A majority digraph is a finite simple digraph G = (V,→) such that there exist finite sets Av for the vertices v ∈ V with the following property: u → v if and only if “more than half of the Au are Av”. That is, u → v if and only if (formula presented). We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change to any real number α ∈ (0, 1), we obtain the same class of digraphs. We apply the characterization result to obtain a result on the logic of assertions “most X are Y ” and the standard connectives of propositional logic.
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This work was partially supported by a grant from the Simons Foundation (#245591 to the third author).