The evidence for Joseph is not as clear as some of the other children, but I believe it is still pretty solid. The biography of Wyatt O. Gullion (son of Joseph) published in 1887 provides in part: "Wyant O. GULLION was born on Mill Creek, in Carroll County., near the present village of English, October 11, 1812. His parents were Joseph and Margaret (Wyant) GULLION, who were natives of Pennsylvania, and of Irish-Dutch origin. Jeremiah GULLION, grandfather of the subject of this sketch, settled at Frankfort, Ky., at a very early day. His son, Joseph, the father of Wyant O., settled on the Kentucky River, at the mouth of Eagle Creek, in Carroll County, in 1794." [Kentucky: A History of the State, Battle, Perrin, & Kniffin, 7th ed., 1887, Carroll Co.] Joseph, James and Benjamin were not listed as heirs to the estate of Jeremiah, probably because they had already received land while Jeremiah was living. I know this is the case for James because I have seen the deed transferring the property (transfer of land and other property by Jeremiah and Isabelle to James in 1802 for "love and affection."). Also there is other evidence that clearly shows the Benjamin was a son. For example, his obituary lists George P. Gullion as a brother. There are also a number of other land transactions that clearly show Benjamin is a son.
Joseph and James lived near Jeremiah in the 1810 census. As far as I know, there were no other Gullions other than descendants of Edward (Jeremiah's father) during this period of time.
Most of the children were listed in the application for a pension based in Jeremiah's Rev. War service. Joseph, James and Benjamin were not listed, although other documentation clearly shows that Benjamin and James were sons of Jeremiah. Martha Ann Bower (Conner) Scholle notes in her book, "A Gullion Family History", that the children's birth records were written on two sheets of 4-1/2" X 6-1/2 inch note paper. She speculates that one of the sheets did not survive to be scanned. Given that there were other's who were clearly children of Jeremiah that also were not listed, I think this is probably correct. Of course, the purpose of the pages attached were to establish the dates of birth of Jeremiah and the fact that Isabelle was his wife. It may have been that the sheets containing the names and dates of birth of the other children were not submitted, because they were not needed.
There is probably other evidence that I have not summarized here, but at least as far as I am concerned, Joseph was a son.