A Communication-First Account of Explanation #
@cite{halpern-pearl-2005} @cite{harding-gerstenberg-icard-2025}
Formalization of @cite{harding-gerstenberg-icard-2025}.
Explanation is modeled as an RSA communication game where:
- Worlds are causal situations
(M, u): dynamics + exogenous context - Utterances are explanations: "FACT because X = x"
- Literal meaning = actual causation (X = x causes FACT in
(M, u)) - Decision problem = manipulation game (listener picks variable to intervene on)
- Goodness = Δ expected utility (post-explanation minus baseline)
Key Results #
Classic "explanatory virtues" — sensitivity to background knowledge, preference for invariant relationships, minimality — emerge from pragmatic dynamics rather than needing to be stipulated (contra @cite{halpern-pearl-2005}).
Scenarios #
| Example | Structure | Key prediction |
|---|---|---|
| Late Meeting | M_T vs M_∧ | Citing known cause T=1 is informative (signals M_T) |
| Roof Replacement | M_R/M_D/M_∧/M_∨ | Citing R=1 more useful than D=1 for roof decision |
A causal world: structural equations + exogenous context. Corresponds to a (M, u) pair in the paper.
- dynamics : Core.StructuralEquationModel.CausalDynamics
- context : Core.StructuralEquationModel.Situation
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Literal meaning (Eq. 1): an explanation is true in a causal world iff (1) FACT holds, (2) the cause is present, and (3) the cause manipulates the effect.
We use manipulates (Woodward's interventionist criterion) as our
notion of actual causation, following the paper's egalitarian stance
(§2.3): any account that treats both overdetermining factors as
actual causes is compatible.
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Manipulation reward (Definition 3, binary case).
R(X, M, u) = 1 if intervening on X can flip FACT, 0 otherwise.
The full definition sums over exogenous contexts weighted by P(u'),
but in the binary single-context case it reduces to manipulates.
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- HardingGerstenbergIcard2025.manipulationReward w action effect = if Core.StructuralEquationModel.manipulates w.dynamics w.context action effect = true then 1 else 0
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Late Meeting #
Bob is late to meet Charlie, who is cross. Bob knows T = 1 (he was late) and B = 1 (he forgot Charlie's birthday), but is unsure whether the causal structure is M_T (tardiness alone suffices) or M_∧ (both tardiness AND birthday forgetting are needed).
Alice's choice: cite T = 1 or B = 1 as the cause of C = 1.
Key prediction (§4.2.1): citing the known cause T = 1 is informative because it signals that the speaker chose NOT to cite B = 1, allowing the listener to infer M_T.
In M_T, tardiness manipulates crossness (T → C).
In M_T, birthday does NOT manipulate crossness (no B → C law).
In M_∧, tardiness manipulates crossness (removing T from T∧B→C kills C).
In M_∧, birthday also manipulates crossness (removing B kills C).
⟦"C=1 because T=1"⟧ = {M_T, M_∧}: both worlds survive. L0 assigns equal probability to each (uniform prior).
⟦"C=1 because B=1"⟧ = {M_∧}: only the conjunctive world survives. L0 assigns all mass to M_∧.
In M_T: T manipulates C (direct law T → C).
In M_T: B does NOT manipulate C (no causal path).
In M_∧: T manipulates C (conjunctive law, B present).
In M_∧: B also manipulates C (conjunctive law, T present).
After "T=1", L0 is uniform → listener is indifferent on the apologize-for-both vs apologize-for-tardiness-only decision. After "B=1", L0 concentrates on M_∧ → listener picks a_both.
So U_S("B=1", M_∧) > U_S("T=1", M_∧): the speaker prefers to cite B=1 when the actual world is M_∧.
Known causes can be informative (§4.2.1).
Even though Bob knows T = 1, citing it is informative because the pragmatic listener L reasons: "the speaker said T=1 rather than B=1 → she must be in a world where B=1 is false or less useful → probably M_T."
Roof Replacement #
A house catches fire (F = 1). Two potential causes: thatched roof (R = 1) and recent drought (D = 1). Bob considers four possible causal structures:
- M_R: R → F (roof alone causes fire)
- M_D: D → F (drought alone)
- M_∧: R ∧ D → F (both needed)
- M_∨: R ∨ D → F (either suffices)
Bob's decision: should he replace his thatched roof?
Key prediction (§4.1.1): citing R = 1 is more useful than D = 1 because Bob's decision is sensitive to whether R is a cause of F, even though both explanations are equally informative at the L0 level (each eliminates exactly one of four structures).
⟦"F=1 because R=1"⟧ = {M_R, M_∧}. M_∨ is excluded because with both R=1 and D=1, neither individually manipulates F (overdetermination — removing one leaves the other).
⟦"F=1 because D=1"⟧ = {M_D, M_∧} (symmetric).
Both explanations survive in exactly the same number of worlds (2 of 4) → equally informative at the L0 level.
R manipulates F in M_R and M_∧ (thatched roof matters).
R does NOT manipulate F in M_D or M_∨ (overdetermination in M_∨).
After "R=1", surviving worlds = {M_R, M_∧}: R manipulates F in BOTH. So the listener knows with certainty: replacing the roof matters.
After "D=1", surviving worlds = {M_D, M_∧}: R manipulates F in M_∧ but NOT in M_D — less helpful for the roof decision.
In M_∧ (conjunctive): both R and D manipulate F.
In M_∨ (disjunctive): neither R nor D individually manipulates F because the other cause is present as backup.
Causal selection interacts with structure (§4.3.1).
In conjunctive structures, both causes are individually necessary → both manipulate F. In disjunctive structures, each cause has a backup → neither individually manipulates (overdetermination).
The paper predicts speakers cite abnormal causes in conjunctive structures (where both are actual causes) and normal causes in disjunctive ones (where the manipulation game with varied contexts favors the more probable cause).
Milk Theft #
Bob's milk is depleted (M = 1). Possible culprits: Charlie (C = 1), Dana (D = 1), or both. The causal structure is disjunctive: M_∨ with C ∨ D → M.
Key prediction (§4.4): when both roommates drank the milk, citing both "C = 1, D = 1" is more useful (Bob wants to confront the right people), but longer messages cost more. The tension between utility and cost generates a preference for minimality.
When only Charlie drank, Charlie is the sole cause.
When both drank, neither individually manipulates M (overdetermination): removing one doesn't stop M = 1 because the other is still present.
Neither explanation is literally true when both drank (overdetermination blocks actual causation for individual factors).
This is EX3 from Halpern & Pearl: the HP analysis would require citing both, but our account derives this preference from pragmatic pressure (longer messages are more informative but costlier).
When Charlie is the sole cause, he is both sufficient and necessary. Necessity is tested against {Da=false} (Dana didn't drink) rather than milkBgCharlie (which includes Ch=true): @cite{nadathur-2024} Definition 10b requires the cause not already entailed by s.