Notre Dame Journal of Formal Logic

Noncontractive Classical Logic

Lucas Rosenblatt

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One of the most fruitful applications of substructural logics stems from their capacity to deal with self-referential paradoxes, especially truth-theoretic paradoxes. Both the structural rules of contraction and the rule of cut play a crucial role in typical paradoxical arguments. In this paper I address a number of difficulties affecting noncontractive approaches to paradox that have been discussed in the recent literature. The situation was roughly this: if you decide to go substructural, the nontransitive approach to truth offers a lot of benefits that are not available in the noncontractive account. I sketch a noncontractive theory of truth that has these benefits. In particular, it has both a proof- and a model-theoretic presentation, it can be extended to a first-order language, and it retains every classically valid inference.

Article information

Notre Dame J. Formal Logic, Volume 60, Number 4 (2019), 559-585.

Received: 6 July 2017
Accepted: 24 March 2018
First available in Project Euclid: 6 September 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}
Secondary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}

substructural logic truth paradoxes


Rosenblatt, Lucas. Noncontractive Classical Logic. Notre Dame J. Formal Logic 60 (2019), no. 4, 559--585. doi:10.1215/00294527-2019-0020.

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  • [1] Barrio, E., L. Rosenblatt, and D. Tajer, “Capturing naive validity in the Cut-free approach,” Synthese, published electronically September 1, 2016.
  • [2] Beall, J., and J. Murzi, “Two flavors of Curry’s paradox,” Journal of Philosophy, vol. 110 (2013), pp. 143–65.
  • [3] Caret, C. R., and Z. Weber, “A note on contraction-free logic for validity,” Topoi, vol. 34 (2015), pp. 63–74.
  • [4] Cintula, P., and F. Paoli, “Is multiset consequence trivial?,” Synthese, published electronically September 8, 2016.
  • [5] Cobreros, P., P. Égré, D. Ripley, and R. van Rooij, “Reaching transparent truth,” Mind, vol. 122 (2013), pp. 841–866.
  • [6] Da Ré, B., and L. Rosenblatt, “Contraction, infinitary quantifiers, and omega paradoxes,” Journal of Philosophical Logic, vol. 47 (2018), 611–29.
  • [7] Field, H., Saving Truth from Paradox, Oxford University Press, New York, 2008.
  • [8] Fjellstad, A., “How a semantics for tonk should be,” Review of Symbolic Logic, vol. 8 (2015), pp. 488–505.
  • [9] Fjellstad, A., “$\omega$-inconsistency without cuts and nonstandard models,” Australasian Journal of Logic, vol. 13 (2016), pp. 96–122.
  • [10] Fjellstad, A., “Non-classical elegance for sequent calculus enthusiasts,” Studia Logica, vol. 105 (2017), pp. 93–119.
  • [11] Girard, J. Y., “Linear logic: Its syntax and semantics,” pp. 1–42 in Advances in Linear Logic (Ithaca, NY, 1993), edited by J. Y. Girard, Y. Lafont, and L. Regnier, vol. 222 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1995.
  • [12] Hinnion, R., and T. Libert, “Positive abstraction and extensionality,” Journal of Symbolic Logic, vol. 68 (2003), pp. 828–36.
  • [13] Hjortland, O. T., “Theories of truth and the maxim of minimal mutilation,” Synthese, published electronically November 10, 2017.
  • [14] Humberstone, L., “Heterogeneous logics,” Erkenntnis, vol. 29 (1989), pp. 395–435.
  • [15] Kripke, S., “Outline of a theory of truth,” Journal of Philosophy, vol. 72 (1975), pp. 690–716.
  • [16] Mares, E., and F. Paoli, “Logical consequence and the paradoxes,” Journal of Philosophical Logic, vol. 43 (2014), pp. 439–69.
  • [17] Negri, S., and J. von Plato, Structural Proof Theory, Cambridge University Press, Cambridge, 2001.
  • [18] Priest, G., “The structure of the paradoxes of self-reference,” Mind, vol. 103 (1994), pp. 25–34.
  • [19] Priest, G., An Introduction to Non-Classical Logic: From If to Is, 2nd edition, Cambridge University Press, Cambridge, 2008.
  • [20] Ripley, D., “Conservatively extending classical logic with transparent truth,” Review of Symbolic Logic, vol. 5 (2012), pp. 354–78.
  • [21] Ripley, D., “Paradoxes and failures of cut,” Australasian Journal of Philosophy, vol. 91 (2013), pp. 139–64.
  • [22] Ripley, D., “Anything goes,” Topoi, vol. 34 (2015), pp. 25–36.
  • [23] Ripley, D., “Comparing substructural theories of truth,” Ergo, vol. 2 (2015), pp. 299–328.
  • [24] Ripley, D., “Contraction and closure,” Thought, vol. 4 (2015), pp. 131–38.
  • [25] Rosenblatt, L., “Naive validity, internalization, and substructural approaches to paradox,” Ergo, vol. 4 (2017), pp. 93–120.
  • [26] Rosenblatt, L., “On structural contraction and why it fails,” Synthese, published electronically May 14, 2019.
  • [27] Shapiro, L., “Naive structure, contraction and paradox,” Topoi, vol. 34 (2015), pp. 75–87.
  • [28] Steinberger, F., Harmony and Logical Inferentialism, Ph.D. dissertation, Cambridge University, Cambridge, 2009.
  • [29] Teijeiro, P., “What is tonk?,” preprint, 2018.
  • [30] Troelstra, A. S., and H. Schwichtenberg, Basic Proof Theory, 2nd edition, vol. 43 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2000.
  • [31] Yaqub, A., The Liar Speaks the Truth, Oxford University Press, New York, 1993.
  • [32] Zardini, E., “Truth without contra(di)ction,” Review of Symbolic Logic, vol. 4 (2011), pp. 498–535.
  • [33] Zardini, E., “Breaking the chains: Following-from and transitivity,” pp. 221–75 in Foundations of Logical Consequence, edited by C. R. Caret and O. T. Hjortland, Oxford University Press, Oxford, 2015.