Let $S$ be a set of social states with at least three elements. Let $\mathcal{R}$ be the set of complete and transitive relations on $S$. A function $f:\mathcal{R}^n\to\mathcal{R}$ satisfies independence of irrelevant alternatives if the restriction of $f(R)$ to $A\subseteq S$ depends only on the corresponding restriction of $\mathcal{R}^n$ for every subset $A\subseteq S$. For $R\in\mathcal{R}^n$, and $i=1,\ldots,n$, let $R_i$ be the $i$th entry of $R$.
Wilson's (first) theorem in the paper says that if $f$ satisfies independence of irrelevant alternatives, and if for any $x,y\in S$ there exists some $R\in\mathcal{R}^n$ such that $x f(R) y$, then $f$ either is either constant with universal indifference as its value, or there is some $i$ such that $x f(R)x$$x f(R)y$ holds exactly when $xR_i y$ holds ($i$ is a dictator), or there is some $i$ such that $xf(R)y$ holds exactly when $yR_i x$ holds ($i$ is an inverse dictator).
In Arrow's theorem, the case of constant $f$ or the existence of an inverse dictator is excluded. But the Pareto condition used by Arrow is also stronger than the condition that for any $x,y\in $ there exists some $R\in\mathcal{R}^n$ such that $x f(R) y$.
