Truth Definitions, Filozofia, Filozofia - Artykuły

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Truth Definitions, Skolem Functions and Axiomatic Set Theory
Author(s): Jaakko Hintikka
Source: The Bulletin of Symbolic Logic, Vol. 4, No. 3 (Sep., 1998), pp. 303-337
Published by: Association for Symbolic Logic
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The
Bulletin of Symbolic Logic.
LOGIC
Volume
4,
Number
3,
Sept.
1998
TRUTH
DEFINITIONS,
SKOLEMFUNCTIONS
AND AXIOMATICSETTHEORY
JAAKKO HINTIKKA
?1.
The mission of axiomatic set
theory.
What is set
theory
needed for in
the foundations of mathematics?
Why
cannot we transact whateverfounda-
tional business we have to transact in terms of our
ordinary logic
without
resorting
to set
theory?
There are
many possible answers,
but most of them
are
likely
to be variations of the same theme. The core area of
ordinarylogic
is
by
a
fairly
common
consent the
received
first-order
logic. Why
cannot it
take care of itself? What is it that it cannot do? A
large part
of
every
answer
is
probably
that first-order
logic
cannot handle its own model
theory
and
other
metatheory.
For
instance,
a first-order
language
does not allow the
codification of the most
important
semantical
concept,
viz. the notion of
truth,
for that
language
in that
language itself,
as shown
already
in Tarski
(1935).
In view of such
negative
results it is
generally thought
that one of
the most
important
missions of set
theory
is to
provide
the wherewithalfor a
model
theory
of
logic.
For instance
Gregory
H. Moore
(1994, p. 635)
asserts
in his
encyclopedia
article
"Logic
and set
theory"
that
Set
theory
influenced
logic,
both
through
its
semantics, by
expanding
the
possible
models of various theories and
by
the formal definition
of a
model;
and
through
its
syntax, by allowing
for
logical languages
in which formulas can be infinite in
length
or in which the
number
of
symbols
is uncountable.
The obvious
rejoinder
here is that one can
perfectly
well describe
models of
any
sort and define what it means for a structureto be a model for a
given
sentence without
resorting
to set
theory.
The most obvious candidates for
this role are
higher-orderlogics.
Likewise,
infiniteformulascan be described
by
means of the resources of
higher-order
and sometimes even first-order
logic. Admittedly,
first-order
logic
alone cannot do all the
jobs
that set
theory
was
reputedly
introduced to do. For
instance,
one cannot define
truth for a
given
first-order
language
in that
particular
first-order
language,
but one
might
be able to define it in
another, richer,
language.
Indeed,
Tarski
Received
July29, 1997;
revised
April17,
1998.
)
1998,
Association for
Symbolic Logic
1079-8986/98/0403-0003/$4.50
303
THEBULLEtIN
(F SYMBOLIC
304
JAAKKO HINTIKKA
himself studied the conditions on which such a definition is
possible. (See
Tarski
1935.)
For
instance,
as Tarski in effect
showed,
one can define truth
for a first-order
language
in the
corresponding
second-order
language.
However,
before
we
indulge
in such
"semantic
ascent,"
we should be
clear about the reasons for it. In this
paper,
it will turn out
logicians
have
universally
missed the
true,
exceedinglysimple
feature
of
ordinary
first-order
logic
that makes it
incapable
of
accommodating
its own truth
predicate.
(See
Section 4
below.)
This defect will also be shown to be
easy
to overcome
without
transcending
the first-orderlevel. This eliminates once and for all
the need of set
theory
for the
purposes
of a
metatheory
of
logic.
Behind the idea that we need set
theory
for a
metatheory
of
logic,
there
thus often
lurks
the
assumption
that
only
first-order
logic
is
really logic
and
that
higher-order
logic is,
as
Quine
once
put
it,
"set
theory
in
sheep's
clothing."
