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?

  • $\begingroup$ What's the link with Bertrand paradox (economics) or Bertrand paradox (probability) ? $\endgroup$ – Mauro ALLEGRANZA Mar 7 '16 at 16:05
  • $\begingroup$ Where the heck are you getting 'another this' from? The liars paradox only has a single 'this' and its statement claims that itself is false. A statement claiming to be false would be true if it were false, and false if it were true. $\endgroup$ – fleablood Mar 7 '16 at 16:16
  • $\begingroup$ The negation of "this$_a$ statement is false" is indeed "this$_b$ statement is true", but this$_a$ and this$_b$ refer to two different statements. I can't really follow the questioner's language, but the "liar paradox" isn't really a paradox, it is just a statement that is false, similar to saying $1 = 2$. The reason people think it is paradoxical is because they assume this$_a$ = this$_b$, however, by the very fact of the transformation creating this$_b$ is negation, it is actually the case that this$_a$ = not this$_b$ $\endgroup$ – DanielV Mar 7 '16 at 16:28
  • $\begingroup$ @DanielV: It's not like that actually. Most people understand the 'standard' liar paradox of "This sentence is false." as meaning that "this" refers to that very sentence it occurs in. The error lies solely in assuming existence of something that doesn't exist. It can arise even without self-reference; consider "preceded by its quotation yields falsehood." preceded by its quotation yields falsehood.! This in fact is analogous to Godel's first incompleteness theorem, where quotation corresponds to Godel encoding. $\endgroup$ – user21820 Mar 8 '16 at 0:13

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!

  • $\begingroup$ "Let P be a true/false value such that -P. Obviously that is impossible." Wha...? -(false) is true. $\endgroup$ – fleablood Mar 7 '16 at 16:12
  • $\begingroup$ @fleablood: Sorry for the stupid mistake and thanks for pointing it out. It's supposed to be defined in the same way as the later examples. I'm editing now. $\endgroup$ – user21820 Mar 7 '16 at 23:52
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    $\begingroup$ Ah, much better. $\endgroup$ – fleablood Mar 8 '16 at 0:05
  • $\begingroup$ @fleablood: It's amazing what carelessness can produce (in this case despite knowing the incompleteness theorems inside out). $\endgroup$ – user21820 Mar 8 '16 at 0:15

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.

  • $\begingroup$ if "this statement is false" is a certainty, i fail to understand why the liars paradox should even exist. because as a second step, we question what would happen if "this statement is false" is indeed true, in which case it becomes false(exactly what it initially was). $\endgroup$ – anushka Mar 7 '16 at 15:27
  • $\begingroup$ @anushka Who said "this statement is false" is a certainty? $\endgroup$ – 5xum Mar 7 '16 at 15:33

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