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A logic with higher order probabilities. (English) Zbl 0884.03019
The authors give an extension LPP$$_{\text{ext}}$$ of the propositional probability logic LPP defined by M. Rašković [Publ. Inst. Math., Nouv. Sér. 53, 1-3 (1993; Zbl 0799.03018)]. In the language of LPP$$_{\text{ext}}$$ there are formulas of the form $$P_r(A)$$, with he intended meaning “the probability of $$A$$ is greater than or equal to $$r$$”, where $$A$$ may be a formula of the form $$P_s(B)$$.
In this interesting paper a possible worlds semantics is provided for LPP$$_{\text{ext}}$$. A model is a triple $$\langle W,\text{Prob}, \pi\rangle$$, where $$W$$ is a set of possible worlds, $$\pi(w)$$ is a truth-value assignment to propositional variables, and Prob is a probability assignment that assigns to any $$w\in W$$ a probability space. Hence, for any $$w\in W$$, Prob$$(w)=\langle V(w),H(w),\mu(w)\rangle$$, where $$V(w)\subset W$$, $$H(w)$$ is an algebra of subsets of $$V(w)$$, and $$\mu(w)$$ is a finite additive probability measure with a fixed finite range.
The definition of satisfaction $$\models$$ is as usual except for $$P_r(A)$$, where we have $w\models P_r(A) \text{ iff } \mu(w)(\{u\in V(w):u\models A\})\geq r.$
A complete axiomatization of LPP$$_{\text{ext}}$$ is given and an extended completeness theorem is proved.
Decidability of LPP$$_{\text{ext}}$$ is proved via the finite model property.

##### MSC:
 03B48 Probability and inductive logic 68T27 Logic in artificial intelligence 03C80 Logic with extra quantifiers and operators