It comes down to a paradox we are all comfortable with- that two things can be both alike and different. Its actually a contradiction, but because we are comfortable with it, we call it a paradox. The Law of the Excluded Middle dictates that either p or not-p is true, not both. The question is, when finding something in the world, say an apple, and calling it “p”, where is “not-p”? We could pick up an orange and say this is not-p, but both apples and oranges are fruit. Since “both are fruit” is true, there is a sense that the Law of Excluded Middle fails to be true, so we must keep looking for “not-p”. We could take the compliment of space that the apple takes up and call that “not-p” but the border, the skin of the apple, is a vague grey area where the Law of Excluded Middle fails to be true. The dimension of time adds to the problem. Eventually the apple will disintegrate or be eaten, turning into millions of other things. The apple shares its physical material and energy with the world. Since the apple shares a very real likeness with the world, you cannot claim that “not-p”, a thing that is totally different from the apple, exists. The Law of the Excluded Middle is never true, except perhaps in a relative sense.
Scientists assume that the Law of the Excluded Middle is true, and then go searching for it in matter. The search for the atoms or elements- the things that are not composite and are utterly different (p or not-p) is a continual discovery of how things are composed of each other. The rhetoric about atoms and elements usually conceal this conclusion. A mild example is on http://www.fnal.gov/pub/science/inquiring/matter/ “Particles called quarks and leptons seem to be the fundamental building blocks – but perhaps there is something even smaller.”
The invitation is to either accept the smallest particle is discovered, or look for the next smaller thing. The smaller than smaller things, or that there will always be a “next smaller thing”, (which would mean matter is ultimately composite, both alike and different) is not suggested, even though we keep finding a “next smaller thing”.
Everyone knows that two things can be both alike and different, you won’t be called stupid for thinking it. But if you look at two lines, one clearly shorter than the other, and call the longer line the shorter, some people will not respect that belief. The idea that the longer line is the shorter is no less a contradiction than the idea that two things are both alike and different. The question of intelligence becomes a question of which paradoxes or contradictions happen to be fashionable.
Where do we go from here? Read on to find how questions can replace the void left by disbelieving in Aristotelian logic.
 Thinking of “p” as “being” and “not-p” as “not-being” has its own problems discussed, along with everything in this post, in Plato’s dialogue “The Sophist”.
THEAETETUS: How, Stranger, can I describe an image except as something fashioned in the likeness of the true?
STRANGER: And do you mean this something to be some other true thing, or what do you mean?
THEAETETUS: Certainly not another true thing, but only a resemblance.
STRANGER: And you mean by true that which really is?
STRANGER: And the not true is that which is the opposite of the true?
STRANGER: A resemblance, then, is not really real, if, as you say, not true?
THEAETETUS: Nay, but it is in a certain sense.
STRANGER: You mean to say, not in a true sense?
THEAETETUS: Yes; it is in reality only an image.
STRANGER: Then what we call an image is in reality really unreal.
THEAETETUS: In what a strange complication of being and not-being we are involved!
STRANGER: Strange! I should think so. See how, by his reciprocation of opposites, the many-headed Sophist has compelled us, quite against our will, to admit the existence of not-being.