The Liars's paradox: Is it a paradox? Is it a truly a simplified version of the Bertrand's Paradox?

Question

I must state that I consider myself inadept at using proper math language. Therefore, I must state my thoughts in word statements.

Let us take a statement. We will call this statement 'This'. I now make another statement. Let us call the second statement 'Another This'.

'Another This' = 'This' is false.

I open 'Another This' to assumptions. Thereby, if 'Another This' is true, then 'This' is false. If 'Another This' is false, then 'This' is true.

My question is: Why in the Liar's paradox do we take Another This and This to be the same?

Answer 1

There is no "another this" in the Liar's paradox. You just say "This statement is false", and the 'this' in the statement refers directly to the statement itself.

Indeed, the statement

The statement "this statement is false" is fals

is not a paradoxical statement in itself. It is only an incorrect statement, because it is claiming something which is not true.

Answer 2

The liar paradox boils down to:

Let $P$ be a true/false value such that $P \equiv \neg P$. [Stating: "P = ( P is false )"]

This is obviously impossible. It is the same kind of fallacy as:

Let $x$ be an undetectable cheese-monster on the moon.

Since $x$ is an undetectable cheese-monster on the moon, there is an undetectable cheese-monster on the moon!

Similarly the extended paradox:

The following statement is true.

The previous statement is false.

boils down to:

Let $P,Q$ be true/false values such that $P \equiv Q$ and $Q \equiv \neg P$.

Clearly impossible, again, and still of the same type of fallacy.

Moral of the story

Before you can refer to some object with some properties, you must first prove that such an object exists!