Inverse (logic)

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 converse. The inverse of $P\rightarrow Q$ is thus $\neg P\rightarrow \neg Q$.

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 $\neg P\rightarrow \neg Q$, is $\neg \neg P\rightarrow \neg \neg Q$. Since the double negation of any statement is equivalent to the original in classical logic, the inverse of the inverse is logically equivalent to the original conditional $P\rightarrow Q$. Thus it is permissible to say that $\neg P\rightarrow \neg Q$ and $P\rightarrow Q$ are inverses of each other. Likewise, $P\rightarrow \neg Q$ and $\neg P\rightarrow Q$ 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. 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, where there are four named types of categorical propositions, 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 contradictories and change the quantity from universal to particular.^[1]

• 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

1. Toohey, John Joseph. An Elementary Handbook of Logic. Schwartz, Kirwin and Fauss, 1918