Editing Inverse (logic)

In [[logic]], an '''inverse''' is a type of [[conditional sentence]] which is an [[immediate inference]] made from another conditional sentence. More specifically, given a conditional sentence of the form <math>P \rightarrow Q  </math>, the inverse refers to the sentence <math>\neg P \rightarrow \neg Q  </math>. Since an inverse is the [[Contraposition|contrapositive]] of the [[Converse (logic)|converse]], inverse and converse are logically equivalent to each other.<ref name=":0">{{Cite web|url=https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458|title=What Are the Converse, Contrapositive, and Inverse?|last=Taylor|first=Courtney K.|date=|website=ThoughtCo|language=en|archive-url=|archive-date=|access-date=2019-11-27}}</ref>

For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition

:"If it's raining, then Sam will meet Jack at the movies."

would be

:"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>, and since the [[double negation]] of any statement is equivalent to the original statement in classical logic, 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, <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.<ref name=":0" /> But ''the inverse of a conditional cannot be inferred from the conditional itself'' (e.g., the conditional might be true while its inverse might be false<ref>{{Cite web|url=https://www.mathwords.com/i/inverse_conditional.htm|title=Mathwords: Inverse of a Conditional|website=www.mathwords.com|access-date=2019-11-27}}</ref>). For example, the sentence

:"If it's not raining, Sam will not meet Jack at the movies"

cannot be inferred from the sentence

:"If it's raining, Sam will meet Jack at the movies"

because in the case where it's not raining, additional conditions may still prompt Sam and Jack to meet at the movies, such as:

:"If it's not raining and Jack is craving popcorn, Sam will meet Jack at the movies."

In [[traditional logic]], where there are four named types of [[categorical propositions]], only forms A (i.e., "All ''S'' are ''P"'') and E ("All ''S'' are not ''P"'') have an inverse. To find the inverse of these categorical propositions, one must: replace the subject and the predicate of the inverted by their respective contradictories, and change the quantity from universal to particular.<ref>Toohey, John Joseph. [https://books.google.com/books?id=S6A0AAAAMAAJ&q=inverse An Elementary Handbook of Logic]. Schwartz, Kirwin and Fauss, 1918</ref> That is: 

*"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}}

[[Category:Immediate inference]]


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