I think Ann Logan was formerly Ann Foxley and married John Logan, Joiner in January 1843. In 1851 they are living in Hulme. I cant see any obvious connection between the families, but they may simply have been friends or neighbours.
The link I gave you was to show you where the records are held - they are not available online.
Somebody may be able to do a look-up for you if you post on the Lancashire Look-up boards, or alternatively they offer a research service, although this involves a fee. You would be looking for a bastardy record involving a female with the surname France, and a male child born on or around Feb 1819 in Manchester.
http://www.rootschat.com/links/01jzi/
