Does it make sense for “If $p$ and not p, therefore $q$” to be a valid argument

logicpropositional-calculus

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)?

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Maybe the confusion comes from the fact that in the real world the assumptions $\{p, \neg p\}$ cannot both be true. A cat cannot be both alive and not alive. So this situation simply never occurs. It can occur in mathematics as part of a proof, or when the axiom system is inconsistent. The axiom system of the real world is consistent.

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[Math] Confusion about definition for a valid argument

You can also have a valid argument when one or more of the premises is false. In that case, the truth or falsity of the conclusion changes nothing about the validity of the argument.

Hence, it is correct to say $$(P\land \lnot C) \implies \lnot A.$$

See Wikipedia for more on when an argument is valid, which starts:

In logic, an argument is valid if and only if its conclusion is logically entailed by its premises. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.

Note that the linked entry above makes a distinction between a valid argument and a sound argument:

Validity and soundness

Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:

All animals live on Mars.
All humans are animals.
Therefore, all humans live on Mars.

The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the deduction must be valid and all the premises true.

Invalidity of propositional formula

It certainly makes sense to have terminology to distinguish truth/falsity of propositional statements from validity/invalidity of arguments that purport to be logical reasoning. On the other hand it is very common for authors to use valid to describe a compound formula with assignments of truth and falsity to its atomic propositions when it results in the overall formula being true (resp. invalid when an assignment gives a false result).

Good authors will be careful in the use of terminology (and in the definitions), but informal usage will vary. Ultimately if you want to use the terms valid/invalid for compound propositions, you should provide consistent definitions that alert readers or listeners to your meaning.

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[Math] Confusion about definition for a valid argument

Invalidity of propositional formula

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