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The song's, Auld Lang Syne, Scots title may be translated into standard English as "old long since", or more idiomatically, "long long ago” or  "days gone by" or "old times". Consequently "For auld lang syne", as it appears in the first line of the chorus, might be loosely translated as "for (the sake of) old times".

On the second stanza of the song, they should not forget the cup of kindness they've shared.