[CWS23]: 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)

PMF stack #

Each model is a latent-variable RSA model on mathlib PMF, with the latent type the model's distinctive mechanism (parse / QUD / interpretation). For a fixed config with latent meaning meaning : Latent → Utterance → World → ℝ≥0∞:

L0 lex u : PMF World — RSA.L0OfMeaning (normalise the prior-weighted meaning over worlds), so L0(w | u, lex) ∝ P(w) · ⟦u⟧_lex(w) (eq. 1).
S1 lex w : PMF Utterance — RSA.S1Belief (L0 lex) (fun _ => 1) 2 w (α = 2, no utterance cost): S1(u | w, lex) ∝ L0(w | u, lex)^2.
marginalSpeaker w : PMF Utterance — RSA.marginalizeKernel of S1 against the uniform Latent prior.
L1 u : PMF World — PMF.posterior marginalSpeaker (uniform World) u.

The L1 anti-exhaustivity test is L1(.A) .w_ab > L1(.A) .w_a, which under a uniform world prior reduces (via PMF.posterior_lt_iff_kernel_lt_of_uniform) to marginalSpeaker .w_ab .A > marginalSpeaker .w_a .A — exactly the marginalised speaker comparison T(w) = Σ_l S1(.A | w, l) of the paper (Appendix A.1).

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. For the anti-exhaustivity classification at equal costs (Appendix A.1, eq. A.4) the L1 prior is a constant factor that cancels in the comparison, so we use a uniform world prior at L1 and bake P(w) into meaning for the prior-in-L0 models (baseline, EXH-LU, FREE-LU). For svRSA the prior does NOT enter meaning; under the coarse QUD all A-true worlds collapse into one cell, neutralising the prior's effect on S1 (Appendix A.3), which we encode by unit meaning weights.

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 T(w) = Σ_l S1(A|w,l). With the uniform world prior this reduces to marginalSpeaker .w_ab .A > marginalSpeaker .w_a .A.

Utterance Costs #

The paper includes costs c(A) = 0, c(A∧B) = c(A∧¬B) = c (eq. 2): S1(u|w) ∝ L0(w|u)^α · exp(-λc(u)). With equal costs for A∧B and A∧¬B, the cost terms cancel in the anti-exhaustivity condition (eq. 6b). We set all costs to 0 (costFactor = fun _ => 1); 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?	Listener
1	Baseline RSA	—	Yes	`baselineL1`
2	wRSA	Prior-in-L0 via background	Yes	`wRSAL1`
3	BwRSA	Bayesian wRSA	Yes (= wRSA at L1)	`wRSAL1`
4	svRSA1	QUD blocks at S1	No	`svRSAL1`
5	svRSA2	QUD blocks at S1	No	`svRSAL1`
6	FREE-LU	Free parse choice	Yes	`freeLUL1`
7	EXH-LU	EXH blocks at L0	No	`exhLUL1`
8	RSA-LI1	EXH blocks at L0	No (= EXH-LU at L1)	`rsaLIL1`
9	RSA-LI2	EXH blocks at L0	No (= EXH-LU at L1)	`rsaLIL1`

RSA-LI (Models 8–9) is [FB20b]'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.

wRSA is a joint world–background listener #

wRSA's latent prior is world-dependent: P(w|b)·P(b) ([DTG15]) puts the 3:1 bias on the default_ background and a uniform world prior on wonky (§4.1). Rather than a marginalised per-world speaker, the listener is jointly uncertain about (w, b): its world belief is the World-marginal of the (World × Background) posterior, built with RSA.jointPrior + RSA.jointListener (the world-dependent-latent-prior siblings of marginalizeKernel/PMF.posterior). The anti-exhaustivity test reduces — via RSA.jointListener_lt_iff_sum_score_lt, the marginal normaliser cancelling — to the latent-summed score comparison Σ_b μ(w_ab,b)·S1(.A|w_ab,b) > Σ_b μ(w_a,b)·S1(.A|w_a,b). The per-background speaker reuses the file's proven closed forms: default_ is baselineS1 (biased prior in L0) and wonky is svRSAS1 .coarse (unit meaning). The other four configs have world-independent uniform latent priors and use the marginalizeKernel stack.

Connection to Other Formalizations #

ExhaustivityLimit.lean: proves RSA at α→∞ recovers Fox's exh, using the same IE infrastructure (Exhaustification.Innocent.exhIE) as our exhMeaning.
FrankeBergen2020.lean: formalizes four RSA models (vanilla, LU, LI, GI) for nested quantifiers, using compositional exhaustification.
ScopeExpressivity.lean: demonstrates the scope-blind vs scope-sensitive expressivity gap between standard RSA and compositional EXH.
PottsEtAl2016.lean: the same latent-marginalisation PMF idiom (L0OfMeaning/S1Belief/marginalizeKernel/PMF.posterior) for embedded implicatures under lexical uncertainty.

[CWS23]: Exhaustivity and Anti-Exhaustivity in RSA #

The Symmetry Problem #

Key Result #

Domain #

PMF stack #

Prior Placement #

Anti-Exhaustivity Definition #

Utterance Costs #

Model Classification #

wRSA is a joint world–background listener #

Connection to Other Formalizations #

Domain types #

Literal semantics #

Exhaustification (Innocent Exclusion) #

Parse-dependent meaning #

Prior weight and cost #

Anti-exhaustivity condition (Eq. 6b) #

Shared sum unfolders and α = 2 score reduction #

Baseline RSA (Model 1) #

EXH-LU RSA (Models 7–9) #

svRSA (Models 4–5) — [spector-2017] supervaluationist RSA #

FREE-LU (Model 6) — [BLG16] #

wRSA (Models 2–3) — [DTG15] joint world–background listener #

Predictions #

Classification #