Incompleteness in Quantified Conditional Logic
Alexander W. Kocurek, James Walsh, Yale Weiss
TL;DR
This paper shows that Stalnaker–Thomason’s quantified conditional logic $ obreak\mathsf{QST}$ is not the logic of any class of selection frames in which selection functions operate on propositions. The authors methodically compare set-selection and ordering semantics, define the core logics $\mathsf{QC2}$ and its variants, and establish a robust frame-incompleteness result by constructing a counter-model $\mathcal{K}$ that satisfies $\mathsf{QC2}$ and $\neg{\mathsf{DS}}$, while $\mathsf{DS}$ is valid in all weakly Stalnakerian frames. They then extend the incompleteness to identity-enabled and variable-domain logics, showing $\mathsf{QC2_=}$ and the variable-domain versions, including $\mathsf{QST}$ (via $\mathsf{QC2^v_=}$), are not the logics of any frame class. The work demonstrates that the classical completeness of $\mathsf{QST}$ depends delicately on the specific semantic design and remains fragile under natural semantic refinements, raising open questions about how to salvage a frame-based account of conditionals in quantified settings.
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.