Skip to main content

Non-reflexivity and Revenge

Abstract

We present a revenge argument for non-reflexive theories of semantic notions – theories which restrict the rule of assumption, or (equivalently) initial sequents of the form φφ. Our strategy follows the general template articulated in Murzi and Rossi [21]: we proceed via the definition of a notion of paradoxicality for non-reflexive theories which in turn breeds paradoxes that standard non-reflexive theories are unable to block.

References

  1. Beall, J. (2009). Spandrels of truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  2. Beall, J., & Murzi, J. (2013). Two flavors of Curry’s Paradox. The Journal of Philosophy, CX(3), 143–65.

    Article  Google Scholar 

  3. Beall, J., & Ripley, D. (2014). Non-classical theories of truth. In M. Glanzberg (Ed.) The Oxford Handbook of Truth. Oxford: Oxford University Press.

  4. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–85.

    Article  Google Scholar 

  5. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122, 841–866.

    Article  Google Scholar 

  6. Curry, H. (1942). The inconsistency of certain formal logics. Journal of Symbolic Logic, 7, 115–7.

    Article  Google Scholar 

  7. Field, H. (2008). Saving truth from paradox. Oxford: Oxford University Press.

    Book  Google Scholar 

  8. Field, H. (2017). Disarming a paradox of validity. Notre Dame Journal of Formal Logic, 58(1), 1–19.

    Article  Google Scholar 

  9. Fjellstad, A. (2017). Non-classical elegance for sequent calculus enthusiasts. Studia Logica, 105(1), 93–119.

    Article  Google Scholar 

  10. French, R. (2016). Structural reflexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy, 3.

  11. Halbach, V., & Horsten, L. (2006). Axiomatizing Kripke’s theory of truth. Journal of Symbolic Logic, 71, 677–712.

    Article  Google Scholar 

  12. Heck, Jr. R. G. (2007). Self-reference and the languages of arithmetic. Philosophia Mathematica, 15, 1–29.

  13. Horsten, L. (2009). Levity. Mind, 118(471), 555–581.

    Article  Google Scholar 

  14. Horsten, L. (2012). The Tarskian Turn. Deflationism and axiomatic truth. Cambridge: MIT Press.

    Google Scholar 

  15. Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72, 690–716.

    Article  Google Scholar 

  16. Mares, E., & Paoli, F. (2014). Logical consequence and the paradoxes. Journal of Philosophical Logic, 43, 439–469.

    Article  Google Scholar 

  17. McGee, V. (1991). Truth, vagueness and paradox. Indianapolis: Hackett Publishing Company.

    Google Scholar 

  18. McGee, V. (1992). Maximal consistent sets of instances of Tarski’s schema (T). Journal of Philosophical Logic, 21(3), 235–241.

    Article  Google Scholar 

  19. Meadows, T. (2014). Fixed points for consequence relations. Logique et Analyse, 333–357.

  20. Moschovakis, Y. (1974). Elementary induction on abstract structures. North-Holland.

  21. Murzi, J., & Rossi, L. (2021). Generalised revenge. Australasian Journal of Philosophy, 98(1), 153–177.

    Article  Google Scholar 

  22. Murzi, J., & Rossi, L. (2018). Naïve validity, Synthese. Online first: https://doi.org/10.1007/s11229-017-1541-6.

  23. Nicolai, C., & Rossi, L. (2018). Principles for object-linguistic validity: from logical to irreflexive. Journal of Philosophical Logic, 47, 549–577.

    Article  Google Scholar 

  24. Priest, G. (2006). Doubt Truth to be a Liar. Oxford: Oxford University Press.

    Google Scholar 

  25. Restall, G. (2000). An introduction to substructural logics. New York: Routledge.

    Book  Google Scholar 

  26. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 354–78.

  27. Ripley, D. (2013). Paradoxes and failures of cut. Australasian Journal of Philosophy, 91(1), 139–64.

    Article  Google Scholar 

  28. Ripley, D. (2013). Revising up. Philosophers’ Imprint, 13(5).

  29. Ripley, D. (2015). Comparing substructural theories of truth. Ergo, 2(13), 299–328.

    Google Scholar 

  30. Rosenblatt, L. (2020). Maximal non-trivial sets of instances of your least favorite logical principle. The Journal of Philosophy, 117(1), 30–54.

    Article  Google Scholar 

  31. Schroeder-Heister, P. (1984). A natural extension of natural deduction. Journal of Symbolic Logic, 49, 1284–1299.

    Article  Google Scholar 

  32. Soames, S. (1999). Understanding truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  33. Tennant, N. (2012). Cut for core logic. Review of Symbolic Logic, 5(3), 450–479.

    Article  Google Scholar 

  34. Zardini, E. (2011). Truth without contra(di)ction. Review of Symbolic Logic, 4, 498–535.

    Article  Google Scholar 

  35. Zardini, E. (2013). It is not the case that [p and ‘it is not the case that p’ is true] nor is it the case that [p and ‘p’ is not true]. Thought, 1(4), 309–19.

    Article  Google Scholar 

  36. Zardini, E. (2013). Näive logical properties and structural properties. The Journal of Philosophy, 110(11), 633–44.

    Article  Google Scholar 

  37. Zardini, E. (2014). Naïve truth and naïve logical properties. Review of Symbolic Logic, 7(2), 351–384.

    Article  Google Scholar 

  38. Zardini, E. (2015). Getting one for two, or the contractors’ bad deal. Towards a unified solution to the semantic paradoxes. In T. Achourioti, K. Fujimoto, H. Galinon, & J. Martinez-Fernandez (Eds.) Unifying the Philosophy of Truth (pp. 461–93). Springer.

Download references

Acknowledgements

We wish to thank an anonymous reviewer for helpful comments on a previous draft as well as the FWF (project number 29716-G24) for generous financial support.

Funding

Open access funding provided by Paris Lodron University of Salzburg.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Julien Murzi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Murzi, J., Rossi, L. Non-reflexivity and Revenge. J Philos Logic 51, 201–218 (2022). https://doi.org/10.1007/s10992-021-09625-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-021-09625-5

Keywords

  • Liar paradox
  • Curry’s Paradox
  • Validity Curry
  • Non-reflexive logics
  • Revenge