@cite{konuk-et-al-2026}: Plural Causes #
@cite{konuk-et-al-2026}
Formalizes Konuk, Quillien & Mascarenhas (2026) "Plural causes," Open Mind.
Core Contributions #
- Compound causes: A∧B is treated as a single compound binary variable for causal selection, not decomposed into individual contributions.
- Necessity-Sufficiency Model (NSM):
NSM(C) = P(C)·Suf(C) + (1-P(C))·Nec(C)from @cite{icard-et-al-2017}, applied to compound causes. - Anti-linearity: NSM(INT∧HIGH) > NSM(LOW∧INT) even though LOW and HIGH have comparable individual causal strength (Experiment 1).
- Homogeneous loss: Loss judgments follow LOSS_strong = ¬A∧¬B∧¬C∧¬D, not classical ¬((A∧B)∨(C∧D)) (Experiment 2), mixed with classical via fitted parameter w ≈ 0.77.
- Crossing avoidance: Within-disjunct plural causes (A∧B) preferred over cross-disjunct (A∧C) when the rule is (A∧B)∨(C∧D) (Experiment 2, Overdetermined Positive round).
Bridges #
| Concept | Connects to | Module |
|---|---|---|
| Compound sufficiency/necessity | causallySufficient/causallyNecessary | Core.StructuralEquationModel |
| NSM (Nec/Suf weighting) | nsm | Causation.CausalSelection (@cite{icard-et-al-2017}) |
| LOSS_strong (all absent) | noneSatisfy | Plural.Distributivity (@cite{kriz-spector-2021}) |
| Compound sufficiency | allSatisfy | Plural.Distributivity (@cite{kriz-spector-2021}) |
| Loss gap (classical − strong) | inGap | Plural.Distributivity (@cite{kriz-spector-2021}) |
CausalLaw.conjunctive | Threshold/disjunctive models | Core.StructuralEquationModel |
| Crossing avoidance | structural sufficiency gap | Core.StructuralEquationModel |
§ 1. Compound Sufficiency and Necessity #
Extend the SEM's individual-variable causallySufficient/causallyNecessary
to compound (plural) causes. A compound cause C = {v₁,...,vₙ} is sufficient
iff setting all components to true produces the effect, and necessary iff
setting all to false prevents it.
A compound cause is sufficient iff setting all its variables to true produces the effect under normal development.
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A compound cause is necessary iff setting all its variables to false prevents the effect under normal development.
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Singleton compound sufficiency reduces to individual sufficiency.
Singleton compound necessity reduces to the simple but-for test (set variable to false, check effect).
Note: compoundNecessary uses the @cite{nadathur-lauer-2020} but-for
test, while causallyNecessary uses @cite{nadathur-2024} Definition 10b
(with precondition + achievability + supersituation quantification).
The two coincide when the Def 10b precondition passes and the cause is
exogenous, but diverge in general.
§ 1b. Bridge: Compound Sufficiency = allSatisfy #
A compound cause {v₁,...,vₙ} is sufficient iff all its constituent
variables being present suffices for the effect. This is exactly
allSatisfy from @cite{kriz-spector-2021}: plural predication where
every atom satisfies the predicate "is causally active."
Compound sufficiency over a Fin-indexed variable set is equivalent to
allSatisfy applied to the "is present" predicate.
This connects causal cognition to plural semantics: judging a compound cause as sufficient = judging that the plurality "all satisfy" the causal activation predicate.
§ 2. The Necessity-Sufficiency Model (NSM) #
The general NSM from @cite{icard-et-al-2017}: NSM(C) = P(C)·Suf(C) + (1-P(C))·Nec(C).
Imported from Semantics.Causation.CausalSelection.nsm.
§ 3. Experiment 1: Threshold Game #
Three urns — LOW (p=1/20), INTERMEDIATE (p=1/2), HIGH (p=19/20) — with rule WIN := sum ≥ 2. The player draws from all three and wins.
SEM Encoding #
The threshold ≥ 2 rule is encoded as three conjunctive laws: A∧B→WIN, A∧C→WIN, B∧C→WIN. Any pair suffices.
Threshold-≥-2 causal dynamics: any two urns on → WIN.
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Any pair of urns is sufficient (compound sufficiency).
No single urn is sufficient (need 2 for threshold).
In the overdetermined actual world, no individual urn is necessary.
But compound pairs ARE necessary in the actual world.
This is the core insight: individual urns are not necessary (overdetermination), but compound pairs are — removing any pair drops below threshold. This justifies treating A∧B as the unit of causal attribution.
NSM Computation for Experiment 1 #
For compound pair causes at s = 0, any pair is deterministically sufficient (Suf = 1), so NSM(C→WIN) = 1 - P(WIN ∧ ¬C). The residual probability P(WIN ∧ ¬C) is the chance that the remaining single urn (Z) plus exactly one of {X,Y} still meets threshold — but with the compound removed, we need all three absent variables except Z, so P(WIN ∧ ¬C) = P(exactly one of {X,Y} on) × P(Z on).
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NSM for a compound pair {X,Y} in the threshold-≥-2 game (Suf=1).
NSM = 1 - P(WIN ∧ ¬C), where P(WIN ∧ ¬C) is the probability that exactly one of {X,Y} is on AND the third urn Z is also on.
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NSM({INT, HIGH}) = 39/40.
NSM({LOW, INT}) = 21/40.
Anti-linearity: INT∧HIGH has strictly higher NSM than LOW∧INT.
