We present the directed simplicial complex: this is an extension of a simplicial complex with a direction on its faces. This enables us to model a local notion of belief in a three-valued semantics, in which formulas may be true, false, or undefined. Of particular interest is a class of directed simplicial complexes called cups, in which any maximal set of vertices of different colors is a facet. Cups permit a more concise relational representation. We give examples and applications of our logical semantics. We then provide a sound and complete axiomatization for the class of directed simplicial complexes and extend it to an axiomatization for the class of cups. We include a detailed technical comparison with other works on belief on complexes. Our work generalizes a prior axiomatization for knowledge on (undirected) impure simplicial complexes.

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