Probabilistic Conditional Semantics #
[Ada65] [JE91] [Dou08] [Kau05a] [Kau13] [Pea13]
The conditional if φ then γ as the expected value of γ given φ.
[Ada65] proposes that the assertability of an indicative
conditional if φ then ψ equals the conditional probability
P(ψ ∣ φ). [JE91] push this further: the
conditional denotation is P(ψ ∣ φ). [Dou08] provides
the evidential support theory variant; [Kau05a] [Kau13] extend to causal premise semantics; [Pea13]
develops the structural-counterfactual face.
[CM23] eq. 44 promotes the consequent from a
proposition to a measure function γ : W → ℝ≥0∞:
⟦if φ, γ⟧^w = E_w[γ ∣ φ]
When γ is the indicator of ψ this reduces to the
[Ada65]/[JE91] conditional
probability (condIf_propositional). When γ is the
count-of-relevant-propositions μ_R(w) = |{r ∈ R ∣ r true at w}|,
it yields C&M's "explanatory value" / "expected utility". One
operator, two flavors.
condIf is an abbrev for PMF.condExpect: the wrapping is for
linguistic naming only; the equality is definitional. The Lewis 1976
triviality results bear on identifying the conditional with a
proposition in some space; we are content to identify it with a
number (an expected value), so they do not bite.
Out of scope #
Counterfactual conditionals (intervention semantics [Pea13];
see Core/Causal/SEM/Counterfactual.lean) and the Stalnaker
selection-function tradition
(Semantics/Conditionals/SelectionFunction.lean) are formalized
elsewhere. Their relation to the probabilistic conditional is itself a
research question ([Sch11b],
[CZC18]).
[CM23] compositional conditional:
⟦if φ, γ⟧^w = E_w[γ ∣ φ]. Definitionally condExpect.
Equations
- Semantics.Conditionals.Probabilistic.condIf p φ γ = p.condExpect φ γ
Instances For
[Ada65]: when the consequent is the indicator of ψ, the
conditional reduces to the conditional probability P(ψ ∣ φ).