A famous example is the Liar paradox: “This is a lie.”
If that is a lie, then it is a lie that it is a lie, so it is not a lie.
But if it is not a lie, then what it says is false, so it is a lie.
Whereas, if it is not a lie, then it is not the case that it is a lie.
Contradiction! So, our logic gives us paradoxes. But, so what?
Even highly evolved apes would hardly have a perfect logic.
Most modern thinkers think of themselves as highly evolved apes, in a purely material world that just happens to exist. They/we think so because they/we have taken logical looks at the evidence; but, what happens to our image of ourselves as scientific if we can play fast and loose with logic? We want to be very careful in any choice to embrace illogicality in our thinking; we want, ironically, to make a very logical choice about any such thing.
In taking logical looks at the world, we may well have given low prior probabilities to the existence of a Creator, maybe following Richard Dawkins; but, what if there is a logical proof that there is a transcendent Creator? That would change everything! Thoughts that such a proof could not be possible are naturally based on those very low priors, and at the end of the day there is such a proof. Still, were we to simply refuse to countenance the possibility of a transcendent Creator, then any such proof would become just another logical paradox; and such simple refusals are not necessarily illogical:
I see a tree, so I know it is a tree; that is certainly rational. I cannot rule out its being an alien quasi-stick-insect of a very convincing kind, but so what? I have been assuming that it is no such thing; and even now, after thinking of this particular possibility, I still have no idea how unlikely, or likely, it really is, and so I still cannot do any better than to continue to make that assumption. Making it makes my knowledge a sort of gamble, but such is human knowledge in the real world.And yet, where do we draw the line? If we had a proof that the tree was really an alien quasi-stick-insect, then surely that assumption would then be illogical. What if you have a very good argument for something that I really do not like; can I take that dislike to trump your argument? Surely not. My dislike can of course motivate me to believe that there is probably a fatal flaw in your argument, but I really should be bothered by the excellence of your argument. Surely I should not just exhibit my dislike, and observe that to err is human. Surely we should all assume logic. Even if it is flawed, it is our logic, and so assuming it would just be the most human error; and maybe our logic is not that bad. Let us look again at the Liar paradox:
“The assertion you are currently considering is not true.”
Let that assertion be called “L” so that: L is true if, and only if, L is not true.
Were “true” a vague predicate, L would be true insofar as L was not true,
from which it would follow logically that L was as true as not.
It follows logically that if “true” could be a vague predicate,
then the Liar paradox is actually a proof by reductio ad absurdum
that it is a vague predicate: then, and only then, is there no contradiction.
Is it only then? That is, after all, why this is a paradox. You could say that the meaning of “true” rules out truth being vague; and of course, truth itself is not normally vague, far from it: to want the truth is to want things to be made clear. But Liar sentences are deliberately constructed to be paradoxical, when they are not simply mistakes that should be rewritten to make them clearer. And consider the following, which is similarly far removed from our normal uses of language:
Consider “It is an apple”
as an answer to the question “What is A?”
where A is originally an apple,
but has its molecules replaced, one by one,
with molecules of beetroot.
Originally, “It is an apple” is a correct answer (or in other words, it is true that it is an apple), but eventually it is not. If that answer must be either correct or else not (if that proposition must be either true or else not true), then an apple can be turned into something else (presumably a mixture of apple and beetroot) by replacing just one of its original molecules with a molecule of beetroot, which certainly seems absurd. It is surely possible, since it does seem more plausible, that A is, at such a stage, no less an apple than it is apple/beetroot mix; that it is, at such a stage, as much an apple as not, so that the proposition that it is an apple is as true as not; what else could it be?