Incompleteness in Quantified Conditional Logic

TL;DR

This paper investigates the completeness of a conditional logic with first-order quantifiers under a proposition-based selection function semantics, revealing fundamental incompleteness issues and their philosophical implications.

Contribution

It demonstrates that the known completeness result does not extend to proposition-based selection functions, establishing frame incompleteness for the logic.

Findings

The logic is frame incomplete under proposition-based semantics.

Invariance of incompleteness across various semantic choices.

Raises questions about the philosophical interpretation of conditionals.

Abstract

Stalnaker and Thomason famously proved that the conditional logic \textsf{C2} with first-order quantifiers is complete with respect to a selection function semantics. However, the selection functions used in this completeness result take formulas, rather than propositions (i.e., sets of worlds), as arguments. Yet Stalnaker has repeatedly emphasized the philosophical importance of viewing selection functions as functions on propositions, and many of the applications of his theory require this. Can their completeness result be extended to a selection function semantics in which the functions take propositions as arguments? We prove the answer is negative: Their logic is frame incomplete. Moreover, this result is invariant with respect to many choice points regarding the semantics, such as variable vs.~constant domains or whether to include an identity or existence predicate. We conclude by discussing some of the important and difficult questions for the philosophical and logical study of conditionals that our results raise.