Additive Particles: too, also, either #
@cite{heim-1992} @cite{kripke-2009} @cite{thomas-2026}
Felicity conditions for additive particles following @cite{thomas-2026} "A probabilistic, question-based approach to additivity".
Related modules #
Semantics.FocusParticles(Theories/Semantics/Focus/Particles.lean): Traditional focus particle semantics (EVEN, only). @cite{thomas-2026} §6 argues that Def. 64 subsumes @cite{heim-1992}'s individual-based presupposition as a special case of the standard focus-alternative use.Phenomena.Focus.AdditiveParticles.Studies.Ahn2015: Ahn's ⊓/⊔ Boolean semantics for too/either. Independent analysis; Thomas does not discuss it.ProbabilisticAnswerhood: Core definitions (Defs. 61–63) used here.
Heim 1992 Subsumption #
@cite{thomas-2026} §6 argues that Def. 64 subsumes @cite{heim-1992}'s
individual-based additive presupposition as a special case. This is
proved formally by heim_subsumption: when ANT and π each entail
distinct alternatives of RQ (the Heim setup), the Antecedent and
Conjunction Conditions (Def. 64a-b) hold automatically.
Insight: Argument-Building Use #
The novel contribution of @cite{thomas-2026} is explaining the "argument-building" use of "too" where the antecedent and prejacent aren't focus alternatives but jointly build an argument for some conclusion.
Example (@cite{thomas-2026}, ex. 1c/14c/65): "A room just opened up at this hotel. It looks kind of fancy, too."
- ANT = "room just opened up", π = "looks fancy" are not focus alternatives
- Both contribute toward conclusion: "This hotel would be a good place to stay"
Definition 64: Felicity Conditions for TOO(π) #
Given resolved question RQ and antecedent ANT:
- Antecedent Condition: ANT probabilistically answers RQ
- Conjunction Condition: ANT ∧ ⟦π⟧ evidences some resolution A more strongly than ANT alone
- Prejacent Conditions:
- ⟦π⟧ doesn't entail the evidenced resolution
- No weaker proposition works as well
Standard vs Argument-Building Uses #
Standard use: ANT and π are focus alternatives
- "John came. Mary came too."
- ANT = "John came", π = "Mary came", RQ = "Who came?"
Argument-building use: ANT and π jointly support a conclusion
- "A room just opened up at this hotel. It looks kind of fancy, too." (@cite{thomas-2026}, ex. 1c/14c/65)
- ANT = "room just opened up", π = "looks fancy"
- RQ = "What would be a good hotel?" (implicit)
- Together they evidence "This hotel would be a good place to stay"
Types of additive particles.
- too : AdditiveParticle
- also : AdditiveParticle
- either : AdditiveParticle
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Polarity requirement for additive particles.
"too" and "also" require positive polarity contexts. "either" requires negative polarity contexts.
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Check if particle is licensed in a given polarity context.
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- p.isLicensed positive = (p.requiresPositive == positive)
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Discourse context for evaluating additive particle felicity.
@cite{thomas-2026} requires:
- A resolved question (RQ) in the discourse
- An antecedent proposition (ANT) that answers RQ
- A prior probability distribution
- resolvedQuestion : Discourse.Issue W
The resolved question (RQ)
- antecedent : W → Bool
The antecedent proposition (ANT) - what was just established
- prior : Questions.ProbabilisticAnswerhood.Prior W
Prior probability distribution over worlds
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Create a context from a polar question.
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- Semantics.Lexical.Particle.Additive.AdditiveContext.fromPolar p ant prior = { resolvedQuestion := Discourse.Issue.polar p, antecedent := ant, prior := prior }
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Create a context from a list of alternatives.
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- Semantics.Lexical.Particle.Additive.AdditiveContext.fromAlternatives alts ant prior = { resolvedQuestion := Discourse.Issue.ofAlternatives alts, antecedent := ant, prior := prior }
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Condition 1: Antecedent probabilistically answers RQ (Def. 64a).
The antecedent must raise the probability of some resolution.
Uses probAnswers (condition (a) of Def. 62 only), which checks that some
alternative's conditional probability exceeds its prior. For polar/binary
QUDs, this is equivalent to the full Def. 62 (probAnswersFull) by
probAnswersFull_eq_simple_binary. For non-binary questions, the full
definition additionally requires Bayes-factor dominance (condition (b)).
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Condition 2: Conjunction answers RQ and evidences more strongly.
