# Is Tarski's derivation of the Liar paradox valid?

First a link to his derivation: http://www.jfsowa.com/logic/tarski.htm

Its a famous essay so you really should read all of it but at the moment its enough if you read section 7 where Tarski derives a contradiction.

Alfred Tarski says in there that we can empirically establish the fact that

1) "The sentence printed in this paper on p. 347, l. 31, is not true." is identical with the sentence printed in this paper on p. 347, l. 31.

And that claim is Tarski's error because by Leibniz law we get

2) "The sentence printed in this paper on p. 347, l. 31, is not true." IS TRUE IFF The sentence printed in this paper on p. 347, l. 31, IS TRUE.

Now we use Aristotle's Rule to simplify line 2 to get

3) The sentence printed in this paper on p. 347, l. 31, is not true. IFF The sentence printed in this paper on p. 347, l. 31, IS TRUE.

This is a contradiction so we have a final truth on the matter!

4) IT IS NOT TRUE THAT "The sentence printed in this paper on p. 347, l. 31, is not true." is identical with the sentence printed in this paper on p. 347, l. 31.

Tarski claims a logically false sentence to be empirically true.

That makes his proof not valid. (QED)

PS: I decided to edit out everything else but Tarskis mistake :)

• Not clear... Tarski applies the (negated) truth-predicate "... is not true" to a sentence getting a new sentence : "The sentence printed ... is not true". May 21, 2018 at 6:16
• In (1) Tarski refers to the sentence "The sentence printed in this paper on p. 347,l. 31, is not true." with the name : "s". Thus, the tuth-condition become : (1) "s" is true IFF ... May 21, 2018 at 7:40
• Relevant comments! Thank you. I will consider them and extend my post later today. May 23, 2018 at 10:48
• I just prove sentences like (i) to be false! I assume them to be true, then I apply Leibniz law on them, simplify and then they become a contradiction! Nothing similar can be done on the alternative: The T-sentence. So WHY I should believe the T-sentence to be false when the identity statement clearly is false? Hmm... I think this matter needs a posting of its own! May 23, 2018 at 19:53
• Possible duplicate of What formal logical systems "resolve" the Liar Paradox? May 23, 2018 at 21:17

Johannes has said:

// ... in the first derivation you derive what you say is a contradiction
(x is false AND "x is false" is false) from the two premises (1) and (2):

(1) x is false.

(2) x = "x is false"

(3) x is false AND "x is false" is false (this is the supposed contradiction) //

That ^ is a falsification Johannes! https://en.wikipedia.org/wiki/Straw_man
Compare line (3) above with line (3) below.

Anyone can verify that I wrote the following:
//
First I assume that

1) x is false.

And that

2) x = "x is false"

BUT: I dont assume that "x is false" IFF x is false! INSTEAD I substitute from 2 into 1!

3) "x is false" is false

And from 1 and 3 I get a contradiction. (as a truth table shows) //

Observe that HIS line (3) is NOT my line (3)!

I think anyone (But perhaps Johannes?) can see that MY line (3) NEGATES line (1).

PROOF:

In line (1) we find the proposition "x is false" and my Line (3) tells us that that proposition is false.
SINCE the proposition in line (1) says that x IS FALSE we must conclude from line (1)
AND line (3) that (4) x IS TRUE! (QED)

1) x is false (assumption 1)
2) x = "x is false" (assumption 2)
3) "x is false" is false (substitution from 2 into 1)
4) x is true (proof from 1 and 3)
5) x is false AND x is true (conjunction of 1 and 4)

And since we have derived a contradiction we must deny at least one assumption...