Also "X implies Y" isn't commutative, so even if Y is true X may not be.

Plus colloquially if asked a two part "or" question, there is some ambiguity in a single part answer, but usually a "yes" will refer to the second part.

"Shall we go to the shop or the café?" "Yes." -> café.

This drives computer programmers nuts because we're so used to thinking in boolean logic terms but it's very common.