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.
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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.
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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
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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