elliott mendelson - foundations logic, language and mathematics

8/18/2019 Elliott Mendelson - Foundations Logic, Language and Mathematics

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FOUNDATIONS:

LOGIC,

LANGUAGE,

AND MATHEMATICS


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FOUNDATIONS:

LOGIC LANGUAGE

AND

MATHEMATICS

Edited by

HUGUES LEBLANC, ELLIOTT MENDELSON,

and

ALEX ORENSTEIN

Reprinted from

Synthese,

Vol. 60 Nos. 1 and 2

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.


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TSBN 978-90-481-8406-4 ISBN 978-94-017-1592-8 (eBook)

DOI 10

.1007/978-94-017-1592-8

AlI Rights Reserved

Copyright © 1984 by Springer Science+Business Media Dordrecht

Originally published by D. Reidel Publishing Company. Dordrecht. Holland in 1984

Softcover reprint

of

the hardcover 1st edition 1984

No

part of the material protected by this copyright notice may be reproduced or

utilized in any form or by any means, electronic or mechanical,

inc1uding photocopying, record ing

or

by any information storage and

retrieval system, without written permission from the copyright owner


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TABLE

OF CONTENTS

Series Editors' Preface 1

Preface 3

MELVIN

FITTING I

A Symmetric Approach to Axiomatizing Quan-

tifiers and Modalities 5

NICOLAS

D .

GOODMAN

I

The Knowing Mathematician 21

RAYMOND

D.

GUMB

I "Conservative" Kripke Closures 39

HENRY HIZ

I

Frege, Lesniewski, and Information Semantics on the

Resolution

of Antinomies

51

RICHARD

JEFFREY

I

De Finetti's Probabilism 73

HUGUES L E BLANC

and CHARLES

G . MORGAN I

Probability

Functions and Their Assumption Sets

-The

Binary Case 91

GILBERT HARMAN I Logic and Reasoning 107

JOHN MYHILL I Paradoxes 129

ALEX

ORENSTEIN I Referential and Nonreferential Substitutional

Quantifiers 145

WILFRIED SIEG

1

Foundations for Analysis and Proof Theory 159

RAYMOND

M.

SMULLY

AN

I

Chameleonic Languages 201

RAPHAEL STERN

I

Relational Model Systems: The Craft

of

Logic 225

J. DILLER and

A. S. TROELSTRA I

Realizability and lntuitionistic

Logic 253


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CITY

COLLEGE

STUDIES

I

N TH

E H

IST

OR

Y AN

D PH

ILO

SOP

HY

O

F SC

IEN

CE

AN

D T

ECH

NOL

OG

Y:

SER

IES

E

DIT

ORS

'

P

REF

AC

E

R

ecen t

year

s hav

e see

n the

em e

rgenc

e of

severa

l new

app

roach

es to

the history and philosophy of science and technology. For one, w hat

w

ere p

erceiv

ed b

y man

y as

separ

ate, t

houg

h per

haps

relate

d, fie

lds o

f

i

nquiry

h av

e com

e to b

e reg

arded

by m

ore a

nd m

ore sc

holar

s as a

singl

e

d

iscipl

ine w

ith d

iffer

ent a

reas

of em

phas

is. I

n thi

s dis

ciplin

e any

p

rofou

nd un

ders

tandin

g so

deep

ly int

ertwi

nes h

istory

and

philo

soph

y

t

hat i

t mig

ht

b

e

said

, to

parap

hras

e K an

t, th

at ph

iloso

phy w

itho

ut

his

tory

i

s

em p

ty an

d his

tory w

ithou

t phi

losop

hy

is

blind

.

Ano t

her co

ntem

porar

y tre

nd in

the h

istory

and p

hilos

ophy

of sc

ience

and

tech

nolog

y ha

s bee

n to

bring

toge

ther

the E

nglis

h-spe

aking

and

continental traditions in philosophy. The views of those who do analytic

philos

ophy

and

the v

iews

of t

he he

rmen

eutici

sts h

ave c

om b

ined

to

in f

luenc

e the

thin

king

of so

m e p

hiloso

pher

s in t

he E

nglish

-spea

king

wo

rld,

and o

ver

the la

st de

cade

that

influ

ence

has b

een

felt i

n the

histo

ry an

d ph

iloso

phy o

f sci

ence.

T

here

h

as als

o be

en th

e lon

g

s

tandin

g in

fluenc

e in

the

West

of c

ontin

ental

think

ers w

orki

ng on

prob

lems

in

the p

hilos

ophy

of t

echno

logy.

Thi

s syn

thesis

of

two

trad

itions

has m

ade

for a

riche

r fund

of id

eas a

nd ap

proa

ches

that m

ay

change our conception of science and technology.

S

till an

other

tren

d tha t

is

in

some

ways

a com

bina

tion o

f the

previ

ous

tw o

cons

ists o

f the

work

of tho

se ch

aract

erize

d by s

ome

as the

"Fri

ends

of D

isco

very"

and

by ot

hers

as the

brin

gers

of the

New

Fu

zzine

ss".

This a

ppro

ach t

o the

histor

y and

phil

osoph

y of s

cienc

e and

tech

nolog

y

conc

entra

tes o

n

cha

nge,

prog

ress,

and discov

ery.

It

ha

s rai

sed o

ld

episte

molo

gical

quest

ions u

nder

the g

uise

of the

prob

lem o

f rati

onali

ty

in the

scie

nces.

Altho

ugh

this a

ppro

ach h

as its

origi

ns in

the w

ork

of

Th

omas

Kuh

n in

the U

nite

d Sta

tes, a

t temp

ts to

expr

ess h

is ide

as in

explicit set-theore tical or m odel-theoretic terms are now centered

in

Germ

any.

Synth

ese

60 (1984

) 1

.


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2

S E R I

ES

E DI T OR S PR E

FA C E

The m ore

tradition

al approac

hes to the

history an

d philosop

hy of

science

and techn

ology cont

inue as we

ll, and prob

ab ly will c

ontinue as

long as

there are

skillful p

ractitioners

such as

Carl Hem p

el, Ernest

Nagel

, and t h ~ i

students.

Final

ly, there a

re still o t

her approa

ches that

address so

me of the

technic

al problem

s arising w

hen we try

to provid

e an accou

nt of belief

a

nd of rat

ional choi

ce . - T he

se include

efforts to

provide

logical

framework

s within wh

ich we ca

n make sen

se of these

notions.

This ser

ies

will

at

tempt to b

ring toge

ther work

from all o

f these

app

roaches to

the history

and philo

sophy of sc

ience and

technology

in

the belief that each has something to add to our understanding.

The

vo

lumes of t

his series h

ave emerg

ed either f

rom lectur

es given

by autho

rs while th

ey served

as honorar

y visiting p

rofessors a

t the City

C

olleg e of N

ew York o

r from c on

ferences sp

onsored by

th at instit

ution.

The

City C

ollege Pro

gram in th

e History

and Philos

ophy of Sc

ience

an

d Technolo

gy overse

es and dir

ects these

lectures a

nd confere

nces

wi

th the fina

ncial aid o

f the Ass

ociation fo

r Philosop

hy of Scie

nce,

P

sychothera

phy, and E

thics.

MA

RTIN TAMNY

RA

PHAEL STERN


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PRE

FACE

The

pape

rs in this c

ollection s

tem largely

from the

conferenc

e 'Foun

dat

ions: Log

ic, Langua

ge, and M

athematic

s ' held a t

the Grad

uate

C en

ter of the

City Unive

rsity of Ne

w York on

14

-15 N o

vember 1980

.

T h e

conferen

ce was sp

onsored by

the Phil

osophy Pr

ogram a t

the

G rad

ua te Cen

ter of th

e City U

niversity o

f New Y

ork .and

the

Associatio

n for Philo

sophy of S

cience, Psy

chotherap

y, and Eth

ics. We

w

ish to exp

ress our g

ratitude an

d appreci

ation to th

ese organi

zations

a

nd to than

k the serie

s editors, R

aphael S t

ern and M

artin T am n

y, for

their help with the conference and the preparation of this collection .

