Propositional Logic and Operators

The inverse ($\neg p \to \neg q$) of $p \to q$ is:

If it is not raining, then the ground is not wet.

The inverse, just like the converse, is also not necessarily true. The ground could still be wet from other sources.

Finally, the contrapositive ($\neg q \to \neg p$) of $p \to q$ is:

If the ground is not wet, then it is not raining.

The contrapositive is always logically equivalent to the original implication $p \to q$.

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BTW, in the implication $p \to q$, the proposition $p$ is sometimes called the hypothesis or the premise, while $q$ is called the conclusion. Also, $p$ is called a sufficient condition for $q$, and $q$ is called a necessary condition for $p$.