Ironically,
the
selfsame
inability
of
ordinary
first-order
logic
to
serve as its own
metatheory
has been used
by
Hilary
Putnam
(1971)
as an
argument
against
the identificationof first-order
logic
as the
logic.
Thus there
is
obviously
a
great
deal of
uncertainty
and confusion about the
relationship
of
higher-order
logics
to set
theory
and indeed about the status of these
logics.
It
has
for
instance,
been
argued
in effect
by
Michael Friedman
(1988)
that the
inadequacy
of
ordinary
first-order
logic
as
its
own model
theory
is
the fundamental reason
why Carnaps'sproject
of a
general "logicalsyntax
of
language"
was
doomed
to
fail. What
will
be established
in
this
paper
shows
that both Putnam's and Friedman'slines of
argument
are
inconclusive.
Be its motivation what it
is,
set
theory
is often considered as some sort of
instant model
theory-or perhaps
we should
say,
non-Californians' model
theory.
For
instance,
when the
sharpestphilosophers
of science realizedthat
a
study
of "the
logical syntax
of the
language
of science" was not
enough,
they
resorted to set
theory
for their
conceptualizations. Ironically
some
misguided philosophers
of science have continued to seek salvation in set
theory
even
long
after the
development
of
logical
semantics and
systematic
model
theory.
In a different corner of the
philosophical world, Wittgenstein's
fanatical
hatred of set
theory
can
only
be understood as a
corollary
to his
deep-
seated and total
rejection
of all model-theoretical and other metatheoretical
conceptualizations.
Such ideas of set
theory
as common folks' model
theory
cannot be
shrugged
off as
popular misconceptions.
It is
hard
to
see
what
foundational
interest
set
theory
would have if it could not serve as a universalframeworkof all model
theory.
It even seems
likely
(as
Moore
apparently
suggests)
that
part
of the
early
motivation of set
theory
was to see it as a universalsource of models for
all and
sundry theories, including
those that cannot be
interpreted
in
already
known theories. Such a
conception
of set theoretical universe as "the model
of all models" seems to lurk in
wings
of much of the
early twentieth-century
TRUTH
DEFINITIONS,
SKOLEM FUNCTIONS AND AXIOMATIC SET THEORY
305
foundational discussion. It is even
present
in Tarski'sclassical
monograph
(Tarski 1935). Contrary
to what is often said and
thought,
Tarski does not
there show how to define truth in a structure
(model).
He shows how to
define it in the model. He is there
envisaging only
one
omnicomprehen-
sive structureof structures of which for instance the models of the theories
Hilbert's school were
studying
form a subset
(Tarski 1956,
p. 199).
In orderto avoid
misunderstandings
it is
important
to realizethat
thereis
a
set
theory
and
there is
a
set
theory.
Conceived
of as a
study
of
differentkinds
of infinite cardinals and
ordinals,
set
theory
is of course
unobjectionable,
but it
is then
only
one mathematical
theory among
others,
without a claim
to be a universal frameworkof all model
theory.
Now the current incarnation of the idea of
general
set
theory
is axiomatic
set
theory.
It exists in different
varieties,
but the
differences,
say,
between the
Zermelo-Fraenkel set
theory
and the von
Neumann-Bernays
set
theory
are
not relevantto our
purposes
in this
paper.
What is crucialis that
they
arefirst-
order theories
in
the sense
of
employing ordinary
first-order
logic
as their
sole
logical component.
We
also have to
make the minimal
assumption
that
the set
theory
in
question
is rich
enough
for us to do
elementary
arithmetic
in it.
But if axiomatic set
theory
is
supposed
to be an
implementation
of the
idea of set
theory
as
poor
man's model
theory,
tables can be
turned
on it.
It will be shown that truth can after all be defined for a
suitable first-order
language
in that
language
itself,
but that
truth in a
model of axiomatic set
theory
cannot be defined
in
that
set
theory.