Synthese

60 (1 984) 3.

HUGUE

S

LEBL

ANC

ELLI

OTT ME

NDELSON

ALEX

ORENS

TEIN


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MEL YIN FITTING

A SYMMETRIC APPROACH TO AXIOMATIZING

QUANTIFIERS

AND

MODALITIES

1 . I N T R O D U C T I O N

We present an axiomatization of several of the basic modal logics, with

the idea of giving the two modal operators 0 and 0 equal weight as far

as possible. Then we present a parallel axiomatization of classical

quantification theory, working

our

way up through a sequence of rather

curious subsystems. It will be clear at the end that the essential

difference between quantifiers and modalities

is

amusing in a vacuous

sort of way. Finally

we sketch tableau proof systems for the various

logics we have introduced along the way. Also, the "natural" model

theory for the subsystems of quantification theory that come up

is

somewhat curious. In a sense, it amounts to a "stretching

out"

of the

Henkin-style completeness proof, severing the maximal consistent part

of the construction quite thoroughly from the part of the construction

that takes care of existential-quantifier instances.

2. B A C K G R O U N D

It is contrary to the spirit of what

we

are doing to take one modal

operator as primitive and define the other, or, one quantifier as

primitive and define the other. So, by the same token, we take as

primitive all the standard propositional connectives too. Thus,

we

have

available all of

A,

v,

~ , =>, 0 , 0,

'V,

3.

We also take as primitive a truth

constant

T and a falsehood constant

l..

For our treatment of propositional modal logic we assume

we

have a

countable list of atomic formulas, and that the set of formulas

is

built up

from them in the usual way. We will use the letters "X",

"Y''

, etc., to

denote such formulas.

For quantification theory, we assume we have a countable list of

variables and also a disjoint countable list of parameters. We will use

the letters

"x",

"y",

etc., to denote variables, and

"a",

"b",

etc., to

denote parameters. Formulas are built up in the usual way, with the

understanding that a sentence contains no free variables, though it may

Synthese

60

(1984)

5-

.19. 0039-7857/84/0601-0005 $01.50

© 1984

by D . Reidel Publishing Company


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6

M E

L V I

N F

IT T

IN

G

con

tain

pa

ram

ete

rs. W

e

fo ll

ow

th e

con

ven

tion

th

at if

cp(x

) is

a

form

ula

wit

h on

ly x

fr

ee,

then

cp(

a)

is

th

e re

sult

of

repl

acin

g a

ll fr

ee o

ccu

rre

nce

s

of

x

by occurences of

a

in

cp

W

e w

ill

use

Kr

ipk

e's

mod

el

theo

ry

for

m o

dal

log

ics

(and

an

an

alo g

for

qu

anti

fica

tion

al

theo

rie

s, t

o b

e de

scr

ibed

in

Se

ctio

n 4

). F

o r

us,

a

Kri

pke

m o

del

is

a

qua

drup

le ('

fi,


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A

X IO M A T I

Z I NG

Q U A N

T IFIERS A N D

M O D A

LITIES

7

Notice tha

t (b)-(e) in

the defin

ition of a K

ripke m o

del above

gives us

flf-

a

iff

f

If-

a

1

and flf-

a2

flf-

f

iff

flf- {31 or

f l f {32.

Inde

ed, this co

uld be use

d instead o

f (b)-(e).

W e

extend thi

s uniform

notation to

the modal

operators

(as in Fitti

ng

[1]) by

defining

the v-formu

las (neces

saries) and

1r-formula

s (possible

s)

a

nd their c

omponents

v

0

and 7To,

respectiv

ely, as foll

ows:

v

ox

~

x

v

X

~ x

7T

7To

X

~ x

N ot

ice th at (f

)-(h) in

the definitio

n of a K ri

pke m ode

l above, ta

ken

togethe

r w ith (d),

give us th

e following

equivalen

t conditio

ns:

fo

r every f E

(f')

f

If-

1r but

f

I. L

v;

for every

f

E

§ -

(x)

~

cf>(a)

The

idea is, in

a classical

first-order

model in w

hich the d

omain con

sists

of th

e set of pa

rame ters,

an

d

each

param eter

is inte rpre

ted as nam

ing

itself:

Y is true i

ff for all a ,

Y(a)

is

tr

ue;

o

is true iff

for som e a , o

(a)

is

t

rue.


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8

M E L V I N FI T T I NG

So much for background.

3 . M O D A L L O G I C S , A X I O M A T I C A L L Y

Let

us

assume that we have an axiomatization of the classical pro

positional calculus, with modus ponens as the only rule. We will build

on that in our introduction of rules and axioms pertinent to tfle modal

operators.

Modal logics are often formulated with a rule of necessitation, which

we could give

as

vo

v

But this gives a central role to the v-formulas and, for no reason other

than autocratic whim, we want to develop modal logic as far as possible

giving equal weight to both v-and 1r-formulas, giving to both neces

saries and possibles a fair share. After a certain amount of experimen

tation we hit on the following curious rule:

(M)

1To v Vo

1 T V V

(we use M for modalization). As a matter of fact, rule M is a correct rule

of inference, as the following argument shows.

Suppose

(W,

, A,

If-)

is a Kripke model, and 1r

0

v

v

0

is valid in it

(holds at every possible world). Let

r E

W; we show

flf- 1r

v

v.

Well, if

r E

, that is, if

r

is queer, then

r If-

7T; so, trivially,

r If-

1T v

v.

Otherwise,

f

E

Y

DX::::>DY

X::::>Y

OX::::>OY


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AXI OMAT I Z I NG QUANT I FI E R S

AND

M ODAL I T I E S 9

Then one may show,

in

the usual way, that replacement of proved

equivalences holds as a derived rule:

X = X'

Z = Z '

where Z' results from Z by replacing some occurrences of X in Z by

occurrences of

X' .

Here we have used = as an abbreviation for mutual

implication. (Actually a stronger version may be shown, concerning

"semisubstitutivity" of implication, in which we must take into account

the positiveness and negativeness of occurrences as well.

The

details

needn't concern us here.)

Now go back and look again at the justification we gave for rule

M.

In the argument that 7T v

v

held at the nonqueer world r we never

needed that 7To v v

0

held at every world of the model; we only needed

that it held at every world accessible from f. But to say that

7To

v v

0

holds at every world accessible from r is to say that D(

7To

v v

0

)

holds at

f. Thus, the same argument also shows the validity in all Kripke models

of the schema

(Ml)

D[

7To

v

vo]::) [ 7T

v

v].

Let

us

add

it

as an axiom schema then.

I f we do so, it is not hard to show that we have a complete axiomatic

counterpart of the Kripke model theory as given in section 2.

That is, X

is

provable in the axiomatic system just described iff

X

holds at every

world of every Kripke model.

To

show this, one may show complete

ness directly, using the now-common Lindenbaum-style construction,

or one may show this axiom system is of equal strength with one

standard in the literature, and rely on known completeness results. We

skip the arguments.

The logic axiomatically characterized thus far is called C in Seger

berg [5]. It is the smallest

regular

modal logic (see Segerberg [5]).

Now, in the model theory for C, one can have queer worlds in which

everything is possible but nothing is necessary. Also there are no special

conditions placed on the accessibility relation

rJl,

so there can be

"dead-end" worlds, worlds from which no world is accessible. In such a

world, everything is necessary, nothing

is

possible. So, our next item of

business is

to rule out such strange worlds.


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10

MEL YIN FITTING

Worlds in which nothing is necessary may be eliminated by postulat

ing that something is necessary. Let us, then, add the axiom

(M2) OT.

The model theory appropriate to this (with respect to which one can

prove completeness) is all Kripke models in which there are no queer

worlds, that is , all normal Kripke models. The logic axiomatized thus

far is the smallest normal logic, and is usually called K (see Segerberg

[5]).

