Cao, White and Lassiter (2025): Cause, make, and force as graded causatives #
Cao, White and Lassiter treat English cause, make and force as graded causatives. Where
Nadathur and Lauer give make a categorical truth condition, causal sufficiency, they measure
three quantities in a structural causal model of tic-tac-toe: Pearl's probability of sufficiency
(suf), a simplified version of Halpern and Kleiman-Weiner's degree of intention, and the number
of alternative actions open to the causee. No one quantity determines which verb speakers accept;
each verb has its own set of reliable interactions. As in the model of Cao, Geiger, Kreiss, Icard
and Gerstenberg, the models are time-indexed (TimeIndex) and their agents play a
soft-optimality policy.
The paper's in-text judgments, its examples (3)–(11), are rows in
Data/Examples/CaoWhiteLassiter2025.json; the regression estimates stay in prose.
Main statements #
intentionDegree_eq_one_of_altCount_eq_zero: an action with no alternative comes out maximally intentional, so the simplified INT drops Frankfurt's alternative-possibilities condition and ALT carries it insteadsuf_eq_one_of_make: where Nadathur and Lauer's make holds, SUF is 1 under every distribution over contextsjudgment_differs_make_force: the paper's (8) separates make from forceProbabilisticExample.suf_eq: with an uncertain background, SUF is the background's probability rather than 0 or 1
References #
Soft-optimality policy #
The paper's mechanism for agent moves (§2.1.1). The highest-utility move — minimax over the game
tree, with terminal utility Winner × (EmptySpace + 1) — is taken with probability ρ + (1−ρ)/n
and every other available move with (1−ρ)/n, for n the number of empty spaces. The skill
parameter interpolates between a uniformly random player ("assume that the players are infants"),
under which the paper's worked SUF contrast between two board states collapses, and a
deterministic professional.
A player of skill ρ plays the highest-utility move best with probability ρ and otherwise a
uniform random move.
Equations
- CaoWhiteLassiter2025.softOptimalPolicy best ρ = (1 - ρ) • ProbabilityTheory.uniformOn Set.univ + ρ • MeasureTheory.Measure.dirac best
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The ALT measure #
ALT (§2.2) counts the alternative actions available to the causee, excluding the one actually
taken — ALT(Y₁) = 5 at the paper's fig. 2a board state. ALT = 0 is the Frankfurt-style
could-not-have-done-otherwise configuration.
The number of alternative actions available to the causee is the number of actions of positive probability other than the one taken.
Equations
- CaoWhiteLassiter2025.altCount taken μ = ({a : A | μ {a} ≠ 0} \ {taken}).ncard
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The causee had no alternative exactly when every other action had probability zero.
The INT measure #
The paper's §2.3 displayed equation, a simplified [HKW18] degree of intention:
INT(a) = Pr(A = a ∧ G) · u′(a) / Σ_{a′} Pr(A = a′ ∧ G) · u′(a′)
"the probability that an action performed in a state will result in the desired outcome,
normalized by the probability of all alternative actions that would have resulted in the same
outcome", each term weighted by exponentiated utility u′ = eᵘ, the exponential serving only to
make the weights strictly positive. Here pr a′ is the joint probability that the agent takes
a′ and the goal results, and the weight is abstracted to any w : A → ℝ≥0; modelIntention
instantiates pr over a SEM.
The simplification costs the principle it is motivated by: [HKW18] hold, after [Fra69], that an action an agent could not have avoided is never intentional, while under this measure such an action is maximally intentional, its sole goal-conducive alternative being itself. It is ALT rather than INT that registers the paper's made/forced contrast in (8).
The goal-weighted share of action a among all goal-conducive
alternatives.
Equations
- CaoWhiteLassiter2025.intentionDegree pr w a = pr a * ↑(w a) / ∑ a' : A, pr a' * ↑(w a')
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INT is a share, so it never exceeds 1.
An action that is the only goal-conducive one carries the whole normalized weight.
An agent who could not have done otherwise comes out maximally intentional, since the simplified INT does not carry the alternative-possibilities condition; ALT is what separates made from forced.
The paper's INT over a causal model instantiates intentionDegree with pr a′ the
probability, over contexts drawn from ν, that under the intervention I the action variable
takes the value a′ and the outcome satisfies the goal: the paper's
Pr((M,u⃗) ⊨ A = a⃗′ ∧ G = g⃗).
Equations
- CaoWhiteLassiter2025.modelIntention M ν I act goal w a = CaoWhiteLassiter2025.intentionDegree (fun (a' : α act) => ν {u : U | M.solve I u act = a' ∧ M.solve I u ∈ goal}) w a
Instances For
SUF, Pearl's probability of sufficiency ([Pea19]), is the probability, among the
contexts drawn from ν in which the observation obs holds, that setting c := x makes
e = y.
Equations
- CaoWhiteLassiter2025.suf M ν obs c x e y = ν[{u : U | M.solve (Function.update ⊥ c ↑x) u e = y} | M.contexts obs]
Instances For
With nothing observed, SUF is the probability that the intervention yields the effect.
Deterministic limit #
With a certain context SUF collapses to a {0,1} indicator, and wherever Nadathur and Lauer's categorical make holds of the empty background, SUF is 1 whatever the distribution over contexts. The converse fails: a single context can make the intervention yield the effect without the effect being settled by the strict development, so the categorical make semantics is strictly stronger than maximal graded SUF.
With a certain context, SUF is the indicator of the counterfactual outcome there.
Nadathur and Lauer's make entails maximal SUF. Whenever make holds of the empty background, SUF is 1 under every distribution over contexts.
A time index for a causal model (the paper's definition 1) places each parent exactly one timestep before its child.
- time : V → ℕ
The timestep of each variable.
Parents immediately precede their children.
Instances For
A time-indexed model is acyclic.
The paper's judgment data #
The in-text contrasts are rows in Data.Examples.CaoWhiteLassiter2025: the
non-interchangeability triplets (3)–(4), the gym gradability triplets (5)–(7), the
could-have-done-otherwise pair (8), the intent-denial continuations (9)–(10), and the
make/let sufficiency pair (11).
The paper's (8) separates make from force. In one frame with a could-have-done-otherwise continuation, made tolerates the continuation and forced resists it, a difference in the causee's alternatives, which ALT measures.
The gym triplets (5)–(7) grade the stronger verbs against a constant cause. The same three causing events leave caused acceptable throughout while forced switches, which is why the account measures the causal relation rather than classifying it.
A probabilistic model #
SUF is a probability when the background is uncertain. In a model whose effect holds when the
cause and a background noise both do, and whose noise is true with probability p, setting
the cause makes the effect true with probability p, strictly between the 0 and 1 of the
categorical semantics.
Equations
- CaoWhiteLassiter2025.ProbabilisticExample.instDecidableEqV x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Equations
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The effect reads the cause and the noise.
Equations
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The effect holds when the cause and the noise both do; the context settles the noise, and the cause is off unless set.
Equations
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In the background the noise is true with probability p.
Equations
- CaoWhiteLassiter2025.ProbabilisticExample.background p = p • MeasureTheory.Measure.dirac true + (1 - p) • MeasureTheory.Measure.dirac false
Instances For
SUF is the probability of the noise, graded as the paper's measure requires.