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De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics

Journal of Philosophical Logic volume 50, pages 215–247 (2021)Cite this article

Abstract

In Part I of this paper, we identified and compared various schemes for trivalent truth conditions for indicative conditionals, most notably the proposals by de Finetti (1936) and Reichenbach (1935, 1944) on the one hand, and by Cooper (Inquiry, 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic, 49, 245–260, 2008) on the other. Here we provide the proof theory for the resulting logics DF/TT and CC/TT, using tableau calculi and sequent calculi, and proving soundness and completeness results. Then we turn to the algebraic semantics, where both logics have substantive limitations: DF/TT allows for algebraic completeness, but not for the construction of a canonical model, while CC/TT fails the construction of a Lindenbaum-Tarski algebra. With these results in mind, we draw up the balance and sketch future research projects.

References

  1. Adams, E.W. (1975). The logic of conditionals. Dordrecht: Reidel.

    Book  Google Scholar 

  2. Asenjo, F.G. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 16, 103–105.

    Article  Google Scholar 

  3. Baaz, M., Fermüller, C., & Zach, R. (1992). Dual systems of sequents and tableaux for many-valued logics. Technical Report TUW-E185.2-BFZ.2–92.

  4. Baaz, M., Fermüller, C., & Zach, R. (1993). Systematic construction of natural deduction systems for many-valued logics: Extended report. Technical Report TUW- E185.2-BFZ.1–93.

  5. Baratgin, J., Over, D., & Politzer, G. (2013). Uncertainty and the de Finetti tables. Thinking & Reasoning, 19, 308–328.

    Article  Google Scholar 

  6. Baratgin, J., Politzer, G., Over, D.E., & Takahashi, T. (2018). The psychology of uncertainty and three-valued truth tables. Frontiers in Psychology, 9, 1479.

    Article  Google Scholar 

  7. Beall, J.C. (2009). Spandrels of Truth. Oxford: Oxford University Press.

    Book  Google Scholar 

  8. Blok, W.J., & Pigozzi, D. (1989). Algebraizable Logics. Volume 77 of Memoirs of the American Mathematical Society. American Mathematical Society.

  9. Calabrese, P. (2002). Deduction with uncertain conditionals. Information Sciences, 147, 143–191.

    Article  Google Scholar 

  10. Cantwell, J. (2008). The logic of conditional negation. Notre Dame Journal of Formal Logic, 49, 245–260.

    Article  Google Scholar 

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

  12. Cooper, W.S. (1968). The propositional logic of ordinary discourse. Inquiry, 11, 295–320.

    Article  Google Scholar 

  13. de Finetti, B. (1936). La logique de la probabilité. In Actes du congrès international de philosophie scientifique, (Vol. 4 pp. 1–9): Hermann Editeurs Paris.

  14. Dubois, D., & Prade, H. (1994). Conditional objects as nonmonotonic consequence relationships. IEEE Transactions on Systems, Man, and Cybernetics, 24(12), 1724–1740.

    Article  Google Scholar 

  15. Edgington, D. (1995). On Conditionals. Mind, 104, 235–329.

    Article  Google Scholar 

  16. Farrell, R.J. (1979). Material implication, confirmation, and counterfactuals. Notre Dame Journal of Formal Logic, 20, 383–394.

    Article  Google Scholar 

  17. Font, J.M. (2016). Abstract algebraic logic–an introductory textbook. Studies in Logic: Mathematical Logic and Foundations. London: College Publications.

    Google Scholar 

  18. Goodship, L. (1996). On dialetheism. Australasian Journal of Philosophy, 74, 153–61.

    Article  Google Scholar 

  19. Gottwald, S. (2001). A treatise on many-valued logics. Volume IX of Studies in Logic and Computation. Baldock: Research Studies Press Ltd.