Next, worlds in which nothing is possible may be eliminated

by

postulating that something

is

possible. We take as an axiom

(M3) T.

The model theory appropriate to this is all normal Kripke models in

which every world has some world accessible to it.

The

logic

is

generally called D. (Again, see Segerberg [5]).

Finally, we might want to restrict our attention to models in which

each (normal) world

is accessible to itself (in which the accessibility

relation is reflexive). To do this one adds either (or both) of

(M4)

v ~

vo

7To

~ 7T

The logic thus characterized

is

T.

Note that T T is an instance of the second of these schemas, and

since

T is

a tautology, T follows by modus ponens. Thus with M4

added, M3 becomes redundant.

REMARKS.

One

goes from C to K by adding

OT

as an axiom. It

is

of

some interest to consider a kind of halfway point, where instead of

adding

OT as an

assumed truth,

we take it as a

hypothesis.

That is, form

the set

S

of formulas

X

such that OT ~ X

is

provable

in

C. This set

S

is,

itself, a (rather strangely defined) modal logic, intermediate between C

and K. I t is closed under modus ponens, but not under rule M. Rather, it

is closed under the weaker rule.

0( 7To

v

vo)

0(

7TV

v)

'

which

is

equivalent to Becker's rule. It

is,

in fact, the logic axiomatized

in Lemmon [4]

as

P2, but without his axiom

O X ~ X

(our M4).


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AXI OMAT I Z I NG QU ANT I FI ER S

AND M ODAL I T I E S 11

If

we

strengthen things a bit, by considering those X such that

DT

:::>X

is provable in C plus axiom M4, one gets the Lewis system S2 .

(Again, see Segerberg [5], chapter four.)

One can also play similar games with axiom M3 (0 T) to produce

interesting logics. We know very little about them.

4. QUANT I FI E D L OGI C S , AXI OM AT I C AL L Y

There is an obvious analogy (of sorts) between the modal operators and

quantifiers. What we do in this section is to parallel the development of

section 3, substituting quantifiers for the modal operators to see how far

the analogy extends when things are done the way we did. The idea

is

simple: ('v'x) may behave likeD, (3x) like 0, y like

v

and o ike ?T. This

may be so;

we

will see.

The language now is first order. Once again we assume a pro

positional-logic base with modus ponens as the sole rule.

First, the analog of rule M

is

rule (Q)

o(a)

v y(a)

OV )

And, as a matter of fact, this is a correct rule of inference in classical

first-order logic. This argument is left to the reader.

Using rule

Q,

one may show analogs of the results listed in section 3

based on rule

M.

Thus, one may derive the usual inter definability of the

quantifiers (again, as mutual implication), and one may show that

replacement of proved equivalences holds as a derived rule.

Now, if you actually thought through the "justification" of rule Q,

almost certainly you also showed the validity, in all first-order models,

of the schema

(Ql) ('v'x)[o(x) v y(x)] :::> [o v y].

So let us add it as an axiom schema. We might, by analogy, call the

resulting logic QC. I t thus has rule

Q

and schema Ql.

The following

is

a reasonable question: What

is

an adequate model

theory for QC,

one with respect to which completeness can-be shown?

Well, the following rather curious one

will

do. We simply translate the

corresponding modal model theory, making suitable adjustmer:ts to

take care of the fact that quantified sentences have many instances, but

modalized formulas have single components. We have chosen, for


16/282

12

MEL YIN FI T T I NG

simplicity, to leave out any mention of the notion of an

interpretation

in

a model. A more elaborate treatment would have to include it, but the

following is enough for our purposes.

A model

is

a quintuple

(Cfi, « >, < P,

< fl,

If-)

where: (1) CfJ is a nonempty

set (of possible worlds); (2) « ~ Cfi; (3) < Pis a mapping from members of

CfJ to nonempty sets of parameters; (4) < fl

is

a relation on Cfi; and (5) If-

is

a relation between members of CfJ and sentences such that

for every f E

Cfi,

conditions (a)-( e) as in section 1;

for every f E

« >,

(f) flf-(3x)cp(x) but fi,V-(Vx)cp(x);

for every f E Cfi- « >

(g) flf-(Vx)cp(x) iff for every dE CfJ such that f< /ld, and for

every a E < } (d), d If- cp(a)

(h) f If-: (3x)cp(x) iff for some dE CfJ such that f< fld, and for some

a

E

< P(d),

fl.

If-

cp(a).

Now, a sentence

X is

a theorem of the logic

QC

iff

X

holds at every

world of every such model. We leave the correctness half to the reader.

Note that

if

cp(a) is provable in QC;so

is

the parameter variant cp(b).

This

will

be of use in proving correctness. And we briefly sketch the

completeness half in the next section.

We note that we could restrict models so that, for each world f,

< P(r)

is a singleton. I t makes no difference. Now

we

can continue with

our

development, paralleling that of section 3.

In the quantifier models above there can be

"queer"

worlds (mem

bers of cp) in which everything exists, and hence no universal sentences

hold. Such anomalies can be eliminated by adding the postulate

(02) (Vx)T.

Doing so gives us a logic we may call QK. An adequate model theory

for it

is

one that consists of all models of the sort described above, but

with« always empty, that

is

no "queer" worlds.

Next, there may still be worlds from which no world is accessible. In

such a world, every universal sentence holds, but no existential. They

behave rather like empty-domain models of first-order logic. They may


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A X

IO M ATIZI

N G Q U

NTIFIE

R S

AN

D M O D A L

ITIES

1

3

be ru led out

by add

ing the ax

iom

(03)

(3x)T.

Let

us call th

e resulting

logic QD.

Its m od e

l theory

is

tha

t of QK

with

the addi t

ional requ

irement that

for ever

y w orld r

there m us

t

be

so m e

w

orld L

l

su c

h tha t

f ~ L l .

Finally (?)

we may

add an ana l

og to M 4 ,

namely

(Q 4

) y => y(a)

or

«5(a) > «

5

(

or both).

A s m ight

be

ex pec ted,

Q3

then b ecom e s redun dan t .

We can

call the

logic thus

axiom atize

d

QT.

Its

m o d el the

ory is that

of Q

D

w

ith the

restr ict ion

tha t the a

ccessibilit

y rela t ion ~

b e reflex

ive.

We

have reac

hed the

end

of

o

ur

parallel

developm

ent (obviou

sly,

since we

have matc

hed every

th ing we d

id in sect io

n 3).

But

we do n

ot

yet

have the u

sual first-

order logic

.

O

ne

d

oesn't

w ant

classical fi

rst

order m

odels with

lots of po

ssible w o r l d ~

in them

: a classic

al model

should be

a one-w

orld m odel

.

N ow

, to only co

nside r one

-wor ld Kr i

p k e m odal

m o d els is

to trivialize

m odal l

ogic; it re

nders the

m o d al ope

ra to rs usel

ess. N eces

sary tru th

b

e c o m es th

e sam e th

ing as tru

th.

T

hat i

s, A => DA

is valid

in all

o n

e -world m

odels. Of c

ourse , this

is no t desi

rable. M od

al operato

rs are

suppo

sed to do

s o m ething

; they are

supposed t

o have an

effect; the

y

ought n

ot be

vacuou

s .

W ell, it i

s precisely

at this poi

n t tha t m o d

al operat

ors

and

qua

ntifiers

diverge.

Quan tifiers

can

be va

cuous.

L

et cp be

a se

ntence (he

nce with

no free va

riables). Then

( Vx)cp o

ught to

mean no t

h ing

mor

e

than cp

itself. So

our

final quantificational axiom

schema

is, simply,

(Q 5)

cp =>

( Vx)c

p,

w here

cp is a sente

nce.

When thi

s is ad d ed ,

convent io

nal c lassica

l first-orde

r logic is

the

result.

5.

