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I'm not completely dumb when it comes to positional play but I never understood what I would get in return for, say, Bd3 Bxd3 in the Caro Kann (e.g. Tal Variation). Or Bd6 Bxd6 in the London. (In the latter case, I could say "one less piece about to sacrifice itself somewhere in my castling position", but the former?)

The short answer might be "A statically bad bishop can still be good dynamically", but please, no stock phrases :-) So what is the exact reason (in the Caro Kann)?

Related but unrelated (I see no tactical reasons concerning the Caro Kann, at least no immediate)

Related but unrelated (So, there they play Be2. Chess is so illogical... :-)

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This is perhaps more of a language problem rather than a chess one: we tend to call bishops that have pawns on their same color as "bad" and those who don't as "good". This correlates most of the time with how strong the bishops actually are, but not always

Taking the London example as reference, how "good" would Black's bishop be if it was passively sitting on e7? Note how Black doesn't try to change the bishop if they decide to develop it via g7 instead.

On the other hand, in a line like 1.d4 d5 2.Nf3 Nf6 3.Bf4 e6 4.e3 Bd6, the bishop trade allows Black to fight for the e5 square. If White takes then the queen goes to d6 already tartetting e5 with an extra piece. If White retreats, then Black wins a tempo it can use to continue with ...c5, ...Qc7 and ...Nd7. Note that if Black is successful in pushing ...e5, suddenly the "bad" bishop on c8 gets a new life.

We could also say that the White bishop on f4 is a strong piece. Even though it "crashes" into its own central pawn chain, it's sitting outside of it, so the only movements it has restricted are Be3, Bd2 and Bc1 (which White would almost never want to do either way). More importantly, it's controlling the e5 square which is critical in all of White's plans.

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