@cite{grusdt-lassiter-franke-2022} #
"Probabilistic modeling of rational communication with conditionals" PLoS ONE 17(7): e0269937.
Overview #
This paper extends RSA to model conditionals by:
- Treating "worlds" as probability distributions (WorldState)
- Using assertability (P(C|A) ≥ θ) as the literal meaning of conditionals
- Having L1 infer both the world state AND the causal structure
The key insight is that the literal meaning of "if A then C" is an assertability condition — P(C|A) ≥ θ — rather than material implication. This grounds RSA's meaning function in conditional probability.
Toy Example (§2.3, Table 2) #
The paper's illustrative example uses 3 states and 4 utterances with θ = 0.9:
| P(A) | P(C) | P(A,C) | P(C|A) | likely C | if A, C | C | A and C | |
|---|---|---|---|---|---|---|---|---|
| s1 | 0.9 | 0.9 | 0.81 | 0.9 | ✓ | ✓ | ✓ | — |
| s2 | 0.65 | 0.65 | 0.6 | 12/13 | ✓ | ✓ | — | — |
| s3 | 0.6 | 0.6 | 0.36 | 0.6 | ✓ | — | — | — |
Key predictions:
- S1 in s1 prefers "C" (most informative true utterance)
- S1 in s2 prefers "if A then C" (only true non-trivial utterance)
- S1 in s3 prefers "likely C" (only true utterance)
- L1 hearing "if A then C" infers s2 > s1 (dependency inference)
- L1 hearing "C" identifies s1 (high marginal probability)
Full Model (§3) #
The full model uses 10,000 sampled world states and three causal relations
(A→C, C→A, A⊥C) as latent variables. We formalize only the toy example
here, which captures all qualitative predictions with finite types amenable
to rsa_predict.
Grounding #
The assertability truth values are grounded in the probabilistic assertability
condition from Semantics.Conditionals.Assertability: a conditional
"if A then C" is assertable when P(A) > 0 and P(C|A) ≥ θ.
Experiments #
Experiment 1: Dependency Inference — Participants hear "if A then C" and judge causal structure. 72% infer A→C, 15% C→A, 10% A⊥C.
Experiment 2: Conditional Perfection — 85% endorse "if ¬A then ¬C" in A→C contexts vs 45% in independent contexts.
Experiment 3: Assertability Thresholds — θ ≈ 0.88 from model fitting.
Utterances from the toy example (Table 2, §2.3).
The paper uses a richer utterance space in the full model, but the toy example uses four utterances ordered by informativity:
- "likely C": assertable when P(C) ≥ 0.5 (weakest, true in all toy states)
- "if A then C": assertable when P(C|A) ≥ 0.9
- "C": assertable when P(C) ≥ 0.9
- "A and C": assertable when P(A∧C) ≥ 0.9
"A and C" is false in all three toy states, making it a vacuous alternative that nonetheless affects the pragmatic competition.
Instances For
Equations
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- One or more equations did not get rendered due to their size.
World states from the toy example (Table 2, §2.3).
Each state is a probability distribution (pA, pC, pAC) representing a different degree of dependence between A and C:
- s1: High marginals, P(C|A) = 0.9 = θ (borderline assertable)
- s2: Moderate marginals, P(C|A) = 12/13 ≈ 0.923 > θ (clearly assertable)
- s3: Moderate marginals, P(C|A) = 0.6 < θ (not assertable)
Instances For
Equations
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- One or more equations did not get rendered due to their size.
WorldState for s1: P(A)=0.9, P(C)=0.9, P(A∧C)=0.81. P(C|A) = 0.81/0.9 = 0.9 = θ.
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.ws1 = { pA := 9 / 10, pC := 9 / 10, pAC := 81 / 100 }
Instances For
WorldState for s2: P(A)=0.65, P(C)=0.65, P(A∧C)=0.6. P(C|A) = 0.6/0.65 = 12/13 ≈ 0.923 > θ.
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.ws2 = { pA := 13 / 20, pC := 13 / 20, pAC := 3 / 5 }
Instances For
WorldState for s3: P(A)=0.6, P(C)=0.6, P(A∧C)=0.36. P(C|A) = 0.36/0.6 = 0.6 < θ.
