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

The deductive closure of a theory ⁠ T {\displaystyle {\mathcal {T}}} ⁠ is often denoted ⁠ Ded ⁡ ( T ) {\displaystyle \operatorname {Ded} ({\mathcal {T}})} ⁠ or ⁠ Th ⁡ ( T ) {\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 ⁠ T {\displaystyle {\mathcal {T}}} ⁠ is exactly the closure of ⁠ T {\displaystyle {\mathcal {T}}} ⁠ with respect to the operation of logical consequence (⁠ ⊢ {\displaystyle \vdash } ⁠).