Objection 1: It seems that Catholicism and Christianity are not necessarily coincidental, for many who are Christian are not Catholic, and many who are Catholic are not Christian. Since the distinction can be observed, it follows by induction that Christianity and Catholicism are not coincidental.
This is already flawed in the frist place, as Catholics are Christians. Now, there might be Catholics who do not live Christian lives, which does not prove the case of Catholics not being Christians, as there are non-Catholic Christians who do not live Christian lives as well. In that alone the very premise of this objection collapses.
Objection 2: Vatican 2 recognizes that many Christians are outside the Catholic Church. Thus, Catholicism and Christianity are not coincidental.
Premise q falls flat because it does not follow. We can make an analogy of this by saying that Sunni Muslims are not real Muslims because they do not belong to the Shi’ite group (or vice versa), but that’s not logically true as both profess to be Muslims, and are indeed practicing Muslims. So saying that many Christians are outside the Catholic Church=the Catholic Church not Christian is absurd. We can further make an analogy by saying that since many are not Fundamentalist Christians, and Fundamentalists profess to be the only true Christians, that those who are not Fundamentalists are not true Christians. Again, flawed logic that’s based on nothing more than bias and a myopic view of things.
Objection 3: Different words have different meanings and connotations. “Catholicism” and “Christianity” are not synonymous. Thus, Catholicism and Christianity are not coincidental.
Perhaps the weakest objection of all; different words might have different meanings, but it does not follow that a different term used would make the other untrue. Again, as analogy, a Presbyterian is called as such; it has a different meaning, and aren’t synonymous. Are they not Christians then? We could further make this case for Lutherans, Methodists, Baptists, etc. Words with different connotations and meanings, yet aren’t they all Christians?
So, 3 objections that don’t add up to anything. No case, no issue. I think this thread is a foregone conclusion.