Terminology 29: A predicate with parameter variables a, b, ..., z is given by a predicate expression \mathfrak{P} (called a predicate symbol if \mathfrak{P} is a single symbol) together with a defining statement \phi such that:
(a) The variables a, b, ..., z are all distinct from one another.
(b) All variables which occur free in \phi are among a, b, ..., z .
If the predicate has no parameters, then we abbreviate the statement \phi by the expression \mathfrak{P} . Otherwise, if there are parameters, then given a list of as many (not necessarily distinct) variables A, B, ..., Z , we let \mathfrak{P}(A, B, ..., Z) \hspace{3mm} \textup{abbreviate} \hspace{3mm} [\phi | a \to A, b \to B, ..., z \to Z], where we are using multiple variable substitution on the right side.