On validity paradoxes and (some of) their solutions


  • Edson Bezerra Sociedad Argentina de Análisis Filosófico




Paradoxes, Predicates, Non-Classical Logic, Logical Validity, Philosophy of Logic


Many semantic theories become trivial when extended with a naïve validity predicate due to the validity paradoxes. The non-classical semantic theories are the ones that allegedly preserve the naïveté of the validity predicate while being capable of avoiding the validity paradoxes. This blocking, on the other hand, usually comes at a high cost. In this paper, we argue that the pre-theoretical notion of validity that the naïve validity predicate intends to capture is unattainable.


Anderson, C. A. 1983. The paradox of the knower. textit{The Journal of Philosophy}, 80(6):338–355.

Andrade-Lotero, E. & Novaes, C. D. 2012. Validity, the squeezing argument and alternative semantic systems: the case of aristotelian syllogistic. textit{Journal of philosophical logic}, 41(2):387–418.

Asenjo, F. G. 1966. A calculus of antinomies. textit{Notre Dame Journal of Formal Logic}, 7(1):103–105.

Barrio, E.; Rosenblatt, L.; Tajer, D. 2015. The logics of strict-tolerant logic. textit{Journal of Philosophical Logic}, 44(5):551–571.

Barrio, E.; Rosenblatt, L.; Tajer, D. 2016. Capturing naive validity in the cut-free approach. textit{Synthese}, pp. 1–17.

Beall, J. & Murzi, J. 2013. Two flavors of curry’s paradox. textit{The Journal of Philosophy}, 110(3):143–165.

Burgess, J. P. 1999. Which modal logic is the right one? textit{Notre Dame Journal of Formal Logic}, 40(1):81–93.

Carnielli, W.; Coniglio, M. E.; Marcos, J. 2007. Logics of formal inconsistency. In textit{Handbook of philosophical logic}, pp. 1–93. Springer.

Cobreros, P.; Égré, P.; Ripley, D.; van Rooij, R. 2012. Tolerant, classical, strict. textit{Journal of Philosophical Logic}, 41(2):347–385.

Cook, R. T. 2014. There is no paradox of logical validity. textit{Logica Universalis}, 8(3-4):447–467.

Crocco, G. 2019. Informal and absolute proofs: some remarks from a G¨odelian perspective. textit{Topoi}, 38(3):561–575.

Dean, W. 2014. Montague’s paradox, informal provability, and explicit modal logic. textit{Notre Dame Journal of Formal Logic}, 55(2):157–196.

Dutilh Novaes, C. & French, R. 2018. Paradoxes and structural rules from a dialogical perspective. textit{Philosophical Issues}, 28(1):129–158.

Égré, P. 2005. The knower paradox in the light of provability interpretations of modal logic. textit{Journal of Logic, Language and Information}, 14(1):13–48.

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

Glanzberg, M. 2015. Logical consequence and natural language. In: textit{Foundations of logical consequence}, pp. 71–120.

Goodship, L. 1996. On dialethism. textit{Australasian Journal of Philosophy}, 74(1):153—-161.

Griffiths, O. 2014. Formal and informal consequence. textit{Thought: A Journal of Philosophy}, 3(1):9–20.

Halbach, V. 2020. The substitutional analysis of logical consequence. textit{Noûs}, 54(2):431–450.

Halldén, S. 1963. A pragmatic approach to modal theory. textit{Acta Philosophica Fennica}, 16:53–63.

Hazen, A. 1984. Modality as many metalinguistic predicates. textit{Philosophical Studies}, 46(2):271–277.

Hlobil, U. 2019. Faithfulness for naive validity. textit{Synthese}, 196(11):4759–4774.

Kennedy, J. & Väänänen, J. 2017. Squeezing arguments and strong logics. In: textit{15th International Congress of Logic, Methodology and Philosophy of Science}. College Publications.

Ketland, J. 2012. Validity as a primitive. textit{Analysis}, 72(3):421–430.

