De Finettian Logics of Indicative Conditionals

TL;DR

This paper investigates trivalent truth conditions for indicative conditionals, analyzing de Finetti's and Reichenbach's approaches, and proposes two logical frameworks addressing validity and Modus Ponens issues.

Contribution

It introduces two novel trivalent logics for indicative conditionals, resolving validity and inference problems in de Finetti's framework.

Findings

Both logics are connexive.

DF/TT preserves non-False values but lacks Modus Ponens.

CC/TT restores Modus Ponens but alters validity considerations.

Abstract

This paper explores trivalent truth conditions for indicative conditionals, examining the "defective" table put forward by de Finetti 1936, as well as Reichenbach 1944, first sketched in Reichenbach 1935. 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 choosing an adequate notion of validity for this conditional. We show that all standard trivalent schemes are problematic, 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 1968 and Cantwell 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.