Английская Википедия:Deductive closure

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Шаблон:Short description

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

The deductive closure of a theory Шаблон:Tmath is often denoted Шаблон:Tmath or Шаблон:Tmath.Шаблон:Citation needed This is a special case of the more general mathematical concept of closure — in particular, the deductive closure of Шаблон:Tmath is exactly the closure of Шаблон:Tmath with respect to the operation of logical consequence (Шаблон: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

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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

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