Most people when they find a contradiction in their lives will fix the error and point themselves in the right direction. The problem comes up when people deny that they have a contradiction in their lives. Let us consider, briefly, a contradiction that exists in many American Catholics today (and most are unaware of it).
If I said that "everyone must listen to the Church" and that "not everyone must listen to the Church", then anyone who knows basic logic will see a flaw. You cannot say "yes" and "no" in the same way about the same thing at the same time. If you break that law, then you are contradicting yourself. After all, that is what the word "contradict" means: "to go against what you are saying".
Thus, when American Catholics insist that all of our politicians must listen to what the Church says (which is a Catholic concept), and at the same time hold to the modern (fairly pagan) explanation of the disestablishment clause in the first amendment, then there is a contradiction. The anti-christian interpretation of the first amendment is that neither the Church nor any of its members, can ever have any influence on the state or any of its members. You cannot tell the state "listen to the Church" and believe also that the state does not have to listen to the Church.
The first amendment was viewed a bit differently when it was first penned. It did not say that the Church cannot influence the state, but that the state cannot prevent the Church from influencing the state (go read it). There are numerous non-catholic aspects to American government and civil law, and yet if we presume that those differences are all acceptable, then we have a big problem when Church teaching conflicts with state teaching (and it does quite often!).
We must have the right interpretation of all American laws and principles, even if those things go against the Catholic faith and force us to reject them. And, in doing so, we must remember that there will be times when we have to choose between what the Church says, and what America says. When that happens, either we agree with the Church and disagree with America, or we agree with America and disagree with the Church; there is no middle ground.