In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge 35. Building on Stalnaker’s core insights, and using frameworks developed in 11 and 3, we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker’s postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.

中文翻译：

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知识、知识和信念的逻辑和拓扑

在最近的工作中，Stalnaker 提出了一个逻辑框架，在该框架中，信念被实现为知识的弱化形式 35。基于 Stalnaker 的核心见解，并使用在 11 和 3 中开发的框架，我们采用拓扑我们认为，改进和改进这种分析的工具。拓扑子集空间的结构允许自然区分什么是已知和（粗略地说）什么是可知的; 我们认为，Stalnaker 系统的基本公理直观地依赖于两个都这些观念中。更准确地说，我们认为 Stalnaker 提出的将知识和信念联系起来的原则的合理性依赖于知识的“手头证据”概念和较弱的“证据在手”概念之间的微妙模糊。可能会出名. 我们的分析导致知识、可知性和信念的三峰逻辑在拓扑子集空间中解释，其中信念可以根据知识和可知性. 我们为此逻辑及其单模态信念片段提供了合理且完整的公理化。然后，我们考虑保留对 Stalnaker 假设的适当翻译的较弱逻辑，但不允许对信念进行任何减少。我们为这些不可约的信念概念提出了新的拓扑语义，概括了我们以前的语义，并为相应的逻辑提供了合理和完整的公理化。