    Google Scholar 

  20. Herrmann, B. (1996). Equivalential and algebraizable logics. Studia Logica, 57, 419–436.

    Article  Google Scholar 

  21. Humberstone, L. (2011). The connectives. Cambridge: MIT Press.

    Book  Google Scholar 

  22. Jeffrey, R.C. (1963). On indeterminate conditionals. Philosophical Studies, 14, 37–43.

    Article  Google Scholar 

  23. Jeffrey, R.C. (1991). Matter-of-Fact Conditionals. Proceedings of the Aristotelian Society, 65, 161–183.

    Article  Google Scholar 

  24. Khoo, J. (2015). On indicative and subjunctive conditionals. Philosopher’s Imprint, 15.

  25. Lewis, D. (1973). Causation. Journal of Philosophy, 70, 556–567.

    Article  Google Scholar 

  26. Lewis, D. (1973). Counterfactuals. Oxford: Basil Blackwell.

    Google Scholar 

  27. Malinowski, G. (1993). Many-valued logics. Oxford: Clarendon Press.

    Google Scholar 

  28. Mandelkern, M. (2020). Import-Export and ‘And’. Philosophy and Phenomenological Research, 100(1), 118–135.

    Article  Google Scholar 

  29. McGee, V. (1985). A counterexample to modus ponens. The Journal of Philosophy, 82, 462–71.

    Article  Google Scholar 

  30. Milne, P. (2004). Algebras of intervals and a logic of conditional assertions. Journal of Philosophical Logic, 33, 497–548.

    Article  Google Scholar 

  31. Moschovakis, Y. (1974). Elementary induction on abstract structures. Amsterdam, London and New York: North-Holland and Elsevier.

    Google Scholar 

  32. Pohlers, W. (2009). Proof theory. The first step into impredicativity. Berlin: Springer.

    Google Scholar 

  33. Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8, 219–241.

    Article  Google Scholar 

  34. Priest, G. (2006). Doubt truth to be a liar. Oxford: Oxford University Press.

    Google Scholar 

  35. Pynko, A. (1995). On Priest’s logic of paradox. Journal of Applied Non-Classical Logics, 5, 219–225.

    Article  Google Scholar 

  36. Reichenbach, H. (1935). Wahrscheinlichkeitslehre. Leiden: Sijthoff.

    Google Scholar 

  37. Reichenbach, H. (1944). Philosophic foundations of quantum mechanics. Berkeley: University of California Press.

    Google Scholar 

  38. Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5, 354–78.

    Article  Google Scholar 

  39. Schütte, K. (1956). Ein System des verknüpfenden Schließens. Archiv für mathematische Logik und Grundlagenforschung, 2, 55–67.

    Article  Google Scholar 

  40. Stern, R., & Hartmann, S. (2018). Two sides of modus ponens. The Journal of Philosophy, 115(11), 605–621.

    Article  Google Scholar 

  41. Troelstra, A.S., & Schwichtenberg, H. (2000). Basic proof theory. Cambridge: Cambridge University Press.

    Book  Google Scholar 

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Open Access funding enabled and organized by Projekt DEAL.

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Authors and Affiliations

  1. Institut Jean-Nicod (CNRS/ENS/EHESS), Département de Philosophie & Département d’études Cognitives, Ecole Normale Supérieure, PSL University, 29 rue d’Ulm, 75005, Paris, France

    Paul Égré

  2. Munich Center for Mathematical Philosophy (MCMP), Fakultät für Philosophie, Wissenschaftstheorie und Religionswissenschaft, Ludwig-Maximilians-Universität München, Geschwister-Scholl-Platz 1, D-80539, München, Germany

    Lorenzo Rossi

  3. Center for Logic, Language and Cognition (LLC), Department of Philosophy and Education Science, Università degli Studi di Torino, Via Sant’Ottavio 20, 10124, Torino, Italy

    Jan Sprenger

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  1. Paul Égré
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  2. Lorenzo Rossi
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  3. Jan Sprenger
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Correspondence to Paul Égré, Lorenzo Rossi or Jan Sprenger.

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We express our thanks to an anonymous referee of this journal as well as to the colleagues acknowledged in Part I of this paper. Funding for this research was provided by grants ANR-14-CE30-0010 (program Trilogmean), ANR-17-EURE-0017 (program FrontCog), and ANR-19-CE28-0004-01 (program Probasem) (P.E.), by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF), grant no. P29716-G24 for research carried out at the University of Salzburg (L.R.), and by the European Research Council (ERC) through Starting Grant No. 640638 (J.S.). The order between the three authors is alphabetical; the authors’ contribution is equal in part I; in the present part II, P.E. and J.S. would like to acknowledge L.R.’s preponderant contribution to sections 3 and 4.

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Égré, P., Rossi, L. & Sprenger, J. De Finettian Logics of Indicative Conditionals Part II: Proof Theory and Algebraic Semantics. J Philos Logic 50, 215–247 (2021). https://doi.org/10.1007/s10992-020-09572-7

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  • DOI: https://doi.org/10.1007/s10992-020-09572-7

Keywords

  • Indicative conditionals
  • Trivalent logics
  • Cooper-Cantwell conditional
  • de Finetti conditional
  • Proof theory
  • Algebraic semantics
  • Connexive logics