The PNC: A property cannot both belong and not belong to a subject at the same time and in the same respect
The PNC is the most certain principle, i.e.:
1) It is not possible to be wrong about it
2) If you know anything, you know it (all statements of knowledge imply it)
But there are those who deny it, including para-consistent logicians. One possible objection is that this law only appears to apply to our use of language, and the way in which we frame our reality. One might also go so far as to say that the principle is, or could be, both true and false. Quantum mechanics might suggest that often we can't know either way, and so must deem the principle to be void and meaningless in many situations.
In response to the kinds of criticisms he could have anticipated or received, Aristotle said that those who deny the principle merely think they deny it, and that since they have knowledge they in fact do believe it. And in response to all those criticisms that still want to retain an understanding of truth and falseness, one can say that by simply asking the question, is the PNC true, you are assuming that it must either be true or false, and so affirming its truth.
But if the principle is so self-evident, and the criticisms so weak, why bother to mention it? Aristotle thinks that not only will it give us a clearer understanding of the role the principle plays in logic and therefore in philosophy, but also once irrefutably established it informs other discussions, in particular:
If the PNC is true, then not everything can be changing in every way, and so radical change must be false (more on this in the next article on change).
If the PNC is true, then relativism as a theory of reality must be false. If Jack believes the sun is shining, and Jill believes it is not shining, then one of the two must be false. This in particular is where the PNC contradicts physics, and so questions either the PNC, the theories of quantum physics, or the nature of the quantum level in relation to the super-quantum level.