Majority digraphs

A majority digraph is a finite simple digraph G = (V,→) such that there exist finite sets A_v for the vertices v ∈ V with the following property: u → v if and only if “more than half of the A_u are A_v”. 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.