C

O M PL E T E

N ESS, HO

W PR O V E

D

In

a sense, t

h e p ro o f o

f the co m p

leteness of

the quant

ifier system

QC

w ith r

espec t to t

h e m odel

theory pr

esented in th

e p revious

sect ion is

a

stret

ching out

of th e usu

al

H

enkin

co m p leten

ess p ro o f f

or first-ord

er

logic

.

Let

us

sketch w hat

we mean

b

y this.

R ecall tha

t th e u

sual Henkin

argume

nt runs as

follows.

Take a


18/282


19/282

AXI OM AT I Z I NG QUANT I FI E R S A N D M ODAL I T I E S 15

model to

F,

since ~ F E r

F ·

Notice that in the proof just sketched, the maximal consistent

construction gives the worlds, while the existential instantiation moves

things from one world to another. This

is

what we meant by "stretching

out" the Henkin construction.

Adding axioms (02)-(04) modifies the construction in obvious ways.

We leave this to the reader.

Now we can ask, What

is

the effect on this construction of also

imposing that final axiom schema (05), cp:::;)

(Vx)cp?

Very simply, it

makes things cumulative. Notice that, in our model, if fq[A, then

moving from

r

to

A,

in effect, drops one quantifier from each sentence.

But cp:::;) (Vx)cp allows us to add one quantifier,

so-the

effect

is

neutralized. Briefly, if we assume (05), then if fll- cp and

fq[A,

then

A II- cp.

For, if fll-

cp,

since also fll- cp:::;)

(Vx)cp,

we must have fii-(Vx)cp.

And since fq[A, then

A II-

some-instance-of-cp. But since the quantifier

was vacuous, this m e a n s ~

II-

cp. Now, that things are cumulative if

(05)

is

imposed means the limit (=chain-union) part of Henkin's proof can

be carried out. And thus a conventional classical model results.

This sketch must suffice. Details, though slightly devious, are far

from devastating.

1. S E M ANT I C T AB L E AUX

We show how the tableau system of Smullyan [7] for propositional logic

may be extended to handle the logics discussed in section 4. We begin

with a brief sketch of the system that suffices for propositional logic.

First, proofs are in tree form (written branching downward). There

are two branch-extension rules:

a

(If

a

occurs on a branch,

a

1

and

a

2

may be added to the end of the

branch.

I f

f3 occurs on a branch, the end of the branch may be split, and

{3

1

added to the end of one fork, {3

2

to the end of the other.)

A branch

is

called closed

if

it contains A and ~ A for some formula

A, or if it contains _1_, or if it o n t a i n s ~

T.

A tree is called closed if every


20/282

16

M EL

VIN FIT TING

branch is

closed. A

closed tre

e with ~

X at the ori

gin

is

, by d

efinition, a

proof

of X.

W e begin

by addin

g to the ab

ove a tab

leau rule to

give the

modal

l

ogic C . In

words, th

e rule is as

follows. If,

on a b

ranch, ther

e are

v-for

mulas, an

d there is

a 7T-for

mula, then t

hat 7T-f

ormula may

be

repla

ced by 1r

0

,

all the v-f

ormulas by

the corre

sponding

v

0

-formula

s,

and a

ll oth er fo

rmulas del

eted.

W

e schemat

ize this as

follows. Fi

rst, if S is a

set of form

ulas, defin

e

S#

=

{

Vo

J v

E S}.

Then

the rule

is

s,

7T

S#,

7To

(provided S#-

4>),

where this

is to be int

erpreted as

follows: if

S U {

1

r} is

the set of f

ormulas

on a bran

ch, it may

be replac

ed

by S

#

U {

7

To} (pro

vided

S

# is

not

empty).

EXAMPLE.

W e show

OX

::

J

~ o

~ X

is

provable using this rule.

The p

roof begin

s

by

puttin

g

~

O X ::J

~ 0 ~ X

) at the ori

gin, then

two a

-rule applicati

ons produ

ce the follo

wing one-

branch tre

e:

~

O X

::J

~ O

~ X)

ox

~ ~

o ~

x

o ~ x .

Now take

S

to consist of the first three formulas, and

7T

to be 0

~ X .

The

n

S#

= {X}

, which is

n

ot em pty,

so the rule

says the se

t of formul

as

on th

e branch m

ay be rep

laced by S#

U {1r

0

}, n

amely

X

~ x

and this is

closed

RE

MARK. Beca

use of the

way trees

are written

, an occu

rrence of a

for

mula may

be com m

on to seve

ral branch

es, b u t

we may wis

h to

m

odify (or d

elete) it u

sing the ab

ove rule o

n only o n

e branch.

T h e n ,

s

imply, first

add new

occurrence

s of the fo

rmula at th

e ends of

all the


21/282

AXI OM AT I Z I NG

QUANT I FI E R S

AND M ODAL I T I E S

17

branches that are not to be modified, then use the above rule on the

branch to be worked on.

Now the other modal logics can be dealt with easily.

The logic K was axiomatized by adding OT. In effect this says there

are always v-formulas available, hence S# can always be considered to

be nonempty. And, in fact, the appropriate tableau system

forK is

the

one above without the provision that S# be nonempty.

The logic D had

T

as an axiom. In effect, this says there

is

always a

7T-formula around, so an explicit occurrence of 7T need not be present to

apply the tableau rule. Properly speaking, a tableau system for D results

from that of K by adding the additional rule:

s

S#

Finally the logic

Thad

as an axiom schema v => v

0

• Well, simply add the

tableau rule

JJ

JJo

(it can easily be shown that the D-rule above is redundant).

For quantifiers

we

proceed analogously of course. Thus, for a

parameter a, and a set S of first-order sentences, let

S(a) = {

y(a) I

y E

S}.

Then a tableau system for QC

is

the Smullyan propositional system

plus the rule (to be read in a similar fashion to the one for C above)

S, 5

S(a), o(a)

provided

(1) a

is

new to the branch;

(2)

S

(a) :f= c/>.

For QK, drop the requirement that S(a) be nonempty.

For

QD

, add the rule

s

S(a)


22/282

1

8

M

E L V

IN F I

T T IN

G

p

rovid

ed a

is ne

w to

the b

ranch

.

Fo

r Q

T, a

dd th

e rule

Y

-y(a)

for

any p

aram

eter a.

(

This

m ake

s the

QD

rule r

edund

ant).

And

fin

ally,

for f i

rst-or

der lo

gic p

roper

we w

ant th

e "cu

m ula

tiven

ess '

tha

t

cp

::::>

(

Vx)cp

bro

ught;

tha t

is, as

we go

on

in a t

ablea

u con

struc

tion ,

sente

nces s

hould

nev

er be

delet

ed. W

ell, w

e cou

ld re

place

the QK

ru

le

by

S

, o

S , S(a

),

o(a)

pro

vided

a is

new

to th

e bra

nch.

T

he onl

y other

qua

ntifie

r ru l

e now

is

Y

-y(a)

It i

s no

t har

d to s

ee th

at the

se ar

e equ

ivale

nt to

the si

m ple

r set

0

o

(a)

a n

ew

Y

-

y(a)'

an

d we

have

exac

tly th

e firs

t-orde

r sys

tem o

f Sm

ullyan

[7].

B IB L IO G R A PH Y

[1

] Fitti

ng, M

.: 1973

, 'Mod

el Exi

ste nce

T heore

m s fo

r M oda

l and

ln tuitio

nistic

Logics

',

Jo

urnal

of

Symb

olic L

ogic

3

8

6

13-627

.

[2]

Kripk

e , S.:

1963,

'S em a

ntical

Analys

is o f

M odal

Logic

I, No

rmal P

ropos

it ional

Cal

culi', Zeit

schrift

fur

math

ematische

Logik u

nd

G

rundla

gen d

er M a t

he m a t

ik

9,

67-9

6 .

[3] K

ripke,

S.: 19

65, 'S

em ant

ic al A

na lysis

of M

odal L

ogic

II , N o

n-norm

al M o

dal

Pro

positi

onal C

alculi '

, The The

ory

o

f

M

od

e l

s, J.

