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

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.

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Open Access funding enabled and organized by Projekt DEAL.

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Correspondence to Paul Égré or 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 (2021). https://doi.org/10.1007/s10992-020-09572-7

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Keywords

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