Intentional Mathematics

INTENSIONAL MATHEMATICS

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS VOLUME 113

Editors J. BARWISE, Stanford D. KAPLAN, LosAngeles H. J. KEISLER, Madison P. SUPPES, Stanford A. S.TROELSTRA, Amsterdam

NORTH-HOLLAND AMSTERDAM 0 NEW YORK 0 OXFORD

INTENSIONAL

m"71mmrc

Edited by

Stewart SHAPIRO The Ohio State University at Newark

Ohio

U.S. A.

1985

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

OELSEVIER SCIENCE PUBLISHERS B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87632 4

Published by: Elsevier Science Publishers B.V. P.O. Box 1991 1OOOBZ Amsterdam The Netherlands Sole distributors for the U.S.A.and Canada: Elsevier Science Publishing Company, Inc. 52VanderbiltAvenue NewYork, N.Y. 10017 U.S.A.

Library of Congmar Cat.1ogi.g In Publlcatlon Data

Main entry under title: Intensional mathematics. (Studies i n logic and the found&Sons of mathematics ;

v. 113)

Bibliography: p; 1. Modality (Logic)--Addresses, essays, lectures.

2. Constructive mathematics--Addresses, essays, lectures. 3. Intuitionistic mathematics--Addresses, essays, lectures. I. Shapiro, Stevart, 195111. Series. QA~.46.157 1985 511 04-10056 ISBN 0-444-87632-4 (U.8. I

.

PRINTED IN THE NETHERLANDS

V

TABLE OF CONTENTS 1. Introduction: Intensional Mathematics and Constructive Mathematics

Stewart Shapiro

1

2. Epistemic and Intuitionistic Arithmetic Stewart Shapiro

11

3. Intensional Set Theory John Myhill

47

4. A Genuinely Intensional Set Theory

Nicholas D. Goodman 5. Extending Gadel’s Modal Interpretation to Type Theory and Set Theory Andrej SZedrov

63 81

6. Church’s Thesis is Consistent with Epistemic Arithmetic Robert C. Flagg

121

7. Calculable Natural Numbers vladimir Lifschitz

173

8. Modality and &&Reference

Raymond M. Smullyan

191

9. Some Principles Related to Lab’s Theorem

Raymond M. Smullyan

213

This Page Intentionally Left Blank

Intensional Mathematics S. Shapfro (Editor] @ Elsevier Science Publishers B. V. (North-Holland), 1985

1

INTRODUCTION--INTENSIONAL MATHEMATICS AND CONSTRUCTIVE MATHEMATICS

S t e w a r t Shapiro

The Ohio S t a t e U n i v e r s i t y a t Newark Newark, Ohio 1T.S .A.

Platonism and i n t u i t i o n i s m are r i v a l p h i l o s o p h i e s of mathematics,

the

former h o l d i n q t h a t the s u b j e c t matter of mathematics c o n s i s t s of a h s t r a c t o b j e c t s whose e x i s t e n c e i s independent of the mathematician, t h e l a t t e r t h a t the s u b j e c t matter c o n s i s t s of mental c o n s t r u c t i o n .

Intuitionistic

mathematics i s o f t e n c a l l e d " c o n s t r u c t i v i s t " while p l a t o n i s t i c mathematics is c a l l e d " n o n - c o n s t r u c t i v i s t "

.

The i n t u i t i o n i s t , €or

example, rejects c e r t a i n n o n - c o n s t r u c t i v e i n f e r e n c e s and p r o p o s i t i o n s as i n c o m p a t i b l e w i t h i n t u i t i o n i s t i c philosophy--as

r e l y i n q on the

independent e x i s t e n c e of mathematical o b j e c t s .

The m o s t n o t a h l e of these

i s the l a w of excluded middle,

AV

7A_

, which

the i n t u i t i o n i s t t a k e s as

a s s e r t i n q t h a t e i t h e r the c o n s t r u c t i o n correspondinq to

A

can he

e f f e c t e d or t h e c o n s t r u c t i o n c o r r e s p o n d i n q to the r e f u t a t i o n of effected.

Another example is

iVg(x)

1 ZI~Z(X) which,

c a n he

i n the c o n t e x t

o f a r i t h m e t i c , the i n t u i t i o n i s t t a k e s as a s s e r t i n q t h a t i f n o t a l l numbers have a p r o p e r t y

,

t h e n one can c o n s t r u c t a numher which l a c k s

P l a t o n i s m and i n t u i t i o n i s m are a l l i e d in the r e s p e c t t h a t tmth views

are i m p l i c i t l y opposed to materialistic a c c o u n t s of mathematics which t a k e t h e s u b j e c t matter of mathematics to c o n s i s t ( i n a d i r e c t way) of

material o b j e c t s .

Perhaps it is f o r this r e a s o n t h a t p l a t o n i s m i s

sometimes c a l l e d " o b j e c t i v e idealism'' and i n t u i t i o n i s m is sometimes c a l l e d "subjective idealism".

Both views hold t h a t mathematical o b j e c t s

are " i d e a l " a t l e a s t i n the s e n s e t h a t t h e y are n o t material.

The

2

S. SHAPIRO

P l a t o n i s t holds t h a t the mathematical " i d e a l s " do not depend on a mind f o r their e x i s t e n c e , the i n t u i t i o n i s t t h a t they do. The two views are p h i l o s o p h i c a l l y incompatible.

Indeed, t h e

e x i s t e n c e of any mentally constructed o b j e c t depends on the mind t h a t c o n s t r u c t s it, and cannot he s a i d to e x i s t independent of t h a t mind. Nevertheless, matters of i n t u i t i o n i s t i c a c c e p t a b i l i t y a r e o f t e n r a i s e d i n non-constructive mathematical contexts.

I t may be asked, i n p a r t i c u l a r ,

whether a c e r t a i n proof is c o n s t r u c t i v e (or can he made c o n s t r u c t i v e ) or whether a c e r t a i n part of a non-constructive proof is c o n s t r u c t i v e can he made c o n s t r u c t i v e ) .

(Or

One does not have to he an i n t u i t i o n i s t , f o r

example, to p o i n t o u t t h a t Peano's theorem on the s o l u t i o n of d i f f e r e n t i a l equations d i f f e r s from P i c a r d ' s i n that t h e former is not c o n s t r u c t i v e , or t h a t the Friedherq-Munchnik s o l u t i o n t o Post' s problem c o n s i s t s of the c o n s t r u c t i o n of an alqorithm, followed by a non-construct i v e proof t h a t t h i s alqorithm r e p r e s e n t s a s o l u t i o n to t h e prohlem. One of the purposes of the f i r s t f i v e papers i n this volume is to formalize the c o n s t r u c t i v e a s p e c t s of c l a s s i c a l mathematical d i s c o u r s e . Each of these papers contains both a non-constructive

lanquage which can

express statements of p a r t i a l or complete c o n s t r u c t i v i t y and a deductive system which can express c o n s t r u c t i v e and non-constructive proofs.

My

own paper and t h e L i f s c h i t z paper concern a r i t h m e t i c , while the Goodman paper, the Myhill paper, and t h e Scedrov paper concern set theory, the l a t t e r a l s o s t u d i e s type theory. I n this Introduction,

I propose a conceptual l i n k between the

i n t u i t i o n i s t i c c o n s t r u c t i o n processes and the c l a s s i c a l epistemic processes.

This l i n k , i n t u r n , provides the p h i l o s o p h i c a l hacksround

f o r c o n s t r u c t i v i s t i c concerns i n non-constructive c o n t e x t s and, t h e r e f o r e , the motivation f o r my c o n t r i b u t i o n to this volume.

Althouqh

t h e o t h e r authors do not ( n e c e s s a r i l y ) share the presented view, t h e i r work is h r i e f l y discussed i n Liqht of it.

Intensional Mathematics and Constructive Mathematics

3

I t w i l l be u s e f u l here t o b r i e f l y r e c o n s t r u c t t h e development of

extreme s u b j e c t i v e idealism i n the c o n t e x t of qeneral epistomoloqy.

Of

c o u r s e , I do not subscribe to the conclusion of the next paraqraph. Probably the most basic epistemoloqical questions are "What i s t h e source of knowledqe?" and "What i s the qround of t r u t h of p r o p o s i t i o n s known?"

Descartes a s s e r t e d t h a t the source of a person's knowledqe i s

s o l e l y h i s own expreience (excludinq, f o r example, the pronouncement of a u t h o r i t y as a source of knowledqe).

This discovery led to a study of

experience and i t s r e l a t i o n to knowledqe.

The qround of t r u t h of a

p r o p o s i t i o n known must l i e i n t h e s u b j e c t matter of the p r o p o s i t i o n .

It

follows t h a t the qround of knowledqe lies i n what our experience is

of. Althouqh we experience of an o u t s i d e

experience

seem compelled to b e l i e v e t h a t our experience

is

world, we have no d i r e c t l i n k with t h i s w o r l d

e x c e p t throuqh our senses. n o t the o u t s i d e world.

The content of sense experience, however,

is

If one s t a n d s c l o s e to and f a r from the same

o b j e c t , he w i l l have d i f f e r e n t sense imaqes.

(For example, i n one of

them, the o b j e c t w i l l occupy more of the f i e l d of vision.)

Thus, t h e r e

seems to be a permanent epistemic qap between knowledqe-experience and t h e o u t s i d e world.

The problem i s t h a t d e s p i t e our s t r o n q conviction t h a t

t h e qround of t r u t h of our b e l i e f s is e x t e r n a l t o us, we are not a h l e t o transcend both our experience and i t s qround t o v e r i f y t h i s . cannot know t h a t our experience is experience

of an

That is, w e

o u t s i d e world.

Since

what we know i s based e n t i r e l y on experience and s i n c e t h e o u t s i d e world

i s a c t u a l l y not a c o n s t i t u e n t of experience, an a p p l i c a t i o n of Ockham's r a z o r seems i n order.

Not t h a t the r e a l i t y of t h e o u t s i d e world is

o u t r i q h t l y denied, but rather it is noted that, so f a r a s w e know, the o u t s i d e world does not f i q u r e i n anythinq we know--we know anythinq ahout it.

do not know t h a t we

Hence, we do not t a l k ahout i t l i t e r a l l y .

On

t h i s view, the whole of the o u t s i d e world is reduced to a supposition t h a t orders our experience.

S. SHAPIRO

4

There is a r a t h e r s e r i o u s d i v e r q e n c e between ( 1 ) p e r c e p t i o n / t h o u q h t

as conceived by such an extreme s u b j e c t i v e i d e a l i s t and ( 2 ) p e r c e p t i o n / t h o u q h t as conceived by t h o s e who hold on to the e x i s t e n c e of the o u t s i d e 1

world--its

e x i s t e n c e independent of p e r c e p t i o n .

The l a t t e r have t h e

( a t l e a s t i m p l i c i t ) p r e s u p p o s i t i o n t h a t part of the e x t e r n a l world i s r e p r e s e n t e d more o r less a c c u r a t e l y i n p e r c e p t i o n .

For example, it is

presumed t h a t correspondinq t o o n e ' s p e r c e p t i o n of a pen i s t h e a c t u a l o b j e c t , t h e pen. perception.

There i s no such presumption i n s u b j e c t i v i s t

On the basis of these p r e s u p p o s i t i o n s , the n o n - s u b j e c t i v i s t

makes c e r t a i n i n f e r e n c e s which may n o t be s a n c t i o n e d by extreme s u h j e c t i v e idealism.

For example, i f a n o n - s u b j e c t i v i s t

sees a b a s e b a l l

s a i l over a f e n c e and o u t of s i q h t i n t o some bushes, he h a s the b e l i e f t h a t t h e b a s e b a l l s t i l l e x i s t s and i s i n the bushes.

Furthermore, he can

make p l a n s t o r e t r i e v e t h e b a s e b a l l and f i n i s h the qame.

Such an

i n f e r e n c e does n o t seem t o be j u s t i f i e d i n s u b j e c t i v i s t thouqht.

It is

n o t hard to imaqine a s u b j e c t i v e i d e a l i s t who a r q u e s t h a t p l a n s a b o u t unperceived baseballs are w i t h o u t f o u n d a t i o n . I n t h e mathematical s i t u a t i o n , a similar d e s i r e t o e x c l u d e presumptions of an o u t s i d e world from d i s c u s s i o n m o t i v a t e s i n t u i t i o n i s m . The o h j e c t i v e r e a l i t y of t h e mathematical u n i v e r s e i s d e n i e d by the i n t u i t i o n i s t i n the same s e n s e t h a t the o u t s i d e world is denied by the subjectivist.

I n p a r t i c u l a r , the i n t u i t i o n i s t does n o t c l a i m o u t r i q h t

t h a t t h e e x i s t e n c e of the mathematical u n i v e r s e depends on t h e mathematician's mind.

R a t h e r he p o i n t s o u t t h a t a l l mathematical

knowledqe i s based on mental a c t i v i t y .

