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 $P\; \backslash rightarrow\; Q$, the inverse refers to the sentence $\backslash neg\; P\; \backslash rightarrow\; \backslash neg\; Q$. Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other.{{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}}

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 $\backslash neg\; P\; \backslash rightarrow\; \backslash neg\; Q$, is $\backslash neg\; \backslash neg\; P\; \backslash rightarrow\; \backslash neg\; \backslash neg\; Q$, 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 $P\; \backslash rightarrow\; Q$. Thus it is permissible to say that $\backslash neg\; P\; \backslash rightarrow\; \backslash neg\; Q$ and $P\; \backslash rightarrow\; Q$ are inverses of each other. Likewise, $P\; \backslash rightarrow\; \backslash neg\; Q$ and $\backslash neg\; P\; \backslash rightarrow\; Q$ 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 cannot be inferred from the conditional itself (e.g., the conditional might be true while its inverse might be false{{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}}). 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.Toohey, John Joseph. [https://books.google.com/books?id=S6A0AAAAMAAJ&q=inverse An Elementary Handbook of Logic]. Schwartz, Kirwin and Fauss, 1918 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".