@cite{cremers-wilcox-spector-2023}: Exhaustivity and Anti-Exhaustivity in RSA #

"Exhaustivity and Anti-Exhaustivity in the RSA Framework: Testing the Effect of Prior Beliefs." Cognitive Science 47(5), e13286.

The Symmetry Problem #

In baseline RSA, hearing "A" can increase belief in w_ab (where both A and B are true) when priors are biased toward w_ab. This "anti-exhaustive" behavior contradicts human interpretation — humans always exhaust "A" to mean "A and not B."

Key Result #

Models with grammatical exhaustification (EXH-LU, RSA-LI) block anti-exhaustivity regardless of prior bias. The EXH parse makes A false in w_ab, so S1 never uses A to convey w_ab under that parse. The prior bias is overridden by the structural asymmetry in meaning.

The paper's experimental data (comprehension task) found no evidence of anti-exhaustivity: listeners consistently infer "A and not B" from "A", regardless of prior bias. This supports grammatical models (EXH-LU, RSA-LI, svRSA) over baseline RSA and wRSA.

Domain #

Worlds: w_a (only A true), w_ab (A and B both true)
Utterances: A (cost 0), A∧B (cost c), A∧¬B (cost c)
Parses (EXH-LU): literal (A true in both worlds) vs exh (A true only in w_a)

Prior Placement #

The paper's L0 includes the prior (eq. 1): L0(w|u) ∝ P(w) · ⟦u⟧(w). The paper also has P(w) at L1 (eq. 3): L1(w|u) ∝ P(w) · S1(u|w). So P(w) enters twice — in L0 and L1.

For models where prior biases L0 (baseline, EXH-LU, wRSA, FREE-LU), we bake P(w) into meaning and set worldPrior = 1. This is equivalent to double-counting for anti-exhaustivity classification (Appendix A.1, eq. A.4): L1(w_ab|A) > P(w_ab) ⟺ T(w_ab) > T(w_a) where T(w) = Σ_l P(l)·S1(A|w,l), regardless of worldPrior. With worldPrior = 1, L1_score(w) = T(w), so the comparison L1(.A, .w_ab) > L1(.A, .w_a) IS the anti-exhaustivity test.

For svRSA, the prior does NOT enter meaning. Under Q_A (coarse QUD), QUD projection aggregates all A-true worlds into one cell, making L0(Q_A(w)|A) = 1 for all w. This neutralizes the prior's effect on S1, giving world-independent S1 under Q_A (Appendix A.3). Our encoding captures this by using uniform weights in meaning under both QUDs: L0(w|A, coarse) = 1/2 for both worlds.

Anti-Exhaustivity Definition #

The paper's definition (Eq. 6b, equal costs): L1(w_ab|A) > P(w_ab), equivalently T(w_ab) > T(w_a) (Appendix A.1, eq. A.4). With our encoding (worldPrior = 1), this reduces to L1(.A, .w_ab) > L1(.A, .w_a).

Utterance Costs #

The paper includes costs c(A) = 0, c(A∧B) = c(A∧¬B) = c (eq. 2): S1(u|w) ∝ exp(λ(log L0(w|u) - c(u))) = L0(w|u)^λ · exp(-λc(u)). With equal costs for A∧B and A∧¬B, the cost terms cancel in the anti-exhaustivity condition (eq. 6b): the classification depends only on the log-prior ratio. We set all costs to 0 in our configs; the analytic condition in antiExhaustivityCondition handles the general case.

Model Classification #

The paper's 9 models fall into two groups by anti-exhaustivity behavior:

#	Model	Mechanism	Anti-exhaustive?	Config
1	Baseline RSA	—	Yes	`baselineBiased`
2	wRSA	Prior-in-L0 via background	Yes	`wRSABiased`
3	BwRSA	Bayesian wRSA	Yes (= wRSA at L1)	`wRSABiased`
4	svRSA1	QUD blocks at S1	No	`svRSABiased`
5	svRSA2	QUD blocks at S1	No	`svRSABiased`
6	FREE-LU	Free parse choice	Yes	`freeLUBiased`
7	EXH-LU	EXH blocks at L0	No	`exhLUBiased`
8	RSA-LI1	EXH blocks at L0	No (= EXH-LU at L1)	`rsaLIBiased`
9	RSA-LI2	EXH blocks at L0	No (= EXH-LU at L1)	`rsaLIBiased`

All 9 models are encoded as RSAConfig instances. Each model's distinctive mechanism maps to a different Latent type and meaning. Costs enter S1 via utteranceCost (= 0, so exp(-λ·0) = 1 is implicit); the analytic condition antiExhaustivityCondition handles general costs.

RSA-LI (Models 8–9) is @cite{franke-bergen-2020}'s Lexical Intentions model; at L1 with uniform P(i) and equal costs it equals EXH-LU (Table 1). wRSA and BwRSA are identical at L1 (BwRSA adds a Bayesian S2 layer). svRSA uses no prior in L0 meaning — QUD projection neutralizes it (Appendix A.3), giving categorical blocking.

Connection to Other Formalizations #

CompareExhaustivity.lean: proves RSA at α→∞ recovers Fox's exh, using the same IE infrastructure (applyIEBool) as our exhMeaning.
FrankeBergen2020.lean: formalizes four RSA models (vanilla, LU, LI, GI) for nested quantifiers, using compositional exhaustification.
CompareRSAExh.lean: demonstrates the scope-blind vs scope-sensitive expressivity gap between standard RSA and compositional EXH.