T h i s mental a c t i v i t y is

apprehended d i r e c t l y , t h e ( s o - c a l l e d ) mathematical u n i v e r s e is n o t .

The

m e n t a l a c t i v i t y of mathematicians, t h e n , i s t a k e n to be the s u b j e c t matter of mathematics-questions

a r e n o t t o he c o n s i d e r e d . called "constructions".

of an o b j e c t i v e mathematical u n i v e r s e

Correspondinq t o s e n s e imaqes are what are The i n t u i t i o n i s t Heytinq once wrote:

Intensional Mathematics and Constructive Mathematics

...

5

...

Brouwer's proqram c o n s i s t e d i n the i n v e s t i q a t i o n of mental mathematical c o n s t r u c t i o n as s u c h , w i t h o u t r e f e r e n c e t o q u e s t i o n s r e q a r d i n q the n a t u r e of the c o n s t r u c t e d o b j e c t s , such as whether these o b j e c t s e x i s t i n d e p e n d e n t l y of o u r knowledqe of them a mathematical theorem e x p r e s s e s a p u r e l y e m p i r i c a l In fact, f a c t , namelv the s u c c e s s of a c e r t a i n c o n s t r u c t i o n mathematics, from the i n t u i t i o n i s t p o i n t of view, is a s t u d y of c e r t a i n f u n c t i o n s of t h e human mind

...

...

.*

As i n d i c a t e d ahove, the same term " c o n s t r u c t i o n " a l s o o c c u r s i n

classical, non-constructive contexts. " s u b j e c t i v i s t perception", d i f f e r e n t contexts. is adopted:

As w i t h " p e r c e p t i o n " and

t h e word has d i f f e r e n t meaninqs i n t h e

To avoid a c o n f u s i o n of terminoloqy, t h e f o l l o w i n q

The a d j e c t i v e " c o n s t r u c t i v e " and the noun " c o n s t r u c t i o n " are

l e f t to the i n t u i t i o n i s t s .

Whenever these words are used i n the s e q u e l

( i n this i n t r o d u c t i o n ) , t h e y are taken t o mean what t h e i n t u i t i o n i s t s mean by them.

The pair " e f f e c t i v e " and " c o n s t r u c t " are used t o r e f e r to

t h e correspondinq c l a s s i c a l e p i s t e m i c p r o c e s s e s . From the p r e s e n t p o i n t of view, the main d i f f e r e n c e between the

c l a s s i c a l e f f e c t i v e mode of t h o u q h t and t h e i n t u i t i o n i s t i c c o n s t r u c t i v e mode is t h a t the former presupposes t h a t there is an e x t e r n a l mathematical world t h a t qrounds o u r c o n s t r u c t s .

c l a s s i c a l v i e w , the c o n s t r u c t d e s c r i b e d by

t o i t s e l f t h r e e times" c o r r e s p o n d s t o the u n i v e r s e expressed by

'I

32 = 3

+

3

+

3

"

32

fact

'I.

For example, on the is o b t a i n e d hy addinq

3

i n the mathematical

As w i t h non-subjectivism,

supposition allows c e r t a i n inferences--precisely

this

the n o n - c o n s t r u c t i v e

p a r t s of mathematical p r a c t i c e r e j e c t e d by the i n t u i t i o n i s t s .

For

example, i f a classical mathematician proves t h a t n o t a l l n a t u r a l numbers have a c e r t a i n p r o p e r t y , he c a n t h e n i n f e r the e x i s t e n c e of a n a t u r a l numher l a c k i n q this p r o p e r t y .

a number i s n o t known--even

This i n f e r e n c e can be made even i f such

i f t h e matematician does n o t know

n u m e r a l s it d e n o t e s such a number.

an e x a c t

An i n t u i t i o n i s t d e n i e s t h i s

i n f e r e n c e because he h e l i e v e s t h a t it relies on the independent, o b j e c t i v e e x i s t e n c e of t h e n a t u r a l numbers.

For an i n t u i t i o n i s t , each

6

S. SHAPIRO

a s s e r t i o n must r e p o r t a c o n s t r u c t i o n .

I n the p r e s e n t example, he would

c l a i m that the e x i s t e n c e of a n a t u r a l numher with the s a i d p r o p e r t y c a n n o t he a s s e r t e d because such a numher w a s n o t c o n s t r u c t e d .

A classical

mathematician may wonder whether such a number can be c o n s t r u c t e d - whether he can know of a s p e c i f i c numeral t h a t d e n o t e s such a number-h u t t h e l a c k of a c o n s t r u c t does n o t p r e v e n t the i n f e r e n c e . Accordinq t o the p r e s e n t a c c o u n t , then, both the c o n s t r u c t i v e mode o f t h o u q h t and the e f f e c t i v e mode of t h o u q h t are r e l a t e d t o e p i s t e m i c

matters.

That is, to a s k f o r a numher with a c e r t a i n p r o p e r t y to he

c o n s t r u c t e d is to ask i f there is a numher which can he known t o have this property.

I f this account i s p l a u s i b l e , t h e n the " c o n s t r u c t i v e " a s p e c t s

o € classical mathematics can be e x p r e s s e d i n a formal lanquaqe which

c o n t a i n s e p i s t e m i c terminoloqy.

T h i s i s t h e approach of the f i r s t f i v e

p a p e r s i n t h e p r e s e n t volume. K

I n my c o n t r i h u t i o n , a n e p i s t e m i c o p e r a t o r lanquaqe of arithmetic. mean

"

A

If

A

is a formula, t h e n

i s i d e a l l y o r p o t e n t i a l l y knowable".

a x i o m a t i z a t i o n e q u i v a l e n t t o t h e modal l o q i c i n this c o n t e x t .

As

suqqested,

i s added t o the K(A)

is taken to

I arque that a n S4

is appropriate f o r

K

35K(A(5)) i s taken as amountinq t o

" t h e r e e f f e c t i v e l y e x i s t s a numher s a t i s f y i n q

A

". The lanquaqe

of

i n t u i t i o n i s t i c a r i t h m e t i c is t h e n " t r a n s l a t e d " i n t o this e p i s t e m i c lanquaqe.

Followinq the i n t u i t i o n i s t i c r e j e c t i o n of non-epistemic

m a t t e r s , the ranqe of this t r a n s l a t i o n c o n t a i n s formulas which have, i n some s e n s e , o n l y e p i s t e m i c components.

S e v e r a l common p r o p e r t i e s of

i n t u i t i o n i s t i c d e d u c t i v e systems are o b t a i n e d f o r the e p i s t e m i c parts o f

mv d e d u c t i v e system (which i n c l u d e s t h e ranqe of the ahove t r a n s l a t i o n ) . The Flaqq paper develops a r e a l i z a h i l i t y i n t e r p r e t a t i o n for t h e lanquaqe of my system and, thereby, s h e d s l i q h t on i t s proof theory.

The Mvhill paper and the Goodman paper c o n t a i n e x t e n s i o n s of my lanquaqe and d e d u c t i v e system to set theory.

~ o t hlanquaqes c o n t a i n a

7

Intensional Mathematics and Constructive Mathematics

s e n t e n t i a l o p e r a t o r analoqous t o my

"K"

.

The lanquaqe i n M y h i l l ' s

p a p e r c o n t a i n s t w o sorts of v a r i a b l e s , one r a n q i n q over sets i n q e n e r a l ( c o n s i d e r e d e x t e n s i o n a l l y ) and one ranqinq o v e r " e x p l i c i t l y q i v e n h e r e d i t a r i l y f i n i t e sets".

The l a t t e r i n c l u d e s , f o r example, e x p l i c i t l y

q i v e n n a t u r a l numbers and e x p l i c i t l y qiven r a t i o n a l numhers.

In the

lanquaqe of Goodman's paper, a l l v a r i a h l e s range over i n t e n s i o n a l "set Althouqh s e t p r o p e r t i e s are n o t e x t e n s i o n a l , c l a s s i c a l

properties".

( e x t e n s i o n a l ) s e t theory can be i n t e r p r e t e d i n Goodman's system i n a s t r a i q h t f o r w a r d manner.

The %edrov paper p r o v i d e s a " t r a n s l a t i o n " of

i n t u i t i o n i s t i c t y p e t h e o r y i n t o a modal t y p e t h e o r y ( a l s o hased on

54)

and a " t r a n s l a t i o n " of i n t u i t i o n i s t i c set t h e o r y i n t o a modal s e t t h e o r y which employs the lanquaqe of Goodman's paper ( b u t h a s a s t r o n q e r Both t r a n s l a t i o n s are q u i t e similar t o t h e

deductive system).

t r a n s l a t i o n of i n t u i t i o n i s t i c a r i t h m e t i c i n my paper. The system developed i n t h e L i f s c h i t z c o n t r i b u t i o n i n v o l v e s a d i f f e r e n t u n d e r s t a n d i n q of t h e e p i s t e m i c i n t e r p r e t a t i o n of constructivity.

i s employed.

I n s t e a d of a n e p i s t e m i c o p e r a t o r , an e p i s t e m i c p r e d i c a t e

T is a v a r i a b l e , then

If

constructed".

K(x)

is t a k e n as

"

x

-

t Ktn)

f o r a l l numerals

t h e set of a l l n a t u r a l numhers. s e m a n t i c s of the paper.

-

fi ,

However,

t h e e x t e n s i o n of

VxK(x)

A(5))

K

would he

1 s u q q e s t t h a t the p r e c i s e meaninq of

K

is

For example,

i s t a k e n as amountinq t o " t h e r e e f f e c t i v e l y e x i s t s a

number s a t i s f y i n q " f o r any given

I f it d i d ,

i s f a l s e i n the

d e t e r m i n e d , i n part, by the c o n t e x t i n which i t o c c u r s .

35(K(x) &

K

I t is i m p o r t a n t t o n o t e t h a t t h e e p i s t e m i c p r e d i c a t e

d o e s n o t have a d e t e r m i n a t e e x t e n s i o n i n the n a t u r a l numbers. then since

can he

A "

2 ,

and

Vz(K(5)+ A ( 5 ) ) i s t a k e n as amountinq t o

A(x) ".

The formulas of i n t u i t i o n i s t i c a r i t h m e t i c

are i n t e r p r e t e d i n this lanquaqe as those formulas whose q u a n t i f i e r s are a l l restricted to

K

.

Althouqh f a i t h f u l n e s s of t h i s t r a n s l a t i o n is

open, s e v e r a l s u q q e s t i v e r e s u l t s are o h t a i n e d .

S. SHAPIRO

8

The systems i n t h e f i r s t f o u r p a p e r s of this volume b e a r a t l e a s t a s u p e r f i c i a l resemblance t o t h o s e developed i n some r e c e n t work by G. Roolos, R. Solovay and

other^.^

There are, however,

important

The l a t t e r systems c o n t a i n a modal o p e r a t o r 0

differences. i s taken a s

"

p

i s provable i n Peano arithmetic".

,

where

up

I n t h a t work,

i t e r a t e d modal o p e r a t o r s are understood a s i n v o l v i n q a r i t h m e t i z a t i o n . For example,

ocp

i s taken as

Bew( IEewrgll )

, where

is the

Bew

p r o v a b i l i t y p r e d i c a t e i n Peano a r i t h m e t i c and, f o r any formula i s t h e & d e l number of

A

.

The modal o p e r a t o r s i n t h e f i r s t f o u r p a p e r s

o f this volume c a n n o t be s i m i l a r l y i n t e r p r e t e d . example, t h e o p e r a t o r

K

I n my system, f o r

is i n t e r p r e t e d a s " p r o v a b i l i t y i n p r i n c i p l e " and

is thereby not r e s t r i c t e d to

Peano a r i t h m e t i c ) .

11 ,

any

p a r t i c u l a r d e d u c t i v e system ( s u c h as

For example, the " e x t e n s i o n " of

Contains n o t o n l y

K

formulas provable i n c l a s s i c a l Peano a r i t h m e t i c , h u t also formulas p r o v a h l e i n the system of my paper.

The o p e r a t o r

R

in M y h i l l ' s p a p e r

is i n t e r p r e t e d as p r o v a b i l i t y i n t h e s y s t e m of t h a t paper and, t h e r e f o r e , i s n o t r e s t r i c t e d t o p r o v a b i l i t y i n c l a s s i c a l s e t theory.

These

i n t e r p r e t a t i o n s of t h e modal o p e r a t o r s e l i m i n a t e t h e need f o r a r i t h m e t i z a t i o n t o understand formulas with i t e r a t e d o p e r a t o r s . M y h i l l ' s system, f o r example, provable".

-

BR(&)

i s simply taken as

"

B(A)

*

In

is

The p r e s e n t a u t h o r s s u q q e s t t h a t the b r o a d e r u n d e r s t a n d i n q of

t h e o p e r a t o r s f a c i l i t a t e s t h e i n t e r p r e t a t i o n of c o n s t r u c t i v e mathematics i n c l a s s i c a l modal systems. R.

