[DokEnkephalin]

Originally Posted by haguruma

They would never mistake a different person for Kanon.
Kanon's name can only be wielded by the owner of said name. No other person can carry this name.

No person would mistake Ushiromiya Kinzo by sight.
But they could lie, couldn't they? It's not a mistake if it's intentional.

Spoiler:

Left off that Rosa didn't find conclusive proof Kanon was dead. If Kanon were alive, then Genji's testimony couldn't be trusted, but Battler's spirited defense of him became much more suspicious. If Kanon really is dead then the choice between Genji and Battler is more pragmatic; which of them would be more supportive of Rosa's accusations?

Both conditionals of the xor premise are satisfied in the conclusion. This is another theme of the story, and the way magic works within the gameboard, and the reason why every fantasy being must have a vessel. Is it fantasy or mystery? The complete mathematician's answer is "yes". There are propositional chains from both conditions that will lead to the same conclusion.

EDIT: Just realized the hole in that Blue.

Spoiler:

If Kanon were alive, she wouldn't have reason to suspect Genji at all, but still more reason to suspect Battler. If Kanon were dead then Genji would be suspicious and Battler much less so for sticking up for the dead. Battler still wouldn't be unconditionally supportive with his testimony, but by now Rosa believes she knows what Genji could or could not be trusted with, his principles are very apparent and reliable. So Rosa didn't come to a clean xor, but gambled on the most likely. Taking safe bets is counter to another of the story's themes, which is why she had to be punished by conclusion.