pp 1–17 | Cite as

Capturing naive validity in the Cut-free approach

  • Eduardo BarrioEmail author
  • Lucas Rosenblatt
  • Diego Tajer
S.I. : Substructural Approaches to Paradox


Rejecting the Cut rule has been proposed as a strategy to avoid both the usual semantic paradoxes and the so-called v-Curry paradox. In this paper we consider if a Cut-free theory is capable of accurately representing its own notion of validity. We claim that the standard rules governing the validity predicate are too weak for this purpose and we show that although it is possible to strengthen these rules, the most obvious way of doing so brings with it a serious problem: an internalized version of Cut can be proved for a Curry-like sentence. We also evaluate a number of possible ways of escaping this difficulty.


Validity Paradoxes Strict-tolerant logic Substructural logics Cut 



We are very grateful to Elia Zardini, Dave Ripley and two anonymous referees for extremely helpful comments on previous versions of this paper. Some of this material was presented at conferences in Campinas (CLE) and Buenos Aires (Buenos Aires Logic Group). We also owe thanks to the members of these audiences for their valuable feedback.


  1. Barrio, E., Rosenblatt, L., & Tajer, D. (2015). The logics of strict-tolerant logic. Journal of Philosophical Logic, 44(5), 551–571.Google Scholar
  2. Beall, J. C. (2009). Spandrels of truth. Oxford: Oxford University Press.CrossRefGoogle Scholar
  3. Beall, J. C., & Murzi, J. (2013). Two flavors of Curry’s paradox. The Journal of Philosophy, 110, 143–165.CrossRefGoogle Scholar
  4. Caret, C., & Weber, Z. (2015). A note on Contraction-free logic for validity. Topoi, 31(1), 63–74.CrossRefGoogle Scholar
  5. Cook, R. (2014). There is no paradox of logical validity. Logica Universalis, 8(3–4), 447–467.CrossRefGoogle Scholar
  6. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012a). Tolerance and mixed consequence in the S’valuationist setting. Studia Logica, 100(4), 855–877.CrossRefGoogle Scholar
  7. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012b). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.CrossRefGoogle Scholar
  8. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In T. Achourioti, H. Galinon, K. Fujimoto, & J. Martínez-Fernández (Eds.), Unifying the philosophy of truth. Dordrecht: Springer.Google Scholar
  9. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.CrossRefGoogle Scholar
  10. Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  11. Field, H., Disarming a paradox of validity. Notre Dame Journal of Formal Logic, forthcoming.Google Scholar
  12. Fitch, F. (1964). Universal metalanguages for philosophy. The Review of Metaphysics, 17(3), 396–402.Google Scholar
  13. Fjellstad, A. (2016). Naive modus ponens and failure of transitivity. Journal of Philosophical Logic, 45(1), 65–72.CrossRefGoogle Scholar
  14. Ketland, J. (2012). Validity as a primitive. Analysis, 72(3), 421–430.CrossRefGoogle Scholar
  15. Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72, 690–716.CrossRefGoogle Scholar
  16. Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43(2–3), 439–469.CrossRefGoogle Scholar
  17. Murzi, J., & Shapiro, L. (2015). Validity and truth-preservation. In H. Achourioti, F. Fujimoto, & J. Martínez-Fernández (Eds.), Unifying the philosophy of truth. Dordrecht: Springer.Google Scholar
  18. Negri, S. (2011). Proof theory for modal logic. Philosophy Compass, 6(8), 523–538.CrossRefGoogle Scholar
  19. Negri, S., & von Plato, J. (2001). Structural proof theory. New York: Cambridge University Press.CrossRefGoogle Scholar
  20. Prawitz, D. (1974). Natural deduction. A proof-theoretical study. Stockholm: Almquist and Wiksell.Google Scholar
  21. Priest, G., & Wansing, H. (2015). External curries. Journal of Philosophical Logic, 44(4), 453–471.CrossRefGoogle Scholar
  22. Priest, G. (2015). Fusion and confusion. Topoi, 34, 55–61.CrossRefGoogle Scholar
  23. Priest, G. (2006). In contradiction (2nd ed.). Oxford: Oxford University Press.CrossRefGoogle Scholar
  24. Restall, G. (1993). On logics without contraction, Ph.D. Thesis, University of Queensland.Google Scholar
  25. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. The Review of Symbolic Logic, 5, 354–378.CrossRefGoogle Scholar
  26. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–164.CrossRefGoogle Scholar
  27. Shapiro, L. (2015). Naive structure, contraction and paradox. Topoi, 34(1), 75–87.CrossRefGoogle Scholar
  28. Zardini, E. (2011). Truth without contra(dic)ction. The Review of Symbolic Logic, 4, 498–535.CrossRefGoogle Scholar
  29. Zardini, E. (2013). Naive modus ponens. Journal of Philosophical Logic, 42(4), 575–593.CrossRefGoogle Scholar
  30. Zardini, E. (2014). Naive truth and naive logical properties. The Review of Symbolic Logic, 7(2), 351–384.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Conicet - University of Buenos AiresCiudad Autónoma de Buenos AiresArgentina

Personalised recommendations