Kreisel, G. 1967. Informal rigour and completeness proofs. In: textit{Studies in Logic and the Foundations of Mathematics}, volume 47, pp. 138–186. Elsevier.

Leitgeb, H. 2009. On formal and informal provability. In: textit{New waves in philosophy of mathematics}, pp. 263–299. Springer.

Löb, M. H. 1955. Solution of a problem of Leon Henkin 1. textit{The Journal of Symbolic Logic}, 20(2):115–118.

MacFarlane, J. G. 2000. textit{What does it mean to say that logic is formal?}. University of Pittsburgh.

Meadows, T. 2014. Fixed points for consequence relations. textit{Logique et Analyse}, pp. 333–357.

Montague, R. 1963. Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. textit{Acta Philosophica Fennica}.

Murzi, J. & Carrara, M. 2015. Paradox and logical revision. a short introduction. textit{Topoi}, 34(1):7–14.

Murzi, J. & Rossi, L. 2017. Naïve validity. textit{Synthese}, pp. 1–23.

Myhill, J. 1960. Some remarks on the notion of proof. textit{The Journal of Philosophy}, 57(14):461–471.

Otte, R. 1982. Modality as a metalinguistic predicate. textit{Philosophical Studies}, 41(2):153–159.

Pailos, F. M. 2020. Validity, dialetheism and self-reference. textit{Synthese}, 197(2):773–792.

Picollo, L. 2020. Truth in a logic of formal inconsistency: How classical can it get? textit{Logic Journal of the IGPL}, 28(5):771–806.

Priest, G. 1979. The logic of paradox. textit{Journal of Philosophical Logic}, 8(1):219–241.

Priest, G. 2008. textit{An introduction to non-classical logic: From if to is}. Cambridge University Press.

Quine, W. V. O. 1966. Three grades of modal involvement (1953). In: Quine, W. V. O. (ed.) textit{The Ways of Paradox and Other Essays}. New York: Random House.

Ripley, D. 2012. Conservatively extending classical logic with transparent truth. textit{The Review of Symbolic Logic}, 5(2):354–378.

Ripley, D. 2013. Paradoxes and failures of cut. textit{Australasian Journal of Philosophy}, 91(1):139–164.

Rosenblatt, L. 2017. Naive validity, internalization, and substructural approaches to paradox. textit{Ergo, an Open Access Journal of Philosophy}, 4.

Sette, A. M. 1973. On the propositional calculus P1. textit{Mathematica Japonicae}, 18:173–180.

Shapiro, L. & Beall, J. 2018. Curry’s paradox. In: E.N. Zalta (ed.), textit{The Stanford Encyclopedia of Philosophy}. Stanford University, summer 2018 edition.

Shapiro, S. 2005. Logical Consequence, Proof Theory, and Model Theory. In: textit{The Oxford Handbook of Philosophy of Mathematics and Logic}, pp. 651–670. Oxford University Press.

Skyrms, B. 1978. An immaculate conception of modality or how to confuse use and mention. textit{The Journal of Philosophy}, 75(7):368–387.

Smiley, T. 1998. Conceptions of consequence. textit{Routledge encyclopedia of philosophy}. London: Routledge.

Smith, P. 2011. Squeezing arguments. textit{Analysis}, 71(1):22–30.

Stern, J. 2015. textit{Toward predicate approaches to modality}, volume 44. Springer.

Tarski, A. 1956. textit{Logic, semantics, metamathematics: papers from 1923 to 1938}. Oxford Clarendon Press.

Terzian, G. 2015. Norms of truth and logical revision. textit{Topoi}, 34(1):15–23.

Varzi, A. C. 2002. On logical relativity. textit{Philosophical Issues}, 12(1):197–219.

Weber, Z. 2014. Naive validity. textit{The Philosophical Quarterly}, 64(254):99–114.

Whittle, B. 2004. Dialetheism, logical consequence and hierarchy. textit{Analysis}, 64(4):318–326.

Zardini, E. 2013. Naive logical properties and structural properties. textit{The Journal of Philosophy}, 110(11):633–644.