Proof by contradiction Because the falsum $\bot$ is logically equivalent to the logical formula $\psi\land \neg\,\psi$, for every formula $\psi$, you will also encounter the following notation for the negation reasoning rule.

You hypothesize $\varphi$ and then demonstrate that for a given formula $\psi$ both $\psi$ and $\neg\,\psi$ ccan be derived, which is a contradiction. The hypothesis is evidently false, so you can conclude $\neg\,\varphi$. This line of reasoning is also known as proof by contradiction or reductio ad absurdum: you assume something to ultimately derive a contradiction. We will revisit this concept later when discussing the double negation reasoning rule.

above because it is a bit strange to derive two subconclusions in a subproof. We prefer a single subconclusion, namely the falsum $\bot$, which is equivalent to the conjunction $\psi\land \neg\,\psi$. We are of opinion that this formula is appropriately placed at the end of the subproof in the alternative notation above.