Buddhist logic and Nagarjuna

In this article http://aeon.co/magazine/world-views/logic-of-buddhist-philosophy/ Philosopher/Logician Graham Priest talks about some of the differences that Western logic has with Buddhist logic. Quickly said: Western/Aristotelian logic doesn't tolerate contradictions (principle of non-contradiction) and everything has to be either true or false (principle of the excluded middle), but Buddhist logic follows a system called the catuskoti which implies that statements can be true, false, true and false, or neither true nor false. Tibetan philosopher Gorampa even included the 'ineffable'. My question is this: In buddhist logic, where does "hey man, that's illogical!" fit in? I mean, it pretty much seems everything can be valid in Buddhist logic, and there's just no place for the illogical, which pretty much points to the inefficacy of that system.