’s calculi LK and LJ. Memoirs of the American Mathematical Society, 10, 126.Google Scholar
Kremer, M. (1988). Kripke and the logic of truth. Journal of Philosophical Logic, 17(3), 225278.Google Scholar
Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43(23), 439469.Google Scholar
McCune, W. (2005–2010). Prover9 and Mace4. Available fromhttp://www.cs.unm.edu/∼mccune/prover9/.Google Scholar
Meyer, R. K., Routley, R., & Dunn, J. M. (1979). Curry’s paradox. Analysis, 39(3), 124128.CrossRefGoogle Scholar
Paoli, F. (2002). Substructural Logics: A Primer. Dordrecht, The Netherlands: Kluwer Academic Publishing.Google Scholar
Paoli, F. (2005). The ambiguity of quantifiers. Philosophical Studies, 124(3), 313330.Google Scholar
Petersen, U. (2000). Logic without contraction as based on inclusion and unrestricted abstraction. Studia Logica, 64(3), 365403.CrossRefGoogle Scholar
Priest, G. (2006). In Contradiction. Oxford: Oxford University Press.CrossRefGoogle Scholar
Priest, G. (2010). Hopes fade for saving truth. Philosophy, 85(1), 109140.Google Scholar
Priest, G. and Routley, R. (1982). Lessons from Pseudo-Scotus. Philosophical Studies, 42(2), 189199.Google Scholar
Priest, G. (forthcoming). Fusion and Confusion. Topoi.Google Scholar
Restall, G. (1994). On Logics Without Contraction. PhD thesis, The University of Queensland.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logics. London: Routledge.CrossRefGoogle Scholar
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354378.Google Scholar
Ripley, D. (2013a). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139164.Google Scholar
Ripley, D. (2013b). Revising up: Strengthening classical logic in the face of paradox. Philosophers’ Imprint, 13(5), 113.Google Scholar
Ripley, D. (2014). Uncut (working title). Ms.Google Scholar
Scharp, K. (2013). Replacing Truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
Schroeder-Heister, P. (2012). The categorical and the hypothetical: A critique of some fundamental assumptions of standard semantics. Synthese, 187(3), 925942.Google Scholar
Shapiro, L. (2011). Deflating logical consequence. Philosophical Quarterly, 61(243), 320342.Google Scholar
Tennant, N. (1982). Proof and paradox. Dialectica, 36(23), 265296.Google Scholar
Weir, A. (2005). Naïve truth and sophisticated logic. In Beall, J. and Armour-Garb, B., editors. Deflationism and Paradox. Oxford: Oxford University Press, pp. 218249.Google Scholar
Zardini, E. (2011). Truth without contra(di)ction. Review of Symbolic Logic, 4(4), 498535.Google Scholar