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Completeness theorems for \(\sigma \)-additive probabilistic semantics. (English) Zbl 07160777
This paper on logics with languages containing probability operators is the latest installment of a steady stream of research outputs from Belgrade on such logics. This paper studies propositional logics to which operators of the form \(P_{\geq r}\) have been added, the intended interpretation is that the probability of the sentence following this operator is greater or equal than \(r\). Furthermore, conjunctions of infinite length shorter than a specified cardinal \(\kappa\) are allowed. The logic under consideration is \(LPP_{\kappa,\lambda}\), it has \(\lambda\) propositional variables, allows conjunctions of \(<\kappa\)-many formulae, and allows iterations of probability operators.
The first two of the six sections of this paper introduce the topic, discuss related work and introduce the formal apparatus (notation, syntax, semantics, etc.).
The third section is start of the paper proper explaining the non-compactness of \(LPP\)-logics and also providing a reason for this phenomenon (non-Archimedean order types). The authors thus strive for an axiomatisation that is strongly complete with respect to the class of the \(\sigma\)-additive models on as many fragments of \(LPP\) as possible. The remainder of this section is devoted to showing that \(LPP_{\kappa,\lambda}\)-logics are not countably compact for all \(\lambda\geq\omega_1\) and that the class of \(\sigma\)-additive models satisfies a continuity property.
Section 4 presents axioms and inference rules which are shown to be sound. It furthermore contains a deduction theorem and demonstrates the equivalence of Hoover’s and Goldblatt’s approaches to capturing \(\sigma\)-additivity.
Section 5 begins with a demonstration that if iterations of probability operators are not allowed, then the resulting logic is strongly complete with respect to the class of all \(\sigma\)-additive models. If iterations of probability operators are allowed, things become more messy, and the authors thus distinguish between strong and weak completeness. Theorem 5.5 is a derivation of strong completeness for “admissible” fragments of \(LPP_{\omega_1,\omega}\).
The final section lists the theorems proved and neatly displays them in a table.
MSC:
03B42 Logics of knowledge and belief (including belief change)
03B48 Probability and inductive logic
03C10 Quantifier elimination, model completeness and related topics
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