Stalnaker's logical problem of conditionals is unsolvable

Abstract

The logical problem of conditionals, as conceived by Stalnaker, amounts to axiomatizing a particular semantics for conditionals which utilizes selection functions that take propositions (i.e., sets of worlds) as arguments. While the sentential form of this semantics is recursively axiomatizable, we prove that its enrichment with first-order quantifiers is not---that is, we show that Stalnaker's logical problem of conditionals is unsolvable in the language with first-order quantifiers. We demonstrate this by showing how to interpret arithmetic in the logic. In the conclusion, we discuss the implications of this result for the study of conditional logic.