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One Step is Enough

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Abstract

The recent development and exploration of mixed metainferential logics is a breakthrough in our understanding of nontransitive and nonreflexive logics. Moreover, this exploration poses a new challenge to theorists like me, who have appealed to similarities to classical logic in defending the logic ST, since some mixed metainferential logics seem to bear even more similarities to classical logic than ST does. There is a whole ST-based hierarchy, of which ST itself is only the first step, that seems to become more and more classical at each level. I think this seeming is misleading: for certain purposes, anyhow, metainferential hierarchies give us no reason to move on from ST. ST is indeed only the first step on a grand metainferential adventure; but one step is enough. This paper aims to explain and defend that claim. Along the way, I take the opportunity also to develop some formal tools and results for thinking about metainferential logics more generally.

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References

  1. Barrio, E., Pailos, F., & Szmuc, D. (2020a). A recovery operator for nontransitive approaches. Review of Symbolic Logic, 13(1), 80–104.

    Article  Google Scholar 

  2. Barrio, E. A., Pailos, F., & Szmuc, D. (2019). (Meta)inferential levels of entailment beyond the Tarskian paradigm. Synthese. To appear.

  3. Barrio, E. A., Pailos, F., & Szmuc, D. (2020b). A hierarchy of classical and paraconsistent logics. Journal of Philosophical Logic, 49(1), 93–120.

    Article  Google Scholar 

  4. Beall, J., & van Fraassen, B. C. (2003). Possibilities and Paradox: An introduction to modal and many-valued logic. Oxford: Oxford University Press.

    Google Scholar 

  5. Bell, J. L. (2011). Set theory: Boolean-valued models and independence proofs Vol. 47. Oxford: Oxford University Press.

    Google Scholar 

  6. Blamey, S. (2002). Partial logic. In D.M. Gabbay F. Guenthner (Eds.) Handbook of philosophical logic. 2nd edn., (Vol. 5 pp. 261–353). Springer.

  7. Blasio, C., Marcos, J.., & Wansing, H. (2017). An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic, 46(3/4), 233–262.

    Article  Google Scholar 

  8. Brady, R. T. (1971). The consistency of the axioms of abstraction and extensionality in a three-valued logic. Notre Dame Journal of Formal Logic, 12(4), 447–453.

    Article  Google Scholar 

  9. Chemla, E., & Égré, P. (2019a). From many-valued consequence to many-valued connectives. Synthese. https://doi.org/10.1007/s11229-019-02344-0. To appear.

  10. Chemla, E., & Égré, P. (2019b). Suszko’s problem: mixed consequence and compositionality. Review of Symbolic Logic, 12(4), 736–767.

    Article  Google Scholar 

  11. Chemla, E., Égré, P., & Spector, B. (2017). Characterizing logical consequence in many-valued logic. Journal of Logic and Computation, 27(7), 2193–2226.

    Google Scholar 

  12. Davey, B., A, & Priestley, H., A. (2002). Introduction to lattices and order. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  13. Dittrich, J. (2020). Paradox, arithmetic, and nontransitive logic. PhD thesis, Ludwig Maximilians Universität München.

  14. Dunn, M. J., & Hardegree, G. M. (2001). Algebraic methods in philosophical logic. Oxford: Oxford University Press.

    Google Scholar 

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

    Book  Google Scholar 

  16. Frankowski, S. (2004). Formalization of a plausible inference. Bulletin of the Section of Logic, 33, 41–52.

    Google Scholar 

  17. Girard, J. Y. (1987). Proof theory and logical complexity Vol. I. Napoli: Bibliopolis.

    Google Scholar 

  18. Gupta, A. (1982). Truth and paradox. Journal of Philosophical Logic, 11(1), 1–60.

    Article  Google Scholar 

  19. Haack, S. (1974). Deviant logic. Cambridge: Cambridge University Press.

    Google Scholar 

  20. Haack, S. (1978). Philosophy of logics. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  21. Halbach, V., & Horsten, L. (2006). Axiomatizing Kripke’s theory of truth. Journal of Symbolic Logic, 71(2), 677–712.

