# Deductive 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`.^{[1]} 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 (“”), then provided that proposition `p` is a member of `O` and `p` is `R`-related to `q` (i.e., p q), `q` is also a member of `O`.

## 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.

## References[edit]

**^**Peter D. Klein,*Closure*,*The Cambridge Dictionary of Philosophy (second edition)*