In
mathematics, a
predicate is commonly understood to be a
Boolean-valued function P:
X→ {true, false}, called the predicate on
X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a
relation, then a predicate is simply the
characteristic function or the
indicator function of a relation. However, not all theories have relations, or are founded on
set theory, and so one must be careful with the proper definition and semantic interpretation of a predicate.