FOM: Tarski and Truth = correspondence to the facts

[Jeffrey Ketland]

Let's discuss the concept of truth seriously. Modern theories of truth start
with a Polish mathematician called Alfred Tarski. As Tarski explained
(following Aristotle, Russell, etc.), truth means *correspondence to the
facts* (see below).

Around 1931 Tarski wrote a classic article, which was published in Polish in
1933 (in Warsaw), and again in German in 1935/36. It was translated into
English in 1956.

(The first major advance on Tarski's work is Kripke's famous paper "An
Outline of a Theory of Truth", Journal of Philosophy 1975).

Tarski's article is:

Tarski, A. 1935/36: "Der Wahrheitsbegriff in den formaliserten Sprachen",
Studia Philosophica I. (Translated by J.H. Woodger as "The Concept of Truth
in Formalized Languages", in Tarski, A. 1956: Logic, Semantics and
Metamathematics. Oxford)

Here is a sample of the start of Tarski's article:
-------
The present article is almost wholly devoted to a single problem---the
definition of truth. Its task is to construct---with reference to a given
language--a materially adequate and formally correct definition of the term
"true sentence". (Tarski 1956, p. 152)
-------
A thorough analysis of the meaning current in everyday life of the term
"true" is not intended here. Every reader possesses in greater or lesser
degree an intuitive knowledge of the concept of truth and he can find
detailed discussions on it in works on the theory of knowledge. I would only
mention that throughout this work I shall be concerned exclusively with
grasping the intentions which are contained in the so-called *classical*
conception of truth ("true--corresponding to the facts") in contrast, for
example, with the utiliarian conception ("true--in a certain respect
useful"). (Tarski 1956, p. 153).
-------

Tarski proceeds to discuss

Section 1. The concept of "true sentence" in everyday or colloquial language
- in particular, he introduces what has come to called the Disquotation
Scheme
DS       "A" is true if and only if A
 e.g., "it is snowing" is a true sentence if and only if it is snowing
- discusses the liar paradox, and "semantic closure"

Section 2. Formalized languages, especially the language of the calculus of
classes
- primarily a detailed axiomatic discussion of syntax and provability.

Section 3. The concept of true sentence in the language of the calculus of
classes
- this is an absolutely crucial section.
- contains Convention T (pp. 187-188).
-----------
CONVENTION T: A formally correct definition of the symbol "Tr", formulated
in the metalanguage, will be called an adequate definition of truth if it
has the following consequences:
(alpha) all sentences of the form "x is true iff p", where "p" is replaced
by the translation of some sentence A of the object language and "x" is
replaced by a name of A.
(beta) the sentence "for all x, if x is in Tr, then x is in S".
------------

- Tarski introduces the *semantic* notion of "satisfies" with examples as
follows:

for every a, a satisfies the sentential function "x is white" if and only if
a is white
for every a, a satisfies the sentential function "for all y, x subset y" if
and only if, for all classes b, we have a subset b.

- Definition 22: inductive definition of "satisfies" for the language of
classes
- Definition 23: explicit definition of "true sentence".

- NB. Definitions 22 and 23 makes NO MENTION of PROVABILITY.
- Tarski defines the set Pr of provable sentences in the previous section.
- Detailed demonstration of several important properties of this definition,
including that it satisfies Convention T.
- Detailed proofs of the following theorems

THEOREM 4: The class Tr of true sentences is a consistent and complete
deductive system.
THEOREM 5: Every provable sentence is a true sentence.
THEOREM 6: There exist true sentences which are not provable.
THEOREM 7: The class Pr is a consistent but not a complete deductive system.

Section 4. The concept of true sentence in languages of finite order

Section 5. The concept of true sentence in languages of infinite order
In particular,
THEOREM I is Tarski's Indefinablity Theorem (in its first published form, I
think).
THEOREM II is roughly a theorem to the effect that truth for formulas of the
nth-order can be defined at the next level up.
THEOREM III is, effectively, the conservativeness theorem (adding the scheme
Tr(#A) <-> A to any rich enough syntactical meta-theory is a conservative
extension).

(A number of truth researchers, including myself, have discussed these
results, and proved new ones, in recent years).

Section 6. Summary

Section 7. Postscript
Contains a discussion of the omega-rule, of the idea that truth can only be
defined in a fundamentally richer metalanguage, and the determination of the
truth value of Goedel sentences using the truth-theoretic metatheory.
In particular,
--------
Goedel has given a method for constructing sentences which---assuming the
theory concerned to be consistent---cannot be decided in any direction in
this theory. All sentences constructed according to Goedel's method possess
the property that it can be established whether they are true or false on
the basis of the metatheory of higher order having a correct definition of
truth. Consequently, it is possible to reach a decision regarding these
sentences, i.e., they can be either proved or disproved. (Tarski 1956, p.
274).
---------

Unlike Kanovei, at no point does Tarski confuse truth with provability (in a
fixed formal system). Truth is one thing; provability another. Whether they
coincide is another matter (THEY DON'T - this is a well-known mathematical
fact). Tarski was defining TRUTH (i.e., correspondence to the facts) for
mathematical sentences.

Truth belongs to a family of concepts: names, designates, denotes, refers,
satisfies, defines, etc. These are called semantic concepts. (They are not
connected to epistemology - not without further argument, anyhow).

For a simple example, in the language of physics:
    An object a satisfies "x is a black hole" iff a is a black hole
    "There is x st. x is an F" is true iff there is something which
satisfies "x is an F"
from which we can derive the T-sentence:
    "There is a black hole" is true iff there is a black hole
This has nothing to do with provability, or knowledge, or evidence, or
anything epistemological. Whether a statement is true depends upon the
FACTS.

As I stressed in my *first* message on this topic, it is now well-known how
to construct materially adequate truth definitions for the first-order
language of arithmetic. The predicate defined is a Pi^1_1 formula Tr(x)
whose extension is exactly the hyperarithmetic set of (the codes of) all the
true arithmetic sentences.
It is also a well-known fact that:

(1) The set of arithmetic truths is not axiomatizable

This shows that arithmetic truth is distinct from provability (in any fixed
sound system for arithmetic). Joe Shipman has also recently stressed this
point.

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
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~