# Deductive closure

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

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

## 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.