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A predicate is the formalization of the mathematical concept of statement. A statement is commonly understood as an assertion (declaration) with a truth value (true or false), depending on the values of its constituent variables.
A predicate is a wff 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 wff of some mathematical theory.
The main logical connectives are
negation (not or ¬)
logical conjunction (and or ∧)
logical disjunction (or or ∨)
existential quantification (∃)
universal quantification (∀)
the predicate 'always true', denoted true or ⟙
the predicate 'always false', denoted false or ⟘
The last two are commonly considered also as logical connectives.
A predicate without quantifiers 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. That is, in FOL predicates range over individuals only, and in HOL over other predicates.