Smullyan's f i r s t paper below can be seen as a s t u d y of t h e above

extended n o t i o n of p r o v a b i l i t y i n a more q e n e r a l s e t t i n q .

p

developed i n t h a t paper h a s a p r e d i c a t e e x p r e s s i o n s of t h e same lanquaqe. lanquaqe and as

"

'A1

If

a name of formula

A i s provable i n -

(p

i n which e v e r y theorem of

'I.

@

The lanquaqe

r a n q i n q over names of

(9

is a

d e d u c t i v e system on t h i s

A ,

then

prA1

c a n he i n t e r p r e t e d

Concern i s with those d e d u c t i v e systems

is t r u e under t h e i n t e r p r e t a t i o n of

p

as

Intensional Mathematicsand Constructive Mathematics

provability i n

8

.

9

Such d e d u c t i v e systems are c a l l e d " s e l f -

r e f e r e n t i a l l y correct". Smullyan's second paper, a s e q u e l t o the f i r s t , c o n c e r n s p r o v a h i l i t y i n a s t i l l more g e n e r a l s e t t i n q .

The r e s u l t s a p p l y t o a n y lanquaqe and

d e d u c t i v e system w i t h a ( m e t a - l i n q u i s t i c ) p r o v a h i l i t y f u n c t i o n s a t i s f y i n q t h e Hilbert-Bernays d e r i v a h i l i t y c o n d i t i o n s .

This i n c l u d e s , f o r example,

t h e systems of t h e f i r s t f o u r p a p e r s of this volume, t h e systems i n Smullyan's f i r s t paper and t h e systems i n , s a y , Boolos' work.

Concern i s

w i t h c o n d i t i o n s under which & d e l l s second incompleteness theorem and a " l o c a l i z e d " v e r s i o n of Lgh's theorem apply. I t s h o u l d he p o i n t e d o u t t h a t t h e a u t h o r s of t h e papers i n t h i s

volume do n o t completely s h a r e their p h i l o s o p h i c a l views and m o t i v a t i o n s . In p a r t i c u l a r , the p h i l o s o p h i c a l remarks i n t h i s I n t r o d u c t i o n e x p r e s s o n l y my views.

The disaqreements amonq t h e a u t h o r s are r e f l e c t e d i n p a r t

h v t h e mutual criticism c o n t a i n e d i n t h e f o l l o w i n s p a p e r s . I would l i k e t o thank John Mvhill and Ray Gumh f o r t h e i d e a of

c o l l e c t i n q papers on this s u h j e d t and t o thank John f o r encouraqinq t h e a u t h o r s to work on the project.

S p e c i a l t h a n k s to t h e e d i t o r i a l s t a f f a t

North Holland, e s p e c i a l l y D r . S e v e n s t e r , f o r t h e prompt and p r o f e s s i o n a l manner i n which the volume w a s handled. t h i s a l l t h e more.

Experience makes m e a p p r e c i a t e

S. SHAPIRO

10

Notes 1.

The word " p r e c e p t i o n "

( s i m p l i c i t e r ) i s used h e r e o n l y t o r e f e r

t o p e r c e p t i o n viewed w i t h t h e p r e s u p p o s i t i o n t h a t t h e r e i s a p e r c e i v e d e x t e r n a l world.

" S u h j e c t i v i s t p e r c e p t i o n " is t o r e f e r t o p e r c e p t i o n as

c o n c e i v e d by an e x t r e m e s u b j e c t i v e i d e a l i s t .

S i m i l a r for " t h o u q h t " and

" s u b j e c t i v i s t thouqht". 2.

A.

Heytinq,

Intuitionism,

Holland P u h l i s h i n q Company, 1956, pp.

3.

See, f o r example, G.

Boolos,

I n t r o d u c t i o n , Amsterdam, North 1 , 8 , 10.

llnprovahility

Camhridqe, Camhridqe D n i v e r s i t y P r e s s , 1979.

of C o n s i s t e n c y ,

In tensional Mathematics S. Shapiro (Editor) 0 Elsevier Science Publishers B. V. (North-Holland), 198.5

11

EPISTEMIC AND INTUITIONISTIC ARITHMETIC Stewart Shapiro The Ohio State University at Newark Newark, Ohio U.S.A.

Introduction. In this paper a language and deductive system of epistemic logic for arithmetic are developed.

In addition to the usual

connectives and quantifiers, the language contains an epistemic operator K. If

5

is a formula, then K ( A ) is taken to mean ,'

knowable"

.

is ideally, or potentially,

In addition to whatever intrinsic interest the presented system may have, I suggest that it can help illuminate the understanding and formalization of mathematical practice.

For example, even though the

underlying logic of the system is classical, it is shown that the language of intuitionistic arithmetic can be translated into the language, and thus, that the present system is capable of expressing formulas of both classical and intuitionistic arithmetic, as well as formulas of mixed constructivity. This indicates that the present system can contribute to an understanding of the difference between classical and constructive arithmetic, and, moreover, that it can account for and help understand the constructive and epistemic aspects of normal, non-intuitionistic mathematical practice. As a preliminary to the technical development, in section 1 below possible understandings and interpretations of the operator K are discussed.

Section 2 contains the basic details of the first-order

version of the language and deductive system. Section 3 concerns the possibility of interpreting K as "deducibility" in a particular deductive system. Although such an interpretation is developed, there are serious philosophical (and technical) limitations to it.

In the following section

4, certain properties of the deductive system are established (through the reinterpretation of section 3 ) .

Section 5 is a brief note on the

possibility of applying Hintikka's [71 semantics for ideal knowledge to the present language.

It is shown that although the present deductive

system is sound in Hintikka's semantics, his notion of "epistemic alternative" is not appropriate to arithmetic and, moreover, that this

S. SHAPIRO

12

shortcoming is shared by many semantics for (ideal) knowledge.

Section 6

contains the translation of the language of intuitionistic arithmetic and related matters.

In section I the present language and deductive system

are extended to include higher-order quantification. There are several interesting features of the expanded systems that are not shared by the first-order version.

Among these are technical counterparts of some well-

known problems in epistemology and the philosophy of language. In the final section 8 , some further applications of our language (and deductive system) to the formalization of mathematical practice are developed. 1.

The Epistemic

Operator. Knowledge, of course, involves a knower.

Thus, any epistemic operator must refer to the knowledge of a particular subjective being, such as a person or, perhaps, a community.

An

epistemic

language (such as that of [ 7 ] ) which involves more than one knower should have a different operator for each. The applications of the present deductive system, however, suggest that the added technical complications of the extra operators are not necessary.

Hence our single operator K.

The intended interpretation of K is not actual knowledge, but rather what may be called "ideal knowledge" or "knowability". It is assumed, in particular, that the "extension" of K is deductively closed:

If KQ)

and

.

The "knower" involved with K is taken to be an ideal then K ( B ) mathematical community. Informally, the preferred reading of K ( 5 ) is ''2 is

A FBI

knowable".

In short, everything known by the community is knowable and

anything that follows from knowable premises is knowable. other possible readings of K ( A ) are "it is possible to come to know community knows", (see [121).l

"5is

verifiable", and

"5 is

A

given what the

(informally) provable"

It is admitted, of course, that these readings are vague and,

perhaps, obscure.

It is hoped that the foregoing analysis will provide the

necessary precision to the present operator K and, derivatively, to the concepts involved in the above readings. It might be noted that the idealization of the present "knowability" is similar to that of other mathematical properties whose terms also have the suffix "-able". Examples include computability, decidability, solvability, definability, and even deducibility. The treatment of computability, for instance, is not made relative to the computation abilities of this or that computist, but rather involves computation ability as such.

Moreover, computability does not concern feasibility--no

(finite) bounds are placed on the memory, materials, life-span, etc. required for a computation. Here. there is only one "knower" and no bounds

Epistemic and lntuitionistic Arithmetic

13

are placed on the ability of the community to obtain the consequences of its knowledge. I return to the "-able" properties in section 8 below. Actual knowledge, of course, is time-dependent in the sense that the knowledge of a given person varies with time. At best, consideration of ideal knowledge only partially removes the time-dependence.

Indeed, even

if, say, the theorems of a deductive system are knowable simultaneously with the axioms, the possible discovery of knowledge through new axioms remains a time-dependent activity. Thus, here I do not envision the possibility of an absolute, time-independent concept of knowability. The operator K may be taken as referring to knowability at a fixed, but unspecified time. Finally, in ordinary language, it is both grammatical and semantically meaningful for an epistemic statement to occur within the scope of another epistemic operator. For a given sentence that it is known that then & is known.

A

&, for example, one can state is true

is unknown or that it is known that if

Moreover, some sentences like this are true and some are

false. Thus, in ordinary language, "knowledge" represents what may be called a (non-trivial) iterative concept. Because the present K is a sentential operator (and not a predicate) it also is iterative. There are well-formed formulas, for example, of the form K(-K(&)) and K ( A + K ( & ) ) . Formally, the operator K can be interpreted, or reinterpreted, as any iterative concept that does not apply to false propositions and is deductively closed.

(To be a sound interpretation vis-a-vis the present

treatment, the property should be closed in the present deductive system.) On the surface, at least, this rules out "deducibility in a particular deductive system" as an interpretation of K, because deducibility is not iterative. Such statements as c e &are usually ill-formed. As is well-known, however, this "surface" may be shallow. At least two attempts around this difficulty suggest themselves. The first is to invoke an arithmetization. If T is a deductive system for arithmetic, PrT the proof predicate for T, and

& a formula of arithmetic, then

be interpreted as PrT ( r&') and K (K@) ) as PrT ( 'PrT

( '&')')

.

K ( A ) might

This

possibility is explored extensively in the literature (see, for example, [2] and R. Smullyan's contributions to the present volume), but it is clear

almost at the outset that this interpretation will not do here. if

fi is

knowable, then & is true.

K(K(A)+;);

Thus, K(&)

-+Ais a

Informally,

correct scheme, as is

both are theorems of the present deductive system. However,

S. SHAPIRO

14

by L & ' s

theorem, PrT(f&7)

+fA

is not a theorem of T unless

5

is a theorem

of T. A second possibility, perhaps, would be to let T' be an extension of

the present deductive system and to relate K1A) to F

T'-A, K(Kf&)) to FT,K(A), etc. With this program, K is interpreted, not as deducibility-inarithmetic, but as deducibility-in-epistemic-arithmetic.

An interpretation

of the present system along these lines is developed in section 3 below. Philosophically, however, it is a =interpretation.

It is shown there that

besides the obvious circularity, there are serious (and insightful) philosophical and technical limitations to such an interpretation. 2.

The Basic Deductive System.

We begin with a standard, first-order

system for arithmetic. The language L has connectives

7,

v,

&,+,-++:

quantifiers V, 3 , and names for all of the usual primitive recursive 2

functions and relations, including the Kleene T-predicates and U-function. We employ a natural deduction system D which operates through the introduction and discharge of assumptions.

The details are routine.

To

note one example, the so-called "deduction theorem" is a rule of inference--the rule of arrow introduction: If F,A

+I:

kg then

F k A+B

-.

Dwill be abbreviated I-. D A new (sentential) operator K is added to L with the formation rule:

In what follows, If

A

is a wff then K ( A ) is a wff.

Parentheses are omitted when unnecessary for clarity. We call a formula ontic if it contains no occurences of K. -

The reason for this designation

is that such formulas do not concern knowledge--they involve only the natural numbers and the interrelations thereof. Moreover, the truth values of ontic formulas are independent of any actual or ideal knowledge. We call a formula epistemic if it is in the form K(A) for some formula

A.

Two rules of inference are added to D for the new operator, an "elimination rule" and an "introduction rule": KE:

K(A) /-

KI:

If F

t

A. A and

every formula in ?? is epistemic, then I?

+

K(A).

On the intended (informal) interpretation, KE is a correct rule simply because only true statements are knowable; KI is a statement of the deductive closure of knowability. Notice that the result of erasing all occurences of K from a theorem of D is itself a theorem of arithmetic. Hence, D is consistent.

15

Epistemic and lntuitionistic Arithmetic

The following theorem schemes have trivial proofs: TO:

K(5)

+A.

TI: K ( 5 ) -+KK(A). T2:

KK(&).

K(&)++

A

T3:

If

T4:

K(&-+B)

C

then -+

k K(A).

'

(K(A) - + K ( g ) ) .

The main theorem of 131 indicates that the logical (i.e., non-arithmetic) part of D is equivalent to the modal logic S4. Theorem T1 may be called a reflection principle. An informal justification for this scheme may be given in terms of a similar principle for actual knowledge. Assume, then, that if or at least knowable, that

A

A

is known then it is known,

is known. This seems plausible if "knowable"

is understood in terms of some sort of (informal) "provability". Suppose that

A

is knowable. Then, of course, A could become known.

By the

assumption, when Abecomes known, it becomes knowable that fi is known. Hence, the knowability of

is knowable.

Notice that a similar line of reasoning fails to justify the scheme If &is simply not known, then (perhaps) one can come to

-K(fi)-+K(lK(&)).

know that TK(_A)

A

means

is not known by self-reflection. This does not matter.

