Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Deductive closure
open-in-new
<p>In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a>, a set T {\displaystyle {\mathcal {T}}} of <a href="/facts/Well-formed_formula/YpTVlzuj">logical formulae</a> is deductively closed if it contains every formula φ {\displaystyle \varphi } that can be <a href="/facts/Deductive_reasoning/HadA1zWS">logically deduced</a> 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 <a href="/facts/Superset/SJGERmbA">superset</a> that is deductively closed. </p><p>The deductive closure of a <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">theory</a> T {\displaystyle {\mathcal {T}}} is often denoted Ded ( T ) {\displaystyle \operatorname {Ded} ({\mathcal {T}})} or Th ( T ) {\displaystyle \operatorname {Th} ({\mathcal {T}})} . Some authors do not define a theory as deductively closed (thus, a theory is defined as any set of <a href="/facts/Sentence_(logic)/lnsLzvTx">sentences</a>), but such theories can always be 'extended' to a deductively closed set. A theory may be referred to as a <i>deductively closed theory</i> to emphasize it is defined as a deductively closed set. </p><p>Deductive closure is a special case of the more general mathematical concept of <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closure</a> — 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 <a href="/facts/Logical_consequence/AyDXp4dn">logical consequence</a> ( ⊢ {\displaystyle \vdash } ). </p>