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 .[citation needed] 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 ().
Examples
[edit]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
[edit]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.