Tarski gave three sufficient conditions in his 1944 paper The Semantic Conception of Truth for the Liar paradox to occur:

  1. The language in which the Liar sentence is stated in is semantically closed, meaning that it has a truth predicate True(x) such that it can refer to its own semantics (where x is the name of the true sentence), and it can refer to its own expressions - for example, a sentence P has the name "P".
  2. Laws of logic hold, including the law of excluded middle (LEM).
  3. Self-reference in this language is possible, e.g. by denoting a sentence with a name

But then Tarski claims that condition 3 is actually not needed, as there are ways to deduce a contradiction from the Liar with just the first 2. (I don't quite get the example Tarski gave, but I take it that something like the following also work: 'Alan: What Boris said is true.', 'Boris: What Alan said is false.')

My questions are as follows:

a) Did I understand condition 3 correctly? (i.e. Cond 3 is indeed about self reference being possible?)

b) If I understand cond 3 correctly, isn't it implied by cond 1? If a language has names for its own expressions, surely that is already sufficient for expressions to refer to themselves?

In case I interpreted Tarski wrongly, here are the excerpts: enter image description here enter image description here

  • 1
    TARSKI'S III is a little bit different for your 3 above, and yes, the last line of page 348 asserts: "It turns out that the assumption (III) is not essential..." Jun 21 '21 at 11:41
  • 1
    But the issue is about a "formal derivation" of the Liar and not an intuitive one. See Monika Gruber, Alfred Tarski and The Concept of Truth in Formalized Languages (Springer, 2016), page 30-on. Jun 21 '21 at 11:46
  • There are popular languages for FOL in which you do not have the ability to spontaneously add symbols, including names for specific things. You have the equivalent of 'subscripted' symbols for a variable and a constant, that get indexed with 'sequence numbers' as they appear in some ordering of the global set of proofs. It is not clear that a sentence should be able to derive its own position in the global set of proofs. So self-references are not an automatic part of I. On the other hand, Goedel numbers exist. So, in effect, you can always derive a name for every sentence. Jun 21 '21 at 16:25
  • Tarski's claim of sufficiency is very dubious. Intuitively it's not clear why the property that a language contains a name for each of its sentences yields the kind of self-reference needed to produce the standard liar paradox. For what we need for that is a fixed point lemma and that requires that the language can represent the recursive functions. That Tarski's claim is demonstrably false was shown by Gupta in his 1982 paper: There are consistent languages that are semantically closed in Tarski's sense.
    – sequitur
    Jun 21 '21 at 16:49
  • Thank you all for your comments; but could anyone let me know if I interpreted Tarski's condition 3 correctly please? Jun 22 '21 at 9:15

Maybe we have to consider what Alfred Tarski is trying to achieve [ref. to the English edition of “Der Wahrheitsbegriff in den formalisierten Sprachen” (1935)].

The purpose is (page 154) to show the difficulties involved into the definition "of truth in the colloquial language".

Tarski intorduce the well-know schema (page 155):

x is a true sentence iff p

with the example (page 156):

"it is snowing" is a true sentence iff it is snowing.

Then he shows (page 157-on) how to reproduce the Liar starting from the above definition of "... it is true" and unobjectionable logical moves (like substitution) that we can perform in current logical systems, like predicate logic or the common (at Tarski's time) foundational system called type theory.

The starting point is to build the well-know self-referential statement, denoted with 'c' in the 1935 monograph:

"the sentence printed on this page,..."

and named (2) in the 1944 essay:

"'s' is identical with the sentence printed in this paper..."

With them Tarski build in 1935 the statement:

for all p, if c is identical with the sentence 'p', then not p,

and in 1944:

's' is true if, and only if, 's' is not true.

The general formulation that Tarski achieves is formalized into Monika Gruber, Alfred Tarski and The Concept of Truth in Formalized Languages (Springer, 2016), page 23:

‘∀p(c =‘p’→¬p)’ = c

from which a contradiction follows:

∀p(c =‘p’→¬p) ⇐⇒ ¬∀p(c =‘p’→¬p).

In both versions, Tarski add the seemingly puzzling note:

We have assumed that we can formulate and assert in our language an empirical premise such as the statement (2) [in 1944 or (c) in 1935] which has occurred in our argument.

What does it mean? that we can formulate in exact and general terms the self-referential sentence avoiding reference to line numbers of a specific text, that obviously change already from edition to edition.

Also a proposal based on the use of "human index", like Alan and Boris or Epimenides will be subject to the same limitations.

  • Thank you so much, I was kinda confused in what sense is the statement empirical, your explanation cleared that up Jun 22 '21 at 15:34

Saying that something is true and not true at the same time requires two different notions of truth. I can utter non-true statements while at the same time being true. The statements themselves can be true or not true if you don't know yet if they are true. I can be a liar and not lie at the same time. If I always lie then you must conclude that I lie when I say that I speak the truth. If I always lie I would't be able to say that I always lie. I must say that I never lie. And from the knowledge that I always lie you must conclude that I'm a liar.

You could propose a new kind of logic in which a statement xan be true and non true at the same time. For the same kind of truth. This is what Tarski does. So if I say that I'm a liar I can be both right and wrong. If I'm a liar this will be true and if I'm not it will be false. The sentence on it's own can be true or false and is not self contradictionary.

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