- Descartes's "I think: therefore I am" presupposes that he cannot be and not be simultaneously.
- Husserl's phenomenological reduction relies on being able to distinguish that which can be doubted from that which cannot be doubted--and furthermore presupposes that certitude is a more valid ground on which to build knowledge than doubt.
- Even Wittgenstein's verifiability principle must take as axiomatic the law of non-contradiction (which itself is not verifiable) in order for the process of verification to proceed.
That a thing is what it is; that a thing cannot be and not be simultaneously; that a cause exists for every effect--no culture has ever existed which did not, explicitly or implicitly, reason in accordance with these laws. Our remotest ancestors reasoned in this way. They built their mud huts--and perhaps observed that one of the huts collapsed. Whereas we would now attribute the collapse to bad geometry, they perhaps attributed the collapse to the displeasure of a god.
Regardless of whose interpretation is correct, the laws of thought remain the same. The hut did not collapse without a cause. (That is the law of causality.) To build the same hut, in the same place, under the same conditions, will bring the same result. (That is the law of identity.) The next hut will either collapse or not collapse. (That is the law of excluded middle.) But it cannot do both simultaneously. (That is the law of non-contradiction.) The rational inklings that inspired Cro-Magnons out of their caves became, in the course of time, the methodology of Aristotle: it became, simply, logic. What Cro-Magnon Man intuited, Postmodern Man has come to disavow. The schism is not merely academic but evolutionary.