What is the situation called where a strong party uses logical reasoning based on a flawed foundation?
A partial solution:

As for the logical fallacy (unrelated to "strong" and "weak" parties), you seem to be describing a "Fallacy of Presumption," where one makes an argument based off of a false/questionable/undefended presumption. There are several more specific cases of this type of fallacy. A website that describes a few is listed below (found via a Google search).

http://www.philosophypages.com/lg/e06b.htm