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{{Other uses|Closure (disambiguation){{!}}Closure}} | |||
'''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. | |||
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>. | |||
== Epistemic closure == | |||
{{main|Epistemic closure}} | |||
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. | |||
==References== | |||
{{reflist}} | |||
[[Category:Concepts in logic]] | |||
[[Category:Deductive reasoning|Closure]] | |||
[[Category:Logical consequence]] | |||
[[Category:Propositional calculus]] | |||
[[Category:Set theory]] | |||
Revision as of 11:32, 22 April 2013
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Deductive closure is a property of a set of objects (usually the objects in question are statements). A set of objects, O, is said to exhibit closure or to be closed under a given operation, R, provided that for every object, x, if x is a member of O and x is R-related to any object, y, then y is a member of O.[1] In the context of statements, a deductive closure is the set of all the statements that can be deduced from a given set of statements.
In propositional logic, the set of all true propositions exhibits deductive closure: if set O is the set of true propositions, and operation R is logical consequence (“”), then provided that proposition p is a member of O and p is R-related to q (i.e., p q), q is also a member of O.
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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- ↑ Peter D. Klein, Closure, The Cambridge Dictionary of Philosophy (second edition)