The government doesn't think inflation is bad

There is no contradiction within the banking system because there is no reason it would make sense. (There is no reason for it to be logical.) If people (banks) can print money this is not a contradiction because there is no reason for it to be consistent. If a shop lets people steal its produce this is not a contradiction in itself... even if the shop is unaware that people are stealing from it still there is no contradiction. It is only possible for there to be a contradiction if the shop (the government) disagrees with something that is happening. If the shop objects to the theft then there is potentially a contradiction to be resolved. If the shop is fine with people stealing from it this is a stable condition... where everyone is being consistent.

This is the vulnerability of paper money... either banks or the government itself can print it and there is nothing to be done (no valid complaint). Paper money would be less 'bad' if banks cannot print it but even if just the government can print it paper money is still bad. It makes no sense that the government would let the banks print money (it serves neither the government or the people) but that is not an absolute contradiction only a subjective (in reference to the government) contradiction. If it is good to preserve the currency by preventing private counterfeiting then it would seem consistent to also prevent banks from doing so. Only if inflation is not bad (and the currency is not worth preserving) would it make sense to allow banks to print money.