Skip to main content
Download PDF

De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity

Journal of Philosophical Logic volume 50, pages 187–213 (2021)Cite this article

Abstract

This paper explores trivalent truth conditions for indicative conditionals, examining the “defective” truth table proposed by de Finetti (1936) and Reichenbach (1935, 1944). On their approach, a conditional takes the value of its consequent whenever its antecedent is true, and the value Indeterminate otherwise. Here we deal with the problem of selecting an adequate notion of validity for this conditional. We show that all standard validity schemes based on de Finetti’s table come with some problems, and highlight two ways out of the predicament: one pairs de Finetti’s conditional (DF) with validity as the preservation of non-false values (TT-validity), but at the expense of Modus Ponens; the other modifies de Finetti’s table to restore Modus Ponens. In Part I of this paper, we present both alternatives, with specific attention to a variant of de Finetti’s table (CC) proposed by Cooper (Inquiry 11, 295–320, 1968) and Cantwell (Notre Dame Journal of Formal Logic 49, 245–260, 2008). In Part II, we give an in-depth treatment of the proof theory of the resulting logics, DF/TT and CC/TT: both are connexive logics, but with significantly different algebraic properties.

References

  1. Adams, E.W. (1965). The logic of conditionals. Inquiry, 8, 166–197.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  4. Beaver, D., & Krahmer, E. (2001). A partial account of presupposition projection. Journal of Logic Language and Information, 10, 147.

    Google Scholar 

  5. Belnap, N.D. (1970). Conditional assertion and restricted quantification. Nouŝ, 68, 1–12.

    Google Scholar 

  6. Blamey, S. (1986). Partial logic. In Gabbay, D., & Guentner, F. (Eds.) Handbook of philosophical logic, vol. 5, pp. 1–70. Springer.

  7. Bledin, J. (2015). Modus ponens defended. Journal of Philosophy, 112, 57–83.

    Google Scholar 

  8. Bochvar, D. A. (1981). [On a three-valued calculus and its applications to the paradoxes of the classical extended functional calculus]. Mathematicheskii Sbornik 4, 287–308. English translation by M. Bergmann in History and Philosophy of Logic, 2, 87–112.

    Google Scholar 

  9. Bradley, R. (2002). Indicative conditionals. Erkenntnis, 56, 345–378.

    Google Scholar 

  10. Calabrese, P. (1987). An algebraic synthesis of the foundations of logic and probability. Information Sciences, 42, 187–237.

    Google Scholar 

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

    Google Scholar 

  12. Cantwell, J. (2006). The laws of non-bivalent probability. Logic and Logical Philosophy, 15, 163–171.

    Google Scholar 

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

    Google Scholar 

  14. Cariani, F., & Goldstein, S. (2018). Conditional Heresies. Philosophy and Phenomenological Research. https://doi.org/10.1111/phpr.12565.

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

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

    Google Scholar 

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

    Google Scholar 

  18. Ciardelli, I. (2019). Indicative conditionals and graded information. Journal of Philosophical Logic, https://doi.org/10.1007/s10992-019-09528-6.

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

    Google Scholar 

  20. Cobreros, P., Egré, P., Ripley, D., & van Rooij, R. (2015). Vagueness, truth and permissive consequence. In Achouriotti, D., Galinon, H., & Martinez, J. (Eds.) Unifying the philosophy of truth (pp. 409–430): Springer.

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

    Google Scholar 

  22. Crupi, V., & Iacona, A. (2019). The evidential conditional. Preprint.

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

  24. Douven, I. (2016). The epistemology of indicative conditionals. Cambridge: Cambridge University Press.

    Google Scholar 

  25. Douven, I., & Verbrugge, S. (2010). The Adams family. Cognition, 117, 302–318.

    Google Scholar 

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

    Google Scholar 

  27. Dummett, M. (1959). Truth. In Proceedings of the Aristotelian Society, LIX, (pp. 141–162).

  28. Edgington, D. (1995). On conditionals. Mind, 104, 235–329.

    Google Scholar 

  29. Égré, P., & Cozic, M. (2011). If-clauses and probability operators. Topoi, 30, 17–29.

    Google Scholar 

  30. Égré, P., & Cozic, M. (2016). Conditionals. In Aloni, M., & Dekker, P. (Eds.) The Cambridge handbook of formal semantics (pp. 490–524). Cambridge University Press.

  31. Égré, P., & Politzer, G. (2013). On the negation of indicative conditionals. In Aloni, M., Franke, M., & Roelofsen, F. (Eds.) Proceedings of the 19th Amsterdam colloquium (pp. 10–18).

  32. Égré, P., Rossi, L., & Sprenger, J. (2020). Gibbardian collapse and trivalent conditionals. Forthcoming in Kaufmann, S., Over, D., Sharma, G. (Eds.) Conditionals: Logic, Linguistics and Psychology. Palgrave MacMillan.

