## 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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