Does it make sense for "If $p$ and not p, therefore $q$" to be a valid argument? In propositional logic, an argument is invalid iff there is any instance where all the premises are true and the conclusion is false, if the set of premises is $$\left\{\:p,\:\neg p\:\right\}$$
then the argument is true for any conclusion $q$, this doesn't make sense in the real world (at least compared to other valid arguments), is there anything that sets this argument apart from other valid ones (that make sense to be valid)?

Assume otherwise. Then $\lnot((p\land \lnot p)\to q)$ holds, which is only true when  $p\land \lnot p$ is true while $q$ is false. But $p\land \lnot p$ is true exactly when both $p$ and $\lnot p$ are true, a contradiction. Thus $(p\land \lnot p)\to q$.
