FOM: Semantics and the problem of reference
Jeffrey Ketland
ketland at ketland.fsnet.co.uk
Wed Oct 25 07:01:14 EDT 2000
This is great - I'm sure the philosophers on FOM list will have something to
say!
Professor Tait said:
>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.
I agree: Ramsey (1927: "Facts and Propositions") wanted to say that "A is
true" and A have the same sense. (Ramsey was talking about propositions,
though). But this cannot always be quite right. Let A be any liar sentence.
Then "A is not true" and A have the same sense!
I.e., disquotation is not valid for liar sentences. If some self-reference
or diagonalization is present, then no consistent truth theory can contain
*all* disquotational T-sentences Tr("A") <--> A. You have to drop
disquotation somehow, or modify it.
(Kripke's 1975 construction shows how to generate fixed point models where
Tr("A") and A always have the same truth value - even if A contains the
truth predicate as well!! But if A is ungrounded (e.g., a liar sentence),
then both Tr("A") and A get the value "ungrounded" or "undefined", instead
of "true" or "false").
However, the definition of *truth-in-a-model* simply isn't what Tarski did
in his 1935/36 paper. Tarski considers the language of class inclusion,
writes down a disquotational "induction definition" of "satisfies" and shows
(in effect) that the T-sentences are provable (i.e., Convention T holds).
(I.e., Tarski's writes down:
"x_n is a subset of x_m" is satisfied by a sequence s iff s_n is subset
of s_m.
Actually, Tarski uses a complex way of making names of formulas, but that's
not too important. The quotation name above is OK).
This has nothing to do with models. They are not mentioned. Disquotational
truth definitions (as given by Tarski in 1935/36) have nothing to do with
models. Tarski's long 1935/36 paper "Der Wahrheitsbegriff" was published in
1933 in Polish.
Models appeared in a later paper 1935 "On the Concept of Logical
Consequence".
>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?
I think you said something like this in your 1986 paper "Truth and Proof -
The Platonism of Mathematics"? But I don't understand this at all. Tarski's
definition of truth in 1935/36 ("Der Wahrheitsbegriff") has nothing to do
with models. He shows how to take a sentence of the form Tr("A") and using
the explicit definition of Tr(x) (in terms of "satisfies"), how to *prove*
A. In other words, how to get from Tr("A") to A and vice versa. I.e., he
shows how to eliminate the truth predicate. Tarski's theory shows how to
prove all the T-sentences Tr("A") <--> A, where A is a sentence from the
base/object language.
My view is that this is much better than Ramsey's (and Frege's) suggestion,
because Tarski shows the *exact mechanism* involved in deductively moving
from Tr("A") to A.
(Alternatively: you can simply introduce a rule of inference: from A infer
Tr("A") (and vice versa). But this leads to the liar problems again).
More generally, Tarski's theory shows how to *extend* a base/object language
L, by adding a system of axioms and new primitive predicate Tr(x), such that
in the extended system, we have, for each sentence A of the base language L,
that Tr("A") <--> A is provable. That's Convention T. The predicate Tr(x)
means "x is a true sentence of L".
For example (this sort of thing has been recently studied in depth by the
likes of Sol Feferman (1991: "Reflecting on Incompleteness", JSL), Volker
Halbach, and others, including me a bit):
The general pattern is this. Take PA and add a primitive predicate
"satisfies". Then add Tarski's disquotational axioms for "satisfies". You
get a formal system which model theorists call PA(S) (see Richard Kaye 1991
"Models of Peano Arithmetic", Chapter 15). This formal system satisfies
Convention T - it contains a materially adequate definition of truth for
arithmetic sentences.
One can easily connect this work on axiomatic Tarskian truth theories to
model-theoretic matters. Call the extended language L+. Let M+ = (M, T) be a
model of PA(S), where the reduct M is a structure for the language L of
arithmetic, and where T is the extension of the truth predicate Tr(x).
Obviously, M must be a model of PA. Then it *follows* from the fact that all
the Tarski T-sentences are provable in PA(S) that,
for any sentence A of L, the code of A an element of T iff M |= A.
There is a small area of mathematical study here. They're called
"Satisfaction Classes" (this is the topic of Kaye's chapter 15, cited
above).
>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.
The relevance of Tarski's theory for the concept of truth is exactly the
same as the relevance of Einstein's theory for spacetime. He showed how to
make precise scientific sense of the concept, after hundreds of years of
insane gibberish by pragmatists, idealists, subjectivists, etc. Tarski's
theory conforms exactly to what Aristotle intended (in Metaphysics) and is
(in my view, and Popper's too) a refined, precisely articulated version of
the correspondence theory of truth.
For example, thankws to Tarski, we know exactly what we mean when we talk
about all the true sentences of arithmetic. This set is *definable* (in Z_2,
say), but not *computable* (it's hyperarithmetic). We also know from Godel's
1st incompleteness theorem that this set is not axiomatizable, so that no
sound proof procedure will generate all and only the truths of arithmetic.
Tarski brought no confusion to philosophy. Quite the opposite. His article
is one of the most important contributions to human ideas in the 20th
century. The sentence "snow is white" is a true sentence of English if snow
is white (and not true otherwise). Similarly, the sentence "there are
infinitely many prime numbers" is a true sentence of arithmetic iff there
are infinitely many prime numbers.
If Tarski's work can be extended to natural languages (as many philosophers,
following Davidson, hope), then a materially adequate definition of truth
for consistent English (i.e., not containing semantic predicates) should
prove each disquotational T-sentence. There are problems with indexicals,
vagueness, intensional predicates, etc. .... Still: we're optimistic.
>`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.
I agree. But a sentence can be *true* even if we are not *warranted in
asserting* it.
E.g., if W is the set of sentences we are warranted in asserting, then some
version of the Church-Turing thesis suggests that W will be (at worst) r.e.
(Putnam made the same point round about 1965, I think). If that's right,
then it follows from Goedel's 1st incompleteness theorem that we shall never
be warranted in asserting all and only the truths. That's why Goedel wanted
to reject mechanism for the mind.
Raining in Nottingham.
Best - Jeff
~~~~~~~~~~~ Jeffrey Ketland ~~~~~~~~~
Dept of Philosophy, University of Nottingham
Nottingham NG7 2RD United Kingdom
Tel: 0115 951 5843
Home: 0115 922 3978
E-mail: jeffrey.ketland at nottingham.ac.uk
Home: ketland at ketland.fsnet.co.uk
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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