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.ws3 = { pA := 3 / 5, pC := 3 / 5, pAC := 9 / 25 }
Instances For
Map states to their WorldState representations.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Assertability threshold θ = 0.9 from the paper.
Equations
Instances For
Assertability truth table for the toy example (Table 2).
Defines when each utterance is assertable in each state. The paper uses
P(C|A) ≥ θ (non-strict), while Assertability.assertable uses > θ.
For the toy example, s1 has P(C|A) = 0.9 = θ exactly, so assertability
under ≥ is true but under > is false. We define the truth table directly
to match the paper's non-strict threshold.
Truth values from Table 2:
- likely C (P(C) ≥ 0.5): s1 ✓, s2 ✓, s3 ✓ (all states have P(C) ≥ 0.6 > 0.5)
- if A then C (P(C|A) ≥ 0.9): s1 ✓, s2 ✓, s3 ✗
- C (P(C) ≥ 0.9): s1 ✓, s2 ✗, s3 ✗
- A and C (P(A∧C) ≥ 0.9): s1 ✗, s2 ✗, s3 ✗
Equations
- One or more equations did not get rendered due to their size.
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.assertable' Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.Utt.likelyC x✝ = true
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.assertable' Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.Utt.conjAC x✝ = false
Instances For
Verify that the assertability truth values are grounded in the actual conditional/marginal probabilities of the WorldStates. These connect the directly-defined truth table to the assertability theory.
s1 has P(C|A) = 9/10 ≥ θ.
s2 has P(C|A) = 12/13 > θ.
s3 has P(C|A) = 3/5 < θ.
s1 has P(C) = 9/10 ≥ 0.9 (so "C" is assertable).
s2 has P(C) = 13/20 ≥ 0.5 (so "likely C" is assertable).
s3 has P(C) = 3/5 ≥ 0.5 (so "likely C" is assertable).
s1 has P(A∧C) = 81/100 < 0.9 (so "A and C" is not assertable).
The forward conditional is assertable (using strict >) only for s2,
not s1, because Assertability.assertable uses strict >. This
demonstrates the ≥ vs > mismatch that motivates the direct truth table.
@cite{grusdt-lassiter-franke-2022} toy example as RSAConfig.
Meaning: Boolean assertability (1 if assertable, 0 if not). World prior: uniform over the 3 states. S1 score: belief-based (rpow): score = L0(w|u)^α. α = 1 (rationality parameter from the paper's toy example).
Equations
- One or more equations did not get rendered due to their size.
Instances For
S1 predictions from the toy example (Table 2) #
The pragmatic speaker in each state prefers the most informative true utterance. Informativity is measured by L0's posterior concentration: utterances that are true in fewer states are more informative.
- s1: "C" > "conditional" > "likely C" (C is true in 1 state, conditional in 2, likely C in 3)
- s2: "conditional" > "likely C" (conditional in 2 states, likely C in 3; C and conjAC are false)
- s3: "likely C" dominates (only true utterance; conditional, C, and conjAC are all false)
The ordering follows informativity: utterances true in fewer states give L0 a sharper posterior, yielding higher S1 scores.
s1: S1 prefers "C" over "if A then C."
In s1, both "C" and "conditional" are true, but "C" is true only in s1 while "conditional" is true in both s1 and s2. So "C" is more informative. S1(C|s1) = 6/11, S1(conditional|s1) = 3/11.
s1: S1 prefers "if A then C" over "likely C."
"conditional" is true in 2 states vs "likely C" in all 3. S1(conditional|s1) = 3/11, S1(likelyC|s1) = 2/11.
s2: S1 prefers "if A then C" over "likely C."
In s2, "conditional" is true in 2 states while "likely C" is true in all 3. S1(conditional|s2) = 3/5, S1(likelyC|s2) = 2/5.
s3: "likely C" dominates — no other utterance beats it.
"likely C" is the only true utterance in s3. The conditional, C, and conjAC are all false, so they get zero S1 score.
L1 predictions: the core dependency inference result #
The central prediction: hearing "if A then C" makes the pragmatic listener infer s2 (moderate dependence) over s1 (high marginals). This is because S1 in s1 would have used the more informative "C" instead of the conditional.
