I'm not sure Gödel's incompleteness theorems have anything to do with this. After all, they don't show that any system of reasoning is inconsistent - just that it can't prove its own consistency and that there are statements that can be neither true nor false. Even then, it only applies when your system is able to express arithmetic - for basic syllogisms (like "all men are mortal. socrates is a man. therefore socrates is mortal.") this doesn't apply, as far as I understand.