"11

Here,

is not knowable" and the unknowable formulas cannot be

determined by self-reflection even in principle. 3.

Comparison offand Deducibility. Proofs in axiomatic deductive

systems are sometimes thought to reflect, at least to some extent, the process of coming to know mathematical facts.

Thus, there seems to be at

least a similarity between formal deducibility and the intended interpretation of K.

In fact, later in this section an interesting and

fruitful metatheorem is obtained by partially interpreting K along these lines. Presently, however, it is shown that there is a serious logical and philosophical limitation to thinking of K as meanine "deducibility in a particular deductive system." 3.1.

In 141 a d e l presents and briefly discusses an axiomatization

equivalent to propositioned S4 (see section 6 below for the details). suggests that K(A) might be taken to mean

"A is provable", but

if so, "provable" must mean "provable in principle" and not

K(5)

cannot mean

''A is deducible in this

deductive system."

I-

He

adds that

--that is Concerning

the present system, the extent of the possible correlation between K and formal deducibility (or any arithmetic property) is born out in an

S. SHAF'IRO

16

interesting metatheorem suggested by some comments in Myhill 1121 on Gadel' s work. A fixed arithmetization of the formulas of L is assumed.

For each

natural number & let B be the formula with W d el number fl. Notice that -n_ if K is interpreted as deducibility in a particular deductive system, then (the set of Giidel numbers of) the extension of K would be recursively enumerable and, therefore, arithmetic. Suppose, then, that there is a formula

--

E(x) of L with one free variable

which is thought to represent the extension of K. sentence

s, suppose that g(E)++K ( B

simplicity, it may be assumed that the technical results.)

)

is true.

That is, for each (For conceptual

E is ontic, but this does not affect

It follows (under the assumption that nothing

knowable is false) that there are true, but unknowable sentences. This, of course, is no surprise. can be --

Our first metatheorem is that one such sentence

--

constructed from E.

{ g @ + + K(B

)

-n_ TA:

I

For this, let

(*E) be the set of sentences

is a sentence).

B

There is a sentence E of L such that

(*E) Proof:

f-

7

K(F).

Moreover, if

E

(*El

l-

and

is ontic, then so is

F.

Let d be the name in L of the diagonal function, the

-"_

primitive recursive function which assigns to each number E, the Wdel number of B (;I.

--

Let 2 be the Wdel number of

(*E) IK ( E ) and, hence, (*El I- F.

TE(d5) and let F be -$(dr~). It follows that From KE we have (*El I- 1

E++7K(E)

Notice that this result does not entail the inconsistency of (*El.

(*E) are not epistemic, the rule K I cannot be (*E) I- K ( F ) from (*E) I- F.) Indeed, in the next

(Since the formulas in invoked to produce

subsection it is shown that

(*g) is

in fact consistent for many formulas

-E.

Two corrolaries of TA, however, indicate philosophical and technical restrictions on the attempt to consider E as representing the extension

Of K. Notice, first, that if, for some formula

E ( x ) ,the

sentences in

(*E)

are all true, then one might think it consistent to add them as axioms of D.

This, however, is not the case.

Let D E be the deductive system

obtained from D by adding each instance of (*El as an axiom. CA1:

The deductive system DE is inconsistent.

Proof: CA2:

From TA, F D E 1 K ( F ) and kD&.

The set of formulas { K ( B )

By KI, J-

1-g~(*E) 1

K(E).

DE

is inconsistent with D.

Epistemic and Intuitionistic Arithmetic

17

That is, there is a sentence G E (*g) such that CD-,K(G) (and FDK(-,K(G)1 1 . One moral of these considerations is that it is not sufficient for new axioms of the deductive system to be true--new axioms should also not be unknowable. If there is such a concept as absolute, time-independent arithmetic knowability and if the deductive system 5 is sound for such a knowability, then theorem TA has a rather succinct interpretation. Suppose that there is a formula E(5) that represents the extension of absolute knowability. Then there is a sentence B unknowable that

E(5)

-n-

such that it is (knowable that it is) absolutely

and E ( B

are materially equivalent.

)

-”_

Thus, either the

extension of absolute arithmetic knowability is not arithmetic or the 4 extension is, in some sense, absolutely unknowable. As noted, however, the intended interpretation of K is not absolute knowability, but knowability-at-a-fixed-time, sat rule out the possibility that (at is represented by some formula

E.

g.

Theorem TA does not

z) the extension of K is arithmetic and The corollary CA2, however, indicates

G that 5 represents the in (*El is true, but

that for each formula E it cannot be known (or even knowable) at the extension of K is represented by extension of knowability at

g,

g.

In short, if

then every formula

some of these formulas are unknowable at

2.

Under these circumstances, it remains possible that it could become known at some later time, say represented by

E.

c,that knowability at g

is (or was)

This knowledge might be obtained, for example, by

reflecting on the epistemic processes available at

G.

CA2

shows that

this knowledge is genuinely new knowledge which was not available at In such a case, some of the formulas in the form

K ( E ( i ) *K(B

true only if the outermost K is interpreted as knowable-at

))

g.

would be

-n_ and the

inner K is interpreted as knowable-at 2. If K is completely reinterpreted as knowable-at 2, then some of the formulas of

-

false. An example of such a formula is g(dE) *K(%m). the extension of knowability at

--

hence, K ( B 3.2.

)

If E represents

2, then E ( d i ) (whicE is equivalent to

of TA) asserts that the sentence with a d e l number knowable-at g . This is false.

(*g) become

However, B

is true as reinterpreted.

-dE

7c

(i.e., F) is

is knowable-at &2- and,

It remains to be seen whether it is formally consistent for the

extension of K to correspond to that of deducibility in a particular deductive system.

For this, an interesting metatheorem is obtained by

S. SHAPIRO

18

adapting, simplifying and extending the method of Kleene 191 to our language and deductive system. A

relation D'] & between extensions D' of D and sentences A of L is

introduced. Roughly, D'I

5

may be thought of as "if K means kD,, then

is true".

A

I

If & is atomic, then D' & iff kD,&.

D'

I A&B iff D' I and D' I B. I fivg iff D'I 5 or D * I B. I A+g iff either D' ,j' A or D'I

D'

I &++B

D' D'

D'

I -, I~ ~

D'

I~xA(x)

D'

I K(A)

D'

g. B or

D' 1 5 and D' 1 g . 1 A. 1 iff 1 ) D *I ~ ( ifor ) every numeral i.

iff either D'I & and D'I

iff D'

(

iff

D'I ~ ( ifor ) some numeral i.

iff D'I & and kD,&.

In what follows, D I & is abbreviated

I A.

Under the assumption that D' is consistent, the following lemmas

A:

follow from the definition of D'I L1:

If

s

and t are closed terms and if s=t is true, then FD,

s=t and, hence, D' free, then D' I L2:

If

s

&(s)

s=t. Moreover, if

A(x) has only 5

++&(t).

is a closed term, then there is a unique numberal

such that f-.,,s=i L3:

I

and, hence, D'

I s=i. 1 A iff 5

If A is an ontic sentence, then D'

i

is true.

The proof of the following metatheorem is straightforward, but tedious. TB:

Let C be a set of sentences of L such that DUG is consistent and (DUC) I

g

for each

FEZ.

If

5 is any

I A.

sentence of L such that I-DUC-A, then (DUS) If the set C of additional sentences is empty, then TB is TB':

If

is a sentence of L and k &, then

I A.

The first corollary of TB is that it is consistent for the extension of K to correspond to that of any arithmetically definable, consistent extension D' of D which satisfies the premises of TB.

Let E ( x ) be any

ontic formula (with only one free variable) which expresses deducibility in D'.

That is, let CB1: Proof:

(*E)

g(i) be

B D'-n-' is consistent with D'. true iff f-

That is,

(*E)

--

D,O=l.

From the lemmas and TB, we have the following equivalences:

Epistemic and Intuitionistic Arithmetic

D' D'

19

I g(G) iff E ( i ) is true iff t= D-"_ ,B iff ( t ,B and -D'"_ I B ) iff D'I K(B ) . Therefore, I)' I (E(n)*K(S)).

?! That is, D'

n -

15 for

each G E

(*El. If (*El k,,;=i,then

it would follow (from TB and the rule of arrow introduction) that D'

I ;=i,a contradiction.

Combining this proof with corollary CA2, notice that for each appropriate deductive system D', there is a sentence 5 (in

(*E)) such that D' 15,

so

b(D17g,but kD,lK(g). Notice also that it is consistent for the extension of K to correspond to that of any arithmetically definable extension of D obtained by adding true ontic sentences. 4. Properties of the Deductive System.

This section focuses on

several corollaries of TB which, I suggest, correspond to important Let D' be any consistent,

properties of knowledge and knowability.

recursively enumerable extension of D which satisfies the premises of TB. 4.1. Disjunction @ Existential Quantification. On the present interpretation of K, there is a difference between K(5vE) and K@)VK(B). Consider, for example, an instance of these formulas in which g is the negation of 5. Notice first that K(Av75) amounts to the truism "it is

A

is either true or false". On the other hand, K(A)vK(~&) A is knowable. If 5 is any ontic says that either 5 is knowable or . sentence whose truth value is unknowable, then K(fiv~&) is true (and

knowable that

provable) but K(&)vK(~A) is not.

The next corollary to TB bears out the

difference between K (fivg) and K (5)VK (E) CB2:

If ;and either

.

are sentences of L and if kD,K@)vK(F)

then

F D I Cor kD,g.

If I-DIK(&)~K(g) then, by TB, D' I K(A)vK@). Therefore, A or kD,B. Hence, either +either D ' I K(5) or D' K@). D'1 . If 5 There is a similar difference between K(3g(x)) and 35K(5(5) Proof:

I

is ontic, for example, the former amounts to only "it is knowable that the extension of the property represented by 5 is not empty". The latter, however, is the stronger statement that there is a (particular) number such that it is knowable that x satisifes the property represented by

A.

This difference is born out by an analogous corollary: free and if kD,35K(5(5)), CB3: If A(x) is a formula with only

x

--

then there is a numeral

1 such that tD,5(;).

The proof of CB3 is similar to that of CB2. undecidable sentence and let g(&) be

For illustration, let

(s=?kg) v(z=i&Tg).

is a logical truth and, thus I-KdjxB(x) )

.

g be any

Of course, 35B(xf

However, if I-35(K(B(z) 1 , then,

20

S. SHAPIRO

by CB3, there would be a numeral however, entails that either

tg

--1 such that c(n=O&G) v (i=i&,g).This, -12, which

or

contradicts the assumption

of undecidability. 4.2.

Universal Quantification--The Barcan Formula.

the similar pair of formulas V g ( & ( x ) ) and K ( V & ( x ) ) . least, the two formulas are not equivalent.

&(x) is knowable

instance of

We next consider

On the surface at

The former asserts that each

(perhaps separately) while the latter is the

statement that it is knowable that & holds universally. is provable in D.

expected, K ( V g ( x ) ) + V g ( & ( x ) )

Barcan scheme, V g ( & ( x ) ) + K ( V S ( x ) ) .

As

might be

The converse is the

One might argue that this scheme

expresses a correct principle by appealing to a strong "reflection" Suppose that V g ( & ( x ) ) is true.

principle:

Then each instance of & could

become known.

If each instance of &did become known, then one could come

to know V&(x)

by a survey of knowledge. Hence, K ( V S ( x ) ) is true.

I

would suggest, however, that since the process of reflection involved here is infinite, the argument fails.

In the scenario, one does not realize

that & holds universally until after a survey of all of the numbers is completed. Such "procedures" are not legitimate even in the present context of ideal knowledge (in which no finite constraints are placed on The following corollary to TB shows that the Barcan

knowledge ability).

formula scheme is not derivable in D ' .

CB4:

There is a formula B ( x ) , with one free variable, such that j'D,Vg(B(x)

Proof:

Let

.

1 -+K(Vg(z)1 the primitive recursive predicate which

g ( x ) be

expresses "5 is not the G6del number of a proof in D ' of For each numeral D'

I

we have CD,B(i),D' I

Therefore, D' 1

K ( g ( Z ) 1.

Vg(g(x)1 .

second incompleteness theorem, Hence, by TB,

;=I."

B(E), and, hence, However, by W e l ' s

kD,Ve(x)and, so D '

,#K(Vg(x)

1.

VxK(B(5)) - + K ( V S ( X ) ) .

D' For contrast, notice the following:

-

FK(V+(A(x)))

+K(VS(x))

If t V s K ( A ( 5 ) ) then t K ( V g ( x ) ) That is, if it is knowable (or provable) that each instance of

is

knowable, then it is knowable (or provable) that &holds universally. 4.3.

Negation.

In L four types of negation can be formulated.

Classical negation

+, of course, simply anrounts to the

The stronger K(-&

says that the falsehood of

,K(&)

falsehold of &. & is knowable; the weaker

says that & itself is not knowable. Finally, the "intuitionistic"

21

Epistemic and Intuitionistic Arithmetic

K(,K(A))

says that the unknowability of

A

is knowable.

