Deductively closed

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]} 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 (⁠${\displaystyle \vdash }$⁠).

Examples

In propositional logic, the set of all true propositions is deductively closed. This is to say that only true statements are derivable from other true statements.

Epistemic closure

In epistemology, many philosophers have and continue to debate whether particular subsets of propositions—especially ones ascribing knowledge or justification of a belief to a subject—are closed under deduction.

References

1. First-order theory at PlanetMath.