Faithful and invariant conditional probability in Łukasiewicz logic
pp. 213-232
in: David Makinson, Jacek Malinowski, Heinrich Wansing (eds), Towards mathematical philosophy, Berlin, Springer, 2009Abstract
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.
