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First-order logic —also called predicate logic, predicate calculus, quantificational 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 ...
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols [clarification needed] in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form , where is a relation symbol ...
Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable. The validities of monadic predicate calculus with identity are decidable, however. This system is first-order logic restricted to those signatures that have no function symbols and whose relation symbols other than equality never take ...
By contrast, for the three-variable fragment of first-order logic without function symbols, satisfiability is undecidable. Counting quantifiers. The two-variable fragment of first-order logic with no function symbols is known to be decidable even with the addition of counting quantifiers, and thus of uniqueness quantification. This is a more ...
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic . The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of ...
Definition as FOL theory. The pure theory of equality contains formulas of first-order logic with equality, where the only predicate symbol is equality itself and there are no function symbols. Consequently, the only form of an atomic formula is where are (possibly identical) variables. Syntactically more complex formulas can be built as usual ...
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using logical rules and axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable using the rules of logic.