Majority Digraphs

A majority digraph is a finite simple digraph $G=(V,\to )$ such that there exist finite sets $A_v$ for the vertices $v\in V$ with the following property: $u\to v$ if and only if ``more than half of the $A_u$ are $A_v$''. That is, $u\to v$ if and only if $\vert A_u \cap A_v \vert > \frac {1}{2} \cdot \vert A_u\vert$. We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change $\frac {1}{2}$ to any real number $\alpha \in (0,1)$, 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.