We provide a novel semantics for belief using simplicial complexes. In our framework, belief satisfies the \textsf{KD45} axioms and rules as well as the ``knowledge implies belief'' axiom ($Kφ\lthen Bφ$); in addition, we adopt the (standard) assumption that each facet in our simplicial models has exactly one vertex of every color. No existing model of belief in simplicial complexes that we are aware of is able to satisfy all of these conditions without trivializing belief to coincide with knowledge. We also address the common technical assumption of ``properness'' for relational structures made in the simplicial semantics literature, namely, that no two worlds fall into the same knowledge cell for all agents; we argue that there are conceptually sensible belief frames in which this assumption is violated, and use the result of ``A Note on Proper Relational Structures'' to bypass this restriction. We conclude with a discussion of how an alternative ``simplicial sets'' framework could allow us to bypass properness altogether and perhaps provide a more streamlined simplicial framework for representing belief.

翻译：我们提出了一种基于单纯复形的新颖信念语义学框架。在该框架中，信念满足 \\textsf{KD45} 公理与推理规则，同时满足“知识蕴含信念”公理（$Kφ\\lthen Bφ$）；此外，我们采用（标准）假设，即单纯模型中的每个面片恰好包含每种颜色的一个顶点。据我们所知，现有单纯复形中的信念模型均无法同时满足所有这些条件，除非将信念平凡化为与知识完全等同。我们还探讨了单纯语义学文献中关系结构“恰当性”这一常见技术假设，即不存在两个世界在所有智能体的知识单元中完全重合；我们认为存在概念上合理的信念框架会违反该假设，并利用《关于恰当关系结构的注记》一文的结果绕过此限制。最后，我们讨论了替代性的“单纯集”框架如何可能完全规避恰当性要求，从而为信念表示提供更简明的单纯框架。