@cite{champollion-alsop-grosu-2019} — Free Choice Disjunction as RSA #
@cite{champollion-alsop-grosu-2019} @cite{bergen-levy-goodman-2016} @cite{fox-2007} @cite{franke-2011}"Free choice disjunction as a rational speech act" Proceedings of SALT 29: 238-257.
The Model #
Domain: "You may take an apple or a pear" with 2 items {A, B}. 5 states based on permission structure. 4 utterances. 2 interpretation functions (I₁ literal vs I₂ exhaustified), following @cite{bergen-levy-goodman-2016}.
- L0: L0(w|u,I) ∝ I(u,w) (meaning under interpretation I)
- S1: S1(u|w,I) ∝ L0(w|u,I)^α (rpow belief-based)
- L1: L1(w|u) ∝ P(w) · Σ_I S1(u|w,I)
Parameters: α = 2 (paper uses α = 100; qualitative predictions hold at α = 2, where "L1 assigns only 70% probability to the FCI states" — p. 249).
Key Innovation #
Standard RSA cannot derive free choice because disjunction is always less informative than its disjuncts. Adding semantic uncertainty — speakers and listeners reason about which interpretation function is being used — creates an avoidance pattern that drives the inference.
The two interpretation functions represent optional exhaustification:
- I₁: Literal meanings (unexhaustified)
- I₂: Strengthened meanings (exhaustified)
How Free Choice Emerges #
- S1 wants to convey "Only One" (each item individually permitted)
- If S1 says "You may A", L0 might interpret via I₂ as "Only A"
- To avoid this misunderstanding, S1 uses disjunction
- L1 reasons: "S1 chose Or to prevent me thinking Only A or Only B"
- L1 infers: Only One or Any Number → Free choice
Qualitative Findings #
| # | Finding | Theorem |
|---|---|---|
| 1 | FCI derived (uniform prior) | fci_derived |
| 2 | FCI robust to biased prior | fci_robust_to_prior |
| 3 | EI holds under uniform prior | ei_uniform |
| 4 | EI weakened under biased prior | ei_prior_sensitive |
State Space (Table 2) #
| State | ◇A | ◇B | ◇(A∧B) | FCI? | EI? |
|---|---|---|---|---|---|
| Only A | T | F | F | no | yes |
| Only B | F | T | F | no | yes |
| Only One | T | T | F | yes | yes |
| Any Number | T | T | T | yes | no |
| Only Both | T | T | T | no | no |
Note: "Only Both" means you may ONLY take both together (not either alone).
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- RSA.FreeChoice.instDecidableEqFCState x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- RSA.FreeChoice.instReprFCState = { reprPrec := RSA.FreeChoice.instReprFCState.repr }
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- RSA.FreeChoice.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- RSA.FreeChoice.instReprUtterance = { reprPrec := RSA.FreeChoice.instReprUtterance.repr }
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Two interpretation functions representing optional exhaustification.
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- RSA.FreeChoice.instDecidableEqInterp x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- RSA.FreeChoice.instReprInterp = { reprPrec := RSA.FreeChoice.instReprInterp.repr }
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- RSA.FreeChoice.instFintypeInterp = { elems := {RSA.FreeChoice.Interp.literal, RSA.FreeChoice.Interp.exhaustified}, complete := RSA.FreeChoice.instFintypeInterp._proof_1 }
Free choice inference: each item is individually permitted. ◇(A∧¬B) ∧ ◇(B∧¬A). True at {onlyOne, anyNumber}.
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Exclusivity inference: taking both is not permitted. ¬◇(A∧B). True at {onlyA, onlyB, onlyOne}.
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Interpretation function I₁ (literal/unexhaustified) from (6).
- ⟦A⟧^I₁ = {Only A, Only One, Any Number, Only Both}
- ⟦B⟧^I₁ = {Only B, Only One, Any Number, Only Both}
- ⟦Or⟧^I₁ = all 5 states
- ⟦And⟧^I₁ = {Any Number, Only Both}
Equations
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.a RSA.FreeChoice.FCState.onlyB = false
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.a x✝ = true
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.b RSA.FreeChoice.FCState.onlyA = false
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.b x✝ = true
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.or_ x✝ = true
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.and_ RSA.FreeChoice.FCState.anyNumber = true
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.and_ RSA.FreeChoice.FCState.onlyBoth = true
- RSA.FreeChoice.I1 RSA.FreeChoice.Utterance.and_ x✝ = false
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Interpretation function I₂ (exhaustified) from (7). Strengthened via innocent exclusion:
- ⟦A⟧^I₂ = {Only A}
- ⟦B⟧^I₂ = {Only B}
- ⟦Or⟧^I₂ = {Only A, Only B, Only One, Any Number}
- ⟦And⟧^I₂ = {Only Both}
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- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.a RSA.FreeChoice.FCState.onlyA = true
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.a x✝ = false
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.b RSA.FreeChoice.FCState.onlyB = true
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.b x✝ = false
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.or_ RSA.FreeChoice.FCState.onlyBoth = false
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.or_ x✝ = true
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.and_ RSA.FreeChoice.FCState.onlyBoth = true
- RSA.FreeChoice.I2 RSA.FreeChoice.Utterance.and_ x✝ = false
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Combined meaning function indexed by interpretation.