W.

Addiso

n, L.

H enki

n and

A .

Tarsk

i (eds.}

, N orth

-H olla

nd Pu

blishin

g Co.,

A mste

rdam,

pp. 2 0

6 -2 2 0 .

[4) Lemm on, E.: 1957, 'N ew Foundati ons for Lew is M odal Systems',

Journal

of

Symbolic

Logic

22

176-

186 .

[

5] Seg

erberg

, K .: 1

971, An E

ssay i

n Clas

sical M

odal L

ogic,

U p

psala.


23/282

A

X

IO

MA

T

IZI

NG

Q

U

AN

T

IFI

ER

S

AN

D

M

O D

A

L I

T IE

S

1

9

[

6]

Sm

ully

an,

R

.: 19

63,

A

Un

ify

ing

Pri

nci

ple

in Q

ua

nti f

ic a

tio n

T h

eo

ry

, P

roc

eed

ing

s of

th

e N

atio

na

l A

cad

em

y of

Sc

ien

ces.

[7) Smullyan,

R.:

1968,

First

Order

Logic,

Springer-V erlag , Berlin .

D e

p t.

of

M a

the

m at

ics

L

ehm

an

C o

lle

ge

B

ro

nx, N

Y

1

046

8

U

.S

.A .


24/282

NICOLAS

D. G O O D M A N

THE

KNOWING MATHEMATIC

IAN

1.

Mathematics is at the beginning of a new foundational crisis. Twenty

years ago there was a firm consensus that mathematics

is

set theory and

that set theory

is

Zermelo-Fraenkel set theory (ZF). That consensus

is

breaking down.

It

is

breaking down for two quite different reasons. One

of these

is

a turning away from the excesses of the tendency toward

abstraction in post-war mathematics. Many mathematicians feel that

the power of the method of abstraction and generalization has, for the

time being, exhausted itself. We have done about as much as can be

done now by these means, and it is time to return once more to hard

work on particular examples. (For this point of view see Mac Lane [16,

pp. 37-38].) Another reason for this turning back from abstraction is

the economic fact that society is less prepared now than it was twenty

years ago to support abstract intellectual activity pursued for its own

sake. Those who support research are asking more searching questions

than formerly about the utility of the results that can reasonably be

expected from projects proposed. Mathematicians, moreover, are in

creasingly obliged to seek employment not in departments emphasizing

pure mathematics, but in departments of computer science or statistics,

or even in industry. Thus there

is

a heightened interest in applied and

applicable mathematics and an increased tendency to reject as abstract

nonsense what

our

teachers considered an intellectually satisfying level

of generality. The set-theoretic account of the foundations of mathe

matics, however,

is

inextricably linked with just this tendency to

abstraction for its own sake. Mathematics, on that account, is about

abstract structures which, at best, may happen to be isomorphic to

structures found in the physical world, but which are themselves most

definitely not in the physical world. Thus as mathematicians turn away

from pure abstraction, they also become increasingly dissatisfied with

the doctrine that mathematics is set theory ·and nothing else.

There is also another reason for the breakdown of what we may call

the set-theoretic consensus on the foundations of mathematics. That

is

Synthese

60

(1984) 21-38. 0039-7857/84/0601-0021 $01.80

©

1984 by D. Reidel Publishing Company


25/282

2

2

N IC

O L AS

D. G O

O D M A N

the b

reakdow n

of the con

sensus with

in set theo

ry. Th

e

w

ork of G od

el

and

Cohen and

Solovay a

nd the rest

has shown

that Zerm

elo-Fraen

kel

is a

n astonishi

ngly weak t

heory, whi

ch settles f

ew of the is

sues of cen

tral

con

cern to the

set theori

st. Not on

ly does it n

ot settle t

he continu

um

problem

, but it also

does not

settle the S

ouslin hyp

othesis, th

e Kurepa

hyp

othesis, th

e structur

e of the

analytic h

ierarchy,

or the ga

p 2

conjectu

re. A con

temporary

set theori

st, faced

with a dee

p-look ing

problem

, asks first

n ot for a p

roof or a co

unterexam

ple, but fo

r a model.

H e doe

s not exp

ect to prov

e or refut

e a co njec

ture, but

to prove it

indepe

ndent. The

num ber

of indepen

dent set-th

eoretic ax

ioms grow

s

alarmingly. More and m ore exotic large cardinals are invented and

studie

d, though

none of th

em can be

proved to

exist. M o

re and m o

re

comp

lex combi

natorial pr

inciples ar

e extracte

d from the

st ructure

of

G odel's c

onstructib

le universe

or from th

e techniqu

e of some

intricate

forcing a r

g u m ent an

d are then

shown to h

old in som e

models bu

t not in

others. I f

mathemati

cs is set t

heory, wh

ich set the

ory is it?

No one

knows ho

w to choo

se am ong

these man

y conflicti

ng princip

les. This

situation

, moreove

r, has gone

beyond t

he point w

here it is o

f interest

on

ly to logi

cians. A lg

ebraists in

terested in

the struc

ture of in

finite

abelian g

roups must

watch the

ir set theor

y (see [10]

and [24]

). Analysts

inte

rested in th

e structur

e of topolog

ical algeb

ras must do

the same

(see

[27

]

).

Increas

ingly it se

ems that e

very m athe

matic ian w

hose inter

ests

are at a

ll abstract

is

going t

o be faced

with the p

roblem of

choosing

w

hich set-th

eoretic ax

ioms to wo

rk with. B

ut it also

seems clea

r that

th

ere is no w

ay within

the presen

t fram ewo

rk to distin

guish whi

ch of

these a

lternative

set theorie

s is the tr

ue one - if

that que

stion even

m akes

sense any

longer. Ea

ch m athe

matic ian m

ust rely o

n his own

,

increasingly bewildered, intuition

or

taste. Evidently it

is

time to try to

f

ind a new

framework

.

2.

I suggest

that both

of the ab

ove difficu

lties with

the set-th

eoretical

fou

ndational

consensus

arise from

the sam

e source

- namely,

its

str

ongly redu

ctionistic

tendency.

Most math

ematical o

bjects, as

they

o

riginally pr

esent them

selves to u

s, are not s

ets. A natu

ral number

is

not

a transi

tive set line

arly o rder

ed by the m

embership

relation.

An ordered

pair is

not a doub

leton of a s

ingleton an

d a double

ton. A fun

ction is no

t

a

set of o r

dered pairs

. A real .n

u m b er is n

ot an equ

ivalence c

lass of


26/282


27/282

24

N IC

O L

A S D .

G

O O

D M

A N

di

sjo

int f

rom

th

e un

ive

rse

of d

isco

urs

e of

the

res

t of

sci

enc

e. In

th

inki

ng

abo

u t

pur

e m

athe

m a

tics

w

e en

d up

no

lo

nge

r th

inki

ng

abo

ut a

pa

rt o

f

scie

nce

, bu

t ra

the

r ab

out

a b

eau

ti fu

l, au

ste

re s

ubs

titu

te fo

r sc

ien

ce.

Thi

s

dev

elo

pm

ent

has

me

ant

a

prog

res

sive

im

po

veri

shm

ent

of

the

ma

the

m

at

icia

n 's

in tu

itio

n.

