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

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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

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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.

References

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