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What is a paradox?
ONE OFTEN comes across statements about something being “paradoxical.” What, exactly, is a paradox?
Generally, a paradox is something which is deemed to be contrary to reason. What, then, is reasoning?
Well, it is simply a process leading from valid basic ideas to a valid conclusion. On the face of it, then, there should be no paradoxes.
It may often happen that something seems paradoxical, but on deeper consideration, it turns out not to run counter to what should have been expected. So, can there really be statements that are logically inconsistent internally?
There were many apparent paradoxes which Aristotle identified as logical fallacies. These are not called paradoxes. However, there were some that were not so easily dealt with.
Perhaps the oldest in this category is what was once known as the Cretan, or liar, paradox. The way it goes is that a Cretan says: “All Cretans always lie.” If we assume that he is telling the truth then this statement should be a lie and hence, false.
Contrariwise, if we assume that he is lying then this statement must be a lie and so he was telling the truth. Either assumption leads to the opposite result. This is a paradox and it remained unresolved from Greek times until the start of the 20th century, when Bertrand Russell resolved it.
Russell reduced this statement to its logical components as a set of two inter-referential statements:
1. The next statement is true;
2. The previous statement is false.
It is easy to see that if the first statement is taken to be true it must be false and if it is taken to be false then it must be true. To resolve the paradox he asked the question: “Should a library catalogue list itself in the library?”
Since the purpose of the catalogue is to guide the library user to the required books, there is no point in listing it. However, if it is not listed the catalogue is incomplete!
The answer is that the catalogue cannot refer to itself. This is the problem with the pair of inter-referential statements. As a pair they are self-referential.
Why can there be no self-reference? To answer this I need to explain Russell’s theory of classes. Russell defined a statement about a concrete object to be of “class 1”; a statement about such a statement to be of “class 2” and so on. For consistency he defined concrete objects to be “statements of class zero.” Thus the chair you are sitting on is a statement of class 0. This statement is of class 1. The new statement is of class 2 and so on.
Let us apply this numbering to the liar paradox as reduced by Russell. Suppose that the second statement is of class n. Then the first must be of one higher class. But the first one refers to the second. This would require that the second one should be higher still, that is of class (n+2). Since we started with the assumption that it is of class n, we see where the paradox comes from — self-reference. Hence we cannot allow inter-referential statements.
This problem of self-reference can be applied to another paradox which goes as follows: “There is a barber in a town who shaves all those men, and only those men, who do not shave themselves.” The paradox is: “Then who shaves the barber?” For, if he shaves himself, he is one of those who shave themselves. So he cannot shave himself.
But if he does not shave himself, he is one of those who do not shave themselves. So he should shave himself. The problem is resolved by realizing that the barber is serving in two capacities: a man of the town, and an operator, namely a barber. If he shaves himself in his private capacity he is one of those who shave themselves and there is no need for him to shave himself in his capacity as a barber.
If he shaves himself as part of his professional work then he is shaved by the barber. In the latter case he would be liable to income and sales tax on his daily shave of himself, while in the former he would not.
Russell’s resolution of the liar paradox seems so satisfactory that one does not detect errors in it at first. However, the fact of the matter is it is not quite correct.
Kurt Godel studied how arithmetical statements can be made in terms of a language composed of a finite set of symbols with a grammar consisting of given rules of combining the symbols. He found that given any finite set of axioms stated in that language, there are always statements that are not inconsistent with the given set of axioms but cannot be derived from them. These statements cannot be assigned any classes and so they violate Russell’s theory of classes.
Godel’s theorem leads to some very interesting consequences. It means that no arithmetical theory can be complete. Because of this theorem the attempt to develop numbers axiomatically was dropped.
Since arithmetic systems are better defined than scientific theories, this also means that no scientific theory can be complete. There will always be some statements that cannot be derived from a scientific theory.
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