# Deductive closure

Short description: Set of logical formulae containing all formulae able to be deduced from itself

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] This 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

Main page: Philosophy:Epistemic closure

In epistemology, many philosophers have and continue to debate whether particular subsets of propositionsâ€”especially ones ascribing knowledge or justification of a belief to a subjectâ€”are closed under deduction.