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 ${\displaystyle P\rightarrow Q}$ is thus ${\displaystyle \neg P\rightarrow \neg Q}$.

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 ${\displaystyle \neg P\rightarrow \neg Q}$, is ${\displaystyle \neg \neg P\rightarrow \neg \neg Q}$. 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 ${\displaystyle P\rightarrow Q}$. Thus it is permissible to say that ${\displaystyle \neg P\rightarrow \neg Q}$ and ${\displaystyle P\rightarrow Q}$ are inverses of each other. Likewise, ${\displaystyle P\rightarrow \neg Q}$ and ${\displaystyle \neg P\rightarrow Q}$ are inverses of each other.

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