Moreover,
this
failureis
not
just
a
corollary
to the deductive
incompleteness
of axiomatic set
theory,
which
has
necessitated
a search for
new,
stronger
axioms.
No,
it will be shown
that
any attempt
to define set-theoretical truth in axiomatic set
theory yields
wrong
results. Our
argument
does not turn on
any requirement,either,
that
the models of axiomatic set
theory
be standard in
something
like Henkin's
(1960)
sense.
Since truth is the central
concept
of all
semantics,
it will thus be
argued
that,
while at least some of the central
parts
of the model
theory (semantics)
of first-order
logic
can be
expressed
in a first-order
language,
the semantics
of axiomatic set
theory
cannot in
general
be dealt with in such a set
theory.
It is hard not to consider this as a failure of its
original
mission.
?2.
Truth
predicates
for
arithmetical
languages.
How
can all this be shown?
The
positive
part
is
relativelyeasy.
Since
essentially
the same
story
has been
told
elsewhere, perhaps
we can afford to be brief. We will consider first a
first-orderarithmetical
language
L-more
generally,
a first-order
language
that includes
elementary
number
theory-which
has as its twin the corre-
sponding
second-order
language
L(2). In such a first-order
language
we
can
carry
out a G6del
numbering
for all formulas both for that first-order
306
JAAKKO HINTIKKA
language
L and for its second-order extension L(2).
Actually,
we can restrict
our attention to the
E1M
fragment
L(*) of L() It is assumed that the
logical
constants of
L
are =,
,
&,
V,
(3x), (Vy).
How can we set
up
a truth-definition for L in L(*)? We will use an idea
that
perhaps
can be considered as a mirror
image
of Tarski's
(1956,
Section
3)
recursivemethod of
truth-definition. We
will consider the
characteristic
properties
that
any
truth
predicate
say
X will have to
satisfy.
(This
X
will
of
course be a
complex
numerical
predicate
of
natural
numbers,
including
Godel
numbers
of
formulas.)
If we succeed in
expressing enough
of
those
characteristics
to make
sure that it
behaves
in
the
right way,
we can then
attribute truth to the Godel number
g(S)
of a sentence S
by simply saying
that there exists such a
predicate
and that
g(S)
has it. If
Tr[X]
is a
summary
of the characteristicsin
question,
then the truth
predicate
will be
(2.1)
(3X)(Tr[X]
&
X(y))
where
y ranges
over natural
numbers,including
Godel numbersof sentences.
If
Tr[X]
is a E.M
formula,
then so will
obviously
be
(2.1).
What the differentclauses of
Tr[X]
will
say
can be
expressedby
a
conjunc-
tion of conditionals and
biconditionals.
In the
following
table,
the left
hand
side describes the antecedent of each conditional while the
right
hand
side
describes the
corresponding consequent.
(2.2) (S1
&
52)
is true
S1
and
S2
are
true
More
literally,
this can be
expressed
as follows:
(2.3)
X
applies
to the Godel X
applies
to the Godel numbers
number of
(S1
&
S2)
of
SI
and of
S2
Denoting
"GSdel number of"
by g,
we can write:
(2.4) X(g(Si
&
S2)) (X(g(S1))
&
X(g(S2)))
It is well known
(and
fairly obvious)
that
g(S1)
and
g(S2)
are recursive
functions of
g
((S1
&
S2))
and hence
representable
in the
language
of elemen-
tary
arithmetic.
(More generally,
for
questions
of
representability
of differ-
ent relations in
elementary arithmetic,
see
any
standard
exposition
of
Godel
numbering,
and Godel's
incompleteness theorem,
for instance Mendelson
1987,
pp. 149-168.)
This means that the
general
conditional from
(2.5) X(g(S1
&
S2))
to
(2.6)
(X(g(S1)
&
X(g(S2)))
is
expressible
in the
language
in
question.
This conditional is one of the
conjuncts
in
Tr[X].
Likewise,
we can write:
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