I actually live in Massachusetts but my husband grew up on a farm that spanned Posey and Gibson Counties so I checkthe forums there often.I don't know anyone there to talk to about vital records myself but you might try the Random Acts of Genealogical Kindness site, which has at least one volunteer from Gibson County who will do vital records lookups.These volunteers work for free but may charge if they have to pay for parking or copies.Generally the lookups cost nothing if you don't need copies of the records.Here's the site:
Arthur's father certainly could have died or divorced Anna but the census did say that she was in her first marriage and Mr. Middleton indicates he was on his second marriage.Even if Arthur was illegitimate, his mother might have used the father's last name.And, if Anna's maiden name was Martin, then that could be why Arthur is listed as Martin - he took his mother's last name at the time of birth. Your best bet would be to get a copy of his birth certificate - I think if you have a one or two year span to search the volunteer or the clerk should be able to find it easily.They might also be willing to look for any record of a marriagefor Anna to a Mr. Martin if the birth record gives his name.
I'll try to look further for you in census records when I have a little more time this week. Nancy