@cite{thomas-2026} Def. 64b: ANT ∩ ⟦π⟧ Answers RQ, AND RQ|_{ANT∩⟦π⟧} is Evidenced more strongly by ANT ∩ ⟦π⟧ than by ANT.
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Get the resolution(s) that are strengthened by the conjunction.
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Condition 3a: Prejacent doesn't entail the evidenced resolution.
π shouldn't make the conclusion trivially certain. Uses a provided world list for computability.
TODO: This checks .any over all strengthened alternatives, but should check
against the specific Q|_{ANT∩π} from Def. 62. For singleton resolutions (all
paper examples), the behavior is identical.
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Condition 3b: No weaker proposition works as well (Def. 64c.ii).
For every strengthened resolution R, no strict weakening S ⊃ ⟦π⟧ should evidence R at least as strongly as π does given ANT.
Over finite types this reduces to an entailment check: for every world w with positive prior, ANT(w) ∧ R(w) → π(w). If this holds, then for any S ⊃ ⟦π⟧, since ANT ∩ S ⊇ ANT ∩ π, conditioning on the larger set ANT ∩ S can only dilute the probability of R (subset dilution of conditional probability), so P(R | ANT ∩ π) ≥ P(R | ANT ∩ S), preserving strict evidence advantage.
TODO: Like nonTrivialityConditionWith, this uses all strengthened
alternatives instead of the specific Q|_{ANT∩π}. Correct for singleton
resolutions (all paper examples).
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Full felicity conditions for TOO(π) with explicit world list.
Definition 64 from @cite{thomas-2026}.
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Noncomputable version using Fintype.elems.
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- Semantics.Lexical.Particle.Additive.tooFelicitous ctx prejacent = Semantics.Lexical.Particle.Additive.tooFelicitousWith ctx prejacent Fintype.elems.toList
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Felicity for "also" — same core conditions as "too".
@cite{thomas-2026} §6 (ex. 86) observes that sentence-initial "also" is
NOT subject to the Prejacent Condition part (ii) (maximality):
"Also, Bailey plays the cello" is acceptable where "#Bailey plays the
cello, too" is not. This implementation does not model syntactic position
and applies full felicity conditions regardless. A position-sensitive
version would skip maximalityCondition for sentence-initial "also".
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- Semantics.Lexical.Particle.Additive.alsoFelicitous ctx prejacent = Semantics.Lexical.Particle.Additive.tooFelicitous ctx prejacent
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Felicity for "either" — placeholder, gates on negative polarity.
@cite{thomas-2026} explicitly defers a precise characterization of "either" to future work (footnote 9). This placeholder simply gates on polarity; the actual felicity conditions for "either" likely involve additional constraints (see @cite{ahn-2015} for an alternative account using ⊔ in a Boolean algebra).
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- Semantics.Lexical.Particle.Additive.eitherFelicitous ctx prejacent inNegativeContext = (inNegativeContext && Semantics.Lexical.Particle.Additive.tooFelicitous ctx prejacent)
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Types of additive particle uses (@cite{thomas-2026}).
Thomas's framework is binary: either ANT and π are focus alternatives (standard) or they are not (argument-building).
- standard : AdditiveUseType
- argumentBuilding : AdditiveUseType
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Heuristic: check if ANT and π each overlap some alternative of RQ.
This is a NECESSARY but not SUFFICIENT condition for focus-alternative use. It only checks that both ANT and π are non-trivially consistent with some RQ alternative — this is much weaker than actual focus-alternative structure (which would require ANT and π to be related by substitution of the focused constituent). Every pair of non-trivial propositions passes this check for sufficiently fine-grained questions.
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Noncomputable version using Fintype.
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- Semantics.Lexical.Particle.Additive.isFocusAlternativeUse ant prejacent rq = Semantics.Lexical.Particle.Additive.isFocusAlternativeUseWith ant prejacent rq Fintype.elems.toList
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Check if this is an argument-building use.
In argument-building use:
- ANT and π are NOT focus alternatives
- Together they provide cumulative evidence for some conclusion
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Classify the use type of an additive particle.
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Central Theorems #
These theorems capture the key linguistic insights of @cite{thomas-2026}:
Standard Use Reduction: When π directly determines an alternative of RQ, the probabilistic felicity conditions reduce to the traditional analysis.
Argument-Building Characterization: Argument-building arises exactly when neither ANT nor π directly determines an alternative, but together they provide cumulative evidence.
Cumulative Evidence Necessity: If π provides no additional evidence beyond ANT, "too" is infelicitous.
A proposition "directly determines" an alternative if it entails that alternative.
When p directly determines alt, learning p makes P(alt) = 1.
