Paraconsistent Logic

So, this is just going to be a sort of mental note to myself, but I've recently been interested in the following problem:

Can you convince a person, of any given thing, if they show no commitment to logical consistency?

Let me give an example. For this example, let us assume that "if Z, then X" is actually the case in this world; that is, X really follows from Z. Your interlocutor has granted to you that Z is true, but fails to acknowledge X.
You tell a person, "Oh, but that's not right, the truth is actually that X. See, it follows naturally from 'if Z, then X'."
The person replies, "No, it is the case that not-X."
"But," you say, "You have previously acknowledged that not-X is in direct contradiction of this other belief that you have, Z. And you also acknowledge that if Z, then X. I have given you, and you accept, Z. Why won't you accept X? Isn't that inconsistent?"
And they reply, "Oh I know it's inconsistent, but I really don't care about that."

How would you reply to this person? Would you characterize them as irrational, or just arational?
If they do not at all care about consistency, the whole logical apparatus falls apart. If they're okay with inconsistency, it means that " P & not-P " can be the case. Usually this would work to produce a reductio ad absurdum, but if your interlocutor is unperturbed, well, by golly! what was the point of trying to convince them in the first place?

As it turns out, there are a couple of people (most notably in Australia) working on something called Paraconsistent Logic, which is logic that can deal with contradictions.
I wonder, though, for this logic -
a) Does *everything* follow from P & not-P? Or how do you go about discriminating about what should and should not follow?
b) Would it still work for convincing someone who doesn't care about consistency?

This is an area to look into.