Advertisement

Generalised Tarski’s Thesis Hits Substructure

  • Elia Zardini
Chapter
Part of the Palgrave Innovations in Philosophy book series (PIIP)

Abstract

At the core of JC Beall and Greg Restall’s brand of logical pluralism is Generalised Tarski’s Thesis, according to which an argument is valid iff, in every relevant case where every premise is true, so is the conclusion. I argue that the thesis implies that many philosophically interesting substructural logics are not legitimate relations of logical consequence. I then diagnose the clash as due to the fact that, in important ways, the thesis is not sensitive to intensional connections and to plurality in occurrences, values and models. Next, I extend the argument to the effect that the more general conception of logical consequence as guaranteed truth preservation clashes with substructurality. I conclude with a proposal as to how, for the substructural logics in question, we can still uphold a broadly semantic conception of logical consequence: given any such logic L, we can reinterpret truth-preservation conditionals with the notions of conjunction and implication available in L, and say that the fact that, in L, φ0, φ1, φ2 …, φi entail ψ is grounded in the fact that, in L, the conditional ‘If “φ0” is true and “φ1” is true and “φ2” is true … and “φi” is true, “ψ” is true’ is a logical truth. On this proposal, contrary to the contemporary Tarskian vulgate, it is logical consequence that is grounded in logical truth rather than vice versa.

Keywords

Commutativity Contraction Generalised Tarski’s Thesis Logical consequence Logical pluralism Logical truth Monotonicity Reflexivity Substructural logics Transitivity Truth preservation 

