May 1, 2011

Logical Paradoxes

I have not written anything for a while, and part of the reason is that I have not received much feedback on what I have written so far; if you like this blog and want me to write more blog entries, please let me know.

In the hope that it might interest people, I decided to write a blog post on logical paradoxes such as the following one:
Sentence S: Sentence S is not true.
Suppose S is true, then it is not true. Suppose instead it is not true, then it must be true...

Or take another example:
Sentence T: If sentence T is true, then 1+1=3.
Suppose T were true, then (think carefully about it) 1+1=3. But from what I just said it follows that T is true (why?) and so it must be that 1+1=3 (why?).

By reflecting upon sentences such as S and T one can (as you hopefully just saw) come to absurd conclusions. In practice, everyone sees that something must have gone wrong and forgets about the whole piece of reasoning, but one might want an explanation of what exactly it is that goes wrong.

It seems to me that the paradoxical aspects of S and T can also be found in paradoxical descriptions such as the following one:
Description D: The natural number 1 – δ(D).
To understand this description you first have to know what I mean by δ:
Definition of δ:
 δ(x) = 1 in case description x describes the natural number 1, and
 δ(x) = 0 in all other cases.

(Why is D paradoxical? Suppose D describes the natural number 1. Then δ(D) = 1. So D simplifies to "The natural number 1 – 1". Which simplifies to "The natural number 0". But then D describes 0, contrary to our assumption. Suppose instead that D does not describe the natural number 1 and you will get another contradiction in much the same way.)
With sentence S one might think that the problem lies with the word "not" or the word "true", and with sentence T one might think the problem is related to the construction "if ... then ...", but one can hardly argue anything similar with D. Yet, it seems to me that the paradoxical aspects of S are also found in D, so what exactly is wrong with S, T, and D?

One question we have not considered so far is whether we are at all able to make sense of S, T and D. Like the mathematical expression "0 / 0" they combine familiar things in an unusual way, and since 0 / 0 is often said to be "meaningless", perhaps we should not be so quick to think that we can give a meaning to S, T, or D. I personally think 0 / 0 can be given a meaning (see http://en.wikipedia.org/wiki/Wheel_theory) but if you ask me whether S (or T) is true or what natural number description D describes, then I have to say I do not know how to make sense of the question (in the above I was reasoning in a naive way for demonstrative purposes). If you think you can make sense of these questions, would you please tell me what their answers are? And if you say you do not know the answers, could you at least give some indication on how one might go about and find them?

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