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.