The Paradox
Quack, quack, quack!
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Okay, I'll bite, if it's not a pair of ducks, what is it?
By one dictionary, a paradox is "a seemingly contradictory statement that nonetheless may be true: the paradox that standing is more tiring than walking." (This is just the appearance of contradiction, because you might reasonably think you expend more energy walking, when in fact keeping your balance requires more.) Also, a paradox is "an assertion that's self-contradictory, though based on a valid deduction."
For comparison, here's what antinomy means: A contradiction or opposition between two laws or rules. It is "a contradiction between two principles or conclusions that seem equally necessary and reasonable; a paradox." So we have contradiction, again, and paradox.
Another step takes us to self-referential, which means "referring to oneself or itself, like: the biographer's account of the poet's life was surprisingly self-referential." I take this to mean the biographer could have been talking about himself.
The idea of referring to oneself or itself is obviously important, but I'm still not satisfied with our progress. So let's take yet one more step and look at an example I had in mind:
This statement is false.
Now that's what I call a paradox! The statement refers to itself and says it itself is false. But if the sentence is true, it is both false and true. Yet if it's false, it must be true. So it is both true and false. And that gives us a bloody self-contradiction!
Or is the contradiction only imaginary?
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When faced with self-contradiction, like This statement is false, it's reasonable to question whether the statement is really meaningful. There are those who believe it's absolute nonsense, if not rubbish, and dismiss the problem out of hand.
Another self-referencing statement is: Suppose Shaver is a barber who shaves all and only those men in the village who don't shave themselves. This yields the following paradox:
Shaver shaves himself.
If he shaves himself, he is not one of the men he shaves, so the statement is both true and false. But if he doesn't shave himself, then he is one of the men he shaves, so the statement, again, is both true and false.
If sentences of this nature are meaningful, and it's hard to argue that they're not, because we seem to understand them, a next step may be to interpret them to eliminate the contradiction. This is what Alfred North Whitehead and Bertrand Russell (philosopher-mathematicians) did when they developed what's known as the Theory of Types.
Expressed by Stephen C Kleene, in his Introduction to Metamathematics, it says:
The primary objects or individuals are assigned to one type (say type 0). [Next] properties of the objects or individuals are assigned to the second level, or type 1. Then properties of properties are assigned as type 2. Etc. [Furthermore,] no properties are admitted that don't fall into either of these logical types.
The Theory of Types resolves self-contradiction by eliminating it, by distinguishing between different levels of sentence types. This statement is false is a sentence on one level, and the sentence to which the statement refers is on another level. The sentences are of different types and not just one sentence. So there's no self-contradiction.
The question is whether the Theory of Types is slight of hand. Russell himself wasn't satisfied with the development, and in fact was pleased to hear that G. Spencer Brown dealt with the matter in a different way. As Brown writes:
It was with some trepidation that I approached [Russell] in 1967 with the proof that [the theory of types] was unnecessary. To my relief he was delighted. The theory was, he said, the most arbitrary thing he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved. [Read about Brown's work,
here.]
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Is Self-Contradiction Imaginary?
Problems relating to the self-referencing paradox were once present in ordinary algebra, but were solved in a highly imaginary(!) way. The problems had to do with the equation
x2 + 1 = 0,
known not to have solutions. That is, no number for x would yield a true statement, because the square of any number was known to be positive. Adding 1 to any positive number can't yield 0.
The problem was solved by creating the imaginary number, i, defined as the square root of -1. By redefining our notion of numbers and adding i to the class of possible solutions to algebraic equations, we can now get the desired two solutions for a quadratic equation, namely +i and -i, the square of each of which yields -1 and satisfies the equation. Is that sleight of hand, or what!
But what does this have to do with self-referencing and paradoxes?
Here's a hint. The answer is in Brown's book, Laws of Form. I'll give you a moment to try to puzzle it out for yourself.
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Okay, that's enough time. Brown answers by l reshaping the quadratic. First he transposes it:
x2 = -1
and divides both sides by x, to get:
x = -1/x.
This expression is self-referential, because the value for x we're looking for "has to be put back into the expression from which we seek it." (Close to this in math is the recursive function, such as: xn = xn-1 + a. There are also self-similar things like fractals.))
By inspection we can tell that x has to have some form of unity, otherwise the equation wouldn't balance. Other than for x = 1, the value on one side would be greater than 1 and on the other side would be less than 1. If we put in either +1 or -1 for x, we'd get +1 = -1, or -1 = +1, respectively, both of which are contradictory. Either way, it's a paradox.
The algebra paradox was solved by including imaginary numbers in the number system. What Brown does, by analogy, is to extend the idea to Boolean algebra, producing a system that lets a valid argument "contain not just three classes of statement, but four, namely true, false, meaningless, and imaginary." In other words, he extends the traditional types of statement to include imaginary statements.
Wow! That's what I call imaginary thinking! Or is that imaginative thinking?
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