The 2nd law is an empirical statement, based on observations like heat is never observed to flow spontaneously from a lower to a higher temperature.
No logics involved in that. Statistical physics has come a long way in explaining the 2nd law (including the increase of entropy in closed systems) on the basis of probabilistic arguments, but a full understanding, let alone a mathematical proof is still missing.
All sciences must abide by logical laws, and the second law of thermodynamics cannot be an exception. The second law of thermodynamics violates empirical principles.