Gibbardian Collapse and Trivalent Conditionals

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.