Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations. A novel notion of probabilistic entailment enjoying desirable properties of logical consequence is proposed and shown to collapse into the classical entailment when the language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the decidability of the theory of real closed ordered fields.

中文翻译：

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关于概率和逻辑

在经典命题逻辑中，将概率分配给公式被证明等同于将概率分配给估值。提出了一种新的概率蕴涵概念，该概念具有合乎需要的逻辑结果特性，并表明当语言保持不变时，它会崩溃为经典蕴涵。受这一结果的启发，通过为允许约束公式概率的新语言构造提供适当的语义，提出了命题逻辑的可判定保守丰富。使用实闭有序场理论的可判定性提供了一个健全的弱完全公理化。