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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.

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