A: Its only a surface-level problem. The people in John 1 werent asking the John the Baptist the right question. They asked, simply, Are you Elijah? John answered No, because the question seems to have been asked if John was Elijah in a literal sense -- i.e., they were asking if John was a re-appearance of Elijah, the Old Testament prophet last seen going up to heaven in a whirlwind and a chariot of fire. John the Baptist was not literally Elijah; he was John ben-Zechariah, locally born and reared.
But if they had asked, Are you the antitype to which Elijah was the prototype? the answer would have been Yes. And that is the sense which Jesus uses to identify John as Elijah. Malachi 4:5-6 looms large in the background here. John /did/ acknowledge that was a forerunner -- that someone much greater was coming after him. And this forerunner-role interlocks with Malachi 3:1a. It looks like some people were interpreting Malachi to mean that Elijah would literally return to prepare the way for the Messiah, rather than interpreting Malachi to mean that someone would arrive on the scene who would be reminiscent of Ellijah. And thats understandable, since technically the text of Malachi 3:1a and 4:5 is open to either a literal or a figurative interpretation (or both).
So: was John the Baptist literally Elijah the Tishbite, who had parked the chariot in the wilderness and just showed up in the wilderness of Judea? No. But, if one is willing to look at things non-literally, in terms of typology and patterns, did John the Baptist fulfill Malachi 3:1 and 4:5? Yes.
(A less theological solution, though at the expense of John the Baptist's comprehension of his role, is to posit that John's questioners in John 1 were asking him if he was in the "Elijah" role that Malachi mentioned, and John honestly, but incorrectly, answered that he was not.)
Yours in Christ,
Waterrock
