Erotetic Logic (Logic of Questions)

Origin. Nuel Belnap and Thomas Steel, The Logic of Questions and Answers (1976), building on Belnap's An Analysis of Questions (1963); Hamblin (1958) and Åqvist's imperative-epistemic reduction (1965) are the other founding lines. Andrzej Wiśniewski's The Posing of Questions (1995) and Questions, Inferences, and Scenarios (2013) give inferential erotetic logic, in which questions stand in consequence relations to each other.

Models. A question is not a proposition and has no truth value, so the standard apparatus does not apply — and yet questions are answered well or badly, arise from what one knows, and follow from other questions. Erotetic logic takes the second fact seriously: it defines consequence relations with questions as terms, so that "Is it p or q?" together with "Not p" yields "Is it q?" as a matter of logic rather than of pragmatics.

Formalism.

Questions as sets of answers (Hamblin, Belnap–Steel): ?{A₁, ..., Aₙ} — the question whose direct answers are the Aᵢ. "Who came?" = {came(a), came(b), ...} Belnap's request/presupposition analysis: a question requests a selection and presupposes at least one answer is true.

Presupposition: PresQ = the disjunction of the direct answers. A question is sound in M iff PresQ is true in M. "Have you stopped beating your wife?" is unsound where the presupposition fails — the failure is a property of the question, not of an answer.

Evocation (Wiśniewski): X evokes Q iff X entails PresQ, and X entails no direct answer of Q. The premises make the question arise and do not settle it. {p ∨ q} evokes ?{p, q}. {p} does not — it settles it.

Erotetic implication: Q implies Q′ relative to X iff (i) every direct answer to Q, with X, gives a sound Q′; and (ii) every direct answer to Q′, with X, narrows Q. ?{p, q, r} together with {¬r} erotetically implies ?{p, q}. This is a consequence relation whose relata include questions.

Erotetic search scenarios: A tree whose nodes are questions and declaratives, generated by evocation and erotetic implication. The formal model of inquiry: what to ask next, given what you have.

Contrast with inquisitive semantics: Inquisitive semantics makes questions and statements one type — propositions as sets of information states — and recovers questions as non-informative content. Erotetic logic keeps them distinct types and gives the relations between them. The two answer different questions about questions.

Symbols.

Metatheory. Wiśniewski's contribution is that evocation and erotetic implication are genuine consequence relations — defined semantically, with soundness properties, and not reducible to the declarative consequence of the answers. The reductive programmes (Åqvist's "bring it about that I know whether", Hintikka's epistemic-imperative analysis) treat a question as an imperative about knowledge and get the logic from deontic and epistemic logic; erotetic logic's case against them is that the reduction loses evocation, since an imperative does not arise from premises the way a question does. Whether questions are a genuine semantic type or an illocutionary use of a declarative is the standing division, and the two formal traditions line up with the two answers.

Applies to. The semantics of interrogatives. Models of inquiry and question-directed search. Dialogue systems and information-seeking agents. Proof search read as a scenario of questions. Belnap's analysis remains the reference for question presupposition in linguistics.

Limitations. The answer-set analysis presupposes the direct answers are given, and for most natural questions they are not — "Why did it fail?" has no antecedently fixed answer set, and the theory says nothing about how one is determined. Erotetic implication is defined relative to a background X, so nothing is a consequence of a question simpliciter. And the field has stayed small: it has no automated implementation, no standard proof theory, and its results have not transferred to the linguistic semantics of questions, which went the inquisitive way.

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