Deductive closure

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

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

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


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 propositions—especially ones ascribing knowledge or justification of a belief to a subject—are closed under deduction.