Predicate (mathematical logic)

In mathematical logic, a predicate is the formalization of the mathematical concept of statement. A "statement" is commonly understood as an assertion that may be "true" or "false", depending on the values of the variables that occur in it. A "predicate" is a well-formed formula that can be evaluated to "true" or "false" in function of the values of the variables that occur in it. It can thus be considered as a Boolean-valued function.
A predicate consists of atomic formulas connected with logical connectives. An "atomic" formula is a well-formed formula of some mathematical theory. The main logical connectives are negation ("not" or ), logical conjunction ("and" or ), logical disjunction ("or" or ), existential quantification () and universal quantification (); the predicates "always true" (denoted "true" or ) and "always false" (denoted "false" or ) are commonly considered also as logical connectives.
A predicate that does not contain any quantifier ( or ), is called a propositional formula. A predicate whose quantifiers all apply to individual elements, and not to sets or predicates, is called a first-order predicate.
Simplified overview.
Informally, a predicate, often denoted by capital roman letters such as formula_1, formula_2 and formula_3, is a statement that may be true or false depending on the values of its variables. It can be thought of as an operator or function, that returns a value that is either true or false depending on its input. For example, predicates are sometimes used to indicate set membership: when talking about sets, it is sometimes inconvenient or impossible to describe a set by listing all of its elements. Thus, a predicate "P"("x") will be true or false, depending on whether "x" belongs to a set or not.
A predicate can be a proposition if the placeholder "x" is defined by domain or selection.
Predicates are also commonly used to talk about the properties of objects, by defining the set of all objects that have some property in common. For example, when "P" is a predicate on "X", one might sometimes say "P" is a property of "X". Similarly, the notation "P"("x") is used to denote a sentence or statement "P" concerning the variable object "x". The set defined by "P"("x"), also called the extension of "P", is written as {"x" | "P"("x")}, and is the set of objects for which "P" is true.
For instance, {"x" | "x" is a positive integer less than 4} is the set {1,2,3}.
If "t" is an element of the set {"x" | "P"("x")}, then the statement "P"("t") is "true".
Here, "P"("x") is referred to as the "predicate", and "x" the "placeholder" of the "proposition". Sometimes, "P"("x") is also called a (template in the role of) propositional function, as each choice of the placeholder "x" produces a proposition.
A simple form of predicate is a Boolean expression, in which case the inputs to the expression are themselves Boolean values, combined using Boolean operations. Similarly, a Boolean expression with inputs predicates is itself a more complex predicate.
Formal definition.
The precise semantic interpretation of an atomic formula and an atomic sentence will vary from theory to theory.