Making Sense

Why should the world "make sense?"
Why should we expect that logic ought to apply to the real world?
It does not seem possible for it to be otherwise—but I wonder.

Perhaps logic is not applicable to the world. If that were so, we would abandon it, right? But, if we have to investigate the world to determine whether logic is useful or not, surely that revokes its a priori status.

Or, perhaps, the internal consistency of logic may retain apriority, while the applicability-to-the-real-world is what requires a posteriori verification. That is, suppose it is a necessary truth that is true within a given system, but perhaps we need to then go out and examine the world to tell whether the world agrees with that statement. Here is another way to think of it is: necessarily, logic is consistent within itself; contingently, the universe is such that logic may be fruitfully applied to it.

Could there be a world where the Law of Excluded Middle (LEM) does not hold? What would that be like? I cannot even imagine how something can be at once true and false.

(1) It is true that at this moment, at that location, there is an apple.
(2) It is false that at this moment, at that location, there is an apple.

Presumably these statements mean something like, "There is (is not) a spatiotemporal region constituted in such a way that it accords with an instantiation of our concept 'apple'." Could there be a world where a spatial region could be arranged in two substantially different ways simultaneously? If we try to imagine such a region, do we automatically start thinking of two separate spatial regions, and thus defeat ourselves by splitting into two an entity that ought to have stayed as one?

(3) The square is exhaustively red.
(4) The square is exhaustively blue.

If we accept (3), have we not already denied any possibility of it being otherwise? How could there be a world in which a square is both red all over and blue all over?

There is the occasional speculation that perhaps quantum mechanics has shown that contradictions inhere in reality—that particles really can occupy two places at once, or whatever. But this needs to be looked in to more before I say more on the subject.