deductive closure
{{Short description|Set of logical formulae containing all formulae able to be deduced from itself}}
In mathematical logic, a set {{tmath|\mathcal{T} }} of logical formulae is deductively closed if it contains every formula {{tmath|\varphi}} that can be logically deduced from {{tmath|\mathcal{T} }}; formally, if {{tmath|\mathcal{T} \vdash \varphi}} always implies {{tmath|\varphi \in \mathcal{T} }}. If {{tmath|T}} is a set of formulae, the deductive closure of {{tmath|T}} is its smallest superset that is deductively closed.
The deductive closure of a theory {{tmath|\mathcal{T} }} is often denoted {{tmath|\operatorname{Ded}(\mathcal{T})}} or {{tmath|\operatorname{Th}(\mathcal{T})}}.{{citation needed|date=March 2020}} 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.{{planetmath|urlname=firstordertheory|title=First-order theory}}
Deductive closure is a special case of the more general mathematical concept of closure — in particular, the deductive closure of {{tmath|\mathcal{T} }} is exactly the closure of {{tmath|\mathcal{T} }} with respect to the operation of logical consequence ({{tmath|\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|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.