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.