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 formulae contain variables which can be quantified. Two common quantifiers are the existential ∃ ("there exists") and universal ∀ ("for all") quantifiers. The variables could be elements in the universe under discussion, or perhaps relations or functions over that universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function". The foundations of predicate logic were developed independently by Gottlob Frege and Charles Peirce.
In informal usage, the term "predicate logic" occasionally refers to first-order logic. Some authors consider the predicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development.
See also 
- Eric M. Hammer: Semantics for Existential Graphs, Journal of Philosophical Logic, Volume 27, Issue 5 (Oktober 1998), page 489: "Development of first-order logic independently of Frege, anticipating prenex and Skolem normal forms"
- Among these authors is Stolyar, p. 166. Hamilton considers both to be calculi but divides them into an informal calculus and a formal calculus.
- A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK ISBN 0-521-21838-1
- Abram Aronovic Stolyar 1970, Introduction to Elementary Mathematical Logic, Dover Publications, Inc. NY. ISBN 0-486-645614
- George F Luger, Artificial Intelligence, Pearson Education, ISBN 978-81-317-2327-2
- Hazewinkel, Michiel, ed. (2001), "Predicate calculus", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4