 First-order logic also known as first-order predicate calculus and predicate logic is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form there exists X such that X is Socrates and X is a man and there exists as a quantifier while X is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations. The theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates define on that domain, and a set of axioms believed to hold for those things. Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. The adjective first-order distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets. There are many deductive systems for first-order logic which are both sound-all-provable statements are true in all models and complete all statements which are true in all models are provable. Although the logical consequence relation is only semi-decitable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several methodological theorems that make it amenable to analysis and proof theory, such as the Laoh and Homs-Golem theorem and the Compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peno-Harrif-Mettichanser-Mello-Friegel set theory are axiomatizations of number theory and set theory, respectively, into first-order logic. No first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axiom systems that do fully describe these two structures that is, categorical axiom systems can be obtained in stronger logic such as second-order logic. The foundations of first-order logic were developed independently by Gottelab Fridge and Charles Sanders Pierce. For a history of first-order logic and how it came to dominate formal logic, see Jose Ferreros 2001.