Abstract This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in that context. In particular we introduce a construction which defines a (finite) Boolean algebra of conditionals from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.

中文翻译：

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条件、概率和逻辑的布尔代数

摘要 本文研究了条件事件的结构以及在这种情况下自然出现的概率测度。特别地，我们引入了一个构造，它定义了来自任何（有限）事件布尔代数的条件的（有限）布尔代数。通过这样做，我们区分了依赖于概率的条件事件的属性和条件事件的逻辑代数结构所固有的属性。我们的主要结果提供了一种将标准的两位条件概率视为条件事件上的一位概率函数的方法。我们还考虑了我们的条件布尔代数的逻辑对应物，其中包含与非单调推理的优先后果关系的链接。