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I₁(Or) is true everywhere (maximally uninformative).
I₂(Or) excludes exactly onlyBoth.
I₂(A) singles out exactly onlyA.
@cite{champollion-alsop-grosu-2019} RSA model with semantic uncertainty. Two interpretation functions serve as latent variables. S1 score is rpow(L0, α) — standard belief-based RSA.
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Uniform prior: all states equally likely.
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Biased prior: P(anyNumber) = 3, others = 1. Models a context where taking any combination is a priori more likely, testing prior sensitivity of FCI vs EI.
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FCI is derived: L1 assigns more mass to FCI states (Only One + Any Number) than non-FCI states upon hearing "Or". This is the central result of the paper.
FCI is robust to prior manipulation: holds even when anyNumber is a priori 3× more likely.
EI holds under uniform prior.
EI is prior-sensitive: a prior biased toward anyNumber defeats EI.
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- RSA.FreeChoice.instDecidableEqFinding x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- RSA.FreeChoice.instReprFinding = { reprPrec := RSA.FreeChoice.instReprFinding.repr }
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Map each finding to its RSA formalization.
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The RSA model accounts for all 4 findings from @cite{champollion-alsop-grosu-2019}.
The model derives FCI even without the conjunction alternative. This requires either removing the Only Both state or adding a null utterance. We define the null utterance version (Table 8).
Utterances with null option (no conjunction).
- a : UtteranceWithNull
- b : UtteranceWithNull
- or_ : UtteranceWithNull
- null : UtteranceWithNull
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- RSA.FreeChoice.instDecidableEqUtteranceWithNull x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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I₁ with null (true everywhere).
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- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.a RSA.FreeChoice.FCState.onlyB = false
- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.a x✝ = true
- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.b RSA.FreeChoice.FCState.onlyA = false
- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.b x✝ = true
- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.or_ x✝ = true
- RSA.FreeChoice.I1_null RSA.FreeChoice.UtteranceWithNull.null x✝ = true
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I₂ with null.
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- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.a RSA.FreeChoice.FCState.onlyA = true
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.a x✝ = false
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.b RSA.FreeChoice.FCState.onlyB = true
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.b x✝ = false
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.or_ RSA.FreeChoice.FCState.onlyBoth = false
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.or_ x✝ = true
- RSA.FreeChoice.I2_null RSA.FreeChoice.UtteranceWithNull.null x✝ = true
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Combined meaning for the null-utterance model.
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RSAConfig for the model without conjunction (Table 8). Replaces "and" with a null utterance (true everywhere under both interpretations).
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FCI is derived even without the conjunction alternative (Tables 7-8). The avoidance mechanism between A/B and Or is sufficient — the conjunction alternative is not essential.
Bridge content (merged from RSA_ChampollionAlsopGrosu2019Bridge.lean) #
Bridge: RSA Free Choice Disjunction → Phenomena Data #
@cite{champollion-alsop-grosu-2019}
Connects the RSA free choice model from @cite{champollion-alsop-grosu-2019}
to empirical data in Phenomena.FreeChoice.
Bridge Theorems #
predicts_free_choice: L1 free choice prediction matches datafc_not_semantic: Free choice is pragmatic, not semantic
Connection to Empirical Data #
The model predicts the patterns in Phenomena.FreeChoice:
Free Choice Permission (
coffeeOrTea):- "You may have coffee or tea" → "You may have coffee AND you may have tea"
- Derived: L1 assigns ~100% to FCI states
Exclusivity Cancelability:
- EI ("not both") is sensitive to world knowledge
- FCI is robust across priors
Ross's Paradox (
postOrBurn):- "Post the letter" semantically entails "Post or burn"
- But pragmatically, adding "or burn" triggers free choice
- The asymmetry comes from the alternative structure