Punctuation is supposed to remove ambiguities, and help you read and understand more easily. In your example
He said, "I've been outside.".
the collection of punctuation marks does very little for making the sentence more understandable, and (IMHO) looks ugly. The name "logical punctuation" is just a name for the style, do not take the word "logical" too literally.
The Rise of "Logical Punctuation".
and
He said, "I've been outside".
?
To me, "logical" punctuation in the first case would be as written, and in the second case would be:
He said, "I've been outside.".
indicating that both the enclosing and enclosed sentence is complete.