FOM: Semantics and the problem of reference
William Tait
wwtx at midway.uchicago.edu
Tue Oct 24 16:01:40 EDT 2000
Jeffrey Ketland (Oct 22) wrote
>4. What does it mean to say that a sentence of mathematics is true?
>
>This question is bothering Kanovei. As Tait (following Frege, Ramsey,
>Tarski, Quine, etc.) pointed out, "CH is true" means the same as CH. That
>is, CH is true just in case any infinite subset of |R is either denumerable
>or equipollent to |R. It's hard to see how anyone could be confused about
>this, given that Tarski's work is well-known.
>In particular, the following T-sentence can be *proved* from the definition
>(v) and axioms (i)-(iv):
I don't know about all the others cited, but, when I said that A and
A is true mean the same thing, I don't mean simply that they have the
same truth-values, but that they have the same sense. Without
checking, I think that this is what Ramsey meant, too. It is not
something to establish by a theorem, e.g. like Tarski's proof of the
material condition for truth for his definition of truth in a model
of a sentence of a formal language. There is no definition of truth
for mathematical English. E.g. what would be the metalanguage?
Tarski's definition yields a `translation' I(phi) of the (formal)
sentence phi of the object language into the metalanguage. That is
what you get when you unpack his definition of `phi is true': you get
a sentence I(phi) of the metalanguage. If A is an ordinary
arithmetical sentence and phi is its formalization in the language of
PA, then relative to the intended model of PA, I(phi) is (or is
synonymous with) A. What should I(A) be?
>5. Does mathematical truth mean "having a mathematical proof"? (Kanovei)
>
>Obviously NOT. The truth set for (|N, 0, s, +, x) is not axiomatizable. This
>was discovered by Kurt Goedel in 1930. N.B. The definition (v) of truth (for
>sentences of the lang of set theory) above makes no mention of *proof*.
`A is true' means A. We are warranted in asserting that A is true,
i.e. in asserting A, when we have a proof of it. We will agree about
the warrant only if we agree about the axioms.
The relevance of Goedel's incompleteness theorems is that when we
speak of proof in this way, we cannot (except is particular cases
such as sentences of elementary Euclidean geometry) be speaking of
proof from some particular given set of axioms. In set theory, for
example, we are, as Goedel emphasized, led always to new axioms.
The relevance of Tarski's truth definition is I DON'T KNOW WHAT. I
have always felt that Tarski, not in his original paper on the
concept of truth, but in his later essays on the subject, has a lot
to answer for in the confusion he brought to philosophy.
From ominously warm Chicago,
Bill Tait
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