Probability logic of finitely additive beliefs
This paper investigates a probability logic \(\Sigma_+\) which resembles certain modal logics. The main focus is on questions related to completeness and compactness. The author enriches propositional logic by adding probabilistic operators \(L_r\) with \(r\in[0,1]\cup \mathbb Q\) to the language. The intended interpretation of \(L_r\varphi\) is that the agents' belief in the event \(\varphi\) is at least \(r.\) As \(\varphi\) is a sentence of this logic, the \(L_r\)-operator possibly occurs multiple times in \(\varphi.\) Next, the notions of finitely additive probability model and finitely additive type space are defined, for which a satisfaction relation \(\models\) is introduced. The author obtains the probability logic \(\Sigma_+\) as the smallest set of formulae that is closed under a set of rules. To prove the truth lemma and Lindenbaum property for \(\Sigma_+\) an auxiliary deducibility relation, \(\Vdash_{\Sigma_+}\) is defined. Using this relation, strong completeness of \(\Sigma_+\) is proved. The brief third section shows how the above results are used to obtain that the axiom system \(\Sigma_H:=\Sigma_+\cup\{L_r\varphi\rightarrow L_1L_r\varphi\}\cup\{\neg L_r\varphi\rightarrow L_1\neg L_r\varphi\}\) is sound and complete with respect to Harsanyi type spaces. Section 4 investigates universal type spaces. The author proves that any canonical finitely additive type space is not universal in the class of finitely additive type spaces. The last section contains a non-compactness theorem for \(\Sigma_+\) and the class of finitely additive type spaces. Hence establishing that probability indices cause the failure of compactness in probability logic.
scientific article; zbMATH DE number 5796446
scientific article