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Decidability (logic)
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==Decidability of a logical system== Each [[logical system]] comes with both a [[Syntax (logic)|syntactic component]], which among other things determines the notion of [[formal proof|provability]], and a [[Formal semantics (logic)|semantic component]], which determines the notion of [[logical validity]]. The logically valid formulas of a system are sometimes called the '''theorems''' of the system, especially in the context of first-order logic where [[Gödel's completeness theorem]] establishes the equivalence of semantic and syntactic consequence. In other settings, such as [[linear logic]], the syntactic consequence (provability) relation may be used to define the theorems of a system. A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, [[propositional logic]] is decidable, because the [[truth table|truth-table]] method can be used to determine whether an arbitrary [[propositional formula]] is logically valid. [[First-order logic]] is not decidable in general; in particular, the set of logical validities in any [[signature (logic)|signature]] that includes equality and at least one other predicate with two or more arguments is not decidable.<ref>{{cite journal|journal=Doklady AN SSSR |volume=88|year=1953|pages=935–956|title=On recursive separability |language=russian|author=[[Boris Trakhtenbrot]]}}</ref> 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 more than one argument. Some logical systems are not adequately represented by the set of theorems alone. (For example, [[ternary logic|Kleene's logic]] has no theorems at all.) In such cases, alternative definitions of decidability of a logical system are often used, which ask for an effective method for determining something more general than just validity of formulas; for instance, validity of [[sequent]]s, or the [[logical consequence|consequence relation]] {(Г, ''A'') | Г ⊧ ''A''} of the logic.
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