Could someone please explain if Inference 2 is valid or not? I thought it would be logically valid to conclude anything based on the principle of explosion, but I'm not sure if I'm right.
I'm sure that Inference 1 is invalid, but the other inferences seem confusing. Am I on the right track with this question?
Thanks
You're right that inference 1 is blatantly incorrect ($p \lor q$ and $\lnot q$ imply $p$, not $\lnot p$) and inference 2 is correct by explosion.
For inference 3, one approach might be to introduce the assumption $\lnot r$, see where that leads in terms of $q$, and then applying a principle to remove the assumption.
As for inference 4, can you find a condition under which $p \land \lnot r$ holds? If so, then that's enough to disprove it.
You could also construct a truth table to verify whether the inference is valid.
For example, the third argument/inference is valid if and only if, the oval column below is populated only with $T$s, which is to say, if and only if its conditional $(→)$ is a tautology.
