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Consistency
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For other uses, see
Consistency (disambiguation).
In The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model; this is the sense used in traditional
Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if there is no
formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.
If these semantic and syntactic definitions are equivalent for a particular logic, the logic is The completeness of and while the completeness of and consistency proofs for arithmetics restricted with respect to the Stronger logics, such as
second-order logic, are not complete.
A consistency proof is a
mathematical proof that a particular theory is consistent. The early development of mathematical
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