Logic is the only rational tool we have. Logic will show you the truth out of data. It is not mathematics. Mathematics is just a tool to help you think logically and clearly. At the end of the day, mathematics is just subservient to the logic. Too many people rely on mathematics more than their logic in a such a way that the servant have become the masters of the the original master, forgetting the original design. [A good analogy is in a democratic society government (Servants) becomes the master of the people (the Masters) while the original design was upside down].

If logical conclusion leads to some paradox then you still have to solve it logically. The best thing is to check the assumptions for their validity. It still involves logic but again that's the only tool you have. You have to use you all logical resources before drawing your penultimate, (if not the ultimate) conclusion. Its hard and time & effort consuming, I know! But still it worth it because that's our job, to "think". Even after exhausting all of your logical resources, if you still left with paradoxes, then don't blame the logic per se. That's the nature of reality. Learn how to deal with it, don't just complain about the best tool on the universe known to exist. And remember, Heroes never complain.

Anyway, logic will lead you to even paradoxes but it will not compromise in detecting truth, given you tried enough. Sometimes logic will say that both are true even they are mutually exclusive. This is the birth of paradoxes. Don't fall in "the danger of taking one as true but because they are mutually exclusive the other must be false" trap. Don't overuse logic. Don't overshoot. If logic tells you both are true, then keep that in the same way, of course after exhaustive use of logic and available evidences. Don't just be biased to take one as true over the other for your false comfort. It will strike you back. Learn how to live in full of paradoxes, because life itself is a paradox. When somebody knows something as true as fully as he knows himself, but still can not prove it due to logical impossibilities, then the only way he can convey the message by making up some stories, often fantastic stories. This is the birth of myth. Although the denotation of myth itself is false, but it points towards the connotation of some underlying truth to it. If you have an inquisitive and imaginative mind then you will see the truth as clearly as daylight behind the veil of myths.

Even with this best use of logic, it cannot guarantee the veracity of the starting assumptions. How can I find genuinely true assumptions? All of the mathematicians, scientists, and logicians are conveniently silent here. It is very uncomfortable for them to talk about those to general public because they don't want to expose their sham starting points. After all they have to have people's faith on them in order to keep them in business. The very starting points they are taking a truth have no scientific ground at all. It's our old good friends namely common senses and gut feelings. Again you have to use logic to probe the assumptions to check for its integrity. But again to check that we need to take even deeper assumption as the new starting point and ad infinitum. At the bottom of such inquiry you will end up with following situation:

"If logic is the tool for ferreting out truth then how can I find the assumption that will validate the logic itself".

The answer of this kind of question will lead to a strange loop which itself is a paradox. As I told you, don't overuse logic and overshoot. This paradox has to be dealt with another complementary paradox. That's the only solution. Just try to imagine why there is such a thing called logic in first place. Logic and assumption are coming from the same source except we see logic and assumption as a concretely different classes of thoughts. Actually they are the different face same coin. Logic is put there just to identify the existence of assumptions but under disguise of being a separate entity. Logic is just put there to acknowledge or appreciate the existence of the veracity of assumptions. This self realization is the marvel of duality logic and its original goal. That's the birth of something new coming out of underlying principle but the new is still part or embedded in the whole without any independent existence! But it looks like they have independent existence and the logic has the total freedom to decide which one is true. Again a paradox. He who understand and master this paradox will feel and appreciate the truth of existence of both sides of the coin, where they meet and thrive.