PHL280: Truth and Reality
Week 5: Tarski on Truth

    Theories of truth for a language

  1. We have been taking propositions to be the primary bearers of truth, and looking at various theories of propositional truth.
  2. But some people think that it is interesting to focus on theories of sentential truth instead (i.e. theories about what it is for a sentence to be true).
  3. One such person is (or was) Alfred Tarski. In a series of papers in the mid-20th century, he developed some ideas about what we should be looking for when trying to develop a theory of truth for the sentences of a language.
  4. He restricted his attention to formal languages (such as the ones used by philosophers - the propositional and predicate calculi), taking natural languages to be too vague and unsystematic to be fruitfully studied.

    But a lot of work is being done these days applying Tarskian methods to natural languages, and much of it seems to be very fruitful. And the methods are not just used to develop a theory of truth for these languages, but rather a theory of meaning (see Donald Davidson, 'Truth and Meaning').

  5. Tarski-style theories of truth for a language

  6. What would it take to give a theory of truth for the sentences of a language? Tarski proposed an answer.
  7. Note that there are up to two languages involved in giving a theory of truth for a language. First, there is the language for which we would like to give a theory of truth - this is called the object language. Second, there is the language that we will use to give the theory of truth - this is called the metalanguage. Very often the object language is some sub-language of English (i.e. a fragment of English), and the metalanguage is English. But they could be other languages (examples below).
  8. Tarski proposed two requirements on any theory worthy of being called a theory of truth for a language, L: that it be formally correct (i.e. have the right form), and that it be materially adequate:

    For the theory to be formally correct is for it to have the following form (or at least be expressible in this form): For all sentences s in L, s is true iff ...

    (Where the gap '...' is not to contain the word ‘true’.)

    For the theory to be materially adequate is for it to entail every instance of the following schema: 'S' is true iff R.

    (Where instances are obtained by replacing 'S' by a sentence of the object language, L, and replacing 'R' by a synonymous sentence of the metalanguage; if the metalanguage contains the object language then 'S' and 'R' can be replaced by the same sentence.)

  9. This schema is sometmes called Convention T. Its instances are sometimes called Tarski biconditionals.
  10. Here are some examples of Tarski biconditionals:
  11. Note that in order to use the metalanguage M to express a theory of truth for the object language L we need M to be expressive enough to be able to refer to each sentence of L, and to be able to express the same thing as each sentence of L. M also needs to have a truth predicate - an expression that is synonymous with 'true'.
  12. Tarski thought that in order to avoid the Liar paradox, the metalanguage needs to be strictly stronger than the object language (more about this next week).
  13. Tarski called this approach to truth the semantic conception of truth.
  14. A simple example

  15. Take the object language, L, to be the following fragment of English:
  16. Here is a theory of truth for this language:

    For all s in L: s is true iff s is 'Grass is green' and grass is green, or s is 'Snow is white' and snow is white.

  17. This theory is formally correct - it has the form "For all s in L: s is true iff ...", where the gap '...' does not contain the word 'true'.
  18. And it is materially adequate - it entails every Tarski biconditional for L, i.e. the following two:

    Here is a derivation of the first:

    According to the theory, for all s in L: s is true iff s is 'Grass is green' and grass is green, or s is 'Snow is white' and snow is white. So, in particular, 'Grass is green' is true iff 'Grass is green' is 'Grass is green' and grass is green, or 'Grass is green' is 'Snow is white' and snow is white. But 'Grass is green' is 'Grass is green' and grass is green, or 'Grass is green' is 'Snow is white' and snow is white iff grass is green. So, 'Grass is green' is true iff grass is green.

  19. So this theory satisfies Tarski's two requirements on being a theory of truth for L.
  20. Whenever L contains only finitely many sentences then we can come up with a Tarskian theory of truth for L in this way.
  21. A more complex example

  22. What about an object language that has infinitely many sentences?
  23. Let L be the following fragment of English:
  24. Here is a theory of truth for this language:
  25. Is this theory formally correct? Yes - we can express it in the right form, as follows:

    For all sentences s in L, s is true iff: s is 'Grass is green' and grass is green, or s is 'Snow is white' and snow is white, or there is a sentence t in L such that s = 'It is not the case that ' + t and it is not the case that t is true, or there are sentences t and u in L such that s = t + ' or ' + u and t is true or u is true.

  26. Is it materially adequate? Yes - we can derive from it the Tarski biconditional for every sentence of the object language (and there are infinitely many).

    Example. Let's derive the following Tarski biconditional: 'Snow is white or it is not the case that grass is green' is true iff snow is white or it is not the case that grass is green.

    Consider the sentence 'Snow is white or it is not the case that grass is green'. 'Snow is white or it is not the case that grass is green' is true iff 'Snow is white' is true or 'It is not the case that grass is green' is true. 'Snow is white' is true iff snow is white; 'It is not the case that grass is green' is true iff it is not the case that 'grass is green' is true, i.e. iff it is not the case that grass is green. So, 'Snow is white or it is not the case that grass is green' is true iff snow is white or it is not the case that grass is green.

  27. So this theory satisfies Tarski's two requirements on being a theory of truth for L.
  28. An even more complex example

  29. If time: consider the fragment of English which has two names, ‘grass’ and ‘snow’, and two predicates, ‘is green’ and ‘is white’.
  30. Some considerations

  31. We have been proceeding on the assumption that the sentences of a language have truth values. But is this right? What is the truth value of 'He is a student'? Or 'Today is Wednesday'? Either we need to exclude these from our attention and focus just on context-insensitive (i.e. eternal) sentences (if there are any). Or we need to talk instead about the truth value of sentences relative to a context of use.
  32. This approach to truth feels a bit like a correspondence theory. Is it?
  33. Tarski took the two requirements, being formally correct and being materially adequate, to be necessary conditions on a theory of truth for a language. But he also took them to be sufficient. Perhaps this makes his approach to truth more of a deflationary approach (more about this later).

Further reading

Exercise

Read Tarski (1969) (the paper is longish, but a fairly easy read, and we'll be using for next week's exercise as well). What exactly is Tarski's interest in truth (what does he take his aim to be)?