M an

y m

ath

em

atic

ian

s w

ho

kno

w s

om

e c

om p

lex

an

alys

is d

o

not

kno

w,

or

do

not

th

in k

of,

the

co

nn

ecti

on

bet

w ee

n

a

nal

ytic

fun

cti

ons

and

th

e flow

o

f a

flu

id. W

he

n I

wa

s ta

ugh

t co

m p

lex

an

alys

is in

g

rad

uate

sc

hoo

l, I

wa

s no

t ta

ugh

t t

hat

con

nec

tion

. H

en

ce

w

hen

th

ink

ing

abo

u t

ana

ly ti

c fu

nct

ions

th

ese

ma

the

m at

icia

ns

can

no t

rel

y o

n th

eir

com

m o

nse

nse

in

sig h

t in

to t

he

beh

avi

or o

f w

ate r

. A

gai

n,

many mathem aticians who know some of the theory of rings and ideals

do

no

t k

now

th

e a

lgor

ithm

s w

or

ked

out

by

K

ron

eck

er a

nd

oth

ers

to

co

m p

ute

suc

h id

eals

. W

hen

I w

as

taug

h t

rin g

the

ory

in

grad

uat

e sc

hoo

l,

I

was

not

tau

gh

t tho

se

alg o

rith

ms

. F o

r th

ese

ma

the

m at

icia

ns a

n i

deal

i

s

not

a c

om p

uta

tion

al

obje

ct

but

mer

ely

an

abs

trac

t se

t sa

tisfy

ing

cer

tain

clo

sure

co

ndit

ion

s. T

hus

the

y fi

nd i

t dif

ficu

lt e

ven

to

con

side

r no

ntr

iv ia

l

exa

m p

les

of t

he

abs

trac

t th

eor

y th

ey

hav

e le

arn

ed.

A n

y in

tu it

ion

the

y

ma

y h

ave

mu

st b

e p

urel

y fo

rm

al.

T

he

se t

-the

ore

tic

red

ucti

onis

ts h

av

e ex

pla

ined

aw

ay

the

ob

ject

s

we

w

ere

try

in g

to s

tu dy

, a

nd n

ow

tho

se o

bje

cts

are

no

long

er

ther

e to

gu

ide

u

s. P

erh

aps

i

t is

tim

e

to b

ring

so

me

of t

hem

ba

ck.

3.

T h

rou

g h

ou t

mo

st o

f th

e tw

ent

ieth

cen

tur

y ph

ysi

cis t

s ha

ve

held

th

a t it

is

no

t p

ossi

ble

ade

qua

tely

to

de

scri

be

the

phy

sica

l w

orld

w

itho

ut t

aki

ng

in

to a

cco

un t

the

ob

serv

er w

ho

col

lect

s th

e d

ata

that

t he

ph

ysic

al t

heo

ry

is

in tended to predict and explain. B oth the theory of relativity and

qu

an

tum

m

ech

anic

s a

re i

n v

ery

la

rge

par

t th

eor

ies

of

the

re

lati

on

bet

w ee

n th

e o

bse

rve

r an

d th

e p

hys

ical

rea

lity

tha

t he

ob

serv

es.

N ei

the

r

the

ory

, ho

we

ver,

de

nies

th

e re

alit

y o

r th

e o

bjec

tivi

ty

of t

he

exte

rna

l

rea

li ty

th

at is

b

ein

g o

bse

rved

. A

lth

oug

h

bo t

h t

heo

ries

in

volv

e

a

f

ar-r

each

ing

ad

m ix

tur

e o

f ep

iste

m o

log

y in

th

eir

bas

ic d

esc

ript

ion

of

p

hys

ical

rea

li ty

, ne

ithe

r th

eor

y c

an l

egit

im a

tely

be

acc

use

d o

f id

eali

sm.

T

he

ob

serv

er

and

th

e

exte

rna

l p

hys

ical

re

ality

w

ith

wh

ich

he

is

c

on

fron

ted

ar

e b

o th

ir

redu

cib

ly

pres

upp

ose

d.

In

thi

s se

nse

th ese

th

eori

es a

re

fund

am

ent

ally

dua

list

ic .

T w

enti

eth

-cen

tur

y m

ath

em a

tici

ans

, on

th

e o

th er

ha

nd,

hav

e s

oug

ht

to

ma

inta

in

the

mo

nis

tic

cha

rac

te r

of

thei

r di

scip

lin

e.

Th

e cla

ssic

al


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29/282

26

NICOLAS

D.

G O O D M A N

extensive or meaningful mathematics. Nevertheless, the constructive

tradition has the very great merit of having emphasized and studied the

epistemic aspects of mathematics. Those aspects have been largely

ignored by classical mathematicians.

The debate between the classical and intuitionistic positions on the

foundations of mathematics can thus be viewed

as

a disagreement

between two opposed, mutually exclusive, monisms. Each denies that

the other's reality is of any fundamental significance.

It

is like a debate

on the foundations of physics between a strict Newtonian who denies

that observability is of any fundamental consequence and a strict

phenomenalist who denies that our sense experiences refer to any

knowable reality outside of ourselves. Obviously such a debate is

unlikely to be fruitful. In physics the fruitful step was the step to a

dualistic view which emphasized both the role of the observer and the

role of the reality being observed. It is the thesis of the present essay

that a similar dualistic view on the foundations of mathematics is both

possible and desirable.

4 .

The observer in quantum mechanics or in the theory of relativity is a

very highly idealized physicist. He has no subjective bias. He does not

forget anything. He has no deficiencies of experimental technique. His

attention never wavers. He never sleeps. In our theory, then, we may

expect that the knowing mathematician will be similarly idealized. In

particular, we

will

assume that his powers of concentration are poten

tially infinite.

There

is to be no finite bound on the complexity of the

computations he can carry out

or

on the length of the proofs he can

construct. As I have argued in another place (see [9]), this is a very

considerable idealization of what is the case for human mathematicians

or even for the human race viewed collectively as a single mathemati

cian. Thus the knowing mathematician of our theory is himself to be

conceived as a mathematical abstraction. His introduction into the

theory is a step not toward a constructivistic impoverishment of

mathematics, motivated by doubt about the metaphysical underpin

nings of set theory, but rather toward an enrichment of classical

mathematics by the introduction of a new and more extensive vocabu

lary.

The

point

is

not to consider only those objects which can be

known, but rather to consider what aspects of arbitrary objects are


30/282

T H E KNOWI NG MATHEMATIC IAN

27

knowable

in

principle.

Thus, although our knowing mathematician

is

to be thought of as

only potentially infinite, he

is

not to be thought of as somehow situated

in

the physical world. All he does

is

mathematics. He has no properties

that do not follow from his being an idealized mathematician. In this

respect he resembles the idealized physicist of quantum mechanics

or

of

the theory of relativity.

Although our goal is to reverse the set-theoretic reductionism of the

last few decades, it

is

clear that we cannot do that in one fell swoop. It

makes no sense to write down a theory having as its primitives all the

notions which, in an unanalyzed form, have

ever

played a role in

mathematical practice. Such a theory would be ugly and, presumably, a

conservative extension of its set-theoretic fragment. It would give no

new insight. The point is not to reintroduce old notions for the sake of

not explaining them away.

The

point, rather,

is

to try to rebuild our

mathematical intuition by gradually enriching it with notions and

principles that are not known to be reducible to set theory.

Thus

I

suggest that our first draft of such a theory should be a set theory, but a

set theory enriched with intensional epistemic notions.

The

first theory

of the sort I have

in

mind was based on arithmetic and

is

to be found

in

Shapiro [23].

Then

Myhill in [17] proposed a theory based on set theory

but

in

which the arithmetic part was still, so to speak, singled

out

in the

very syntax of the theory. Finally, in [8], I proposed a theory that

is

strictly set theoretic. A similarly motivated but independent and

formally very different theory can be found in the recent work of

Lifschitz (see [14] and [15]).

The theory I will discuss here is the theory of my [8]. I t is a modal set

theory, with membership as its only nonlogical primitive, and with

Lewis's S4 as its underlying logic. The modal operator of S4

is

to be

read epistemically. Thus

O

means that is knowable.

5.

In one respect, at least, our theory will resemble quantum mechanics

more than it does the theory of relativity.

