Logic Colloquium: symposium on logic held at Boston, 1972-73

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 453 Logic Colloquium Symposium on Logic Held at Boston, 1...

Author: R. Parikh

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

453 Logic Colloquium Symposium on Logic Held at Boston, 1972-73

Edited by R. Parikh II

III

Inl

Springer-Verlag Berlin. Heidelberg • New York 1975

Prof. Dr. Rohit Parikh Department of Mathematics Boston University College of Liberal Arts Boston Massachusetts 02215/USA

Library of Congress Cataloging in Publication D a t a

Main entry under title: Logic CoLloquium. (Lecture notes in mathematics ; 453) Based on talks at the Boston T~gic Colloquium in

1972-73. Includes bibliographies and index. i. Logic, Symbolic and mathematical--Congresses. I. Parik~ Rohi~ 1936Iio Logic Colloquium, Boston~ 1972.1973. III. Series: Lecture notes in mathematics (Berlin) ~ 453. QA3.L28 no. 453 [QAg] 510'.8s [511'.3] 75-11528

AMS Subject Classifications (1970): 02 B 99, 02 C 1O, 02 C 99, 02 D 99, 02F27, 0 2 G 0 5 ISBN 3-540-07155-5 Springer-Verlag Berlin • Heidelberg. New York ISBN 0-387-07155-5 Springer-Verlag New York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin. Heidelberg 1975. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

PREFACE

The papers in this volume originated as talks given at the Boston Logic Colloquium during the year 1972-73. However, some of them contain more recent developments not originally included in the talks.

They are a l l technical papers

in mathematical logic but with a strong foundational interest.

Thus they should

also be of interest to philosophers. Eleven talks were actually given at the colloquium.

However, the talks

by Abraham Robinson and Alexander Yessenin-Volpin were expository and the one by Ivor Grattan-Guinness was historical.

The eleventh talk, by John Myhill,

on constructive set theory, w i l l appear in the Journal of Symbolic Logic. Funds for the colloquium were provided by the graduate school of Boston University through the Boston Colloquium for the Philosophy of Science. The colloquium was a joint e f f o r t by many and I am grateful to the following for their help.

George Berry, Robert Cohen, David Ellerman, James

Geiser, llona Webb, Judson Webb, and Marx Wartofsky.

August 12, 1974

Rohit Parikh

CONTENTS

1.

J. N. CROSSLEYand ANIL NERODE Combinatorial Functors

2.

H. FRIEDMAN Equality Between Functionals

3.

G. KREISEL, G.E. MINTS and S.G. SIMPSON The Use of Abstract Language in Elementary Metamathematics: Some Pedagogic Examples

4.

7.

132

W.V. QUINE

The Variable 6,

38

A. MEYER Weak Monadic Second Order Theory of Successor Is Not Elementary-Recursive

5.

22

155

R. SUSZKO Abolition of the Fregean Axiom

169

A Realizability Interpretation of the Theory of Species

240

W. TAIT

COMBINATORIAL

J.N.

FUNCTORS

Crossle~ and Anil Nerode

Algebra and model theory deal with properties isomorphism.

We first of all develop the theory of a new sort of con-

tinuous functor and then effectivize veloped greatly facilitate effective

preserved under

everything.

The techniques

the study of properties

preserved u ~ e r

isomorphisms.

The work outlined here will appear in a much more extensive ment in our forthcoming monograph sents the latest developments and recursive principally others

de-

equivalence

[Crossley & Nerode 1974'] •~

in the theory of combinatorial

types which has been extensively

by Dekker, Myhill,

(see the bibliography

Ellentuck,

of Crossley

all these for giving us a foundation acknowledge

financial

Monash University

It reprefunctors

developed

Nerode but also by many [1970]).

We are grateful to

on Which to build.

support from Cornell University,

and The National Science Foundation

We are also grateful

treat-

We also U.C.L.A.,

from 1966 on.

to Liz Wachs and Bill Gross for many improve-

ments to the original version. Introductions

to the special

cases of recursive

types of sets are most readily provided by Dekker's book

[1966] and Dekker and Myhill's pioneering

the set-theoretic

approach see Ellentuck

to 1969 is given in Crossley Peter Aczel's

equivalence useful little

monograph

[1965].

[1960].

A general survey

[1970].

regrettably unpublished

dissertation

[1966] was

the first hint in print that a ftu~ctorial point of view is profitable in RET's.

* The book has already

appeared

For

i.

Categories For simplicity we consider a fixed but arbitrary

which arises

in the following way.

-~

= (U,fi)ic !

where each

to

U

n

for some

fi

depending

set of) the natural numbers structures

E.

The objects

of

and the morphisms

preserving

where

ordered pairs, in

p,

Po =

in

(Do'fo'Co)

Po A Pl =

~

(D o A D I,

We require

@

an inverse

p

If -i

,

(1.4) Inclusions.

