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.
- Eric M. Hammer: Semantics for Existential Graphs, Journal of Philosophical Logic, Volume 27, Issue 5 (October 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