A good example of a refutation from the inconsistent comes in Luke 20:41-44. Jesus is apparently responding to a certain conception of the Messiah as the Son of David. In what I believe is an attempt to confuse his opponents, Jesus refutes the concept of the Messiah as the son of David by pointing out inconsistencies in his opponents narrative.
Luke 20:41 Then he said to them, “How can they say that the Messiah is David’s son?In this case the refutation lies in the inconsistency of David calling his son Lord.
Luke 20:42 For David himself says in the book of Psalms,
‘The Lord said to my Lord,
“Sit at my right hand,
Luke 20:43 until I make your enemies your footstool.” ’
Luke 20:44 David thus calls him Lord; so how can he be his son?”