You'll often hear over-IQ'd teenagers and people emotionally equivalent thereto prate about how something is "logical" or "follows logically," and by this, they mean "can be deduced by first-order logic." They're usually using this on things outside the realm of mathematics, which is the only domain where logic gives any kind of certainty. Even in mathematics, that Gödel dude showed that such certainty is fragile.
What you need in order for logic to be able to function even slightly is a high degree of confidence in the postulates you start with (very hard to obtain in areas like politics, economics, etc.) and a high degree of accuracy with the facts at hand. When either or both of these is missing, logic just causes you to look like a fool. Take the yawpings of any random Marxist or libertarian, for egregious examples.
For the mathematically inclined, however, there is a bigger problem, one which can't be solved by doing extensive research about the universe, and that's the one that Gödel proved. It goes like this: given any finite set of rules which includes the ability to do integer arithmetic, the rules will allow you to construct a statement whose truth or falsity you will not be able to deduce from the rules, no matter how hard you try or how long you take.
Oops.