Evaluation

• Kripke's objections - See LiarKripke
• A consistent universal language is not possible?

@There is, however, no need to use universal languages in all possible situations. In particular, such languages are in general not needed for the purposes of science (and by science I mean here the whole realm of intellectual inquiry)@

@We shall try to find a solution that will keep the classical concept of truth essentially intact. The applicability of the notion of truth will have to undergo some restrictions, but the notion will remain available at least for the purpose of scholarly discourse.@

1. The classical / semantic conception of truth - implies every instance of : "p" is true if and only if p.

1. The classical conception of truth - implies every instance of : "p" is true if and only if p.
2. Suppose sentence S = "S is not true"
3. "S is not true" is true if and only if S is not true.
4. S is true if and only if S is not true.
5. S is both true and not true.

1. p is true and p is not true.

Tarski's explanation of why the paradox comes about : There are two assumptions.

1. Convention T : "____" is true if and only if _____.
2. Suppose p = "p is not true"
3. "p is not true" is true if and only if p is not true.
4. p is true if and only if p is not true.

1. The laws of classical logic.

• [Required] Alfred Tarski (1969) Truth and Proof. Scientific American, June Issue.

• Alfred Tarski (1969) Truth and Proof. Scientific American, June Issue.

Kripke's criticisms

• See LiarKripke

• No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language. (But see Yablo)

• No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language. (But see Yablo)

More details

• No liar paradox will arise in a formalized language L as long as L does not contain its own truth predicate.

• Given a formalized language L, we can define the truth predicate for L in a larger language L' that contains L, but the truth-predicate is not part of the vocabulary of L. L is called the object-language and L' the meta-language.

Category.LogicAndMaths

Tarski on the liar

• Tarski's explanation of why the paradox comes about : There are two assumptions.

1. Natural language is "semantically closed". (a) The language contains both expressions and names of these expressions. (b) The language has a truth-predicate that applies to sentences of the same language.
2. The ordinary laws of logic.

• With (1), it is possible to write down a sentence which says of itself that it is false (or not true). With (2), we can derive a contradiction from the sentence.
• Note: Tarski gives up (1). TV gaps/gluts give up (2).

• Given a formalized language L, we can define the truth predicate for L in a larger language L' that contains L, but the truth-predicate is not part of the vocabulary of L. L is called the object-language and L' the meta-language.
• The proof appears in Tarski (1956).
• No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language.

• Alfred Tarski (1956) The concept of truth in formalized languages. In Logic, Semantics, Metamathematics. Oxford University Press.

Readings

Tarski and the Liar Paradox

• Alfred Tarski (1956) The concept of truth in formalized languages. In Logic, Semantics, Metamathematics. Oxford University Press.

• The proof appears in Tarski (1956).

Reading

Notes

Tarski on truth

• Alfred Tarski (1969) Truth and Proof. Scientific American, June Issue.
• Tarski's explanation of why the paradox comes about : it has to do with the semantic universality of natural languages.
• Such a language allows us to talk about the truth or falsity of any sentence in the very same language.
• So it is possible to write down a sentence which says of itself that it is false (or not true).
• But Tarski is not saying that it is only self-reference that leads to the liar paradox.
• Tarski's suggestions on how to deal with the paradox :
  • There is no need to make use of semantically universal languages in formulating scientific theories (including mathematical and logical ones).
  • No liar paradox will arise in a formalized language L as long as L does not contain its own truth predicate.
  • A formalized language is a language with a clearly defined vocabulary and syntax, and which does not contain any indexical expressions, expressions whose referents depend on the context, e.g. "he", "it", etc..
  • A truth predicate for a language L is a predicate (e.g. "is true") that applies to all and only the true sentences of L.
• Given a formalized language L, we can define the truth predicate for L in a larger language L' that contains L, but the truth-predicate is not part of the vocabulary of L. L is called the object-language and L' the meta-language.
• The proof appears in Alfred Tarski (1956) "The concept of truth in formalized languages" in Logic, Semantics, Metamathematics Oxford University Press. (Very difficult)
• No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language.

2 Tarski on truth

• See excerpt from Alfred Tarski (1969) "Truth and Proof" Scientific American, June Issue. On reserve at the dept main office.
• Tarski's explanation of why the paradox comes about : it has to do with the semantic universality of natural languages.
  • Such a language allows us to talk about the truth or falsity of any sentence in the very same language.
  • So it is possible to write down a sentence which says of itself that it is false (or not true).
    • But Tarski is not saying that it is only self-reference that leads to the liar paradox.
• Tarski's suggestions on how to deal with the paradox :
  • There is no need to make use of semantically universal languages in formulating scientific theories (including mathematical and logical ones).
  • No liar paradox will arise in a formalized language L as long as L does not contain its own truth predicate.
    • A formalized language is a language with a clearly defined vocabulary and syntax, and which does not contain any indexical expressions, expressions whose referents depend on the context, e.g. "he", "it", etc..
    • A truth predicate for a language L is a predicate (e.g. "is true") that applies to all and only the true sentences of L.
    • Given a formalized language L, we can define the truth predicate for L in a larger language L' that contains L, but the truth-predicate is not part of the vocabulary of L. L is called the object-language and L' the meta-language.
      • The proof appears in Alfred Tarski (1956) "The concept of truth in formalized languages" in Logic, Semantics, Metamathematics Oxford University Press. (Very difficult)
    • No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language.