The following

implications are easily derived in D.

CB5:

In the deductive system D', the above diagram represents the only derivable implications among the four negations. That is, neither -&+K(lK(A))

nor K(TK(5)) +$nor

any of

the four converses is generally derivable in D'. Proof:

It suffices to show that neither of the two implications

which involve both classical negation (7A) and intuitionistic negation (K(-,K(:)))

are derivable in D'.

ontic sentence that is not refuted by D'.

(i) Let g b e any false Then D'

]E and

t/

b/D,+C(g).It follows from TB that D , (7g+K(lK(g) 1 ) . (ii) A s noted after the proof of corollary CB1, there is a sentence 5 such that D' by TB, 4.4.

A(g/g)

[ 5 and bD,7K(g). Thus, kD,K(lK(g)) and,

bc,, (K(-/K(G))+-ts).

Substitution of Equivalents.

If

A, g,Cz are

fOnIiukiS, let

be the result of substituting C2 for some (or all) of the

occurrences of

2 as subformulas of A. In ordinary first-order logic, all

instances of the substitutivity of equivalents scheme,

(2-z)

(&(~lJg)ttA),

are logically true and, thus, are theorems.

the case in D.

Pre-formally, notice that the

equivalence of

c.1 and 2 is not

truth of the

+

This is not (material)

sufficient for these formulas to be

intersubstitutable & epistemic contexts.

To illustrate, it is shown that

unrestricted substitutivity of equivalents would entail I-A+K(A):

A A +--f

premise

(o=o) (;=El)

-t

(K(~=O)+-+K(A)

substitutivity of equivalents

K(Z=Z)) : C K * K (A) The following theorems are the correct counterparts of the substitutivity of equivalents theorem. T5: b T6:

K(C1-E)

+

-5) l-A(GClC2) -11

(A(%/=)

If C g + - + g then

Generally, two formulas are intersubstitutable only if their equivalence is knowable.

S. SHAPIRO

22

4.5.

Substitution

x=y+((A(z) ++&(XI),

of Identicals.

The substitutivity of identicals,

is another scheme that ordinarily represents a logical

truth, but can fail in epistemic contexts. Let b and c be two constants. Even if b=c is true, it does not follow that b and c are intersubstitutable. The problem is that the identity may not be known or knowable. To take an example from ordinary language, "the-number-of-planets = 3x3" and "Hegel knew that 3x3 = 9" are both true, yet one would not conclude "Hegel knew that the-number-of-planets = 9 " .

Modifying the above discussion, perhaps

the substitutivity of identicals should be replaced in D by

K(x=y) (A(& ++&(y) -+

.

In the present case, however, this modification is not necessary. The reason is that in elementary arithmetic there are no true, but unknowable identities.

) provable First, the sentence YEV~( ~ = y - + K ( ~ =) ~ is

in D (by induction on 5 and l). Second, notice that all terms of the present language L are constructed from numerals, names of primitive recursive functions, and variables.

Thus, in principle, one can determine

the numerical value of any term (given the values of the variables), and, therefore, the truth value of any identity is knowable. The expanded deductive systems of section 7 below, as well as virtually any epistemic analysis, set theory, etc. do have true, unknowable identities.

In such cases, restrictions on the substitution of identicals

are in order. 5.

A

Note on Semantics.

There is no attempt here to provide a

philosophically correct semantics for the language L.

That is to say, I

do not attempt to codify a justified collection of truth conditions for the sentences of L.

There are several well-known semantics for the notion of

"ideal knowledge", but none (that I am aware of) are appropriate for mathematical knowledge all for more or less the same reason.

I discuss

here the work of Hintikka 171. Instead of the present single operator K, Hintikka has a class of operators

Y, in which

denotes a person. Because present concern is with

the ideal knowledge of a single person or community, the subscript is omitted. It is easily verified that if Hintikka's semantics is formulated on the present language L, then D is sound for it.

I suggest, however, that

the semantics has serious philosophical shortcomings in the context of arithmetic. The major technical artifacts of Hintikka's work are the "epistemic alternatives", possible worlds relative to a given subject's

Epistemic and Intuitionistic Arithmetic

knowledge. Let

5 be

23

any sentence such that both 2 and 75 are unknowable.

In Hintikka's semantics, this would amount to the subject having two epistemic alternatives, one in which

5 is true and one in which 5 is false.

Of course, at least one of these would have to be a non-standard model of arithmetic.

That is to say, in one of these alternatives, "the natural

numbers" would not denote a structure isomrphic with the natural numbers. An epistemic alternative, however, is supposed to be a world

consistent with present knowledge--a world which "might be the case for all the subject knows".

I submit that a non-standard model is not the sort of

thing that can constitute an epistemic alternative.

In arithmetic, all

that is unknown (or unknowable) are certain facts =the

natural

numbers, the very structure of the natural numbers &known.

Mareover, the

structure of the non-standard models is also known, along with the properties of these models that make them non-standard.

That is, it is

known why the non-standard models are not (isomorphic to) the natural numbers.

In short, my thesis is that non-standard models of arithmetic

are not possible given present knowledge and, thus, that they are not adequate candidates for epistemic alternatives. Of course, one might attempt to overcome this problem by using a second-order language or simply requiring that epistemic alternatives contain only standard models.

This would make the present semantics

useless, however, because under either condition, every ontic truth of arithmetic would be true-in-all-epistemic-alternatives and, thus, would be knowable. I suggest, in conclusion, that a semantics for the present language L (and its extensions) should allow the truth values of the ontic sentences to be fixed in advance by the mathematical structure under study, and should not rely on the possible truth values of such sentences in assigning truth conditions to sentences containing the epistemic operator. 6.

Intuitionism. Intuitionism is a philosophy of mathematics which,

in effect, denies the "ontic" aspects of mathematical practice in favor of what is called "the constructive". Concerning ontology, the intuitionist conceives of, say, the natural numbers as the result of a mental act of construction and thereby denies that these numbers have an existence independent of the mathematician.

Accordingly, theorems of arithmetic do

not represent objective facts about independently existing mathematical entities, but, rather, the results of construction. Concerning semantics, the intuitionist Heyting [61 once said that to explain a formula, one does

S. SHAPIRO

24

not give its

truth conditions, but

rather its proof conditions.

In

practice, the intuitionist rejects those classical laws and inferences which, when interpreted, are taken to be incompatible with intuitionistic philosophy. Notable among these are the law of excluded middle, &v-&, which is interpreted by the intuitionist as "either the construction corresponding to &has been effected or the construction corresponding to the refutation of & has been effected". Another example is the quantifier exchange + e - + 3 5 q A , property

which is interpreted as "if not all numbers have a

5, then one can construct a number that lacks i".

In recent years, several non-intuitionists have suggested that the logic of intuitionistic mathematics can be interpreted as a logic of justification or an epistemic 10gic.~ To a classical mathematician, then, intuitionistic mathematics might be understood as a "pure" epistemic mathematics--a mathematics with no non-trivial ontic component. In this section, the proposal is born out by a translation of the language of intuitionistic arithmetic into the present language L.

Intuitionistic

arithmetic is interpreted in L as the arithmetic of "purely" epistemic formulas. 6.1.

Translations. The following are taken to be intuitionistic

connectives and quantifiers:

2,

5, y, 2, T,t,?.

To paraphrase

Heyting [61, the meanings of these are:

-A&B :

AvB : A*:

A * : -., ":

Vz(x):

B. I can prove & or I can prove B, and I know which. I can prove &and I can prove

I have a method which, if given a proof of A, produces a proof of

B.

5

amounts to

(B+&).

amounts to ~ ;6=i. t I have a method which, if given a number 1, produces a proof of

A@.

3xA(x) : I know of a number 1 such that I can prove &(El. --

It is clear that the notion of provability in use here is pre-formal and does not refer to a fixed deductive

Moreover, intuitionistic

proof seems to be ideal in the sense that no finite bound is placed on the length of a proof.

Also,

"provability" here is at least prima facie

iterative. The intuitionistic "I can prove

If",

then, appears to be

clearly allied to the intended interpretation of the present KC;).

The

following translation of the intuitionistic connectives and quantifiers into L is proposed:

7

Epistemic and Intuitionistic Arithmetic

25

the clause "and I know which" has no translation. However, see CB2. this amounts to "it is knowable that knowledge of

(materially) implies

knowledge of

El'. An exact translation

is not possible. this amounts to "it is knowable that knowledge of

is impossible".

again, an exact translation is not possible. see CB3. Notice that if 4 is any formula in the range of the translation (i.e.,

--

ALB, AvB, etc.) then b+K($).

The following scheme for translating

intuitionistic formulas into L is proposed.

- be

For each ontic formula g , let

its intuitionistic translation in L. If

A

-

is atomic, then

ALB: a

-

AvB -:

-

A*:-

A+B

7

-

SrXA: -

>V

15:

3g: We say a formula

4 is K[&).

-E9I -

A :

formula

--AvBALB

$!

3 2 is intuitionistic if B is provably equivalent to a

& for some ontic A.

6.2.

Interpretation

Meta-theorems.

6.2.1.

Deduction Theorem.

There is an interesting difference between the interpretation of classical deductions and the interpretation of intuitionistic deductions.

In the

former, each line is only stated, but in the latter each line is asserted. Suppose, for example, that in the course of a classical deduction, a formula A is introduced as a premise or assumption. read "assume 5'' or, perhaps, "assume A is true".

That line might be

In an intuitionistic

deduction, a similar line would better read "assume A is known" or "assume

A is provable".

The difference is brought out in a single case of arrow

introduction (or deduction theorem) :

from

g,

infer b&+g. Formally,

the rule (or theorem) holds in both contexts, but the interpretation is

26

S. SHAF'IRO

different. Roughly, in classical deductive systems, the rule is taken to mean that if one can derive g after assuming A, then, in effect, one can derive "if & is true, then B is true".

In intuitionistic deductive systems,

on the other hand, the rule is taken to mean that if one can derive assuming that implies B ~-

--

g after

is provable, then one can infer that & intuitionistically

that "one can prove B if given a proof of

6".

To obtain this

conclusion, the assumption of & in the original deduction must be interpreted as involving the provability of that formula.

I conclude that

not only does each formula itself have a different meaning, but also that the

use of a

formula in a deduction has a different meaning in classical

and intuitionistic deductive systems. Thus, since the present language L and deductive system D is, after all, classical, one should not expect a general intuitionistic deduction theorem : from to hold.

r,i

I-

g

r t ++g

infer

A "counterexample" is readily obtained.

generally correct, but

A t E=E;t&

amounts to &

of course, does not generally hold.

A, E=E

KC&) or

1- -A is, of course, t +(&I, which,

The following intuitionistic

deduction theorem is easily verified:

r , K(&) f- B r t 5%. if

TC:

and every formula inr is epistemic, then

Faithfulness of the translation. The soundness of the present

6.2.2.

deductive system for intuitionistic arithmetic is tedious, but straightforward: TD:

For any ontic

A, if 5

arithmetic, then The converse of TD

--

is a theorem of intuitionistic

@.

the completeness of D for intuitionistic arithmetic--

was recently proved by Nicolas Goodman 151. weaker results are presented here.

Some easily obtained, but

Notice, first, that it is easy to see

that many instances of intuitionistic excluded middle, Indeed, it follows from CB2 that

provable in D. or I-,K(&).

I-

&vs

zg,are not only if either I-

The next theorem and corollary concern the completeness of

subsystems of D. Let LD be the logic subsystem of D.

That is, let I1D contain every

axiom and rule of D except the axioms of arithmetic. TE:

For any ontic

A,

-

if kL&,

then &is a theorem of

intuitionistic predicate calculus. Proof outline: Familiarity with the Kripke semantics for both

Epistemic and Intuitionistic Arithmetic

21

the modal logic S4 and the intuitionistic predicate calculus is assumed (see [lo] and 1111).

Let a be a Kripke structure for

intuitionistic predicate calculus. L, then a

If

fi is

an

ontic formula of

Il-A is taken as "& is satisfied by a viewed as a

Kripke structure for intuitionism". If then a f=B is taken as structure for S4".

"g is

B

is any formula of L,

satisfied by a viewed as a Kripke

TE follows from the following lemmas.

L6 is

due to Kripke.

g

is a theorem of ED, then a

kg.

L4:

If

L5:

If A is any ontic formula of L, then a

L6:

If a l h & for every Kripke structure a for intuitionism, then

8 f=ij iff a IF fi.

A is a theorem of intuitionistic predicate calculus. Let D- be the deductive system consisting of all the axioms and rules of D except the non-intuitionistic instances of the induction scheme. CE:

For any ontic

A,

if

kD-5,

then & is a theorem of

intuitionistic arithmetic. Proof:

It is easily verified that all the axioms and rules of D-

are satisfied by every Kripke structure for intuitionistic arithmetic.

The corollary follows from L5 and the appropriate

counterpart to L6. 6.3.

