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Self-Referencing Paradoxes

Posted by Douglas Twitchell on Sep 14, 2013
As a math teacher, I love logic and logical puzzles. There is one kind of logical puzzle that I've always found particularly fascinating: logical paradoxes. Consider a piece of paper on which is written the following statement:
This statement contains threee mistaks.

What are the mistakes? "Three" is spelled incorrectly, and so is "mistakes". That's two mistakes. Oh! So the third mistake is that there are actually only two mistakes. But wait! That means there really are three mistakes. So that means it's not a mistake to say there are three mistakes. Which means there are only two mistakes, which means...

Okay, so consider this one:

This statement is false.

Clearly the statement can't be false, because if it is, that makes it true. But if it's true, that makes it false. Which makes it true, which makes it false, which makes it...

Or how about this one:

Is the answer to this question 'No'?

I'll let you work out for yourself why this is paradoxical.

Do you know what all of these statements have in common? They are self-referencing. They are statements/answers that try to define themselves. In order to guarantee that paradoxes are avoided, nothing should be allowed to define or reference itself. It's a form of circular reasoning and circular defining that we do not allow in mathematics.

The concept of self-referencing paradoxes places a strict limitation on what is knowable by the human race. Science is the creation trying to understand itself. Psychology and sociology are examples of humanity trying to understand and define itself. The truth is, we are ill equipped to understand ourselves or define ourselves. I'm not saying that science is bad - merely that it is limited, and it is important for us to understand this limitation.

For a true understanding of our own nature, we would be wise to get an outside perspective rather than a self-referencing perspective.

Jeremiah 17:9 says exactly this. The word of the Lord says that the human heart is deceitful, and who can know it? Then comes the answer from God himself: "I, the Lord, search the heart and test the mind."

This is no self-referencing analysis of humanity; this is the Creator Himself spelling out His own analysis of the thing He has created.

I don't know about you, but I'd rather take His view over anyone else's!

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