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In "The Empty Hearse", the Holmes brothers play a deduction game. They were trying to deduce the characteristics of the the owner of a woollen bobble hat. They both(?) agree that he is single.

But when Sherlock met him later the cap owner said he has a girlfriend. So how can the master of deduction be wrong?

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Single is often used a status about marriage. Single / Married. One can be single in this sense and still be dating. – Meat Trademark Jan 2 '14 at 16:17
@MeatTrademark IMO in today's time I don't think we use "single" even if someone is dating! – Anirban Nag 'tintinmj' Jan 2 '14 at 16:34
Sherlock actually said "isolated", not "single", and Mycroft questioned that deduction. Which, by the way, seemed to be what Sherlock wanted. - Moments earlier Mycroft basically says that he feels no need to have friends because no one is like him (on an intellectual level). Then, through the deduction game, Sherlock gets Mycroft to say that being different doesn't necessarily mean that one has to be isolated. – Oliver_C Jan 3 '14 at 0:17
up vote 15 down vote accepted

We don't know the brothers were wrong.

At no point do we see any evidence of a girlfriend. The hat's owner claims he has one, but he might do this because he doesn't want to be perceived as a sad loner which is what he looks like. So we actually have no evidence that the brothers' deductions are wrong.

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Another explanation could be that he doesn't want to admit the HE is a fan of Sherlock Holmes, which is why he says "My girlfriend is a big fan of yours." – Oliver_C Jan 2 '14 at 18:24

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