References

  1. Anderson, A., and N. Belnap. 1975. Entailment. Vol. I. Princeton: Princeton University Press.Google Scholar
  2. Asenjo, F. 1966. A Calculus of Antinomies. Notre Dame Journal of Formal Logic 7: 103–105.CrossRefGoogle Scholar
  3. Beall, J., ed. 2007. Revenge of the Liar. Oxford: Oxford University Press.Google Scholar
  4. ———. 2015. Free of Detachment: Logic, Rationality, and Gluts. Noûs 49: 410–423.CrossRefGoogle Scholar
  5. Beall, J., and G. Restall. 2006. Logical Pluralism. Oxford: Oxford University Press.Google Scholar
  6. Bočvar, D. 1938. Ob odnom trexznačnom isčislenii i ego primenenii k analizu paradoksov klassičeskogo rasširennogo funkcional’nogo isčislenija. Matematičeskij sbornik 4: 287–308.Google Scholar
  7. Bolzano, B. 1837. Wissenschaftslehre. Vol. II. Sulzbach: Seidel.Google Scholar
  8. Brouwer, L. 1927. Über Definitionsbereiche von Funktionen. Mathematische Annalen 97: 60–75.CrossRefGoogle Scholar
  9. Chisholm, R. 1966. Freedom and Action. In Freedom and Determinism, ed. Keith Lehrer, 11–44. New York: Random House.Google Scholar
  10. Cobreros, P., P. Égré, D. Ripley, and R. van Rooij. 2012. Tolerant, Classical, Strict. Journal of Philosophical Logic 41: 347–385.CrossRefGoogle Scholar
  11. Dummett, M. 2000. Elements of Intuitionism. 2nd ed. Oxford: Oxford University Press.Google Scholar
  12. Etchemendy, J. 1988. Tarski on Truth and Logical Consequence. The Journal of Symbolic Logic 53: 51–79.CrossRefGoogle Scholar
  13. ———. 1990. The Concept of Logical Consequence. Cambridge, MA: Harvard University Press.Google Scholar
  14. Field, H. 2008. Saving Truth from Paradox. Oxford: Oxford University Press.CrossRefGoogle Scholar
  15. ———. 2015. What Is Logical Validity? In Foundations of Logical Consequence, ed. Colin Caret and Ole Hjortland, 33–69. Oxford: Oxford University Press.Google Scholar
  16. Fine, K. 1975. Vagueness, Truth and Logic. Synthese 30: 265–300.CrossRefGoogle Scholar
  17. Girard, J. 1995. Linear Logic: Its Syntax and Semantics. In Advances in Linear Logic, ed. Jean-Yves Girard, Yves Lafont, and Laurent Regnier, 1–42. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  18. Jaśkowski, S. 1948. Rachunek zdań dla systemów dedukcyjnych sprzecznych. Studia Societatis Scientiarum Torunensis 1: 55–77.Google Scholar
  19. Kaplan, D. 1989. Demonstratives. In Themes from Kaplan, ed. Joseph Almog, John Perry, and Howard Wettstein, 481–563. Oxford: Oxford University Press.Google Scholar
  20. Kleene, S. 1938. On a Notation for Ordinal Numbers. The Journal of Symbolic Logic 3: 150–155.CrossRefGoogle Scholar
  21. Makinson, D. 1965. The Paradox of the Preface. Analysis 25: 205–207.CrossRefGoogle Scholar
  22. Martin, E., and R. Meyer. 1982. S (for Syllogism). ms.Google Scholar
  23. Moruzzi, S., and E. Zardini. 2007. Conseguenza logica. In Filosofia analitica, ed. Annalisa Coliva, 157–194. Rome: Carocci.Google Scholar
  24. Ono, H. 2003. Substructural Logics and Residuated Lattices—An Introduction. In Trends in Logic. 50 Years of Studia Logica, ed. Vincent Hendricks and Jacek Malinowski, 193–228. Dordrecht: Springer.Google Scholar
  25. Paoli, F. 2002. Substructural Logics: A Primer. Dordrecht: Kluwer.CrossRefGoogle Scholar
  26. Popper, K. 1948. On the Theory of Deduction, Part II. The Definitions of Classical and Intuitionist Negation. Indagationes Mathematicae 10: 44–54.Google Scholar
  27. Prawitz, D. 2005. Logical Consequence from a Constructivist Point of View. In The Oxford Handbook of Philosophy of Mathematics and Logic, ed. Stewart Shapiro, 671–695. Oxford: Oxford University Press.CrossRefGoogle Scholar
  28. Priest, G. 2006. In Contradiction. 2nd ed. Oxford: Oxford University Press.Google Scholar
  29. Putnam, H. 1968. Is Logic Empirical? In Proceedings of the Boston Colloquium for the Philosophy of Science 1966/1968, Vol. V of Boston Studies in the Philosophy of Science, ed. Robert Cohen and Marx Wartofsky, 216–241. Dordrecht: Reidel.Google Scholar
  30. Read, S. 1981. Validity and the Intensional Sense of ‘and’. Australasian Journal of Philosophy 59: 301–307.CrossRefGoogle Scholar
  31. ———. 2003. Logical Consequence as Truth-Preservation. Logique et Analyse 183: 479–493.Google Scholar
  32. Restall, G. 2000. An Introduction to Substructural Logics. London: Routledge.CrossRefGoogle Scholar
  33. ———. 2005. Multiple Conclusions. In Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress, ed. Petr Hájek, Luis Valdés Villanueva, and Dag Westerståhl, 189–205. London: College Publications.Google Scholar
  34. Tarski, A. 1930. Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften. I. Monatshefte für Mathematik und Physik 37: 361–404.CrossRefGoogle Scholar
  35. ———. 1936. O pojęciu wynikania logicznego. Przegląd filozoficzny 39: 58–68.Google Scholar
  36. Wittgenstein, L. 1953. Philosophische Untersuchungen. Oxford: Blackwell.Google Scholar
  37. Wright, C. 2000. Cogency and Question-Begging: Some Reflections on McKinsey’s Paradox and Putnam’s Proof. Philosophical Issues 10: 140–163.CrossRefGoogle Scholar
  38. Zardini, E. 2008. A Model of Tolerance. Studia Logica 90: 337–368.CrossRefGoogle Scholar
  39. ———. 2011. Truth without Contra(di)ction. The Review of Symbolic Logic 4: 498–535.CrossRefGoogle Scholar
  40. ———. 2012. Truth Preservation in Context and in Its Place. In Insolubles and Consequences, ed. Catarina Dutilh-Novaes and Ole Hjortland, 249–271. London: College Publications.Google Scholar
  41. ———. 2014a. Context and Consequence. An Intercontextual Substructural Logic. Synthese 191: 3473–3500.CrossRefGoogle Scholar
  42. ———. 2014b. Confirming the Less Likely, Discovering the Unknown. Dogmatisms: Surd and Doubly Surd, Natural, Flat and Doubly Flat. In Scepticism and Perceptual Justification, ed. Dylan Dodd and Elia Zardini, 33–70. Oxford: Oxford University Press.CrossRefGoogle Scholar
  43. ———. 2014c. Evans Tolerated. In Vague Objects and Vague Identity, ed. Kensuke Akiba and Ali Abasnezhad, 327–352. Dordrecht: Springer.CrossRefGoogle Scholar
  44. ———. 2014d. Naive Truth and Naive Logical Properties. The Review of Symbolic Logic 7: 351–384.CrossRefGoogle Scholar
  45. ———. 2015a. Breaking the Chains. Following-from and Transitivity. In Foundations of Logical Consequence, ed. Colin Caret and Ole Hjortland, 221–275. Oxford: Oxford University Press.CrossRefGoogle Scholar
  46. ———. 2015b. The Opacity of Truth. Topoi 34: 37–54.CrossRefGoogle Scholar
  47. ———. 2016. Restriction by Noncontraction. Notre Dame Journal of Formal Logic 57: 287–327.CrossRefGoogle Scholar
  48. ———. 2018a. Forthcoming. Closed without Boundaries. Synthese.Google Scholar
  49. ———. 2018b. Forthcoming. Instability and Contraction. Journal of Philosophical Logic.Google Scholar
  50. ———. 2018c. Forthcoming. The Underdetermination of the Meaning of Logical Words by Rules of Inference. In The A Priori: Its Significance, Grounds, and Extent, ed. Dylan Dodd and Elia Zardini. Oxford: Oxford University Press.Google Scholar
  51. ———. 2018d. Changing without Contra(di)ction. ms.Google Scholar
  52. ———. 2018e. The Bearers of Logical Consequence. ms.Google Scholar
  53. ———. 2018f. Unstable Knowledge. ms.Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Elia Zardini
    • 1
    • 2
  1. 1.LanCog Research Group, Philosophy CentreUniversity of LisbonLisbonPortugal
  2. 2.International Laboratory for Logic, Linguistics and Formal Philosophy, School of PhilosophyNational Research University Higher School of EconomicsMoscowRussian Federation

Personalised recommendations