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Linglib.Phenomena.Questions.Studies.Belnap1982

@cite{belnap-1982}: Approaches to the semantics of questions in natural language #

Single-paper formalisation of the answerhood theorems from @cite{belnap-1982} ("Approaches to the semantics of questions in natural language", in Issues in the Logic of Questions), formulated over the GSQuestion W substrate.

Theorems #

Unique Answer Fallacy (§2.3) #

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theorem Phenomena.Questions.Studies.Belnap1982.unique_answer_fallacy {W : Type u_1} (q : Semantics.Questions.GSQuestion W) (w v : W) (hDiff : ¬QUD.r q w v) :
∃ (u : W), QUD.ans q w u ≠ QUD.ans q v u

@cite{belnap-1982}'s Unique Answer Fallacy: it is a fallacy to assume that each question has a unique complete true answer. In the G&S framework, ans q w varies with the index w — the same question Q yields different complete true answers at different worlds. Direct restatement of ans_situation_dependent.

Distributivity Principle (§2.4) #

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theorem Phenomena.Questions.Studies.Belnap1982.distributivity_principle {W : Type u_1} {E : Type u_2} [DecidableEq E] (domain : List E) (pred : E → W → Bool) (w : W) (epState : W → Bool) :
have q := Semantics.Questions.GSQuestion.ofProject fun (w' : W) => List.map (fun (x : E) => pred x w') domain; (∀ (v : W), epState v = true → QUD.ans q w v = true) ↔ ∀ e ∈ domain, ∀ (v : W), epState v = true → pred e v = pred e w

@cite{belnap-1982}'s Distributivity Principle: knowing the answer to a wh-question is equivalent to knowing, for each individual, whether the predicate holds.

An agent whose epistemic state (the set of worlds they consider possible) is epState knows the answer to Q at w iff their state is a subset of ans(Q, w) — i.e., every world they consider possible agrees with w on the full extension of the predicate.

The Distributivity Principle says this is equivalent to knowing each atomic fact: for every entity e in the domain, the agent knows whether pred e holds. This bridges question-embedding ("knows who walks") and propositional attitudes ("knows that John walks ∧ knows that Mary walks ∧ ...").

Distributivity Test (§2.4, p. 177) #

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def Phenomena.Questions.Studies.Belnap1982.isPartialAnswer {W : Type u_1} [DecidableEq W] (p : W → Bool) (q : Semantics.Questions.GSQuestion W) (worlds : List W) :
Bool

Partial answer: eliminates some cells but not all.

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  • One or more equations did not get rendered due to their size.
Instances For
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    def Phenomena.Questions.Studies.Belnap1982.failsDistributivityTest {W : Type u_1} (p : W → Bool) (q : Semantics.Questions.GSQuestion W) (w : W) (worlds : List W) :
    Bool

    @cite{belnap-1982}'s Distributivity Test (§2.4, p. 177): a negative criterion for ruling out candidate answers. For any proposition P and indirect question IQ, if the following is consistent:

    Sally knows that P, but Sally doesn't know IQ.

    then P is not an answer to IQ. The test "distributes" the know inside the question and onto its answers: if knowing P doesn't suffice to know IQ, then P doesn't answer IQ.

    Formalisation: P fails the test for Q at w if there exists an epistemic state (set of worlds the agent considers possible) that is a subset of P (the agent knows P) but not a subset of ans(Q, w) (the agent doesn't know Q).

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    Instances For
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      theorem Phenomena.Questions.Studies.Belnap1982.passes_test_implies_answer {W : Type u_1} (p : W → Bool) (q : Semantics.Questions.GSQuestion W) (w : W) (worlds : List W) (hPasses : failsDistributivityTest p q w worlds = false) (v : W) :
      v ∈ worlds → p v = true → QUD.ans q w v = true

      If P passes the Distributivity Test (no witnessing world exists), then knowing P implies knowing Q — i.e., P is at least as informative as Q w.r.t. the partition. Contrapositive of the test.

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      theorem Phenomena.Questions.Studies.Belnap1982.distributivity_test_examples :
      have q := QUD.ofProject id; have worlds := [0, 1, 2]; failsDistributivityTest (fun (w : Fin 3) => decide (w = 0)) q 0 worlds = false ∧ failsDistributivityTest (fun (w : Fin 3) => decide (↑w < 2)) q 0 worlds = true

      Concrete demonstration of the Distributivity Test.

      @cite{belnap-1982} §2.4, p. 177: "Peter knows that the person who kicked Sam is John, but Peter doesn't know who kicked Sam." This is inconsistent — so the person who kicked Sam is John IS an answer to who kicked Sam.

      Vs: "Peter knows that China is populous, but Peter doesn't know which person kicked Sam." This IS consistent — so China is populous is NOT an answer.

      We verify on Fin 3 with identity partition (who kicked Sam → full extension):

      • "the answer is 0" passes the test (knowing it entails knowing who kicked Sam);
      • "w.val < 2" (an irrelevant fact) fails the test.