Dear ol' Mycroft's conclusions are usually right, but Sherlock would do everything, actively, to prove his conclusions, while Mycroft would rather consider himself WRONG than to get off his armchair and PROVE it right.
"Everyone will bare witness to the fact that I am acting in self-defense...
No, sir, a gentleman! Only a ruffian deals a blow with the back of the hand,
a gentleman uses the straight left - and I, sir, am a gentleman."