Abian tell us there are no paradoxes in the real world. But the real world and the theories about it is two different thing. It is possible to construct many axiomatic systems, and they are more or less capable to describe the real world. The modern proposition logic is one of those. In the theory itself it is impossible to deduce any paradox.
Paradoxes are problems of our modeling or understanding of a particular real-world situation.
This does no mean they are not important, and we can't easily discount their existence in various formalizations. They are major factors in driving such formalizations forward. Many so-called paradoxes are essentially unsatisfiable propositional formulae. You might look that word up in a dictionary. It has to do with perceptions and understanding.