This is the key mechanism behind dependency inference: conditionals signal that the speaker could not have used a stronger utterance, implicating a state where only the conditional is assertable.
L1 hearing "if A then C": prefers s2 over s1.
The core dependency inference result. S1 in s1 would prefer "C" over
"conditional" (by s1_C_gt_conditional), so hearing "conditional"
makes L1 shift probability toward s2 where "conditional" is the
best available utterance.
L1 hearing "if A then C": prefers s2 over s3.
s3 makes the conditional literally false, so it gets zero L1 weight.
L1 hearing "likely C": prefers s3 over s1.
"likely C" is true in all states, but S1 in s1 prefers "C" and S1 in s2 prefers "conditional," so hearing "likely C" implicates that stronger utterances were unavailable — i.e., the state is s3 where "likely C" is the only option. L1(s3|likelyC) = 55/87 > L1(s1|likelyC) = 10/87.
The conditional's meaning in the RSA model equals the assertability condition: assertable iff P(C|A) ≥ θ. Since we use the direct truth table, we verify consistency with the WorldState probabilities.
"likely C" is assertable in all states (P(C) ≥ 0.5 everywhere).
"A and C" is never assertable (P(A∧C) < 0.9 in all states).
Connection to causal inference #
The toy example does not include causal relations as a latent variable (the full model in §3 does). However, the key qualitative prediction — that conditionals signal dependency between A and C — is captured by the L1 inference: hearing "if A then C" makes the listener prefer s2 (where A and C are strongly correlated: P(C|A) = 12/13 ≈ 0.923) over s1 (where A and C have high marginals but weaker per-unit correlation).
The full model adds CausalRelation (A→C, C→A, A⊥C) as a latent variable, but the dependency inference result already emerges from the simpler model via scalar implicature.
Causal asymmetry detection from assertability patterns.
If the forward conditional "if A then C" is assertable but the reverse
"if C then A" is not, inferCausalRelation returns .ACausesC.
Conditional perfection is NOT a semantic entailment.
There exist world states where "if A then C" is assertable but the converse "if ¬A then ¬C" need not be. This supports the paper's claim that conditional perfection is a pragmatic implicature, not entailment.
Experiment 1 result: causal structure inference from conditionals.
Participants (N≈150) hear "if A then C" and judge causal structure. The asymmetry between forward and reverse conditionals is the key finding.
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.conditionalResult = { utterance := "if A then C", pACausesC := 72 / 100, pCCausesA := 18 / 100, pIndependent := 10 / 100 }
Instances For
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.reverseConditionalResult = { utterance := "if C then A", pACausesC := 15 / 100, pCCausesA := 75 / 100, pIndependent := 10 / 100 }
Instances For
Fitted assertability threshold from Experiment 3.
Equations
Instances For
Experiment 2 perfection rates by causal context.
- causalContext : Core.CausalBayesNet.CausalRelation
- perfectionRate : ℚ
Instances For
Equations
- One or more equations did not get rendered due to their size.
Instances For
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.perfectionInCausal = { causalContext := Core.CausalBayesNet.CausalRelation.ACausesC, perfectionRate := 85 / 100 }
Instances For
Equations
- Phenomena.Conditionals.Studies.GrusdtLassiterFranke2022.perfectionInIndependent = { causalContext := Core.CausalBayesNet.CausalRelation.Independent, perfectionRate := 45 / 100 }
Instances For
Conditional perfection is modulated by causal context: much higher in A→C contexts than in independent contexts.
Key claims supported by the model #
Conditionals communicate dependency: L1 hearing "if A then C" infers a state with high P(C|A) relative to P(C) — i.e., a state where A and C are dependent. This is
l1_conditional_prefers_s2.Conditional perfection is pragmatic: The semantic meaning of conditionals (assertability) does NOT entail perfection (
perfection_not_semantic). Perfection arises via scalar implicature in the full model.Speaker informativity drives inference: S1 in s1 prefers "C" over "conditional" (
s1_C_gt_conditional), so hearing "conditional" implicates that "C" was unavailable (i.e., the state is s2, not s1).Weak states produce weak utterances: In s3, the speaker can only use "likely C" (
s3_likelyC_dominates). Hearing "likely C" makes L1 infer s3 (l1_likelyC_prefers_s3), the state with weakest dependence.