I was arguing against a rhetorical "you", identified with the sceptic, not you personally.
That said, an extreme skepticism is altogether a different game compared to a skepticism about probabilities. The latter is reasonable; the former, although it cannot be falsified, is useless. Reality does not go away when one stops believing into it.
Logical skepticism, on the other hand, is self-defeating. To make a logical argument against the possibility of logical arguments, against the value of reasoning - that is self-contradictory.
The logical sceptic could argue that they are showing that logic is self-defeating- that when logic is taken to its ultimate conclusion it is shown to be false, therefore logically logic should be rejected. This is precisely what I would argue.
As for the matter of reality- if it exists, then of course it doesn't go away when we stop believing it. But how do we know that?
I've raised arguments for philosophical scepticism before, which have mostly been argued against in a Popper-esque manner of arguing that even if we don't know anything with certainty, we can have legitimate knowledge on probabilities.
The problem with this, however, is how you answer a sceptic about the notion of probability having a correlation with reality. Probability depends upon axioms of probability- how are said axioms to be justified? It can't be by definition, or it has no correlation to reality.