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Linglib.Theories.Semantics.Causation.CausalSelection

Causal Selection Models #

@cite{icard-et-al-2017}

Theory-layer definitions for causal selection: the cognitive process of ranking causes by importance, given a causal model.

Two models are formalized:

  1. Necessity-Sufficiency Model (NSM, @cite{icard-et-al-2017}): NSM(C) = P(C) · Suf(C) + (1 − P(C)) · Nec(C)

  2. Sampling propensity (shared by NSM and CESM): SP(V) = s · ⟦V⟧^w@ + (1 − s) · P(V)

These definitions are theory-layer infrastructure imported by study files (e.g., KonukEtAl2026, BellerGerstenberg2025).

Sampling Propensity #

The probability that a variable retains its actual-world value in a counterfactual sample. Interpolates between the actual value (stability parameter s) and the prior probability (1 − s).

source
def Semantics.Causation.CausalSelection.samplingPropensity (s actualValue priorProb : ℚ) :
ℚ

Sampling propensity of a binary variable V.

  • s: stability parameter ∈ [0,1] — probability of copying the actual value
  • actualValue: ⟦V⟧^w@ ∈ {0,1} — the variable's value in the actual world
  • priorProb: P(V) — the prior probability of V being true
Equations
Instances For
    source
    theorem Semantics.Causation.CausalSelection.sp_at_zero (v p : ℚ) :
    samplingPropensity 0 v p = p

    At s = 0, sampling propensity reduces to the prior.

    source
    theorem Semantics.Causation.CausalSelection.sp_at_one (v p : ℚ) :
    samplingPropensity 1 v p = v

    At s = 1, sampling propensity reduces to the actual value.

    Necessity-Sufficiency Model (NSM) #

    The causal strength of a candidate cause C for outcome E, computed as a weighted combination of sufficiency and necessity scores.

    source
    def Semantics.Causation.CausalSelection.nsm (pC suf nec : ℚ) :
    ℚ

    NSM formula: NSM(C) = P(C) · Suf(C) + (1 − P(C)) · Nec(C).

    • pC: prior probability (or sampling propensity) of the cause
    • suf: sufficiency score ∈ [0,1]
    • nec: necessity score ∈ [0,1]
    Equations
    Instances For
      source
      theorem Semantics.Causation.CausalSelection.nsm_suf_one (pC nec : ℚ) :
      nsm pC 1 nec = pC + (1 - pC) * nec

      When Suf = 1, NSM simplifies to P(C) + (1 − P(C)) · Nec.

      source
      theorem Semantics.Causation.CausalSelection.nsm_suf_nec (pC : ℚ) :
      nsm pC 1 1 = 1

      Sufficient and necessary: NSM = 1 regardless of probability.

      source
      theorem Semantics.Causation.CausalSelection.nsm_suf_only (pC : ℚ) :
      nsm pC 1 0 = pC

      Sufficient but not necessary: NSM = P(C).

      source
      theorem Semantics.Causation.CausalSelection.nsm_nec_only (pC : ℚ) :
      nsm pC 0 1 = 1 - pC

      Necessary but not sufficient: NSM = 1 − P(C).

      source
      theorem Semantics.Causation.CausalSelection.nsm_neither (pC : ℚ) :
      nsm pC 0 0 = 0

      Neither sufficient nor necessary: NSM = 0.

      source
      theorem Semantics.Causation.CausalSelection.nsm_mono_suf (pC nec : ℚ) (hpC : 0 ≤ pC) (s₁ s₂ : ℚ) (h : s₁ ≤ s₂) :
      nsm pC s₁ nec ≤ nsm pC s₂ nec

      NSM is monotone in sufficiency.