Fechner / strong utility: there are \(u : X \to \mathbb{R}\) and \(F: \mathbb{R} \to \qty(0,1)\), continuous, strictly increasing, and \(F\qty(-t) = 1 - F\qty(t)\), and

\begin{equation} P\qty(j \succ j’) = F\qty(u_{j} - u_{j’}) \end{equation}

Quadruple Condition: for all choices of \(j, j’, j’’, j’’’\), we have:

\begin{equation} P\qty(j \succ j’) \geq P\qty(j’’ \succ j’’’) \Leftrightarrow P\qty(j \succ j’’) \geq P\qty(j’ \succ j’’’) \end{equation}

Solvability: for \(j, j’, j’’\) and \(\lambda \in [0,1]\) with \(P\qty(j \succ j’) \leq \lambda \leq P\qty(j \succ j’’)\), some \(j’’’\) has \(P\qty(j \succ j’’’) = \lambda\)