  33. Evans, J., Handley, S., Neilens, H., & Over, D. (2007). Thinking about conditionals: A study of individual differences. Memory & Cognition, 35, 1772–1784.

    Google Scholar 

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

    Google Scholar 

  35. Farrell, R.J. (1986). Implication and presupposition. Notre Dame Journal of Formal Logic, 27, 51–61.

    Google Scholar 

  36. Francez, N. (2016). Natural deduction for two connexive logics. IFCoLog Journal of Logics and their Applications, 3, 479–504.

    Google Scholar 

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

    Google Scholar 

  38. Gibbard, A. (1980). Two recent theories of conditionals. In Harper, W (Ed.) Ifs (pp. 211–247): Springer.

  39. Goodman, I.R., Nguyen, H.T., & Walker, E.A. (1991). Conditional inference and logic for intelligent systems. North-Holland: Amsterdam.

    Google Scholar 

  40. Handley, S., Evans, J., & Thompson, V. (2006). The negated conditional: A litmus test for the suppositional conditional?. Journal of Experimental Psychology: Learning, Memory and Cognition, 32, 559–569.

    Google Scholar 

  41. Huitink, J. (2008). Modals, Conditionals and Compositionality: Radboud University Nijmegen PhD dissertation.

  42. Humberstone, L. (2011). The connectives. Cambridge/MA: MIT Press.

    Google Scholar 

  43. Jackson, F. (1979). On assertion and indicative conditionals. Philosophical Review, 88, 565–589.

    Google Scholar 

  44. Jackson, F. (1987). Conditionals. London: Blackwell.

    Google Scholar 

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

    Google Scholar 

  46. Kapsner, A. (2018). Humble connexivity. Logic and Logical Philosophy, 28(3), 513–536.

    Google Scholar 

  47. Khoo, J. (2015). On indicative and subjunctive conditionals. Philosopher’s Imprint, 15, 1–40.

    Google Scholar 

  48. Kneale, W., & Kneale, M. (1962). The development of logic. Oxford: Oxford University Press.

    Google Scholar 

  49. Krzyżanowska, K. (2015). Between “If” and “Then” Towards an empirically informed philosophy of conditionals. PhD dissertation: University of Groningen.

    Google Scholar 

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

    Google Scholar 

  51. Lewis, D. (1986). Philosophical papers Vol. 2. Oxford: Oxford University Press.

    Google Scholar 

  52. Łukasiewicz, J. (1970). [On three-valued logic]. Ruch Filosoficzny, 5, 70–71. English translation in J. Łukasiewicz, Selected Works, L. Borkowski ed.

    Google Scholar 

  53. MacColl, H. (1908). ‘If’ And ‘Imply’. Mind, 17, 453–455.

    Google Scholar 

  54. Malinowski, G. (1990). Q-consequence operation. Reports on Mathematical Logic, 24, 49–54.

    Google Scholar 

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

    Google Scholar 

  56. Manor, R. (1975). Propositional commitment and presuppositions. American Philosophical Quarterly, 12, 141–149.

    Google Scholar 

  57. McDermott, M. (1996). On the truth conditions of certain ‘if’-sentences. The Philosophical Review, 105, 1–37.

    Google Scholar 

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

    Google Scholar 

  59. McGee, V. (1989). Conditional probabilities and compounds of conditionals. The Philosophical Review, 98, 485–541.

    Google Scholar 

  60. Milne, P. (1997). Bruno de Finetti and the logic of conditional events. The British Journal for the Philosophy of Science, 48, 195–232.

    Google Scholar 

  61. Olivier, F. (2018). Les enoncés conditionnels et la négatioń. Master’s Thesis. Paris: Ecole normale supérieure.

    Google Scholar 

  62. Over, D. (2016). Causation and the probability of causal conditionals. In Waldmann, M. (Ed.) Oxford handbook of causal reasoning. Oxford: Oxford University Press.

  63. Over, D., & Baratgin, J. (2017). The “defective” truth table. In Galbraith, N., Lucas, E., & Over, D. (Eds.) The thinking mind: A festschrift for Ken Manktelow (pp. 15–28): Routledge.

  64. Over, D., Hadjichristidis, C., Evans, J., Handley, S., & Sloman, S. (2007). The probability of causal conditionals. Cognitive Psychology, 54, 62–97.

    Google Scholar 

  65. Pizzi, C. (1977). Boethius’ thesis and conditional logic. Journal of Philosophical Logic, 6, 283–302.

    Google Scholar 

  66. Politzer, G. (2009). Could it be the case that if I am right my opponents will be pleased? A rejoinder to Johnson-Laird, Byrne and Girotto. Topoi, 28, 81–85.

