The variable a in the previous formula can be universally quantified, for instance, with the first-order sentence "For every a, if a is a philosopher, then a is a scholar". Quantifiers can be applied to variables in a formula. 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". This formula is a conditional statement with " a is a philosopher" as its hypothesis, and " a is a scholar" as its conclusion. Consider, for example, the first-order formula "if a is a philosopher, then a is a scholar". Relationships between predicates can be stated using logical connectives. While first-order logic allows for the use of predicates, such as "is a philosopher" in this example, propositional logic does not. The variable a is instantiated as "Socrates" in the first sentence, and is instantiated as "Plato" in the second sentence. The predicate "is a philosopher" occurs in both sentences, which have a common structure of " a 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. Consider the two sentences "Socrates is a philosopher" and "Plato is a philosopher". While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates and quantification.Ī predicate takes an entity or entities in the domain of discourse as input while outputs are either True or False. 9 Automated theorem proving and formal methods.8 Restrictions, extensions, and variations.4.2 Hilbert-style systems and natural deduction.3.5 First-order theories, models, and elementary classes.
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. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, into 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.įirst-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics. Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. all statements which are true in all models are provable). There are many deductive systems for first-order logic which are both sound (i.e., all provable statements are true in all models) and complete (i.e. In interpreted higher-order theories, predicates may be interpreted as sets of sets. : 56 In first-order theories, predicates are often associated with sets. 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 predicate quantifiers or function quantifiers or both are permitted. Sometimes, "theory" is understood in a more formal sense, which is just a set of sentences in first-order logic. 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.Ī 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 about them. 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", where "there exists " is a quantifier, while x is a variable.
For logics admitting predicate or function variables, see Higher-order logic.įirst-order logic-also known as predicate logic, quantificational logic, and first-order predicate calculus-is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.
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