Law of Clavius explained

''Clavius' law'' is just a variant of proof by contradiction or reductio ad absurdum: If $\sim\! P$ implies $P$, then $\sim\! P$ is inconsistent (because it then also implies the contradiction $\sim\! P\land P$). Therefore $P$.

Formally: $\sim\! P \Rightarrow P\vdash P$.

You should be able to formulate any proof by contradiction as an instance of Clavius'law, but it might read a bit awkward in some cases.

Cantor's diagonal argument that $\mathbb R$ is not enumerable, written as an instance of Clavius' law:

Assume you have an enumeration of $\mathbb R$. Then, by diagonalization you can construct a real number that is not in that enumeration. Therefore you don't have an enumeration of $\mathbb R$. Therefore: $\mathbb R$ is not enumerable.

(Doubly negate the first premise to exactly match the logical form of Clavius' law, if you are picky.)

Argument against (absolute) Relativism, written as an instance of Clavius' law:

"No sentence is true" $\Rightarrow$ "There are true sentences" (for instance: "No sentence is true"). Therefore there are true sentences. ref