This sounds a lot like Descartes' demon scepticism in the First Meditation. There he worries that he might be mistaken even about things that seem most obvious, like simple mathematical sums. If one suffers from this kind of 'hyperbolic doubt', a doubt that even what seems obviously contradictory might be consistent, and that even what seems obviously consistent might be contradictory, then it is very difficult to see how one could reassure oneself by argument, since one might worry hyperbolically about that very argument.

Perhaps if we did suffer from hyperbolic cognitive disability, then we wouldn't even be capable of thought; but that seems cold comfort. Let me make two slightly more encouraging points. The first is that the case for demon scepticism is not as impressive as the case for dream scepticism. For while the dream sceptic can construct a possible world where we are mistaken about the external world, there is a sense in which a demon sceptic cannot even describe the world she is worried about, since it is a world that seems impossible to her, a world where what seems impossible to her is possible. The second point is that even if, as I fear, there is no justification that would have force against a demon sceptic, it does not follow that we cannot know that p and not-p cannot both be true. For if in fact we can recognize a simple contraction when we see one, that ability may be enough: knowledge may not require that we can show that we have the ability as well.