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Actually, the American approach is, in a sense, technologically
obsolete. In the old days, when type was set from individual pieces
of lead, one for each character, Americans put punctuation inside
quotation marks so it wouldn't fall off the end of the line without
being noticed. (Obviously, much punctuation isn't at the end of the
line, but people like consistency.) The British approach is, as
has already been noted, more logical, and few people are setting
type in that manner these days. Nonetheless, I'm sure I'm not the
only editor to whom punctuation outside the quotes looks wrong.
(Just for good measure, even in the US, question marks and semicolons
that are not part of the quoted material go outside the punctuation.)
Fortunately, my employer agrees with me, so I don't have to spend my
days feeling vaguely uncomfortable, or retrain my reflexes.