Inmathematical logic, a set oflogical formulae isdeductively closed if it contains every formula that can belogically deduced from; formally, if always implies. If is a set of formulae, thedeductive closure of is its smallestsuperset that is deductively closed.
The deductive closure of atheory is often denoted or.[citation needed] Some authors do not define a theory as deductively closed (thus, a theory is defined as any set ofsentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as adeductively closed theory to emphasize it is defined as a deductively closed set.[1]
Deductive closure is a special case of the more general mathematical concept ofclosure — in particular, the deductive closure of is exactly the closure of with respect to the operation oflogical consequence ().
Inpropositional logic, the set of all true propositions is deductively closed. This is to say that only true statements are derivable from other true statements.
Inepistemology, many philosophers have and continue to debate whether particular subsets ofpropositions—especially ones ascribingknowledge orjustification of abelief to a subject—are closed under deduction.
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