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During TAB, Sherlock can be seen as good as well as a bad detective.
Bad detective:
Just about every person from Sherlock´s surroundings considers him a crappy detective in his mind palace.
John boasts that he "persuaded the reading public than an unprincipled drug-addict is some kind of a gentleman hero" and makes many remarks from which we could discern that he consideres Sherlock a looser junkie rather than a detective.
Mary mocks him that he is not the clever one of the two Holmes brothers, openly tells him that she was deployed to guard "Mycroft´s mad brother", solves the case of brides for him etc.
Mxcroft has his usual sneering, malicious fun at Sherlock´s address.
Sherlock, "the most observant man" doesn´t recognise that Molly is a woman there in the morgue.
In the Carmichael´s house, he somehow doesn´t pay any attention to the tracks of boots the perpetrators had to leave on carpets and in the rest of the house (the kitchen where John waited had to be full of them, IMHO), he doesn´t notice the shattered glass on the ground of the courtyard (I cease to believe brides managed to carry it away) etc.
He makes the weird deduction that the bride lurking inside the house can only leave through the window which he himself had broken. While in reality, in a house that big, the "ghost" could unlock any door or a window from inside and escape.
He weirdly leaves aside the fact that both clues discovered only after the bodies were already seen without them (the brides red finger, the "miss me" card) are the clear indicators of "the inside job" - the crime being perpetrated by people working in the morgue and in lady´s house.
Good detective:
Very early in the narrative, Sherlock makes remark that instead of Emilia Riccoletti, mysterious murders are normal domestic murders being blamed on a ghost. This is absolutely correct.
Sherlock immediatelly recognises Mary while John doesn´t.
I have a feeling that after seeing the red stain in a carpet in front of Lady Carmichael, Sherlock immediatelly figures out that she is the perpetrator of the crime.
In the long run, we find out that Sherlock managed to solve the problem of Moriarty´s reappearance in less then ten minutes, while unconscious.
So what is your impression? Was Sherlock presented as a good or a bad detective in the story?
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I felt that Sherlock was presented as confused and out of his depths throughout the episode, both in the Victorian scenes and the present-day scenes (including the present-day MP scenes).
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I didn't think was presented too much as a detective in this episode, but rather as an emotional being.
If I'm being completely honest, I think his deductive skills were not as acute as they are in reality, but I wouldn't think him a lousy detective either, mostly because all these comments are generated by Sherlock himself, who we know has a lot of self-loathing in him.
As for missing the obvious, I find it's also self-depreciation.
I am certain that, had this case happened IRL and not in his mind palace -under drug influence, as well, no matter that they help him focus, if he took too much which he obviously did since he "probably just OD'd" his skills would be very much lessened- Sherlock would not have had any trouble solving it, nor would he have...miscalculated or overlooked things or people.
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I feel like Sherlock was really shown as a more human human being than we've seen him before... that there is an outer shell and an inner being.
Maybe that was why I was crying after the episode was over (other than the fact that Sherlock was back on the sauce); because I got to see how vulnerable he is... There's a side he shows the world... the smug 'bast*rd' who solves everything and everyone in a second and doesn't care whose toes he steps on...
And then there's the Sherlock whose entire world can be rattled by a ghost of his past. Who needs to see himself through his friends eyes because perhaps he can't bear to look at himself through his own.
I think in the real world Sherlock would have solved the case a lot quicker, without detours and distractions... and the need to 'go deeper' and take drugs.
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I think, overall, good.
It's his mind palace - all the deductions are his, including any done by other people. His mind palace John spots Molly, for instance, but Sherlock creates that deduction and watches it in his mind palace. In "real life" there never was a Victorian John spotting a crossdressing Molly. In this strange mind palace, Sherlock seems to be deducing by telling himself the story.
This reveals some things about Sherlock (his insecurity around Mycroft and his view of John, for instance), but does not make him a bad detective - quite the opposite, because he does the whole thing himself with no help from any "real" person.