The whole point of the theory

of relativity is to relate the observations of different observers. In the

mathematical case this does not seem to make much sense. Troelstra in

[26] considers two mathematicians, one of whom communicates num

bers to the other without giving him complete information about how


31/282

28 N IC O L

A S D. G

O O D M AN

those num

bers a re o

btained. The

o th er m

athem atic

ian, then,

is able to

make certa

in predict

ions abou t

the future

behavior

of the first

mathe

matic

ian. This

entire p ic

ture, how

ever, seem

s to m e

more clos

ely

relat

ed to what

happens in

empirical

science th

an to anyth

ing that g

oes

on in

mathemati

cs. M athem

aticians e

xchange c

omplete i

nformation

.

T hey

do no t gen

erally hid

e all or

part of their

algorithms

from eac

h

other. M

oreover,

in contrast

to the situa

tion in phy

sics, I do n

ot see that

it make

s sense t

o suppose

that one

mathem a

tician som

ehow has

preferen

tial access

to some p

art of math

ematical r

eality. On

this basis,

th

en, it seem

s to m e ad

equa te to c

onsider th

e case o f o

nly on

e

kn

ower.

T his

is

analogous to the situation in quantum m echanics, w here

one

does not

usually co

nsider the

effect of

having m o

re than

one experi

m enter.

Thus

w h

en, in

our theory, we

write Ocp,

we take th

is to m ean

that the

f

ormula

cp is know ab

le to our id

ealized m a

them aticia

n . W e say

"know

able" ra th

er than "know

n" bec

ause of th

e logic of S

4. Let us r

ecall the

rules of

that system.

In additi

on to th

e usual rule

s of the c

lassical

p

redica te ca

lculus, we

have the

follow ing m

odal rules

:

1.

Ocp--7cp.

2

. 0cp--?0

0cp.

3.

0 1\ 0

p --7 1/1)

--7

01/J.

4.

From

1-

cp

infer Ocp.

If we tak

e 0 to m e

an 'know n

', then the

third of the

se rules as

serts tha t

whenever

our math

ematician

can make

an infere

nce, he ha

s already

done so.

Even

an idealized mathem atician presumably does not follow

out every p

ossible ch

ain of infe

rence. Thus

we must

ta

ke 0 to

m ean

'

know able ' .

On this re

ading, the

first of ou

r rules asse

rts tha t eve

rything

k

nowable

is true. T

he second a

sserts that

anything

knowable

can be

k

nown to

be knowabl

e.

O

ne m ig

ht argue f

or this by n

oting

that if < > is

ever kn

own, it wi

ll then

be

k

nown t o b

e know n, a

nd hence k

nown to b

e

knowab

le. The t

hird rule a

sserts th a

t if both

a conditio

nal and it

s

an teced

ent ar

e

kn

owable, the

n the cons

equent is k

nowable. F

inally, the

fourth ru

le derives

from our co

nfidence i

n our

very

system o f a

xioms. If

we

actually pr

ove a claim

, then w

e know th a

t claim to

be true, an

d so

tha

t claim

is knowable.

t seems

to have b

een G ode l

in [7] w ho

first obser

ved that S

4 can be


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34/282

T H E KNOWING MAT HE MAT I C

IAN

31

function such that . . . . In order to see that there really are mathemati

cal notions independent of set theory, let us consider a familiar example

that is actually rather closely connected with

our

idea of mathematical

knowability in principle. I am thinking of Church's thesis.

The standard account is that there was a vague and unanalyzed

informal notion of computability in principle, and that

Church, Turing

,

Kleene, and Post offered alternative analyses of this notion in set

theoretic terms that turned out to be equivalent. As usual, let us refer to

a function satisfying the formal definition as recursive. Church's thesis

is

then the vague and premathematical claim that recursiveness coincides

with computability in principle. It has even been urged that the

replacement of computability in principle by recursiveness

is

analogous

to the replacement of the informal eighteenth-century notion of con

tinuity by the formal nineteenth-century

concept

defined using epsilons

and deltas (see Shapiro [22]

).

Note, however, that the two analyses

function quite differently

in

practice.

The

informal notion of continuity

is

used only heuristically to motivate the epsilon-delta definition.

Once

the epsilon-delta definition has been given, no further reference is made

to the informal notion. The practicing analyst who uses the notion of

continuity thinks in terms of the epsilon-delta definition, or in terms of

some other definition equivalent to that definition on the real line.

Every assertion about continuity is explicitly justified by means of the

epsilon-delta definition. The contemporary recursive-function theorist,

on the

other

hand, uses the informal notion of computability constantly.

He thinks in terms of that notion, rather than in terms of one of the

standard formal definitions. Proofs in the theory of recursive functions

usually no longer refer to the formal definitions

at

all.

If

it becomes

relevant to relate the theory to

one

of the formal definitions, this

connection

is

established by a global appeal to

Church's

thesis.

The

analyst says, "This function

is

continuous because I have shown that it

satisfies the epsilon-delta definition."

The

recursion theorist says, "This

function

is

recursive because I have shown how to

compute

it."

The

role of the formal, set-theoretic definition

in

the two cases could not be

more different.

It might be suggested that we are still so close to the time when the

definition of recursiveness was first given that there has not been time

for the formal notion to drive

out

the informal notion it is intended to

replace.

The

historical evidence, however, points in the opposite

direction.

The

early papers in the theory of recursive functions were


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36/282


37/282

34

NICOLAS

D.

G O O D M A N

recursively enumerable set can exhaust the truths of any nontrivial

mathematical theory. Thus, at least as far as we could know, the

problem of intensionality does arise.

9.

When we do set-theoretic mathematics, we reason about sets, not about

descriptions of sets. Nevertheless, when we reason epistemically about

particular sets, asking ourselves questions such as whether we could

actually construct a set with a certain property, we are necessarily

dealing not with the sets themselves, but only with defining criteria. As

a mathematical abstraction, at least, it is

not difficult to construct a

language in which every set has a defining criterion. For, form a

"language" in the logician's sense having a name for every set. Then

the set

A is

defined by the criterion, 'x belongs to A'

.

Of course, in this

language, the set A

will

also have many other defining criteria, some of

which may not be knowably equivalent to this one. Although no human

being could learn this language, we often talk as though we were using a

finite fragment of it. A mathematician who has proved the existence of

a set with a certain property but does not know any actual criterion for

membership

in

such a set will not hesitate to introduce a name for a

particular such set. For example, he may write as follows:

"Thus

we see

that there exists a regular ultrafilter. Let D be such an ultrafilter. We

know that

D

has such and such property. Hence . . . . " As an illustration

of this procedure, let L be a language of the sort described.

Every set has a defining criterion expressible in

L.

These criteria can

be thought of as representatives , of the sets. For all mathematical

purposes, these representatives will do the work of the sets. Member

ship

in

a set

is

just satisfaction of the corresponding membership

criteria. The usual axiom of extensionality, which asserts that two sets

with the same members are identical, tells us that two defining criteria

which are satisfied by the same objects are representatives of the same

set. Thus, in the spirit of the usual set-theoretic reductionism, we may

think of a set as an equivalence class of these defining criteria under the

relation of being satisfied by the same objects. (Actually, as this

construction

is

carried out in my paper [8], it is technically somewhat

more complex. The problem is that some criteria will not be exten

sional. Thus sets should be thought of as equivalence classes of

hereditarily extensional criteria. But these details need not concern us


38/282

T HE

K N O W I N G

M A T H E M A T I C

IAN

35

here.)

The relation of extensional identity is not the only interesting

equivalence

relation

on the

defining criter ia expressible in

our language

L. A

somewhat stronger

relation

is

the relation which holds

between

two of these criteria when it is knowable that they apply to exactly the

same

objects.

Let me

refer to this relation as

modal

extensional identity.

Criteria

that

are modally extensionally identical

can

be thought of as

representatives

of the

same set-theoretic property or

attribute.

More

formally, by a

property

let us

mean

an equivalence class of defining

criteria

under

the relation of modal extensional identity.

Then

we

can

interpret

our

modal

set theory

as being

about

properties.