There are some interesting theorems of D which relate the

intuitionistic connectives and quantifiers to their classical counterparts. The proofs are straightforward. T?: T8:

T9:

(A&B) -+ (A&B) K(&&g) cf f;+ (A€iE)

28

S. SHAPIRO

T18:

(T&)

2 (&I

Theorems T9 and T12 indicate that classical conjunctions and classical universal quantifications are "intuitionisticly equivalent" to their intuitionistic counterparts.

That is, if an instance of one of them is

knowable, then the corresponding instance of the other is knowable.

It is

an easily verified corollary of TB that the converses of T7 and T10 do not generally hold. Theorems T13-Tl6 indicate that intuitionistic disjunctions and existential quantifications are at least as strong as their classical counterparts. Again, none of the converses hold (see, for example, CB2 and CB3).

Theorem T17 asserts that if the classical implication &*-is then (it is knowable that) if

A

is knowable, then

g

knowable,

is knowable.

This

suggests that a classical implication is "intuitionisticly no weaker than" the corresponding intuitionistic implication. Notice that it is a corollary of TB that the similar The latter formula amounts to "if knowable." 6.4.

(&+g)+ &+g is

does not hold generally in D. true and

A

is knowable, then

g

is

The converses are taken up in the next subsection. It is proposed that the present language and deductive system

is useful in understanding the differences between intuitionistic formulas and their classical counterparts. Several examples follow. 6.4.1.

In informal discourse it is often remarked that any formula

with an intuitionistic main connective or quantifier is at least as strong as the similar formula with a classical main connective or quantifier. At least in D, however, this is not the case. example, that

It is a corollary of TB, for

(e) 2 @+g), the converse of T17, is not derivable in D.

illustrate this, let

A(x) be ontic and

formula amounts to "if

notice that

/-&(z) ~ZX&(Z).

A(S) is knowable, then there is a number

A(x) - is knowable". It follows from this and The latter formula amounts to A(5) +@(E). true, then there is a number 5 such that

To

This such that

the converse of T17 that the implausible "if

&(x) is knowable".

&(s)is

similar

remarks apply to negation and the converse of T17. It is easily verified that if & is ontic and kg then

CA.

It may be

thought that each intuitionistic formula is no weaker than its classical counterpart. Another corollary of TB, however, indicates that this also is not the case.

Epistemic and Intuitionistic Arithmetic

CB6:

There is an ontic

Proof:

Let

29

such that/&g.

C(5) be a primitive recursive predicate such that Let g be - j V g ( ~ ) . The

V g ( r ) is true but not deducible.

sentence

is equivalent to K(,K(VS(tl)

)

.

Let D' be the

deductive system formed by adding this sentence to D as an axiom. Notice first that the result of erasing all occurrences of K from a theorem of D' is a theorem of the system consisting of Peano arithmetic plus

B.

It follows from the consistency of the latter

system that D' is consistent. Notice also that D' 1 therefore, that D' satisfies the premises of TB.

fD,E+g.

Therefore, by TB,

A

and,

B.

By L3 D'

fortiori,/&g.

It is well-known that the intuitionistic connectives and

6.4.2.

quantifiers are not interdefinable the same way their classical counterparts are. For example, although classically we have l-(-$vg) I-(&+€€)

c+l(A&,B),

++

(&-+El and

neither inference holds in intuitionistic logic.

Although the two schemes

(e) 2 (&+I and

intuitionistic logic, neither converse is.

(%+I

;t

~(32 are)provable

in

The differences are illustrated

The following are simplifications of the translations:

in L and D.

7AVB:

K(-iK(A)) v K(B)

A+B -7(A&7B): -_T_

K(K(&) -+K(E))

:

K(,(K(&)

&

K(7K(B))))

From T6 again, the following equivalences are derivable t+K(K(iK(&))VK(B))

(A) VK

c+ K (TK

)

-, 7(A&7B) ++ K(7K(&)V,K(-jK(B) 4 -

1)

Temporarily ignoring the o u t e m s t K in each formula, then, the implication

e g

amunts to "either A is unknowable or B is knowable".

-

The disjunction

YAvB amounts to the stronger "either the unknowability of 5

& knowable

or

B is knowable". The negated conjunction z(&&A&g) amounts to the weaker "either A is unknowable or it is not knowable that 6.5.

decidable". be known.

g

is not knowable".

Decidability. We list three possible interpretations of (1) &vz&--either

"A is

& is knowable or it is knowable that 5 can't

Because the truth value of &is not directly referred to here,

5 is 5 is true, then 5 is knowable; if & is false, then it is knowable that 5 is

this is called intuitionistic decidability. knowable or it is knowable that false.

A

is false.

(2) K(fi)vK(7A_)--either

(3)

(~K(A))Q(,(,A))--if

30

S. SHAPIRO

Fsrmula ( 2 ) is equivalent to Av~& and, therefore, implies formula (1). The converse does not hold generally.

It follows (from classical excluded

middle) that formulas ( 2 ) and ( 3 ) are equivalent. Markov's principle, which has caused much debate a m n g intuitionists, is the scheme 2%

(~V-I(~)Z 3 2 ( 5 ) ) . It asserts that if A is intuitionisticly decidable and if it is knowable that & does not universally k

fail to hold, then there is a number 5 such that t ( 5 ) is knowable. Informally, the number 5 is found by checking & ( G I ,

&(I),.

..

possible because & is decidable) until one is found that holds.

(which is It is

well-known that Markov's principal is not derivable in, but is consistent with intuitionistic arithmetic. There is a theorem of D which is a somewhat altered (and weakened) version of Markov's principle. k 3 & . & is strongly decidable and if the extension of & is (knowable to be) not empty, then there is a number 5 such that A(&) is

T19:

K(A)vK(@

T19 says that if

knowable. Notice that both the premise and the antecedent of Markov's principle are somewhat strengthened here (thus weakening the principle). First, the premise of decidability is strengthened and, second, under this premise, the antecedent of T19 entails that of Markov's principle. to say, K(A)vK(lE) I6.6.

That is

3=9V&.

Church's Thesis.

There is a formula scheme of intuitionistic

arithmetic that is sometimes thought to be an analogue of Church's thesis: CT:

v&(x,x).+

3 e v d Y "Tl(g,5,@)(x,U(y)) 1

For each intuitionistic formula

A,

on Heyting's interpretation of the

connectives, CT amounts to "if one has a method which given a number fi produces a number fi such that

&(i,i)is proved, then one knows of a is proved." &(&,r(&)

recursive function g such that, for each g,

It is well-known that the deductive system consisting of intuitionistic arithmetic together with all instances of CT is consistent. It follows from the faithfulness of the translation [ 5 1 that all instances of CT can be consistently added to D. Church's thesis itself says that if there is an algorithm which, given

m, produces f such -

such that

&(i,i)is true, then there is a recursive function

that, for each 2, A ( & G )

is true.

Stated this way, the existential

quantifier in "there is a recursive function" is classical

--

Church's

thesis does not entail that a name of the recursive function is known. Moreover, the restriction to intuitionistic formulas is artificial.

The

Epistemic and Intuitionistic Arithmetic

31

following formula scheme is a weaker version of CT which is closer to Church's thesis in these respects:

zlfi(&,x)-t3~Vx311(Tl(e,x.y)&A_(2,U(~)L)))

CT1: For each formula

of L.

1 It is tedious, but not difficult to verify that ICT It follows that 1 no contradiction can be derived in D from CT As above, however, this

.

.

does not entail that it is consistent to add every instance of CT1 to D. The consistency of CT1 with D is proved in R. Flagg's contribution to this volume. 7.

Second and Higher-Order Extensions.

In this section, the language

and deductive system are extended to include second and higher-order variables and quantifiers. We call the extended language L2 and the Several changes in the deductive system are

extended deductive system D2.

indicated by some problems that do not arise in the context of first-order arithmetic.

The new problems are technical variants of those in

epistemology and philosophy of language. 7.1.

Functions

Function Presentations. There are some well-known

difficulties that result when function names occur in epistemic contexts. To pursue an earlier discussion (section 4 . 5 ) , substitutivity of identicals fails.

it is often remarked that

For example, if r a n d

C J

are the same

function and Harry knows that f.is computable, it does not follow that Harry knows that

C J

is computable. The relevant fact here, of course, is

that Harry may not know that

g

and

C J

are the same function. If there are

two function names which denote the same function but which cannot be known

to denote the

same function, then this problem will occur even in the

present context of ideal knowledge.

This proves to be the case with the

extended language and deductive system. To overcome this difficulty, a distinction between number-theoretic functions and what may be called presentations of number-theoretic functions is in order.

For present purposes, a number-theoretic function

is a set of ordered pairs of natural numbers (containing no two pairs with the same first element).

A

presectation of a function is an interpreted

linguistic expression which denotes a function. For example,'Xx(x+l)2 ' and 1

'X5i;d2i+l)

' are two different presentations of the same function.'

In

general, each function has infinitely many presentations. Because functions are infinite abstract objects, human beings, as knowers, have no (epistemic) access to individual functions independent of their presentations. Modifying a proposal made by Frege, I suggest that

32

S. SHAPIRO

function names which occur in epistemic contexts be considered as denoting function presentations, and not functions themselves. To clarify this, consider the following sentence:

(1) Harry is given a function f and asked if he knows whether f is computable. Since Harry is a person, this can only mean: (1') Harry is given a function presentation 4 and asked if he

knows whether the function described by $ is computable. The connection between ( 1 ) and (1') holds simply because of the human dependence on language--no (non-physical) entity can be given to a person except with a verbal or written presentation.

More can be said about

functions, however, because functions are infinite abstract objects.

To

bear out the distinction, we compare the situation of (1) and (l'), which involves functions (and function presentations), to a similar situation involving numbers (and what may be called number presentations).

Consider

the following: (2)

Harry is given a nuntber

and asked if he knows whether

n is prime. Again, since humans have no access to individual numbers independent of language, this sentence must mean: (2') Harry is given a number

the number denoted by

5 and asked if he knows whether is prime.

There is, however, a clear epistemic difference between numerals and function presentations.

Numerals, of course, are standard canonical names

for individual numbers.

This alone, however, does not go to the heart of

the matter.

The point is that numerals are "transparent", in the sense that

when one is given a numeral in a standard notation (that is understood), one knows what number is denoted--at least in the sense that if one were given another numeral (in the same or another understood notation) one can tell (at least in principle) whether the numbers presented are the same or different and, if different, which number is larger. Function presentations, on the other hand, are not like this.

In the first place,

there is no canonical notation for functions and, even if there were, there is no uniform way of comparing the whole of the function presented by one presentation with the whole of the function presented by another (for example, to check if the functions are identical or if one eventually dominates the other)., Even if one begins with two effective presentations, the "comparison" is an infinite process.

Epistemic and Intuitionistic Arithmetic

33

To carry this further, consider the following sentences (la) Harry is given 'kz(~+l)~'and asked if he knows whether the function so described is computable. (lb) Harry is given 'ki&A+l)

and asked if he knows whether

I

the function so described is computable. (2a) Harry is given '5' and asked if he knows whether the number so denoted is prime. (2b) Harry is given 'the number of different Platonic solids' and asked if he knows whether the number so described is prime. The situations described by (la) and (lb) are instances of (1').

We

believe standard usage has both situations as instances of (1) as well. The situation described by (2a) is literally an instance of (2') and, we suggest, an instance of (2).

The situation described by (2b). however, is

not an instance of (2'), because 'the number of different Platonic solids' The situation is also not an instance of (2) (on the

is not a numeral.

usual meaning of that sentence).

In this case, one might say that Harry

is not given a number, but only a description of a number.

Of course, one

can back up and claim that in the situations described by (la) and (lb) Harry is not given a function, but only a description of a function. agree with this.

I

In the case of functions, however, function presentations

are the best one can do. The question for Harry depicted in (2) is, in some sense, independent of the numerals because numerals are canonical and transparent.

Two

different instances of (2') which contain different numbers for the same number (in different understood notations) would constitute the question for Harry.

same

The question depicted in (l), however, is not

independent of the function presentation.

Indeed, the two instances of (1)

depicted in (la) and (lb) represent different mathematical questions even though the function involved in each is the same. them would not do as an answer to the other.

An answer to one of

This is especially true if

Harry does not know the mathematical fact that the two presentations describe the same function. As above, the problem with function names in epistemic contexts Will

be carried over to the context of ideal knowledge if there are different presentations which describe the same function, but which can't be known to describe the same function.

For the present extended language, this

condition is equivalent to the existence of true but unknowable

34

S. SHAPIRO

propositions. Therefore, the first extensions of L and D contain terminology for function presentations. A later section concerns further extensions of L2 and D2 which contain terminology for functions themselves as well as terminology for other higher-order entities. For the expanded language LZ, then, a new second-order sort,

7.2.

called "function presentations", is added to L .

Capital letters from the

middle of the alphabet are used as variables ranging over function presentations.

