Incompleteness in Quantified Conditional Logic

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.

💡 Research Summary

The paper revisits the celebrated completeness result of Stalnaker and Thomason (1970) for the quantified conditional logic often called QST (or CQ). Stalnaker’s philosophical program treats conditionals as functions that take propositions—identified with sets of possible worlds—and return propositions. However, the original completeness proof uses a selection function that operates on formulas of the object language rather than on propositions. This mismatch raises the question of whether the logic truly captures Stalnaker’s intended semantics.

To address this, the authors introduce a set‑selection‑function frame 𝔽 = ⟨W, R, f, D, d⟩. Here W is a non‑empty set of worlds, R⊆W×W an accessibility relation, f : 𝒫(W)×W→𝒫(W) a selection function defined for all propositions P⊆W, D a global domain of individuals, and d : W→𝒫(D) a local‑domain assignment. The only basic requirement on f is that f(P,w)⊆R(w); additional constraints mirror Stalnaker’s own conditions: Success (f(P,w)⊆P), Weak Centering (if w∈P then w∈f(P,w)), and a monotonicity condition. These frames are “full” in the sense that the selection function is defined for every proposition, unlike the original formula‑based semantics.

The core technical contribution is a frame‑incompleteness theorem for the quantified conditional system QC₂, the simplest extension of C₂ with first‑order quantifiers and without identity or existence predicates. The authors construct a family of infinitary entailments that are valid in every Stalnaker‑compliant selection frame but are not derivable in QC₂. The construction exploits the fact that, when f is required to be defined on all propositions, one can engineer propositions P such that f(P,w)=∅ even though w∈R(w). This creates “non‑representable” worlds that defeat any attempt to capture the logic with a class of Kripke‑style frames. Consequently, QC₂ (and hence QST) lacks strong completeness with respect to the intended proposition‑based semantics; it is frame incomplete.

The authors then show that this incompleteness is robust under a wide range of semantic choices. Whether the domain is variable (actualist) or constant (possibilist), whether the language includes an identity predicate (=) or a primitive existence predicate (E), and whether the Barcan formula or its converse is adopted, the same incompleteness argument can be adapted. In each case the essential obstacle remains the same: the selection function’s obligation to handle every proposition forces the existence of infinitary counter‑examples that no finite axiom system can capture.

The paper situates this result within the broader literature on frame incompleteness. It draws a parallel with the well‑known case of the propositional provability logic GL, which is frame‑complete propositionally but becomes frame‑incomplete once quantifiers are added (Montagna 1984). Similarly, C₂ itself is weakly complete with respect to propositional selection‑function semantics, yet it fails strong completeness, a fact previously observed by van Fraassen (1974) and others. The present work extends that line of inquiry to the quantified setting and to the philosophically motivated proposition‑based semantics.

In the concluding discussion, the authors reflect on the philosophical implications. Since Stalnaker’s original motivation was to model conditionals for non‑linguistic agents—where beliefs, desires, and probabilities are about propositions rather than sentences—the failure of QST to be characterized by any class of proposition‑based frames suggests that the standard quantified conditional logic does not fulfill Stalnaker’s program. This has repercussions for several active research areas: (1) probabilistic accounts of conditionals that treat probabilities as functions on propositions; (2) causal decision theory, where expected utilities depend on the probabilities of counterfactual conditionals; and (3) analyses of laws of nature that aim to avoid linguistic stipulations. The authors propose future work on alternative semantics (e.g., ordering‑based semantics), on identifying restricted frame conditions that might restore strong completeness, and on integrating proposition‑based conditional logic with probabilistic and causal frameworks. In sum, the paper demonstrates that the quantified conditional logic QST is fundamentally frame incomplete when selection functions are required to operate on propositions, thereby challenging the adequacy of the standard formalism for Stalnaker‑style philosophical applications.