Majority Digraphs

TL;DR

This paper characterizes majority digraphs, showing they are precisely those with every directed cycle having a reversal, and applies this to logic involving 'most' assertions and propositional connectives.

Contribution

It provides a complete characterization of majority digraphs and demonstrates their invariance under changing the threshold from 1/2 to any real number in (0,1).

Findings

Majority digraphs are characterized by cycles having reversals.

The class of majority digraphs is invariant under threshold changes in (0,1).

Application to logic of 'most' assertions and propositional connectives.

Abstract

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 $ |A_u \cap A_v | > \frac{1}{2} \cdot |A_u|$. 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.