Faithful and Invariant Conditional Probability in Łukasiewicz Logic

Abstract

To every consistent finite set Θ of conditions, expressed by formulas (equivalently, by one formula) in Łukasiewicz infinite-valued propositional logic, we attach a map ℘ _Θ assigning to each formula ψ a rational number ℘ _Θ (ψ)∈[0,1] that represents “the conditional probability of ψ given Θ”. The value ℘ _Θ (ψ) is effectively computable from Θ and ψ. The map Θ ↦℘ _Θ has the following properties: (i) (Faithfulness): ℘ _Θ (ψ)=1 if and only if Θ ⊢ ψ, where ⊢ is syntactic consequence in Łukasiewicz logic, coinciding with semantic consequence because Θ is finite. (ii) (Additivity): For any two formulas φ and ψ whose conjunction is falsified by Θ, letting χ be their disjunction we have ℘ _Θ (χ)=℘ _Θ (φ)+℘ _Θ (ψ). (iii) (Invariance): Whenever Θ′ is a finitely axiomatizable theory and ι is an isomorphism between the Lindenbaum algebras of Θ and of Θ′, then for any two formulas ψ and ψ′ that correspond via ι we have ℘ _Θ (ψ)=℘ _Θ′(ψ′). (iv) If θ=θ(x ₁,…,x _n ) is a tautology, then for any formula ψ=ψ(x ₁,…,x _n ), the (now unconditional) probability    ℘_{θ}(ψ) is the Lebesgue integral over the n-cube of the McNaughton function represented by ψ.