predicate logic

For the specific term, see First-order logic. In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulas contain variables which can be quantified. Two common quantifiers are the existential and universal quantifiers. The variables could be elements in the universe, or perhaps relations or functions over the universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function".
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Sometimes it is inconvenient or impossible to describe a set by listing all of its elements. Another useful way to define a set is by specifying a property that the elements of the set have in common. We use the notation P(x) to denote a sentence or statement P concerning the variable object x. The set defined by P(x) written {x | P(x)}, is just a collection of all the objects for which P is sensible and true.
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<logic> (Or "predicate calculus") An extension of propositional logic with separate symbols for predicates, subjects, and quantifiers.
For example, where propositional logic might assign a single symbol P to the proposition "All men are mortal", predicate logic can define the predicate M(x) which asserts that the subject, x, is mortal and bind x with the universal quantifier ("For all"):
All x . M(x)
Higher-order predicate logic allows predicates to be the subjects of other predicates.
(2002-05-21)