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Check if a proposition directly determines SOME alternative of an issue.
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Theorem: Standard Use Reduction
When π directly determines an alternative A of the resolved question, and that alternative isn't already certain given ANT, then the conjunction condition is automatically satisfied.
This captures why standard "too" works: if π = "Mary came" and this IS an alternative of "Who came?", then learning π guarantees that alternative, so ANT ∧ π always evidences it more strongly than ANT alone (unless ANT already entailed it).
Linguistically: In standard focus-alternative uses, the general probabilistic conditions REDUCE TO the simple requirement that ANT be true and π be true. The complex probability calculations aren't needed - direct entailment suffices.
Corollary: Standard use satisfies the conjunction condition.
When π directly determines some alternative and that alternative isn't already certain (neither given ANT nor a priori), the conjunction condition holds.
Theorem: Heim 1992 Subsumption (@cite{thomas-2026} §6).
@cite{heim-1992}'s additive presupposition for TOO(π) requires: ∃x ≠ y such that P(x) is established and π = "y P'd".
Under @cite{thomas-2026}'s Def. 64, this falls out as a SPECIAL CASE of the standard focus-alternative use. When ANT and π each entail distinct alternatives of RQ:
- The Antecedent Condition holds because ANT entails an alternative,
raising its probability to 1 (by
probAnswers_when_entailing). - The Conjunction Condition holds because π entails another alternative
that wasn't already certain given ANT, so ANT ∧ π provides new evidence
(by
standard_use_conjunction_condition).
The remaining conditions (non-triviality, maximality) depend on the
question structure and are verified concretely in the pizza/spaghetti
scenario (Thomas2026.pizza_spaghetti_too_felicitous).
This theorem makes precise the claim in @cite{thomas-2026} §6 that Def. 64 subsumes @cite{heim-1992}'s individual-based presupposition: every Heim-felicitous "too" is Thomas-felicitous, but Thomas additionally explains argument-building uses that Heim cannot.
Theorem: Argument-Building Characterization
Argument-building use arises exactly when:
- Neither ANT nor π directly determines any alternative of RQ
- But together, ANT ∧ π evidences some alternative more strongly than ANT alone
This is the DEFINITION of argument-building: ANT and π are not themselves answers to RQ, but jointly serve as EVIDENCE for some answer.
Example (@cite{thomas-2026}, ex. 1c/65): "A room just opened up at this hotel. It looks kind of fancy, too."
- RQ = "What would be a good hotel?" (implicit)
- ANT = "room just opened up" - doesn't determine which hotel is good
- π = "looks fancy" - doesn't determine which hotel is good
- But ANT ∧ π together raise the probability that this hotel is good
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Argument-building requires that the resolved question be about something OTHER than what ANT and π directly assert.
This captures the "implicit QUD" requirement: in argument-building, the question being addressed isn't "Did ANT happen?" or "Did π happen?" but rather some further question that ANT and π provide evidence for.
Two propositions are probabilistically independent given a third if P(A | B ∧ C) = P(A | B).
When π is independent of all alternatives given ANT, π provides no additional evidence - it's irrelevant to the question at hand.
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π is evidentially irrelevant to Q given ANT if π doesn't change the probability of any alternative when we already know ANT.
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- Semantics.Lexical.Particle.Additive.evidentiallyIrrelevant ant prejacent q prior = ∀ alt ∈ q.alternatives, Semantics.Lexical.Particle.Additive.conditionallyIndependent ant prejacent alt prior
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Theorem: Cumulative Evidence Necessity
If π is evidentially irrelevant to RQ given ANT (i.e., π doesn't change the probability of any alternative), then the conjunction condition fails, and "too" is infelicitous.
This explains WHY "too" requires the prejacent to contribute something: if π is just noise that doesn't affect the question at hand, it can't felicitously be marked with "too".
Example of failure:
- "Sue cooks, and she has brown hair, too."
- If hair color is independent of who should host the dinner party, this is infelicitous (or requires a different implicit QUD).
Corollary: If π is irrelevant, "too" is infelicitous.
This follows from cumulative_evidence_necessary plus the fact that conjunctionCondition is required for felicity.
Standard use satisfies felicity when both ANT and π answer RQ.
In standard "too" use where ANT and π are both partial answers to the resolved question, the felicity conditions are satisfied.
Argument-building use is distinct from standard use.
If we have an argument-building use, the antecedent and prejacent are not focus alternatives.
If ANT alone doesn't answer RQ, "too" is infelicitous.
The antecedent must be established as relevant to the resolved question.