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| {{Other uses|Closure (disambiguation){{!}}Closure}}
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| '''Deductive closure''' is a [[property (philosophy)|property]] of a [[set (mathematics)|set]] of [[object (philosophy)|objects]] (usually the objects in question are [[statement (logic)|statement]]s). A [[set (mathematics)|set]] of objects, <var>O</var>, is said to exhibit ''closure'' or to be ''closed'' under a given [[closure operator|operation]], <var>R</var>, provided that for every object, <var>x</var>, if <var>x</var> is a member of <var>O</var> and <var>x</var> is <var>R</var>-related to any object, <var>y</var>, then <var>y</var> is a member of <var>O</var>.<ref>[[Peter D. Klein]], ''Closure'', ''[[The Cambridge Dictionary of Philosophy]] (second edition)</ref> In the context of statements, a deductive closure is the set of all the statements that can be [[Deductive reasoning|deduced]] from a given set of statements.
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| In [[propositional calculus|propositional logic]], the set of all true propositions exhibits '''deductive closure''': if set <var>O</var> is the set of true propositions, and operation <var>R</var> is [[logical consequence]] (“<math>\vdash</math>”), then provided that proposition <var>p</var> is a member of <var>O</var> and <var>p</var> is <var>R</var>-related to <var>q</var> (i.e., p <math>\vdash</math> q), <var>q</var> is also a member of <var>O</var>.
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| == Epistemic closure == | |
| {{main|Epistemic closure}}
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| In [[epistemology]], many philosophers have and continue to debate whether particular subsets of [[proposition]]s—especially ones ascribing [[knowledge]] or [[Theory of justification|justification]] of a [[belief]] to a subject—are closed under deduction.
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| ==References==
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| {{reflist}}
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| [[Category:Concepts in logic]]
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| [[Category:Deductive reasoning|Closure]]
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| [[Category:Logical consequence]]
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| [[Category:Propositional calculus]]
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| [[Category:Set theory]]
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Revision as of 21:24, 28 February 2014
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