Logic is very useful to science, mathematics, and other systems needing rules. We use it everyday in its most simplest form.
However no matter how well versed in logic you are, you will never be able to win an argument or prove a point unless the other person or entity agrees to the same rules of logic as you. This is a failure of logic.
Have you ever had an argument with someone who in your opinion was using faulty logic, yet could not convince them because they didn't agree on the rules?
This problem could also stem from a different interpretation of the rules of logic. This is seen wideley in the courts of law, and most recently in the Supreme Courts decision over the American Presidential Election , which was split over the differences in how Republicans and Democrats interpret the Constitution.
This is not to suggest that logic is not useful, because it is more useful than not, but it also is not totally infallible. This is just a reflection of the creators of logic: humans, who are certainly not perfect in any or anything they do