I've done some searching through newspaper archives and various
Ancestry.com-indexed databases. I haven't found any use of a middle name
spelled out, always F. I even have an image of his draft card and it just
reads "Joseph F. Ossanna, Jr."
However, digging a little deeper finds that his father went by the name
"Joseph FRANK Ossanna" so it's quite likely that this was the son's
middle
name (since he was a "junior"). I suspect he might have avoided spelling
out Frank because people would have difficulty undersding it wasn't short
for Francis or something.