In this subsection we discuss the changes from the previous

system which accomodate the new terminology. The first subsection 7.2.1 deals with additions to L and new axioms for D. while 7.2.2

deals with

actual changes required by the higher-order terminology. 7.2.1.

Formally, function presentations are, in some ways, like

functions. The first new axiom of D2 is a standard function axiom: F1: w z 3 ! y ( e = y ) . To avoid unnecessary complication, concern here is only with total function presentations. An important aspect of function presentations (unlike functions themselves) is that they are not extensional. That is to say, different presentations can have the same values at the same arguments, or, in other words, different presentations can describe the same function. To avoid a troublesome (and fruitless) technical problem, we do not introduce terminology for the identity of function presentations.

Indeed, there seem

to be no obvious criteria for determining whether two presentations are the "same" or, in other words, whether two presentations describe the same function the same way.

For present purposes, it suffices to introduce an

abbreviation to express extensional equivalence: abbreviation of Vx(Fx=Gx)

.

Ezc is taken as an

For the second addition to D2, an axiom scheme, let

&(z,x)be

any

ontic formula which has no second-order terminology. F2: V53!*(3y)

-+

33e(3pX)

This scheme entails that there is a function presentation corresponding to each "description" of a function. The introduction of an abstraction variable-binding-term-operator is thereby justified. If formula which has no second-order terminology and

&(z,x)is an ontic

xIz free, we

let

[ e ( f , 4 1 ) ] be a function presentation. In unambigious contexts, &(i,x)is abbreviated as & and [*(5,2)] added :

as 151.

The following axiom scheme is

35

Epistemic and Intuitionistic Arithmetic

(&(vpw) &‘Jz-($(~,E)- + E ~ E)v(~=~&V~Z(-&(~,~) ) ))

[ e ( z , xIF)-++

F3:

c

In more graphic language, this is equivalent to the least

rgv =

such that

$(v,w) is true,

if there is such a

w

0 otherwise

The restrictions on

5 in

F2 and F3 are, at least in part, for technical

and conceptual simplicity--it should be clear that there is a determinate

[A].

function described by

Moreover, the restrictions also facilitate the

extension of the “provability“ interpretation of K to the extended language. In section 7.4 the possibility of relaxing these restrictions is considered. In the first-order D, the axiom scheme of the substitutivity of

7.2.2.

identicals (for numbers) is

sr: (z=g+ (&(x)t t A ( y )

.

As noted above, substitutivity of identicals often does not hold in

epistemic contexts.

The problem is that true, but unknown (or unknowable)

identities cannot be inter-substituted in epistemic contexts.

In the

first-order case, however, there is no problem since in that case, there are no true, unknowable identities (at least not in L ) .

In the present

second-order L2, on the other hand, there are true, unknowable (numerical] identities.

Moreover, unrestricted substitutivity of identicals would

A-+K(A) for any first-order ontic formula 5. Indeed, let A be such a is free in A), let B be (z=z) & formula and (assuming that neither x nor (l=i)& A. Consider the following deduction: imply

A

premise

[BI5 =i K(i=i)

from axiom F3

c1. K(

[g]a=i)unrestricted Si

K ( [gG=i) K (5)

from axiom F3

It would seem natural at this point to maintain only those instances of SI in which

A (x)

x does not occur free within the scope of a K-operator in

and to add an “epistemic version“ of SI:

K(x=P)

-+

(A(x)++A(x))

This approach, however, fails because, as noted, VzVx(x=y+K(z=l)) provable in D.

is

This sentence, together with the epistemic S1, implies the

original unrestricted Sl. The present dilemma can be resolved by noting that, for similar reasons, D2 should also contain restrictions on the terms that can be interchanged with variables in epistemic contexts. Unlike the second-order

S.SHAF'IRO

36

variables (for function presentations), the first-order variables in L2 range over numbers and not what may be called "number presentations". A free number variable that occurs within the scope of a K-operator denotes an unspecified number rather than an unspecified number description.

Therefore, I propose that in an application of (first-order) universal instantiation or existential generalization, such a variable can be exchanged with a term t only if the denotation of t is knowable--only if RecaLl that there is a number ; such that it is knowable that t denotes the terms of the first-order L have this property: if s is a closed term

.;

of L, then there is a numeral

such that t-K(s=;).

Although it might seem

(at first glance) a bit drastic, the following quantifier rules are proposed : YE:

vs(s)ki(s) V%(x)fi&(t)

where s is a term of L where t is a term of L2 and 5 does not occur free within the scope of a K-operator

31: &(s)l-3g(x)where s is a term of L

A(t)f-3%(5)

where t is a term of L2 and ~f. does not occur free within the scope of a K-operator

To

L.

A(x) -

illustrate the change, let t be a term of L2 that is not a term of

That is, let t be a term which contains abstraction operators. Let be an ontic formula. The restriction of 31 may prevent

K(A(t) )CggK(&(s)).

Indeed, if A(t) is knowable, it does not follow

(automatically)that there is a number x such that &(s)is knowable, because it may not be knowable which number t denotes. However, we do have 1 . This inference only amounts to "if &(t) is knowable K(A(t))cK(3=(x) then it is knowable that the extension of A(@ is not empty". Similarly,

we do not generally have Vs(&(s))l-K(&(t)). that for each number n,

The formula Vg(&(x))

is knowable. That is, Vg(&(x))

says

entails that

each instance of & is knowable (perhaps separately), not that it is knowable that &holds universally. Therefore, if it is not knowable which number is denoted by t, then it cannot be inferred that A(t) is knowable. We do, however, have Vs(&(x))C&(t).

That is, it does follow from

VgC(A(5)) that A(t) is true.

We also have K(V=(x))kK(&(t)): if it is is knowable. knowable that &holds universally, then A(:) To return to the substitution of identicals, let s and t be two terms

of L2. with the restrictions on the quantifier rules, it is easily seen is directly that an instance of the formula scheme (s=t)+(A(s)*A(t))

Epistemic and Intuitionistic Arithmetic

derivable from SI only if either free in

s

and t are terms of L or 5 does not occur

&(x) within the scope of a K-operator.

restrictive.

31

As noted, this may be too

Intuitively, if s and t are known to denote the same number,

then (even if it is not knowable which number that is) they should be substitutable.

All is well, however.

It is straightforward (but tedious)

to see that SI and the quantifier rules imply an epistemic substitutivity scheme: KSI:

K(s=t)

3

(A(s) MA(t))

is a theorem for any (numerical) terms s,t and any formula

A

(provided that

no variables free in s or t occur in &I. It can also be seen that the above restrictions on the quantifier rules are not too harsh.

If t is a term of L2 (not containing 5 free),

then the formula 3lf(t=x) amounts to "there is a number x whose identify with t is knowable" (especially in light of the extension of CB3).

This is

the above informal condition for substitution (and corresponds to a similar requirement in [71). As might be expected, with KSI, VE, and31, the following are derivable in D2: Sg(t=x)

I

&(t) t 3 * ( y

t

axI

is the

iff

i s e q u i v a l e n t t o BX 2 X being provable i n

t h e p r o v a b i l i t y of L i n Qrr

weakly obeys GBdel's second theorem i f f t h e p r o v a b i l i t y

of ^ B 4 i

Y, . ,

x

0.

Therefore

weakly obeys GBdel's second theorem i f f t h e p r o v a b i l i t y i n ff of

BX 3 X implies t h e p r o v a b i l i t y i n ff of X--in

o t h e r words i f f X i s weakly

LBbian f o r W , B > .

Discussion

-

P a r t ( a ) of Theorem 1 i s e s s e n t i a l l y Kripke's argument:

Taking Peano Arithmetic f o r ff, t h e f a c t i s t h a t a l l e x t e n s i o n s of P.A.

do

weakly ( i n f a c t , s t r o n g l y ) obey GBdel's second theorem, and t h e r e f o r e f o r any sentence X , t h e p a i r theorem.

does weakly obey GBdel's second

Therefore every X is weakly LBbian f o r P.A.,

s o LBb's theorem

holds f o r P.A. A s i n d i c a t e d i n t h e i n t r o d u c t i o n , one motivation of t h i s paper was

t h e attempt t o show t h a t t h e above theorem holds i f we r e p l a c e "weakly" by "strongly. "

1.2.

S t r o n g Properties.

The next theorem is b a s i c .

Same Principles Related to Ub's Theorem

Theorem 2

- For

219

any acceptable p a i r d , B > and f o r any sentence X ; ( a ) Bx

i s a p r o v a b i l i t y f u n c t i o n f o r ff. (b) BX i s a p r o v a b i l i t y f u n c t i o n f o r

Ox.

To prove Theorem 2 , we f i r s t l i s t some p r o p e r t i e s of d , B > which a r e

w e l l known i n t h e c o n t e x t of t h e modal system K 4 . provable i n

0,and w e w r i t e

By "provable" we mean

F X t o mean t h a t X i s provable i n ff. Then f o r

any sentences X,Y:

F B ( X 3 (Y 2 2 ) ) 3 (BX 3 (BY 3 BZ))

L1:

I f X 3 Y i s provable i n ff, so i s BX 3 BY

L2:

I f BX 13 Y i s provable i n

L3:

ff, so

i s BX 3 BY

F B ( X 3 Y ) 2 B(BX 3 B Y )

L4:

L is proved by two a p p l i c a t i o n s of condition B2 of t h e d e f i n i t i o n o f 1 a p r o v a b i l i t y function. A s f o r L a , i f X 3 Y i s provable, so i s B(X 3 Y ) ,

and hence so i s Bx 3 BY.

A s f o r L3, i f BX 3 Y i s provable,

so i s

B ( B X 3 Y ) , hence so i s BBX 2 BY, but a l s o BX 3 BBX i s provable, and hence

( s i n c e ff i s t a u t o l o g i c a l l y complete) so i s BX

B(X 3 Y ) i s provable, so i s BX Proof of Theorem 2

- We

3

3

BY.

A s f o r L4,

if

BY, and hence so i s B(BX 3 BY).

f i r s t prove ( a ) .

We must show f o r any sentences

X,Y,Z: (1) I f Y is provable i n ff, so is BX Y ; ( 2 ) Bx(Y 3 Z ) 3 (BxY 3 B x Z ) i s provable i n ff; ( 3 ) B Y 3 B g Y i s provable i n X X R e (1), Y 3 (X 3 Y ) i s a t a u t o l o g y , hence provable ( i n Hence

a;

BY

3 B(X 3 y)

BY.

i s provable (by L 2 ) .

a).

Now suppose Y i s provable.

Hence (by modus ponens) so i s B(X

3 Y)--i.e.,

Then so i s

so i s B Y. X

R e ( 2 ) : The sentence (X 3 (Y 3 Z ) ) 3 ( (X 3 Y ) 3 (X 2 Z ) ) i s a t a u t o -

logy, hence i s provable i n 0.

(B(X 3 Y ) 3 B ( X 3 2 ) ) - - i . e . ,

Then, by L1,

s o i s B ( X 3 ( Y 3 2 )) 3

Bx(Y 3 Z) 3 ( B Y 3 BxZ) i s provable i n X

ax.

Re ( 3 ) : B(X 3 Y) 3 ( X I B ( x 3 Y ) ) i s a tautology, hence provable i n

0. Then by

L ~ B(X , 3 Y ) 3 B(X 3 B(X 3 Y )1 i s provable i n

a, i . e . ,

B ~ 2 Y

BXBXY is provable i n ff. This proves t h a t Bx i s a p r o v a b i l i t y f u n c t i o n f o r

a. (b)

As f o r

ax,w e

a b l e i n ff i s provable i n

f i r s t n o t e t h e t r i v i a l f a c t t h a t anything prov-

ffx (because

Re (1) Suppose Y i s provable i n

i f Y i s provable i n ff, so i s X 3 Y).

ax.

Then X 3 y i s provable i n ff.

Then B ( X 3 Y) i s provable i n ff, which means t h a t BxY i s provable i n hence i n

Ox.

a,and

220

R.M. SMULLYAN

Re (2),

(3):

Since B (Y 3 Z) 2 (BxY 3 B x Z ) and Bx Y 3 B2 X

provable i n ff, they a r e provable i n

ax.

xY

a r e both

This completes t h e proof of

Theorem 2 . For any sentence X I w e temporarily l e t X* be t h e sentence B

W e now know t h a t Be

B-A.

say t h a t < f f , B

^x

M4M1

ff-.

is a p r o v a b i l i t y f u n c t i o n f o r a a n d f o r

= To

> s t r o n g l y obeys GUdel's second theorem i s t o say t h a t X* i s

provable i n ff; t o say t h a t

a.

s t r o n g l y obeys GBdel's second theorem i f f X

( b )

(X V X ) i s provable i n

+ 1. s

s t r o n g l y obeys GOdel's second theorem i f f

a.

