AA World Services first published the big book in 1939. They did not renew the copyright on it and this edition fell into the public domain. In fact I could reprint the entire big book from 1939 if I wanted. AAWS for whatever reason has brought pressure to bear on SR and other sites to have any quotes refer to "big book edition 1", there is obviously a legal reason behind this but I do not know what it is.
Even then, you can take small snippets of the book or paraphrases of the book, even the newest edition from 2004, post them and it is not a copyright violation. It is completely legal under the fair use doctrine which has been all the way to the US Supreme Court.
It is a good thing I don't run this website, I would not comply, but I when I post here and quote the BB, I do.