it constantly boggles me how fans of a show about logical reasoning and deduction often fail to apply those skills to their analysis of the show
(yes this is a deduction post, I do have some ~advice~ at the end)
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Anderson is the one who mentions "rache."
Anderson is a dick to Sherlock, and Sherlock dislikes him - this is the First Episode and those sorts of character relationships need to be established.
Sherlock does not actually think that "obviously the word in blood is not German for revenge, it's the name rachel" - since he literally considers the German possibility first:
The "obviously" is what Sherlock outwardly portrays to others, specifically Anderson - the scene is of course a nod to the original (while switching it up a bit because that's what adaptations are meant to do) but the scene is also part of a larger plot/episode, and it helps establish the sort of relationship Sherlock and Anderson have. His dismissal of the idea is intentional and who he is dismissing is also intentional for purposes beyond being a nod to the original.
So like, someone might be able to come to the conclusion "bbc sherlock was focused on one upping and not showing love to the original" but not using this scene as an example.
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If you look at OP's thoughts as a sort of "deduction" about the show/the intents of the writers, we can note a few deduction lessons here:
1. Don't look at something in a vacumn; think about the wider context surrounding something.
Immediate context: Who is Sherlock speaking to? What is going on inside Sherlock's head surrounding his actual dialogue? (since in this show we often have access to that! use everything you have access to with deductions)
How to apply to real life deduction: look at where someone is, who someone is talking to, etc.
You can go further too - the wider context here is this is a TV show: a specific structure that needs to be followed - characters need to be established, conflict needs to happen, etc. (thus it making sense to conclude that Sherlock was written to react that way to Anderson specifically in order to establish that character relationship)
How to apply to real life deduction: the wider context would be like, what country/city is the person in, what is their culture, what is their life like beyond the instance you're observing?
2. Don't jump to conclusions (or jump to throw evidence at a conclusion). Tread carefully!
Sometimes, you might have a potential deduction, based on a few pieces of evidence, but not quite enough information to be sure. So, you look for other evidence (observations, etc.) that could support that deduction.
I don't recommend making that a habit, but if you're doing that, make sure you review the evidence carefully before using it to support your deduction!
Because here, if OP had considered the context of the scene more, OP would have realized that the scene does not support the conclusion.
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