In this way, interpreting

our

theory as being a

theory

of

properties

rather

than a theory of sets, the problems of interpretation we men

tioned

above

disappear. For, two properties

that can

be known to apply

to the same objects will have exactly the same properties expressible in

the language of

our

theory.

Thus

if we adopt a modal axiom of

extensionality asserting that

properties

that can be known to apply to

the same things are identical, then we will have full substitutivity of

identity.

10.

Let me now suggest that the

above

construction of properties from sets

is exactly backwards. The historical development, at least, went in the

other

direction. Sets first appeared in mathematics in the form of

properties expressed in some language - say mathematical

German.

That is to say, the defining criteria are historically the primary objects .

These

properties

were studied by impredicative

and

nonconstructive

methods, so that it was clear very early that mathematical German was

not

adequate

to formulate all possible such

properties

(for what

amounts to this point , see Borel [3, pp. 109-11 0]).

Thus

sets were

introduced

as

mathematical

abstractions to support a notion of

property

that

had

come detached

from the idea of expressibility in any particular,

or even

any possible, language. The

problem

then

arose about

what

criterion of identity

one

should use for these strange new mathematical

objects

. In

the

positivistic intellectual climate of the turn of the

century,

it was not difficult to arrive at a consensus that extensional identity was

the only possible criterion. As a

matter

of fact, however,

some

sort of

intensional identity may be more suitable for pre-set-theoretic mathe-


39/282

36

NICOLAS D . G O O D M A N

matical practice. Thus I suggest that we should go back to the situation

in, say, 1880, when it was not yet clear what it should mean to say

that

two sets are identical. In

that

situation it makes sense

to

suggest

that

two sets are identical just in case it can be known that they have exactly

the same elements. Given

that

we

have

an adequate logic of knowabil

ity, which we do , this suggestion may be more fruitful than the generally

accepted one. To avoid confusion today, however, we should probably

not refer to the resulting objects as sets, but as properties.

1 1 .

Once

it

is

clear

that the

theory we want to write down

is

a theory of

properties in

the above

sense, it is not difficult to write down the axioms

by imitating ZFC (that is,

Zermelo-Fraenkel

set theory with the axiom

of choice).

For

the details, I refer thti

reader

to [8]. The resulting theory

is as rich as set theory. ZFC is faithfully interpretable in it. As a matter

of fact, the theory is richer than set theory. For example,

in

it we can

express that we have actually constructed an object, rather than merely

proved

it to exist. Most of the basic

constructive

notions are easily

expressible. Of course, the underlying metaphysical position is not at all

intuitionistic,

but

Platonistic.

What remains is to try to develop set theory and analysis in this new

framework, trying to exploit its additional expressive power. As we do

so, we should be able to develop sufficient intuition for the epistemic

component

of the new framework that we

can

begin to think informally

about

the knowing mathematician without having to rely

on

any

formalization, including ours.

Then

we should be led to ask new

questions and to find new

phenomena

that will enrich classical

mathe

matics.

It

seems rational to hope that the resulting sharpening of

our

set-theoretic intuition will

either

lead to new insight into the apparently

unanswerable questions of set theory, or else will enable us to see that

those questions are not really very important or very central after all.

What is more important is that by working with intensional notions

directly connected with

our

actual mathematical experience - by

considering mathematical knowledge, computability, and construction

as ingredients of our universe of discourse rather than as merely

psychological aspects of

our

work - we may restore

some

of the

concrete mathematical intuition that we

have

lost. In this way we may

gradually break down the prestige of

set-theoretic

abstraction and


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41/282

3

8

NI C OL A

S D

.

G O O D

M A N

[1

8] Quine, W

. V.: 1976,

'Three Grades

of Mod al Invo

lvem ent' , in

The Ways

of Paradox

and Other

Essays, rev. e

d. , Harvard

U.P. , Cambri

dge, Mass., p

p. 158-176.

[19] Randolph ,

J.

F.: 1968,

Basic Real and Abstract Analysis

,

Academic Press, New

York.

[20] Roge

rs, H.: 1967

, Theory of

Recursive

Functions an

d Effective C

omputability

,

McGraw-Hil

l, New York .

[21] Ro

senlicht, M.:

1968, Intr

oduction

to

A

nalysis, Scott,

Foresman, G

lenview, III .

[22] S

hapiro, S.: 'O

n Church 's T

hesis ' , Typesc

ript.

[23] Shapi

ro, S.: 1984, '

Epistemic

r i t ~ m e t i c

and

Intuitionistic

Ari thmetic ',

to ap pear in

S. Shapiro (e

d .), Intensio

nal Mathema

tics, North-Ho

lland Pub. C o

., New York.

[2

4] Shelah,

S. : 1979,

'On Well Orderin

g and More o

n Whitehead

's Problem', A

bstract

79T-E47,

Notices Amer

. Math. Soc

., 26, A-442 .

[2 5] Shoenfield , J. R.: 1972,

Degrees

of

Unsolvability,

N orth-Holland , Amsterdam .

[26] T

roelstra,

A .

S

.: 1968, '

The The ory of C h

oice Sequenc

es', in B. Va n

R ootselaar a

nd

J. F

.

Staal (eds.),

Logic , Met

hodology and

Philosophy

of Science I l

l

N

orth-Hollan d

,

A m st

erdam , pp. 20

1 - 223 .

[27]

Williamson,

J

.

H.: 1979,

Review of

Topological

Algebras by E .

Beckenstein

, L.

Na

rici, and

C. Suffel, Bull

. Am

er

. Mat

h. Soc. (New S

eries) 1

,

237-

244.

Dept. of M

athematics

Suny-Buff

alo

A m h

erst, NY 142

60

U .S.A.


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R AYM OND D. GUMB

"CONSERVATIVE"

KRIPKE

CLOSURES*

0. I N T R O D U C T I O N

Computable Kripke closures are properties of relations which have

closures in, roughly speaking, the sense of the transitive closure. They

were introduced in [8] to generalize Kripke-style tableaux con

structions and were studied from a model-theoretic perspective by

Weaver and Gumb [19].

In section 1 of this paper, we review the properties of computable

Kripke closures. In section 2, we state four additional laws that can be

imposed on computable Kripke closures and state properties of the

closures determined by these laws. In a sense, all four of our laws

require closures to be "conservative". However, regarding later sec

tions, it

is

more revealing to classify two of the laws as being

commutative and two as being conservative.

In the remaining sections, we sketch applications of these laws

in

modal logics having a Kripke-style relational semantics: simplifying

Kripke-style tableaux constructions (section 3), proving the Craig

Interpolation Lemma (section 4), establishing Henkin-style complete

ness proofs (section 5), and providing a somewhat plausible prob

abilistic semantics (section 6). At least in modal logic, our laws carve

out natural classes of properties of binary relations.

1. C OM P UT AB L E KR I P KE C L OS UR E S

The presentation of the computable Kripke closures

in

this section

is

much the same as that

in

[8, section 5]. A model-theoretic study of the

first-order Kripke closures using a different notation can be found

in

[19].

A (binary) relational system

is

a pair i i =(a-, r), where a-

is

a nonvoid

set and

r

sa-Xu

is

a (binary) relation on

a-.

Let

BR

be the class of

relational systems. Understand Pr

s;

BR to be a property of relations if Pr

is

closed under isomorphisms. Let Pr be a property of relations, and let

Synthese

60

(1984) 39-49. 0039-7857/84/0601-0039 $01.10

© 1984

by

D.

Reidel Publishing Company


43/282

4

0

R A

YM O

N D D

. G

UM B

i

i = (u,

r

) an

d ii+

= (u+,

r+)

be re

lation

al sy

stems

. W e

say t

hat r+

is a

Pr-re

lation

(on

u+) if

i

i

+

E Pr

. W e

call

r

+

the

Pr

-closu

re

of r (on

u +

) and

writ

er+=

p

ru •

(r) if u s

u +,

r s r+

, a

nd r+

is th

elliott mendelson - foundations logic, language and mathematics

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