Oh my gosh, Nancy..."thank-you" so very much with the info on this. I've been searching for years to find him on ANY census (1910, 1920, etc...).
I would definitely say that was an error.All of the info I have on Arthur Edward Martin (my grandfather), from his war records and 20-years in the Navy, to his death certificate to his parents names on his own marriage certificate, says that his father was a William Martin.
With his name clearly stated as "Martin" in this census you found, it's clear that William MIDDLETON was not his biological father anyway.
Perhaps William Martin married Anna Clark (or Martin) as Ernie Laskey states...and got a divorce, or he died...certainly there must be some court records about this.
Anybody you know "work" Gibson County courthouse(s) ??