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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.

DOI: 10.1007/978-1-4020-9084-4_11

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