The simplest version The Philosopher gave is this:
It is impossible to hold the same thing to be and not to be.
If something is a “Quantum Superposition” it cannot also simultaneously be “Not a Quantum Superposition”. If something is a “House” it cannot also simultaneously be “Not a House.” That is to say, somebody cannot think that a thing is a “Quantum Superposition” or “House” and also think simultaneously that is it “Not a Quantum Superposition” or “Not a House.” Somebody may certainly claim to hold both beliefs simultaneously, but they are either confused about the word simultaneously, or they are lying or switching between beliefs temporally. It is plausible, or at least intelligible, to say that at this instant I think “House” and at the next instant I think “Not a House“, and that I may oscillate between these views, but I may not hold both at the same time.
Another example: UN chief Ban Ki Moon said yesterday, “Freedoms of expression should be and must be guaranteed and protected,” but then he also said, “When some people use this freedom of expression to provoke or humiliate some others’ values and beliefs, then this cannot be protected in such a way.” So he is either deluded (or thinks his audience is), or he is lying or he changed his mind and settled on the position that speech should be regulated by the UN. Time will tell.
Dispute this? Then forget all of mathematics for a start, which depends fundamentally on this principle. Aristotle goes on to say (Metaphysics Book 4, Article 4):
There are some people who, as we have said, both maintain that the same thing can be and not be and say that it is possible to hold this view. This is the view of many who study nature. We have assumed here that it is impossible for the same thing to be and not to be at the same time, and on the basis of this we have shown that of all principles this is the least open to question.
But this principle is one which is not provable: it is one that just is true, or we believe is true, based on no evidence except our introspection.
Some people, through their lack of education, expect this principle, too, to be proved; for it does show a lack of education not to know of what things we ought to seek proof and of what we ought not.
Note to moderns: this is not an ad hominem, but an observation. Here is the kicker:
For it is altogether impossible for there to be proofs of everything; if there were, one would go on to infinity, so that even so one would end up without a proof; and if there are some things of which one should not seek proof, these people cannot name any first principle which has that characteristic more than this.
If you don’t have a ground, then down you must go and forever, as each turtle supports the weight of those above it. You will never have anything to say that anybody should ever believe, and this by your own admission. Every valid argument, on the other hand, is a chain, anchored to a immovable base of truth.
Now, if you are one of the Moons of the world and believe (based on what principle?) the law of non-contradiction false, but do not wish to defend this view, then Aristotle says you are “no better than a vegetable” and shame on you. But if you do try a defense, then Aristotle asks you to “say something that has meaning both for [yourself] and for someone else. For this [you] must do if [you are] to say anything at all.” But in doing this while you are “trying to do away with reason, [you are] also accepting it.” And obviously, if you say Non-contradiction is false, you have “conceded that something is true quite independently of the process of proof.”
Aristotle then goes on to destroy the not-non-contradiction position, adding in the end that
if all contradictory assertions made about the same thing are true [a necessary implication is non-contradiction is false], all things will clearly be one. A trireme, a wall, and a man will all be the same thing, if it is possible to assert or to deny anything of everything…For if anyone thinks that man is not a trireme, according to their theory he clearly is not one; but in this case he also will be a trireme, if the contradictory statement is true.
There is of course more, much more. Best to go to the original.