Editing Predicate variable (section)

== Usage ==

If the predicate variables are not defined as belonging to the vocabulary of the predicate calculus, then they are predicate '''metavariables''', whereas the rest of the predicates are just called "predicate letters". The metavariables are thus understood to be used to code for [[axiom schema]] and theorem schemata (derived from the axiom schemata). 

Whether the "predicate letters" are constants or variables is a subtle point: they are not constants in the same sense that <math> =, \ \in , \ \le,\ <, \ \sub, </math> are predicate constants, or that <math> 1,\ 2,\ 3,\ \sqrt{2},\ \pi,\ e\ </math> are numerical constants.

If "predicate variables" are only allowed to be bound to predicate letters of zero [[arity]] (which have no arguments), where such letters represent [[propositional logic|propositions]], then such variables are ''[[propositional variable|propositional variables]]'', and any predicate logic which allows second-order quantifiers to be used to bind such propositional variables is a second-order predicate calculus, or [[second-order logic]].

If predicate variables are also allowed to be bound to predicate letters which are unary or have higher arity, and when such letters represent ''[[propositional function]]s'', such that the domain of the arguments is mapped to a range of different propositions, and when such variables can be bound by quantifiers to such sets of propositions, then the result is a higher-order predicate calculus, or [[higher-order logic]].