 Second-order relation variable

A second-order relation variable of arity $k\ge 1$ can be assigned a subset of the $k$-fold Cartesian product $D^k$; its application $R(x_1,\dots ,x_k)$ is true exactly when the tuple $(x_1,\dots ,x_k)$ belongs to that subset. Arity is the number of arguments, so a unary second-order relation variable is a predicate on individual objects. For a finite set $D$ of cardinality $n$, there are $n^k$ possible tuples and $2^{n^k}$ possible relations under full semantics for second-order logic. A relation is encoded by $n^k$ membership bits once the objects have been numbered; the number of possible witnesses can be exponential even when each witness has a short encoding.