Consistency

In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory

T

{\displaystyle T}

is consistent if there is no formula

φ

{\displaystyle \varphi }

such that both

φ

{\displaystyle \varphi }

and its negation

¬

φ

{\displaystyle \lnot \varphi }

are elements of the set of consequences of

T

{\displaystyle T}

. Let

A

{\displaystyle A}

be a set of closed sentences (informally "axioms") and

⟨

A

⟩

{\displaystyle \langle A\rangle }

the set of closed sentences provable from

A

{\displaystyle A}

under some (specified, possibly implicitly) formal deductive system. The set of axioms

A

{\displaystyle A}

is consistent when there is no formula

φ

{\displaystyle \varphi }

such that

φ

∈

⟨

A

⟩

{\displaystyle \varphi \in \langle A\rangle }

and

¬

φ

∈

⟨

A

⟩

{\displaystyle \lnot \varphi \in \langle A\rangle }

. A trivial theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a syntactic notion, whose semantic counterpart is satisfiability. A theory is satisfiable if it has a model, i.e., there exists an interpretation under which all axioms in the theory are true. This is what consistent meant in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead.

In a sound formal system, every satisfiable theory is consistent, but the converse does not hold.