In
classical deductive logic, a
consistent theory is one that does not contain a
contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent
if and only if it has a model, i.e., there exists an
interpretation under which all
formulas in the theory are true. 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 and only 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.
