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Mar19-12, 10:26 AM
P: 27
To see what is involved, lets replace "Sentence 1 " with a variable:

1 x is not true.
2 x = " x is not true."

Sentence 1 is then no longer a sentence; its a sentence-function it has no truth value unless x is replaced with a name of a sentence, or a sentence inside quote signs.

But sentence 2 is an identity, and we can get an equivalence:

3 x is true if and only if "x is not true" is true.

Simplifying the right side we get a contradiction:

4 x is true if and only if x is not true.

And we must deny sentence 2:

5 It is not true that x = "x is not true"

Sentence 5 is a logical truth... its the law of identity:

6 x=x (law of identity)

7 -(x = -x) (from 6 by double negation)

Sentences 5 and 7 has the same logical form since (-x) = "x is not true"

Now let us again look at the foundation of the Liar Paradox:

1 Sentence 1 is not true.

2 Sentence 1 = "Sentence 1 is not true"

Sentence 2 is a denial of the law of identity so it is logically false...and empirically true!

This is because we were violating the law of identity when we created sentence 1!

Sentence 1 is identical with its negation thereby making the logically false sentence 2 empirically true!

So we can neither deny nor assert sentence 1 since its very existence is forbidden by Logic!

The Laws of logic are prescriptions that CAN be broken... They are NOT Natural Laws!

SO: Unless you violate the Laws of Logic you cant derive the Liar Paradox!