Deductive closure

Template:Short description In mathematical logic, a set Template:Tmath of logical formulae is deductively closed if it contains every formula Template:Tmath that can be logically deduced from Template:Tmath; formally, if Template:Tmath always implies Template:Tmath. If Template:Tmath is a set of formulae, the deductive closure of Template:Tmath is its smallest superset that is deductively closed.

The deductive closure of a theory Template:Tmath is often denoted Template:Tmath or Template:Tmath. Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of sentences), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a deductively closed theory to emphasize it is defined as a deductively closed set.

Deductive closure is a special case of the more general mathematical concept of closure — in particular, the deductive closure of Template:Tmath is exactly the closure of Template:Tmath with respect to the operation of logical consequence (Template:Tmath).

Examples

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

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.