Belnap (1982): Questions and Answers in Montague Grammar

Belnap's answerhood essay: answers are propositions, a question offers a set of possible answers, and two theses organize the field. The Unique Answer Fallacy (§2.2): it is fallacious to build into the foundations — as Tichy's answer-functions and [Kar77b]'s jointly-constituted denotations do — that every question has exactly one complete true answer; Where is a place at which I can get gas on a Sunday? has several, each full, complete, and true. The Distributivity Principle and Test (§2.4): to know an indirect question is to know some answer to it, so if Sally knows that P but Sally doesn't know IQ is consistent, P is not an answer to IQ — a necessary condition only, since P may entail an answer without being one. Formalized on the Hamblin-style reading Belnap prefers ([Ham73]): the possible answers are the maximal alternatives Question.alt, and an information state knows Q at w when it settles an alternative true at w.

Main statements

• unique_answer_fallacy: a mention-some which question with two distinct complete true answers at a single world; joint_answer_not_an_answer: the jointly-constituted [Kar77b] denotation is not itself among the possible answers there.
• distributivity_test: if some information state entails P without knowing Q at w, then P is not a true answer to Q at w (p. 176–177).
• test_only_necessary: the converse fails — the joint answer entails an answer, so knowing it always yields knowledge of Q, yet it is not an answer (§2.4's caveat).

References

• [Bel82]: Questions and Answers in Montague Grammar.
• [Kar77b]: Syntax and semantics of questions.
• [Ham73]: Questions in Montague English.

def Belnap1982.KnowsAnswer {W : Type u_1} (σ : Set W) (Q : Question W) (w : W) : Prop

An information state σ knows question Q at world w when it settles some alternative true at w — the Distributivity Principle (§2.4), "to know IQ is to know some answer to the question as to IQ", as a semantic rendering of know + indirect question. Dropping the truth conjunct gives resolution of the question, σ ∈ Q.

Equations

• Belnap1982.KnowsAnswer σ Q w = ∃ p ∈ Q.alt, w ∈ p ∧ σ ⊆ p

theorem Belnap1982.KnowsAnswer.resolves {W : Type u_1} {σ : Set W} {Q : Question W} {w : W} (h : KnowsAnswer σ Q w) : σ ∈ Q

Knowing an answer resolves the question.

The Unique Answer Fallacy (§2.2)

Tichy makes the question a function delivering one entity per world; Karttunen has it denote "the set of propositions which in that situation jointly constitute a complete and true answer" — on both, uniqueness is built in, and neither can state a question that asks for examples. On the Hamblin/Bennett reading the members of the set are each answers, and mention-some questions have several true ones at once.

def Belnap1982.stationA : Set (Fin 3)

Station A is open in worlds 0 and 1.

Equations

• Belnap1982.stationA = {w : Fin 3 | ↑w ≤ 1}

def Belnap1982.stationB : Set (Fin 3)

Station B is open in worlds 0 and 2.

Equations

• Belnap1982.stationB = {w : Fin 3 | ↑w ≠ 1}

@[instance_reducible]

instance Belnap1982.instDecidablePredFinOfNatNatMemSetStationA : DecidablePred fun (x : Fin 3) => x ∈ stationA

Equations

• Belnap1982.instDecidablePredFinOfNatNatMemSetStationA w = { decide := (↑w).ble 1, reflects_decide := ⋯ }

@[instance_reducible]

instance Belnap1982.instDecidablePredFinOfNatNatMemSetStationB : DecidablePred fun (x : Fin 3) => x ∈ stationB

Equations

• Belnap1982.instDecidablePredFinOfNatNatMemSetStationB w = { decide := !decide (↑w = 1), reflects_decide := ⋯ }

def Belnap1982.gasQuestion : Question (Fin 3)

Where is a place at which I can get gas on a Sunday? (p. 172) as a mention-some which over the two stations.

Equations

• Belnap1982.gasQuestion = Question.which Set.univ fun (b : Bool) => if b = true then Belnap1982.stationA else Belnap1982.stationB

theorem Belnap1982.stationA_mem_alt : stationA ∈ gasQuestion.alt

theorem Belnap1982.stationB_mem_alt : stationB ∈ gasQuestion.alt

theorem Belnap1982.unique_answer_fallacy : ∃ (p : Set (Fin 3)) (q : Set (Fin 3)), p ∈ gasQuestion.alt ∧ q ∈ gasQuestion.alt ∧ p ≠ q ∧ 0 ∈ p ∧ 0 ∈ q

The Unique Answer Fallacy (§2.2): at world 0 both stations are open, and each is a full, complete, true answer — two distinct complete true answers to one question at one world.

theorem Belnap1982.joint_answer_not_an_answer : stationA ∩ stationB ∉ gasQuestion.alt

The [Kar77b] denotation — the true propositions taken jointly — is here the intersection of the two stations' propositions, and it is not itself among the question's possible answers: uniqueness is an artifact of taking the set itself as the answer.

The Distributivity Test (§2.4, pp. 176–177)

theorem Belnap1982.distributivity_test {W : Type u_1} {p : Set W} {Q : Question W} {w : W} (h : ∃ σ ⊆ p, ¬KnowsAnswer σ Q w) : ¬(p ∈ Q.alt ∧ w ∈ p)

The Distributivity Test: "For any P and IQ, let the following inference fail: Sally knows that P, therefore Sally knows IQ. Then P is not an answer to IQ." A failing instance is an information state that entails P without knowing Q — its existence rules P out as a true answer.

theorem Belnap1982.irrelevant_fact_not_answer : ¬(Set.univ ∈ gasQuestion.alt ∧ 0 ∈ Set.univ)

The p. 177 example pattern: an irrelevant truth (China is populous, here the trivial proposition) supports no alternative, so the test rules it out as an answer.

theorem Belnap1982.test_only_necessary : ∃ (p : Set (Fin 3)), (∀ σ ⊆ p, KnowsAnswer σ gasQuestion 0) ∧ p ∉ gasQuestion.alt

The test is necessary, not sufficient: "P might obviously entail some answer to IQ without itself being such an answer" (p. 177). The joint answer is the witness: any state that knows it knows the question, yet it is not an answer.