Predicate logic

First-order logic—also known as predicate logic and first-order predicate calculus—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 is a quantifier while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. First-order logic—also known as predicate logic and first-order predicate calculus—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 is a quantifier while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic. A 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 defined 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 semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem 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.Peano arithmetic and Zermelo–Fraenkel 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 logics such as second-order logic. The foundations of first-order logic were developed independently by Gottlob Frege and Charles Sanders Peirce. For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001). While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification. A predicate takes an entity or entities in the domain of discourse as input while outputs are either True or False. Consider the two sentences 'Socrates is a philosopher' and 'Plato is a philosopher'. In propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate 'is a philosopher' occurs in both sentences, which have a common structure of 'a is a philosopher'. The variable a is instantiated as 'Socrates' in the first sentence and is instantiated as 'Plato' in the second sentence. While first-order logic allows for the use of predicates, such as 'is a philosopher' in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives. Consider, for example, the first-order formula 'if a is a philosopher, then a is a scholar'. This formula is a conditional statement with 'a is a philosopher' as its hypothesis and 'a is a scholar' as its conclusion. The truth of this formula depends on which object is denoted by a, and on the interpretations of the predicates 'is a philosopher' and 'is a scholar'.