Set of logical formulae containing all formulae able to be deduced from itself
In mathematical logic, a set T {\displaystyle {\mathcal {T}}} of logical formulae is deductively closed if it contains every formula φ {\displaystyle \varphi } that can be logically deduced from T {\displaystyle {\mathcal {T}}} ; formally, if T ⊢ φ {\displaystyle {\mathcal {T}}\vdash \varphi } always implies φ ∈ T {\displaystyle \varphi \in {\mathcal {T}}} . If T {\displaystyle T} is a set of formulae, the deductive closure of T {\displaystyle T} is its smallest superset that is deductively closed.
The deductive closure of a theory T {\displaystyle {\mathcal {T}}} is often denoted Ded ( T ) {\displaystyle \operatorname {Ded} ({\mathcal {T}})} or Th ( T ) {\displaystyle \operatorname {Th} ({\mathcal {T}})} .[citation needed] 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 {\displaystyle {\mathcal {T}}} is exactly the closure of T {\displaystyle {\mathcal {T}}} with respect to the operation of logical consequence ( ⊢ {\displaystyle \vdash } ).
Examples[edit]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[edit]Main article:
Epistemic closureIn 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]This mathematical logic-related article is a stub. You can help Wikipedia by expanding it.
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