Imagine you are a carpenter's apprentice. One day, the carpenter gives you a ruler and tells you to go find a four-inch long piece of stock in a large bin full of cut wood. Your boss insists you must use only the ruler he has given you to measure pieces of wood until you find one that the ruler measures as 4 inches long. However, upon inspection, you discover that the ruler is flawed. Sometimes it reads one measure, sometimes it reads another. It becomes clear that, using the provided ruler, you will never be able to be certain that a given piece of wood has any particular length, 4 inches or otherwise.You go back to the carpenter and show him the flawed ruler. He disagrees with you. The ruler has been passed down to him from every master carpenter since the first master carpenter ever, and it cannot be flawed. But since you have told him you cannot find a 4 inch long piece of stock, he now charges you to prove - using the same ruler - that there exists no 4 inch piece of stock in the bin.
It ought to be clear that this is an unreasonable demand. Your criticism of the instrument did not make any claims about the pieces of stock, so your boss has misrepresented your claim. And, given that the instrument is not sound enough to prove that any piece of wood has any particular length, it is totally unreasonable for anyone to demand you use it to disprove the existence of anything.
Next, suppose your boss tells you to go back to the now-hated bin of wood stock and bring him the most perfect piece of stock in the bin. You bring him piece after piece, but each time he shakes his head, sighs at the ineptitude of his apprentice, and sends you back to the bin to look again. Finally you ask him exactly what he means by perfect. He tells you that you must find the piece of wood that is the essence of wood stock, the piece that is in every other piece of wood, but stands above and outside of them all.
At this point, it becomes clear that the carpenter has once again given you an impossible task. The piece of wood he has asked you to find is not, in fact, a piece of wood at all, but an idea. But he has also charged you to bring him back an object. An abstract idea that is also an object defies basic logic, because it cannot be itself. No objective piece of wood can be the ideal piece of wood, because objects have definite physical proportions, but idealizations of entire categories of objects by definition do not. Idealizations of objects have had the incidental details such as the specifics of their physical dimensions removed in the process of abstraction.
However, in this case, the carpenter has not given you a flawed physical instrument. He has given you a flawed logical instrument. Whereas, in the first case, it might have been possible to create a reliable ruler, because the theory of measurement of length is logically sound, in this case it is not possible - even in principle - to create a physical instrument that can find the ideal piece of wood, because the logical instrument - the theoretical description of the object sought - is not logically consistent. At this point, it ought to be clear that there is no point in returning to the wood bin and looking for the ideal piece of wood. In this case, the apprentice would be justified in claiming, without any further investigation, that the ideal piece of wood does not objectively exist.
So it goes with any claim about the empirical existence of logically impossible beings. Without a well-defined logical instrument to define it, a being cannot exist objectively.
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