Inquisitive Logic

Origin. Groenendijk, Roelofsen (2000s). Questions in logic. Support semantics. Beyond truth conditions. Foundation for question semantics.

Models. Sentences have inquisitive content. Questions are propositions. Support replaces truth. Information states central. Unifies declaratives and interrogatives.

Formalism.

Information states: State σ = set of possible worlds. σ supports φ: σ ⊨ φ. Support replaces truth. Relativized to information.

Inquisitive propositions: Proposition [φ] = {σ : σ ⊨ φ}. Set of sets of worlds. Downward closed under ⊆. Alternatives structure.

Support conditions: σ ⊨ p iff σ ⊆ |p| (atomic) σ ⊨ φ ∧ ψ iff σ ⊨ φ and σ ⊨ ψ σ ⊨ φ ∨ ψ iff σ ⊨ φ or σ ⊨ ψ σ ⊨ ¬φ iff for all τ ⊆ σ: τ ⊨ φ implies τ = ∅

Inquisitive disjunction: φ ∨ ψ raises issue: which? Either φ or ψ settles it. Source of inquisitiveness. Not just truth-functional.

Questions: ?φ := φ ∨ ¬φ "Is φ the case?" Alternatives: φ-worlds or ¬φ-worlds. Settled by either.

Projection operators: !φ (assertion): project to informative content, ignore the issue. ?φ (question): raise the polar issue. Every proposition factors into assertive and inquisitive parts.

Informativeness vs inquisitiveness: Informative: rules out possibilities. Inquisitive: raises issues. Both captured in one framework. Orthogonal dimensions.

Entailment: φ ⊨ ψ iff [φ] ⊆ [ψ]. Every way of supporting φ supports ψ. Questions can entail questions. Cross-type entailment.

Symbols.

Symbol	Unicode	Name	Meaning
σ	U+03C3	Sigma	An information state: a set of possible worlds
τ	U+03C4	Tau	A substate of σ, used in the clause for negation
φ, ψ	U+03C6, U+03C8	Phi, psi	Sentence variables
⊨	U+22A8	Double turnstile	Support: σ ⊨ φ, and also entailment between propositions
⊆	U+2286	Subset	Substate relation and downward closure
∧	U+2227	Conjunction	Supported iff both conjuncts are
∨	U+2228	Inquisitive disjunction	Supported iff one disjunct is; the source of inquisitiveness
¬	U+00AC	Negation	σ ⊨ ¬φ iff no nonempty substate supports φ
∅	U+2205	Empty set	The inconsistent state, which supports everything
?	—	Question operator	?φ := φ ∨ ¬φ, raising the polar issue
!	—	Assertion operator	!φ := ¬¬φ, discarding the issue and keeping the information
[φ]	—	Inquisitive proposition	The downward-closed set of states supporting φ
p	—	Atomic sentence	Its truth set is written with vertical bars

Metatheory. Support is evaluated at information states rather than worlds, and propositions are downward-closed sets of states, so a sentence carries both informative content — the union of its support set — and an issue, the way that content is divided into alternatives. Ciardelli and Roelofsen (Journal of Philosophical Logic 40, 2011) axiomatized the propositional system InqB by adding to intuitionistic logic the Kreisel–Putnam axiom together with the double negation law restricted to atoms, and proved completeness; InqB is decidable, has the disjunction property, and coincides with the negative variant of Medvedev's logic of finite problems. It is not closed under uniform substitution: the classical laws hold for atoms but fail for compounds, which is exactly what distinguishes a declarative from an interrogative. The projection operators factor every proposition into an assertive part !φ = ¬¬φ and an inquisitive part ?φ = φ ∨ ¬φ, and entailment is one relation covering declarative-to-declarative, question-to-question, and mixed cases uniformly.

Applies to. Question semantics. Dialogue logic. Information dynamics. Linguistics.

Limitations. The failure of uniform substitution means InqB has no substitution-closed logic behind it, so standard techniques for intermediate logics do not transfer and the system must be studied on its own. Treating alternatives as the maximal supporting states works only when they exist; in the infinite case a downward-closed set need not have maximal elements, and the alternative-based descriptions of question content then have no object to describe. Disjunction is inquisitive by default, so every plain declarative disjunction needs a stipulated ! to suppress the issue, and nothing in the semantics says where that operator goes. The framework also gives up the partition semantics of Groenendijk and Stokhof, and with it the exhaustive readings of questions embedded under know, which have to be reintroduced separately. Attitude and modal embedding remain the least settled part of the system.