# Faithful and Invariant Conditional Probability in Łukasiewicz Logic

• Daniele Mundici
Chapter
Part of the Trends in Logic book series (TREN, volume 28)

## 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.

## Keywords

conditional conditional probability de Finetti coherence criterion Dutch Book many-valued logic Łukasiewicz logic infinite-valued logic MV-algebra state finitely additive measure subjective probability invariant measure faithful state

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