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!