Gibbardian Collapse and Trivalent Conditionals

TL;DR

This paper explores the limitations of classical truth-functional conditionals highlighted by Gibbard's triviality result and proposes trivalent logics based on Reichenbach and de Finetti to avoid collapse, maintaining Import-Export without triviality.

Contribution

It introduces a trivalent logic framework for indicative conditionals that circumvents Gibbard's triviality result by allowing undefined conditionals with false antecedents.

Findings

Trivalent logics can preserve Import-Export without triviality.

Gibbard's proof relies on implicit assumptions that can be challenged.

Trivalent conditionals based on Reichenbach and de Finetti avoid collapse.

Abstract

This paper discusses the scope and significance of the so-called triviality result stated by Allan Gibbard for indicative conditionals, showing that if a conditional operator satisfies the Law of Import-Export, is supraclassical, and is stronger than the material conditional, then it must collapse to the material conditional. Gibbard's result is taken to pose a dilemma for a truth-functional account of indicative conditionals: give up Import-Export, or embrace the two-valued analysis. We show that this dilemma can be averted in trivalent logics of the conditional based on Reichenbach and de Finetti's idea that a conditional with a false antecedent is undefined. Import-Export and truth-functionality hold without triviality in such logics. We unravel some implicit assumptions in Gibbard's proof, and discuss a recent generalization of Gibbard's result due to Branden Fitelson.