Deductive closure

For other uses, see Closure.

Deductive closure is a property of a set of objects (usually the objects in question are statements). A set of objects, O, is said to exhibit closure or to be closed under a given operation, R, provided that for every object, x, if x is a member of O and x is R-related to any object, y, then y is a member of O. In the context of statements, a deductive closure is the set of all the statements that can be deduced from a given set of statements.

In propositional logic, the set of all true propositions exhibits deductive closure: if set O is the set of true propositions, and operation R is logical consequence (“$\vdash$”), then provided that proposition p is a member of O and p is R-related to q (i.e., p $\vdash$ q), q is also a member of O.

Epistemic closure

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.