Deductive closure

Short description: Set of logical formulae containing all formulae able to be deduced from itself

In mathematical logic, a set 𝒯 of logical formulae is deductively closed if it contains every formula φ that can be logically deduced from 𝒯; formally, if 𝒯⊢φ always implies φ∈𝒯. If T is a set of formulae, the deductive closure of T is its smallest superset that is deductively closed.

The deductive closure of a theory 𝒯 is often denoted Ded⁡(𝒯) or Th⁡(𝒯). 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.[1]

Deductive closure is a special case of the more general mathematical concept of closure -- in particular, the deductive closure of 𝒯 is exactly the closure of 𝒯 with respect to the operation of logical consequence (⊢).

## 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

Main article: Philosophy: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.

## References

1. ↑ First-order theory at PlanetMath.org.

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