    Article  Google Scholar 

  22. Humberstone, L. (1988). Heterogeneous logic. Erkenntnis, 29(3), 395–435.

    Article  Google Scholar 

  23. Humberstone, L. (2012). The Connectives. Cambridge Massachusetts: MIT Press.

    Google Scholar 

  24. Keefe, R. (2000). Theories of vagueness. Cambridge: Cambridge University Press.

    Google Scholar 

  25. Körner, S. (1966). Experience and theory. New York: Humanities Press.

    Google Scholar 

  26. Kremer, M. (1988). Kripke and the logic of truth. Journal of Philosophical Logic, 17(3), 225–278.

    Article  Google Scholar 

  27. Kripke, S. (1975). Outline of a theory of truth. Journal of Philosophy, 72(19), 690–716.

    Article  Google Scholar 

  28. Leitgeb, H. (1999). Truth and the liar in De Morgan-valued models. Notre Dame Journal of Formal Logic, 40(4), 496–514.

    Article  Google Scholar 

  29. Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24, 49–59.

    Google Scholar 

  30. Martin, R. L., & Woodruff, P. W. (1976). On representing ‘true-in-L’ in L. In A. Kasher (Ed.) Language in Focus: Foundations, Methods and Systems. D.Reidel (pp. 113–117).

  31. Martin-Löf, P. (1984). Constructive mathematics and computer programming. Philosophical Transactions of the Royal Society of L. A: Mathematical, Physical and Engineering Sciences, 312(1522), 501–518.

    Article  Google Scholar 

  32. Meadows, T. (2014). Fixed points for consequence relations. Logique et Analyse, 227, 333–357.

    Google Scholar 

  33. Nicolai, C., & Rossi, L. (2017). Principles for object-linguistic consequence: from logical to irreflexive. Journal of Philosophical Logic, 47(3), 549–577.

    Article  Google Scholar 

  34. Pailos, F. (2019). A family of metainferential logics. Journal of Applied Non-Classical Logics, 29(1), 97–120.

    Article  Google Scholar 

  35. Pailos, F. (2020). A fully classical truth theory characterized by substructural means. The Review of Symbolic Logic, 13(2), 249–268.

    Article  Google Scholar 

  36. Plumwood, V. (1993). The politics of reason: Towards a feminist logic. Australasian Journal of Philosophy, 71(4), 436–462.

    Article  Google Scholar 

  37. Priest, G. (1989). Classical logic aufgehoben. In G. Priest, R. Routley, & J. Norman (Eds.) Paraconsistent logic: essays on the inconsistent (pp. 131–150). Munich: Philosophia Verlag.

  38. Priest, G. (2008). An introduction to non-classical logic: From If to Is, 2nd edn. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  39. Priest, G. (2013). Vague inclosures. In K. Tanaka, F. Berto, E. Mares, & F. Paoli (Eds.) Paraconsistency: logic and applications (pp. 367–378). Dordrecht: Springer.

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

    Article  Google Scholar 

  41. Ripley, D. (2017). On the ‘transitivity’ of consequence relations. Journal of Logic and Computation, 28(2), 433–450.

    Article  Google Scholar 

  42. Routley, R., Meyer, R. K., Plumwood, V., & Brady, R. T. (1982). Relevant Logics and their Rivals 1. California: Ridgeview, Atascadero.

    Google Scholar 

  43. Scambler, C. (2020). Classical logic and the strict tolerant hierarchy. Journal of Philosophical Logic, 49(2), 351–370.

    Article  Google Scholar 

  44. Schütte, K. (1960). Syntactical and semantical properties of simple type theory. The Journal of Symbolic Logic, 25(4), 305–326.

    Article  Google Scholar 

  45. Scott, D. (1975). Combinators and classes. In Lambda-calculus and computer science theory: proceedings of the symposium held in Rome March 25–27, 1975 (pp. 1–26). Berlin: Springer.

