There are no known documents that assure the relationship of Francis to John Goodwin. There seems to be more against the fact than for it. While I have found no absolute proof that John and Francis were brothers, I have no absolute proof that they were not.
Francis served in the Revolutionary War and was listed as a "Frontiersman". He was also called an "Indian Spy". He was turned down in his request for a pension. John Goodwin had vouched for him. His application states he was born in the town of Derry, in the County of Londonderry, in the Province of Ulster in that part of Great Britain called Ireland in 1761. Francis says in his application that he is not related to the John Goodwin who vouched for him.
The 1850 Census shows Francis as 90 years old and living in Taylor County, Virginia having been born in Ireland. Living in his household were Nancy Goodwin, 58 years old and born in Virginia,his second wife.. also in his household was Harrrison Goodwin, 25 years old who may have been his grandson, born of Francis II in 1825.
William Harrison Goodwin is listed elsewhere as the son of Francis II.
Francis married Nancy in 1821 when he was 61 and Nancy was 29. If Harrison is Francis' son he was born when Francis was nearly 65 years old and Nancy was 33.
Information on the families of daughters Temperence and Amanda provided by Richard Tull.