en>JaGa: /* See also */ disambiguate per WP:INTDABLINK

2014-12-29T19:56:24Z

See also: disambiguate per WP:INTDABLINK

← Older revision		Revision as of 21:56, 29 December 2014
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			== グッチウォレット財布 ==

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14.139.63.2: /* Curies as a measure of quantity */

2014-02-23T08:04:20Z

Curies as a measure of quantity

Show changes

en>Helihax: Changed descriptors referring to radioactivity in the human body

2013-12-22T03:39:39Z

Changed descriptors referring to radioactivity in the human body

New page

In [[logic]], an '''inverse''' is a type of [[conditional sentence]] which is an [[immediate inference]] made from another conditional sentence. Any conditional sentence has an inverse: the [[contrapositive]] of the [[Conversion (logic)|converse]]. The inverse of <math>P \rightarrow Q</math> is thus <math>\neg P \rightarrow \neg Q </math>.

For example, substituting propositions in natural language for logical variables, the inverse of the conditional proposition, "If it's raining, then Sam will meet Jack at the movies" is "If it's not raining, then Sam will not meet Jack at the movies."

The inverse of the inverse, that is, the inverse of <math>\neg P \rightarrow \neg Q </math>, is <math>\neg \neg P \rightarrow \neg \neg Q </math>. Since a double negation has no logical effect, the inverse of the inverse is logically equivalent to the original conditional <math>P \rightarrow Q </math>. Thus it is permissible to say that <math>\neg P \rightarrow \neg Q </math> and <math>P \rightarrow Q </math> are inverses of each other. Likewise, we may say that <math>P \rightarrow \neg Q </math> and <math>\neg P \rightarrow Q </math> are inverses of each other.

The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. But ''the inverse of a conditional is not inferable from the conditional''. For example, "If it's not raining, then Sam will not meet Jack at the movies" cannot be inferred from "If it's raining, then Sam will meet Jack at the movies." It could easily be the case that Sam and Jack are attending the movies no matter the weather.

In [[traditional logic]] only forms A and E have an inverse. To find the inverse of these [[categorical propositions]] one must: replace the subject and the predicate of the invertend by their respective contraditories and change the quantity from universal to particular<ref>Toohey, John Joseph. An Elementary Handbook of Logic. Schwartz, Kirwin and Fauss, 1918</ref>.
*All ''S'' are ''P'' (''A'' form) becomes Some non-''S'' are non-''P''
*All ''S'' are not ''P'' (''E'' form) becomes Some non-''S'' are not non-''P''

== See also ==
* [[Contraposition]]
* [[Converse (logic)]]
* [[Obversion]]
* [[Transposition (logic)]]

==Notes==
{{Reflist}}

{{DEFAULTSORT:Inverse (Logic)}}
[[Category:Immediate inference]]

{{logic-stub}}

Curie - Revision history

en>JaGa: /* See also */ disambiguate per WP:INTDABLINK

14.139.63.2: /* Curies as a measure of quantity */

en>Helihax: Changed descriptors referring to radioactivity in the human body