 Compared to a monadic vocabulary, the other extreme can be described as follows: Let P be a ternary predicate (a “pairing function”) on a non-empty set S. Suppose that for every \(x,y\in S\) there is \(z\in S\) with \((x, y, z) \in P\) and for every \(z\in S\) there is at most one pair \((x, y)\) with \((x, y, z) \in P\). Then the true (full) second-order theory of S is interpretable in the monadic theory of \((S, P)\) (Gurevich 1985). In other words, in a context where a pairing function is present, as in set theory and number theory, no special advantage arises from limiting second-order logic to its monadic fragment.