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 ⁠${\displaystyle \operatorname {Th} ({\mathcal {T}})}$⁠. Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of sentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a deductively closed theory to emphasize it is defined as a deductively closed set.

Deductive closure 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 (⁠${\displaystyle \vdash }$⁠).