概率稳定修正与比较概率：一个表示定理及应用
Probabilistically stable revision and comparative probability: a representation theorem and applications
AI总结：
本文证明概率稳定信念修正算子的表示定理，给出其完整刻画与选择函数语义，同时得到比较概率理论的两个独立结果，并指出其在投票博弈与显示偏好理论中的应用。
摘要：
由Leitgeb提出的信念稳定性规则（《纯粹与应用逻辑年刊》第164卷，2013年）是一种理性接受规则，它用**概率稳定命题**来刻画绝对信念：即主体赋予其弹性高置信度的命题。该稳定性规则生成了一类**概率稳定信念修正**算子，这类算子刻画了主体通过贝叶斯条件化更新其置信度，同时使其全有或全无信念符合稳定性规则时产生的信念动态。\n本文证明了一个表示定理，该定理给出了这类概率稳定修正算子的完整刻画，并为概率稳定信念修正的（非单调）逻辑提供了一种“定性”选择函数语义。借助比较概率序理论，该结果给出了有限概率空间上一个选择函数可表示为最强稳定集算子的充要条件。由此得到的概率稳定信念修正逻辑具有强单调性，但不满足AGM信念修设公设，且仅满足极弱形式的案例推理。\n在证明主定理的过程中，我们还证明了两个对比较概率理论本身具有独立价值的结果：第一个给出了一对（分别为严格和非严格的）比较概率序联合可表示的充要条件。第二个结果提供了一种公理化比率比较逻辑的方法，这类比率比较的形式为“事件A的可能性至少是事件B的k倍”。除了这些测量论应用外，我们还指出了主结果在简单投票博弈理论和显示偏好理论中的两个应用。
英文摘要：
The stability rule for belief, advocated by Leitgeb [Annals of Pure and Applied Logic 164, 2013], is a rule for rational acceptance that captures categorical belief in terms of $\textit{probabilistically stable propositions}$: propositions to which the agent assigns resiliently high credence. The stability rule generates a class of $\textit{probabilistically stable belief revision}$ operators, which capture the dynamics of belief that result from an agent updating their credences through Bayesian conditioning while complying with the stability rule for their all-or-nothing beliefs. In this paper, we prove a representation theorem that yields a complete characterisation of such probabilistically stable revision operators and provides a `qualitative' selection function semantics for the (non-monotonic) logic of probabilistically stable belief revision. Drawing on the theory of comparative probability orders, this result gives necessary and sufficient conditions for a selection function to be representable as a strongest-stable-set operator on a finite probability space. The resulting logic of probabilistically stable belief revision exhibits strong monotonicity properties while failing the AGM belief revision postulates and satisfying only very weak forms of case reasoning. In showing the main theorem, we prove two results of independent interest to the theory of comparative probability: the first provides necessary and sufficient conditions for the joint representation of a pair of (respectively, strict and non-strict) comparative probability orders. The second result provides a method for axiomatising the logic of ratio comparisons of the form ``event $A$ is at least $k$ times more likely than event $B$''. In addition to these measurement-theoretic applications, we point out two applications of our main result to the theory of simple voting games and to revealed preference theory.
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