In mathematical logic, a set of logical formulae is deductively closed if it contains every formula that can be logically deduced from , formally: if always implies . If is a set of formulae, the deductive closure of is its smallest superset that is deductively closed.
The deductive closure of a theory is often denoted or . This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of is exactly the closure of with respect to the operation of logical consequence ().
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.
Main article: 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.