    Google Scholar 

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

    Google Scholar 

  68. Prince, A., & Smolensky, P. (2008). Optimality theory: Constraint interaction in generative grammar. New York: Wiley.

    Google Scholar 

  69. Quine, W.V.O. (1950). Methods of logic. Cambridge/MA: Harvard University Press.

    Google Scholar 

  70. Quine, W.V.O. (1970). Philosophy of logic. Cambridge/MA: Harvard University Press.

    Google Scholar 

  71. Ramsey, F.P. (1926). Truth and probability. In Mellor, D.H. (Ed.) Philosophical papers (pp. 52–94). Cambridge: Cambridge University Press.

  72. Ramsey, F.P. (1929). General propositions and causality. In Mellor, D.H. (Ed.) Philosophical papers (pp. 237–257). Cambridge: Cambridge University Press.

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

    Google Scholar 

  74. Reichenbach, H. (1944). Philosophic foundations of quantum mechanics. Berkeley and Los Angeles: University of California Press.

    Google Scholar 

  75. Rothschild, D. (2014). Capturing the relationship between conditionals and conditional probability with a trivalent semantics. Journal of Applied Non-Classical Logics, 24, 144–152.

    Google Scholar 

  76. Schay, G. (1968). An algebra of conditional events. Journal of Mathematical Analysis and Applications, 24, 334–344.

    Google Scholar 

  77. Skovgaard-Olsen, N., Collins, P., Krzyżanowska, K., Hahn, U., & Klauer, K.C. (2019). Cancellation, negation, and rejection. Cognitive Psychology, 108, 42–71.

    Google Scholar 

  78. Skovgaard-Olsen, N., Singmann, H., & Klauer, K.C. (2016). The relevance effect and conditionals. Cognition, 150, 26–36.

    Google Scholar 

  79. Spohn, W. (2013). A ranking-theoretic approach to conditionals. Cognitive Science, 37, 1074–1106.

    Google Scholar 

  80. Stalnaker, R. (1968). A theory of conditionals. In Studies in logical theory: American philosophical quarterly monograph series. Oxford: Blackwell.

  81. Stalnaker, R.C. (1970). Probability and conditionals. Philosophy of Science, 37(1), 64–80.

    Google Scholar 

  82. Stalnaker, R. (1975). Indicative conditionals. Philosophia, 5, 269–286.

    Google Scholar 

  83. Stalnaker, R.C. (1980). A defense of conditional excluded middle. In Harper, W. (Ed.) Ifs (pp. 87–104): Springer.

  84. Wansing, H. (2016). Connexive logic. In Zalta, E.N. (Ed.) The stanford encyclopedia of philosophy (Spring 2016 ed.). Metaphysics Research Lab, Stanford University. Retrieved from. https://plato.stanford.edu/archives/spr2016/entries/logic-connexive/.

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

    Google Scholar 

Download references

Funding

Open Access funding provided by Projekt DEAL.

Author information

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 Educational Science, Università degli Studi di Torino, Via Sant’Ottavio 20, 10124, Torino, Italy

    Jan Sprenger

Authors
  1. Paul Égré
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lorenzo Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jan Sprenger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Paul Égré, Lorenzo Rossi or Jan Sprenger.

Additional information

Publisher’s Note

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

We thank Jean Baratgin, Stefano Bonzio, John Cantwell, Emmanuel Chemla, Nicole Cruz de Echeverria, Vincenzo Crupi, Didier Dubois, Luis Estrada-Gonzales, Robert Farrell, Branden Fitelson, Thomas Ferguson, Nissim Francez, Chris Gauker, Andreas Herzig, Andrea Iacona, Andreas Kapsner, Dan Lassiter, Hannes Leitgeb, Christoph Michels, Julien Murzi, Jo Nathan, François Olivier, Hitoshi Omori, David Over, Francesco Paoli, Guy Politzer, Graham Priest, Dave Ripley, Robert van Rooij, Hans Rott, Paolo Santorio, Damian Szmuc, and Heinrich Wansing for various helpful exchanges, as well as audiences in Amsterdam, Bochum, Munich, Regensburg, Dagstuhl, Paris, Turin, and Buenos Aires. We are particularly grateful to Robert Farrell, Andrea Iacona, Paolo Santorio, and an anonymous referee from this journal for detailed comments. 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.). Thanks also to the network European Non-Categorical Thinking (EUNoC), which enabled this collaboration.

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

Égré, P., Rossi, L. & Sprenger, J. De Finettian Logics of Indicative Conditionals Part I: Trivalent Semantics and Validity. J Philos Logic 50, 187–213 (2021). https://doi.org/10.1007/s10992-020-09549-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10992-020-09549-6

Keywords

  • Indicative conditionals
  • De Finetti conditional
  • Cooper-Cantwell conditional
  • Trivalent semantics
  • Trivalent validity
  • Connexive logics