By a GOdel sentence f o r < f f , B > w e mean a sentence S such t h a t t h e sen-

tence S

= wB(S)

i s provable i n ff.W e can a l s o speak of a GBdel sentence

f o r d , B >, meaning a sentence S such t h a t S X

.-B

X

(S) i s provable i n 0, o r

of a GUdel sentence f o r d x , B x > , meaning a sentence S such t h a t S

= hBX(S)

i s provable i n Ox. We s h a l l s t a t e t h e n e x t theorem without proof ( s i n c e it i s w e l l known f o r f i r s t - o r d e r

t h e o r i e s , and a l s o follows from t h e well known f a c t

e) 3 (0- i 3 01) i s provable

t h a t t h e modal sentence o ( p

Theorem 4 [After GOdel]

- For

c o n d i t i o n s a r e equivalent:

i n K4).

any acceptable p a i r d , B > t h e following two

(1) There is a GBdel sentence f o r < a , B > ;

(2)

d , B > s t r o n g l y obeys GOdel's second theorem.

W e now apply Theorem 4 t o t h e p a i r d , B - x . X > , Theorem 5

-

and g e t

A s u f f i c i e n t (and a l s o necessary) condition f o r X t o be strong-

l y LObian f o r < a , B >

i s t h a t t h e r e i s a sentence S such t h a t S z d ( S V X )

is provable i n ff. Proof

- The

tence S

sentence S

2 -B_,(S).

- B ( S V X ) i s provably e q u i v a l e n t i n

Therefore t h e p r o v a b i l i t y of S

e q u i v a l e n t t o S being a GOdel sentence f o r d , B x > .

= H B (S V

ff t o t h e sen-

X ) i n ff is

But by Theorem 4

Some Principles Related to Lab’s Theorem (applied t o t h e p a i r ) t h e r e i s a GOdel sentence f o r i f and

s t r o n g l y obeys GOdel’s second theorem, which i n t u r n i s provable i n

a

(by ( a ) of Theorem 3 ) .

of a sentence S such t h a t S

Therefore t h e e x i s t e n c e

4 ( S V X ) i s provable i n

a is

equivalent t o

X being s t r o n g l y LObian f o r ff.

W e have now proved (3) of Theorem A.

We have not y e t proved t h a t

Theorem 1 holds, r e p l a c i n g “weakly” by “ s t r o n g l y “ , but we w i l l .

I t is per-

haps e a s i e s t t o f i r s t do t h i s i n t h e c o n t e x t of t h e modal s y s t e m K 4 , t o which we now t u r n .

2.

Some P r o p e r t i e s of K4.

We formulate t h e modal system K4 a s i n

[l], except t h a t i n s t e a d of having a r u l e of s u b s t i t u t i o n , w e take t h e axioms a s axiom schemata.

S p e c i f i c a l l y we t a k e t h e following axiom sche-

mata :

A ~ :

A2: A3:

A l l tautologies A l l sentences C ( X 3 Y ) 2 (RX 3 O Y )

All sentences C X 2 O C X

The only i n f e r e n c e r u l e s are modus ponens and n e c e s s i t a t i o n (from X t o infer O X ) . W e t a k e one p a r t i c u l a r p r o p o s i t i o n a l v a r i a b l e q.

tence X we i n d u c t i v e l y d e f i n e t h e q - t r a n s l a t e of X

F o r any modal sen-

- which we

w r i t e q{X}--

by t h e following r u l e s :

[For example, i f X i s t h e formula o p a C ( O r o(q 2 p ) 3 o(q 2 ( o ( q 2 r ) 2 I)1.1 L e t us define

c

X t o be O ( q 3 X ) 4

.

3 I),

then q { X ) i s t h e formula

Then (roughly speaking) q { X } i s

obtained from X by r e p l a c i n g each subformula CY by 0 Y. Q The t r a n s l a t i o n theorem--which we a r e about t o prove--is

that for

any X provable i n K4, i t s q - t r a n s l a t e q { X } i s a l s o provable i n K4. Let O b e t h e axiom systen: K4 and B t h e f u n c t i o n which a s s i g n s t o each modal sentence X t h e sentence O X .

Since B i s a p r o v a b i l i t y f u n c t i o n

222

R.M. SMULLYAN

f o r K4, then Theorem 2 a p p l i e s t o ) we shall mean a function to each propositional variable p a sentence d(p) of of a modal sentence X under

a.

6 which assigns

The translation X d

is defined inductively by the rules:

(2) pd = z(p) (for each propositional variable p);

' I

= I;

xd

B 2Y ;

(ox)+= B ( Xz ).

(4)

(3)

(1)

( X ='YYld

=

BY a translation of x (in ) we mean a

translation under some realization d.

The following fact can be proved as

in Boolos (Th. 1, Ch. 3 ) .

-

Lemma 2

For any acceptable pair

of K4 are provable in

, all

translations of all theorems

a.

Now, by Theorem 8 , the formula -U(p V q ) is a K-formula, which means d

O(q V ( p

s, 2

x

in

a, we

+

is provable in K4. Then given any sentences take any realization which maps p to s and q to X, and Lemma p V g)) ) 3 g

gives:

-+ For any sentences S I X of G? (a) B ( X v (S S - B ( S V X ) ) ) =' X is -+ provable in (b) If x V (S 4 (S V X ) ) is provable in so is X Lemma

3

-

a,

a,

.

226

R.M. SMULLYAN We note t h a t ( b ) follows from ( a ) , because i f X V (S

provable i n

ff, so i s

B (X

v

tence f o r d W x I B

-X

Lemma 4 in

-

(5') i s provable i n

-B

a,,

I f t h e r e i s a GMdel sentence f o r !

is

-B (S v X ) ) 1.

(S

Now, X V (S f -B(S V X ) ) i s provable i n ff i f f -X is provable i n ff, i f f S

-B (S V X ) )

>, then X + i s provable

Now we e a s i l y prove:

Theorem 9

-

s t r o n g l y obeys G8del's second theorem i f and only i f

X i s s t r o n g l y LMbian f o r q , B > .

Proof

-

( a ) I f 'ff-xlB-x>

s t r o n g l y obeys G8del's second theorem, then by

Theorem 4 t h e r e i s a GMdel sentence f o r 'a-xlB-x>I

is s t r o n g l y LMbian f o r - d , B > . X+ i s provable i n

a,

so is X V

and hence by Lemma 4 X

( b ) The converse i s r e l a t i v e l y t r i v i a l :

+ X .

If

Then by ( b ) of Theorem 3 ,

s t r o n g l y obeys GBdel's second theorem. The above theorem i s t h e "strong" v e r s i o n of Theorem 1, and h a s t h e following c o r o l l a r i e s .

Corollary 1 d

-

I f d-xlB-x>

-

I f t h e r e i s a GMdel sentence f o r

s t r o n g l y obeys GMdel's second theorem, so does

r BMX' *

Corollary 2

then t h e r e is a

GMdel sentence f o r q7,BWx>.

Corollary 3

-

If

x

V

x+

is provable i n ff, so i s.'X

Corollary 1 follows, because i f d w x I B cond theorem, then X'

WX

> s t r o n g l y obeys GMdel's se-

i s provable i n ff, which by (a) of Theorem 3 implies Corollary 2 follows

t h a t d , B m X > s t r o n g l y obeys GMdel's second theorem. from Corollary 1 by Theorem 4 . provable i n

a.

second theorem. theorem.

As t o Corollary 3 , suppose (X V X+) i s

Then by (b) of Theorem 3 , -d ,B

-x w >

Then by Corollary 1, g,B,,>

Then by (a) of Theorem 3 , X

+

s t r o n g l y obeys G6del's

s t r o n g l y obeys GMdel's second

i s provable i n

a.

Corollary 3 could have been a l t e r n a t e l y proved from t h e following f a c t about K4:

Some Principles Related to Wb's Theorem T h e o r e m 10 Proof

-

The sentence O ( q V q

- The formula q 3 (04

3

+)

3 q+

227

is provable in K4.

q ) is a tautology, hence O ( q 2 (Oq

3

4 ) ) is

provable in K4, and hence so is Oq I> C(oq 3 q ) . From this it easily folU q , and hence to uq

lows that q+ is provably equivalent to O(0q I> q )

n(cq 3 q ) . From this it follows that ( q V 4') is provably equivalent to q V (nq 5 c(0q 3 q ) ) , and hence that the following is provable in K4: (1) c(q v 4+) 2 0 ( q v (0q

5 O(C4 3

Now, by Theorem 8, C(p 4 ) ) ) 3 q+ is provable in K4.

ing is provable in K4:

3 q)

9))).

is a K-formula--i.e.,

O(q V

(p

=

C(p 2

Then, substituting "tq" for " p " , the follow-

(2) C ( q V (Cq 3 U ( 0 q 3 ql

) ) 3 4.'

Then by (1), ( 2 ) and syllogism, the formula

C(q

v

4')

+ IS .

prov-

3 q

able in K4. Remarks

- From Theorem 10 and Lemma 2 it follows that for any

sen-

. provable in a. Then if x of a, the sentence B(x V x+) 3 X + is (x V x+) is provable in a, so is B(X V X+), and hence by modus ponens, s o is x+. This gives an alternative proof of Corollary 3 of Theorem 9. And

tence

I might mention that this corollary with Theorem 3 immediately gives Theorem 9 , so we could have given an alternative proof of Theorem 9 along these lines. More LL5bian P r i n c i p l e s for K4.

4.

The following two theorems

will be useful in obtaining more L-formulas and G-formulas. T h e o r e m 11

-

(a) If A(p,q) is an L-formula,

so

is 4(-p,q).

(b) If @(p)

is a G-formula, so is -@ ( - p ) .

-

Proof

(a) suppose A(p,q) is an L-formula--i.e.,

provable in K4.

means thatNA(-p,q)

ciple.

2

Substituting -p for p , the formula O(-p

is provable in K4, hence C(p

Remarks

O(p

E

,4(-prq)) 3 q

is an L-formula.

+ 1. s

+ .

A(p,q)) 3 q E A(-p,q))

-

3q

+

provable in K4, which

(b) Proof is similar to (a).

- We are tempted to call the above theorem a

"duality" prin-

Part (a) holds for K-formulas as well as L-formulas.

T h e o r e m 12

1s

If A(p,q) is an L-formula, then A ( p , i ) is a G-formula.

228

R.M. SMULLYAN

+

Proof

- Suppose U ( p = A ( p , q ) )

U(p

A ( p , l ) ) 3 I+ is provable in K4, which means that A ( p , i ) is a G-

I

=I q

is provable in K4.

Taking I for q ,

formula. Remarks

-

Theorem 1 2 is the basis of Kreisel's observation that

Gadel's second theorem can be looked at as a special case of Lob's theorem. In particular, since up G-formula.

Also, C ( p

is a G-formula.

is an L-formula. C p

3 q

3 q)

I--which is + o p - - i s

a

(Alternatively, since -qp is a G-formula, then by Theorem

is a G-formula, hence so is

11,

3

is an L-formula, hence C ( p I3 i)--which ism+--

4p.)

Our next theorem is somewhat in the character of a lemma. Theorem 1 3

-

(C(cq 3 q ) 3

Proof

-

-

in K4:

Then C ( p

5

Substituting Oq for q , the sentence C ( p But (04)'

A ( p , C q ) ) 3 9'

3 q+

+ .

A(p,q)) 2 q

is prov-

A ( p , . @ ) ) 3 (04)'

is provable in K4 by Theorem 13.

is

Therefore

is provable in K4, which means that A ( p , O q ) is an L-

formula. Since U p 3 q and -(D-p Theorem 15

3

If A ( p , q ) is an L-formula, So is A ( p 5 q ) .

Suppose A ( p , q ) is an L-formula.

able in K4.

9

(C(cLq 2 C q ) 3cnq)

nq) is provable in K4.

provable in K4. O(p

is provable in K4--i.e.,

- The following formulas are successively provable

Theorem 14 Proof

( C q ) + 3 q+

3

q ) are L-formulas, Theorem 14 gives

- U p 3 oq and -[nr, 3 w) are L-formulas.

By Theorem 15 and Theorem 12,

Some PrinciplesRelated to ulb's Theorem Theorem 16

- O p 3 OL and -(o.lp

3

229

CL)are G-formulas.

Then by Theorem 16 and Theorem 7 , we have: Theorem 17

- O(p V

q ) 3 Dq

and - ( O ( p

3

q ) 3 t l q ) are L-formulas--in

f a c t K-

formulas. This gives a l l the L-formulas needed for the proof of Theorem A stated i n the introduction.

R.M. SMULLYAN

230

REFERENCES

[l]

The Unprovability of Consistency, Cambridge University

Boolos, G.

Press, 1979. [2]

Jeroslow, R.G.,

"Redundancies in the Hilbert-Bernays Derivability

Conditions for Gddel's Second Incompleteness Theorem," Journal of Symbolic Logic 38 (1973), 359-67. [3]

Ldb, M.H.,

"Solution of a Problem of Leon Henkin," Journal of Sym-

bolic L o g i c 141

Smullyan, R.

20 (1955), pp. 115-118. Theory of Formal Systems, Annals of Mathematics

Studies #47, Princeton University Press, 1959. [5]

Smullyan, R.

"Modality and Self-Reference," this volume.