Deductive closure

In mathematical logic, a set ${\displaystyle {\mathcal {T}}}$ of logical formulae is deductively closed if it contains every formula ${\displaystyle \varphi }$ that can be logically deduced from ${\displaystyle {\mathcal {T}}}$, formally: if ${\displaystyle {\mathcal {T}}\vdash \varphi }$ always implies ${\displaystyle \varphi \in {\mathcal {T}}}$. If ${\displaystyle T}$ is a set of formulae, the deductive closure of ${\displaystyle T}$ is its smallest superset that is deductively closed.

The deductive closure of a theory ${\displaystyle {\mathcal {T}}}$ is often denoted ${\displaystyle \operatorname {Ded} ({\mathcal {T}})}$ or ${\displaystyle \operatorname {Th} ({\mathcal {T}})}$.^{[citation needed]} This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of ${\displaystyle {\mathcal {T}}}$ is exactly the closure of ${\displaystyle {\mathcal {T}}}$ with respect to the operation of logical consequence (${\displaystyle \vdash }$).