The additive hypothesis predicts LOW∧INT ≈ INT∧HIGH (since LOW and HIGH have comparable individual NSM in the threshold game). The holistic NSM gives 39/40 vs 21/40, matching the empirical finding (t(355) = -4.67, p < 0.001).
§ 4. Experiment 2: Disjunctive Rule and LOSS #
WIN := (A∧B) ∨ (C∧D), with P(A)=7/10, P(B)=1/10, P(C)=1/5, P(D)=9/10.
Classical negation: LOSS = ¬(A∧B) ∧ ¬(C∧D). Homogeneous negation: LOSS_strong = ¬A ∧ ¬B ∧ ¬C ∧ ¬D.
Empirical loss judgments match a mixture: w · LOSS_strong + (1-w) · LOSS_classical, with fitted w ≈ 0.77, consistent with the homogeneity property of plural negation (@cite{kriz-spector-2021}).
Experiment 2 causal dynamics: (A∧B)∨(C∧D) → WIN.
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LOSS_strong entails classical LOSS.
Classical LOSS does NOT entail LOSS_strong.
Witness: A=1, B=0, C=0, D=0 — neither A∧B nor C∧D holds (classical LOSS), but A is present (LOSS_strong fails).
Mixture model: w · LOSS_strong + (1-w) · LOSS_classical.
Fitted w ≈ 0.77, reflecting the dominance of the homogeneous reading (neither spoke German) over the classical reading (not both spoke).
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§ 4b. Bridge: Loss Gap = inGap (Homogeneity) #
The gap between lossClassical and lossStrong — valuations where
classical loss holds but homogeneous loss does not — is exactly the
set of worlds in the truth-value gap (inGap) for the "is present"
predicate over the four causal variables. This connects the paper's
w-parameter mixture to the formal semantics of plural homogeneity.
The loss gap (classical but not strong) is exactly inGap for the
"is present" predicate: some but not all variables are false.
The classical negation ¬(A∧B) ∧ ¬(C∧D) allows worlds where some variables are true and others false. The homogeneous negation ¬A∧¬B∧¬C∧¬D requires all false. The gap — where they disagree — is the truth-value gap from @cite{kriz-spector-2021}: worlds where the plurality is neither all-P nor none-P.
§ 5. Experiment 2: Crossing Avoidance (Overdetermined Positive round) #
In (A∧B)∨(C∧D), a compound cause is "within-disjunct" if both variables come from the same conjunct, and "cross-disjunct" otherwise.
Empirical finding: within-disjunct causes are preferred over cross-disjunct ones, even controlling for counterfactual dependence.
Disjunct membership classification for a pair of variables.
- withinAB : DisjunctMembership
- withinCD : DisjunctMembership
- crossDisjunct : DisjunctMembership
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Classify a pair of Experiment 2 variables by disjunct membership. Indices: A=0, B=1 (first conjunct), C=2, D=3 (second conjunct).
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Structural crossing avoidance: within-disjunct compound {A,B} is sufficient for WIN, but cross-disjunct compound {A,C} is NOT.
This is a structural consequence of the causal model: A∧B matches a conjunctive law, so setting A=B=1 fires the law and produces WIN. But A∧C does not match any single law — each needs a different partner (B for A, D for C).
§ 5b. Experiment 2: Triple-1 and Triple-0 Conditions #
Triple-1: A, B, D drawn (colored), C not drawn — John wins. #
The win is via A∧B (purple pair). Urn D (yellow) is idle: it draws a colored ball but has no effect on the outcome because urn C (its partner) does not. The CESM and NSM both predict A∧B rated near ceiling.
Triple-0: A, B, D draw white balls, C draws colored — John loses. #
The mirror image: the loss is driven by the absence of colored balls from A and B (and D), but not C. Under the non-classical (homogeneous) representation of loss, LOSS := ¬A ∧ ¬B ∧ ¬D, and the white ball from urn D is indispensable while A and B are redundant.
Model fits with the w parameter: r = .94 (CESM) and .99 (NSM) for Triple-0 — the paper's most dramatic improvement over the base models (r = .34 and .52 without w).
In Triple-1, the compound A∧B is both sufficient and necessary for WIN.
In Triple-1, the idle variable D is not individually necessary.
In Triple-1, adding D to the A∧B compound does NOT increase sufficiency (already sufficient) but strictly decreases necessity — the triple A∧B∧D is no longer necessary because removing all three prevents WIN trivially, but so does removing just A∧B.
Empirically, participants penalized plurals containing D relative to their D-free counterparts: every plural with D rated lower than the same plural without D.
In Triple-0 (loss), under homogeneous representation LOSS = ¬A∧¬B∧¬D, the white ball from D is indispensable: its absence would break the homogeneous conjunction. Formally, removing D (setting it to true) blocks the loss under the strong reading.
§ 6. Bridge: LOSS_strong = noneSatisfy (Homogeneity) #
LOSS_strong is exactly the noneSatisfy predicate from @cite{kriz-spector-2021}
applied to the four causal variables: every individual variable is false.
In Semantics.Lexical.Plural.Distributivity, noneSatisfy P x w = true
iff ∀ a ∈ x, P a w = false. LOSS_strong instantiates this with the
identity predicate "is present" over the four causal variables, connecting
causal cognition to the homogeneity account of plural negation.
LOSS_strong is exactly noneSatisfy from @cite{kriz-spector-2021}:
"none of {v₁,...,v₄} are present" under the homogeneity account.