Thomas (2026): A Probabilistic, Question-Based Approach to Additivity #
This file formalizes [Tho26]'s felicity conditions for additive too, which unify its canonical use, a second answer to a salient question, (2), with the argument-building use in which the antecedent and the prejacent together argue for a conclusion, (1) and (18). The account is stated in an inquisitive question semantics with a listener's probability measure over worlds. A proposition is relevant to a question if it shifts the probability of one of the question's alternatives, (61); it answers the question if it raises the probability of some resolution, a nonempty set of alternatives, by a factor exceeding that of every resolution not entailed by it, (62); and one proposition evidences a resolution more strongly than another if it gives it the higher conditional probability, (63). Too requires an antecedent proposition and a question relevant to the discourse such that the antecedent answers the question, the conjunction of antecedent and prejacent answers it and evidences its resolution more strongly than the antecedent alone, and the prejacent neither entails that resolution nor could be weakened without weakening the evidence, (64).
The definitions are Relevant, IsResolutionOf, Answers, EvidencesMore and
TooFelicitous. Answering needs a positive prior on the evidence, IsResolutionOf.ne_zero,
and trivial evidence answers nothing, not_answers_univ; the resolutions a proposition
evidences are nested, IsResolutionOf.subset_or_subset, and a singleton resolution makes the
proposition relevant to the question, IsResolutionOf.relevant. The paper's arguments for
its examples reduce to three patterns: an antecedent entailing a resolution gives it
probability one, cond_eq_one_of_subset, so a conjunction entailing what the antecedent
leaves uncertain evidences it more strongly, evidencesMore_of_subset, as in (68); a
prejacent that leaves every resolution's conditional probability unchanged fails the
conjunction condition, not_tooFelicitous_of_cond_eq, as in (25) and (72); a prejacent
entailing the evidenced resolution fails the first prejacent condition,
not_tooFelicitous_of_subset, as in (11) and (29); and evidence raising every candidate
resolution by the same factor answers no question, not_answers_of_impact_eq, as in (71).
Implementation notes #
The listener's belief state is a probability measure, and conditioning is mathlib's
ProbabilityTheory.cond, so the paper's statements hold at every prior. The resolution
evidenced by a proposition, which the paper calls unique, is only unique up to nesting: the
dominance clause of (62) waives resolutions containing the candidate, so the felicity
condition quantifies existentially over the resolution that the conjunction evidences and
the prejacent conditions constrain. Bayesian belief revision through a speaker model, (57)–(59),
is not formalized: the antecedent is the proposition the listener learns. The examples are the
rows of Data.Examples.Thomas2026, with the paper's diagnosis of each too recorded as a
feature.
References #
Relevance and answerhood (section 5.1.3) #
Relevance (61a): a question is relevant to a question if some alternative of the first shifts the probability of some alternative of the second.
Equations
- Thomas2026.Relevant μ R S = ∃ A ∈ R.alt, ∃ A' ∈ S.alt, μ[A' | A] ≠ μ A'
Instances For
The impact of a proposition on a proposition: the factor by which learning the first raises the probability of the second.
Equations
- Thomas2026.impact μ R A = μ[A | R] / μ A
Instances For
Answerhood (62): a nonempty set of alternatives is the resolution of a question evidenced by a proposition when the proposition raises the probability of its conjunction and impacts it more than the conjunction of any set of alternatives not entailed by it.
- subset_alt : 𝒜 ⊆ Q.alt
- nonempty : 𝒜.Nonempty
- raises : μ (⋂₀ 𝒜) < μ[⋂₀ 𝒜 | R]
Instances For
A proposition answers a question when it evidences some resolution of it.
Equations
- Thomas2026.Answers μ R Q = ∃ (𝒜 : Set (Set W)), Thomas2026.IsResolutionOf μ Q 𝒜 R
Instances For
(63): a proposition evidences a resolution more strongly than another proposition.
Equations
- Thomas2026.EvidencesMore μ A R R' = (μ[A | R'] < μ[A | R])
Instances For
The felicity conditions of too (section 5.2) #
(64a)–(64c): the antecedent answers the question; the conjunction of antecedent and prejacent answers it and evidences its resolution more strongly than the antecedent; the prejacent does not entail that resolution, and any weaker prejacent would evidence it less strongly.
Equations
- One or more equations did not get rendered due to their size.
Instances For
(64): too is licensed by an antecedent and a question relevant to a question in the discourse tree.
Equations
- Thomas2026.Too μ π ant rq dq = (Thomas2026.TooFelicitous μ π ant rq ∧ Thomas2026.Relevant μ rq dq)
Instances For
Consequences #
Evidence for a resolution has positive prior probability.
Two resolutions evidenced by the same proposition are nested: the dominance clause waives only resolutions containing the candidate.
An answer through a single alternative is relevant to the question (section 5.1.3).
A prejacent that leaves the conditional probability of every resolution unchanged fails the conjunction condition: dogs are mammals, (72), and I had pancakes for breakfast, (25).
A prejacent entailing every resolution the conjunction evidences fails the first prejacent condition: he's ecstatic, (11), and he stole the cookies, (29b).
Evidence raising every candidate resolution by the same factor answers no question whose candidates are not all nested: she invited Bailey and the mention-two question, (71).
The conjunction of a felicitous too answers the question and has positive prior probability.
Trivial evidence raises nothing and so answers no question.
A proposition entailing another gives it probability one.
A conjunction entailing a resolution that the antecedent leaves uncertain evidences it more strongly, the conjunction condition in (68).