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.
Originally the Catuṣkoṭi or Tetralemma was just an indication of all possible combinations of two predicates, and obviously, if these two predicates don't contradict, there is no problem.

Now, the Buddha himself does make seemingly contradictory statements, like when he refuses to answer the questions of Vacchagotta in SN 44.10, who asks first whether there is a self, then whether there is no self. Questioned by Ananda as to why he did not answer, the Buddha effectively claims, that both answers would have been wrong.

In this case, since there is only one predicate (and its contradiction), there do arise logical problems. The only attempt, to solve these kinds of problems I am aware of is in the article 


  Klaus Butzenberger: Einige Aspekte zur catuskoti unter besonderer Berücksichtigung Nagarjunas, in: Synthesis Philosophica 1990, 567–580.


If you know German, then go read it. If not, let me summarize that it attempts to solve these problems in three ways: by classical logic, non-classical logic and thirdly by admitting its insolubility, but Butzenberger admits, that each of these attempts at solution does remain unconvincing to fruitless.

This might - now here comes my 50 cents - be due to the case, that the Tetralemma is used by a great number of authors in Indian logic and they do not handle it the same way. In the case of the Buddha's refusal to answer, it might be conjectured, that the Buddha is talking in two different levels of truth: a conventional truth and an absolute truth. So in order to remain truthful on the absolute level, we may have to accept contradictions on the conventional level, if - what the Buddha sometimes blames questioners for - the question is posed in a wrong way.