条件信念,知识和概率

J. Eijck, K. Li
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引用次数: 5

摘要

贝叶斯概率论提出了一种表示信念和更新信念的自然方法,其中agent A在P中的信念可以解释为考虑P比不P更有可能。本文试图获得其背后的核心逻辑概念。提出了条件信念和条件知识的健全完备邻域逻辑,并追溯了其与条件信念和条件知识的概率逻辑的联系。本文的关键概念是agent a在有信息Q的情况下有条件地相信P,其中假设Q与a所知道的相容。条件邻域逻辑可以被看作是一个核心系统,用于推理主观似是而非的解释,在数值概率方面。事实上,每一个加权Kripke模型都会产生一个条件邻域模型,反之则不然。我们证明了对加权Kripke模型的条件邻域逻辑的演算是健全的,但并不完整。接下来,我们将展示如何扩展微积分以获得加权Kripke模型类的完备性。条件信念的邻域模型在模型限制下关闭(公告更新),而早期的“下注意愿”信念邻域模型则没有。因此,我们提出的逻辑是对先前的信仰和知识邻域逻辑的改进。我们提出了一种完备的演算方法,用于公开声明和利用约简公理公开揭示命题的真值。这些缩减表明,在语言中加入这些公告操作符并没有增加表达能力。
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Conditional Belief, Knowledge and Probability
A natural way to represent beliefs and the process of updating beliefs is presented by Bayesian probability theory, where belief of an agent a in P can be interpreted as a considering that P is more probable than not P. This paper attempts to get at the core logical notion underlying this. The paper presents a sound and complete neighbourhood logic for conditional belief and knowledge, and traces the connections with probabilistic logics of belief and knowledge. The key notion in this paper is that of an agent a believing P conditionally on having information Q, where it is assumed that Q is compatible with what a knows. Conditional neighbourhood logic can be viewed as a core system for reasoning about subjective plausibility that is not yet committed to an interpretation in terms of numerical probability. Indeed, every weighted Kripke model gives rise to a conditional neighbourhood model, but not vice versa. We show that our calculus for conditional neighbourhood logic is sound but not complete for weighted Kripke models. Next, we show how to extend the calculus to get completeness for the class of weighted Kripke models. Neighbourhood models for conditional belief are closed under model restriction (public announcement update), while earlier neighbourhood models for belief as `willingness to bet' were not. Therefore the logic we present improves on earlier neighbourhood logics for belief and knowledge. We present complete calculi for public announcement and for publicly revealing the truth value of propositions using reduction axioms. The reductions show that adding these announcement operators to the language does not increase expressive power.
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