This is obviously the grandchild of John Richardson (1791-1841) on whose page he is listed, but he is listed on both the 1850 Hardin Co., KY and the 1860 Coles Co., IL Census with the family.
The author favors the possibility that William J. Richardson is the child of Samuel B. and Elizabeth (Funk) Richardson. If it were found that the middle initial "J." stood for "John" then he could have been named after the two grandfathers the same as John William Richardson (1846-1926), the first born of Daniel L. and Lydia A. (Funk) Richardson... Samuel B. Richardson is shown living alone in the 1850 Hardin Co., KY Census. Double cousins i.e. brothers and sister who marry into another family occurred in this and the generations before and since , for example: Elizabeth (Linder) Richardson Sawyer and the Richardson and Peters marriages.
Sam Daugherty says that the Hardin County Marriage books are wrong and Samuel B. Richardson did not marry Elizabeth Funk. Sam believes that Elizabeth Funk married a "Cocanougher" (sic).
Other researchers have noted William J. Richardson, as the possible son of Joseph L. Richardson or other sons of John Richardson, to an unknown wife who name and marriage are not recorded...
Some have even said that this was the child of Joseph L. Richardson with "Sarah". The author checked Broderbund's CD #229 , "Marriages, Southern States Early - 1850 for Kentucky, North Carolina, Tennessee, and Virginia (Including West Virginia)". No Joseph L. Richardson was found. Nor was a Joseph Richardson marring a "Sarah" between 1835 - 1847, which seems the reasonable period for Joseph L. Richardson to have married.
The author also checked the "Hardin County, Kentucky Marriages 1830 -1850" by Mary Josephine Jones (1996) in which there was no Joseph Richardson who was married during that period, but on page 133, the book reads:
Mrs. Jones also has the correct middle initial for Daniel Richardson in the citation for "Daniel L. Richardson with Lydia Ann Funk", in this edition!