  46. Sharvit, Y. (2017). A note on (Strawson) entailment. Semantics & Pragmatics, 10(1), 1–38.

    Article  Google Scholar 

  47. Shoesmith, D. J., & Smiley, T. J. (1978). Multiple-conclusion logic. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  48. Skolem, T. (1960). A set theory based on a certain three-valued logic. Mathematica Scandinavica, 8(1), 127–136.

    Article  Google Scholar 

  49. Skolem, T. (1963). Studies on the axiom of comprehension. Notre Dame Journal of Formal Logic, 4(3), 162–170.

    Article  Google Scholar 

  50. Smith, N. J. J. (2008). Vagueness and degrees of truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  51. Strawson, P. F. (1952). Introduction to logical theory. Methuen & Co.Ltd.

  52. Tait, W. W. (1966). A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic. Bulletin of the American Mathematical Society, 72(6), 980–983.

    Article  Google Scholar 

  53. Tappenden, J. (1993). The liar and sorites paradoxes: Toward a unified treatment. The Journal of Philosophy, 90(11), 551–577.

    Article  Google Scholar 

  54. Terzian, G. (2015). Norms of truth and logical revision. Topoi, 34(1), 15–23.

    Article  Google Scholar 

  55. Tye, M. (1994). Sorites paradoxes and the semantics of vagueness. Philosophical Perspectives, 8, 189–206.

    Article  Google Scholar 

  56. Visser, A. (1984). Four valued semantics and the liar. Journal of Philosophical Logic, 13(2), 181–212.

    Article  Google Scholar 

  57. Von Fintel, K. (1999). NPI licensing, Strawson entailment, and context dependency. Journal of Semantics, 16, 97–148.

    Article  Google Scholar 

  58. Weir, A. (2015). A robust non-transitive logic. Topoi, 34(1), 99–107.

    Article  Google Scholar 

  59. Weirich, S. (2014). Pi-forall: How to use and implement a dependently-typed language. Oregon Programming Languages Summer School. https://www.github.com/sweirich/pi-forall.

  60. Williamson, T. (1994). Vagueness. London: Routledge.

    Google Scholar 

  61. Wintein, S. (2016). On all strong Kleene generalizations of classical logic. Studia Logica, 104(3), 503–545.

    Article  Google Scholar 

  62. Wójcicki, R. (1988). Theory of logical calculi. New York: Springer Science & Business Media.

    Book  Google Scholar 

  63. Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2), 347–385.

    Article  Google Scholar 

  64. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378.

    Article  Google Scholar 

  65. Ripley, D. (2013). Revising up: strengthening classical logic in the face of paradox. Philosophers’ Imprint, 13(5), 1–13.

    Google Scholar 

  66. Ripley, D. (2013). Sorting out the sorites. In K. Tanaka, F. Berto, E. Mares, & F. Paoli (Eds.) Paraconsistency: logic and applications (pp. 329–348). Dordrecht: Springer.

  67. Cobreros, P., Égré, P., Ripley, D., & van Rooij, R. (2013). Reaching transparent truth. Mind, 122(488), 841–866.

    Article  Google Scholar 

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

    Google Scholar 

  69. French, R., & Ripley, D. (2019). Two traditions in abstract model theory. Synthese. To appear.

  70. French, R., & Ripley, D. (2019). Valuations: bi, tri, and tetra. Studia Logica, 107(6), 1313–1346.

    Article  Google Scholar 

  71. Zardini, E. (2008). A model of tolerance. Studia Logica, 90(3), 337–368.

    Article  Google Scholar 

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Funding

Research partially supported by “Logic and Substructurality”, Grant FFI2017-84805-P, Government of Spain, and by “Substructural logics for bounded resources”, FT190100147, Australian Research Council.

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  1. Philosophy Department SOPHIS Building 11, Monash University, 3800, Clayton, VIC, Australia

    David Ripley

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Ripley, D. One Step is Enough. J Philos Logic 51, 1233–1259 (2022). https://doi.org/10.1007/s10992-021-09615-7

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