~

,~

,...,

and

p

f = graph p.

Po ~ Pl

~ Ob(~)

and ~

is a m o r p h i s m Since-~is

a subalgebra

is a m o r p h i s m

We write

of

of

Po A Pl = Po"

conditions.

(The

[1974~.) ~

is a m o r p h i s m Ob(~)

of ordered

If

iff

and of

p,

as a map,

has

~.

for the objects

is a subset o f ~ t h e n of

of

if, but only if, this triple

to satisfy the following

p

objects

of

are maps then we define

C o A CI)

Similarly

then

are subsystems

the set of second elements

(DI'fI'CI)

-!

to

is a single valued set of

contains

fo A fl'

p

is (a sub-

By a map we mean a triple

is taken from Crossley & Nerode

(l.3)inverses.

~

~

is the set of first elements

Pl =

is a map in our sense.

numbering

of ~ .

and we write

and

U

xnu

later but for the present

are one-one maps b e t w e e n

D,C c E

p

from

approach.)

of

D = dom p

@

structure

(The extension from algebras

denoted b y - ~ ,

C = codom p

ordered pairs

function

and the universe

~,

the structure

p = (D,f,C)

pairs

i

is clear and will be assumed

we p r e f e r a p u r e l y algebraic

U

There is a (countable)

is a (partial)

on

category

of

C.

If

the inclusion map

C.

an algebra the intersection

and similarly for monomorphisms.

of subalgebras We require

is again

3

(1.5) Intersection.

If a set of morphisms

(as a set of maps) then that intersection (1.6) Directed unions.

~

is in

has an intersection C.

The union of any set of morphisms

directed under inclusion

is a morphism

This is a variation union of algebras

of

of

of

C,

C.

on the familiar fact that the directed

(of a given sort)

is an algebra.

As usual in

category theory we identify objects with the identity morphisms

on

them. Any object O Z i n ambiguously

is the universe

of a subalgebra.

We

write D'L for this subalgebra.

(1.7) Subcategory. full subcategory

The finitely generated

6-6 ~ dom p

objects

in

{

form a

o@.

(1.8) Restriction.

of

@

If

p

is a morphism

then the bijection

q ~ p

of

{, ~Ze 0b(C)

and

with domain grtis a morphism

~. We write

p]0t

for this morphism.

Note that if g~ge o~

and

p e ~

One of our main techniques from finitely generated So we require

plDZ

e o~.

will involve extending functors

subalgebras

our objects

then

to infinitely

and morphisms

generated

to satisfy

ones.

the next

condition. (1.9) Approximation. p'

of

union

o{

with

If

p~ ~ p

p

is a morphism of

form a directed

set

~

then the morphisms

(under inclusion)

with

p. (1.9) allows us to approximate

The next condition,

"large"

objects

when used in conjunction with

or morphisms.

(1.6)

allows

us to build up large objects morphisms in

~

p,q,..,

such that

is compatible

~)

If

p,q

a least morphism

moreover,

in

@

We say that a set of

if there is a morphism

r

p c r, q c r,...

(i. i0) Compatibility. (under

(and morphisms).

are morphisms r

in

~

of

o~

such that

then there is

p,q ~ r

and,

r e o~.

Induction

immediately yields a generalization

of (i.i0) to any

finite set of morphisms. By

(1.6) and (i.i0)

if

[Pi: i ~ I]

morphisms

~hen there is a least morphism

We write

q =~Pi

: i e I]

erated in the sense that for each and

A,B

are sets of generators

any morphism containing of

q ~ Pi

and similarly

Finally we note that the morphisms

the map

is any compatible

in

r ¢ o~

for all

set of

i e I.

for objects. o~ if

are finitely genr = (J

of the subalgebras

,f, ~ )

~

(A, flA, f(A) U B)

,~

then

is an extension

r. A category

appropriate

of suba!gebras

of the above type is said to be an

category.

It is useful to mention here that if priate then so is (n+l)-tuples performed

@o ×'''× @n

(po,...,pn)

co-ordinatewise.

where

~o,...,~n

whose objects p~ e ~i

I.i:

The category

(morphisms)

We shall not draw attention

$ (sets).

natural numbers and the morphisms

are

and all operations

in the theory below but shall use it implicitly Exa~le

are appro-

are

to this fact

in the examples.

The objects

are sets of

one-maps between sets of natural

numbers, o$ is the full subcategory whose objects are finite set of natural numbers. E~ample 1.2. The e~tegory L (linear orderings). Objects are sets of ~ationals, morphisms are one-one order preserving maps, o~ is the full subcategory

of

finite

sets.

Example. 1.3.

The category

those subsets

of a fixed countably

over a fixed field

K

V

Example

between

of finite dimensional 1.4.

The category

those subsets

infinite

2.

~

are vector

is the full sub-

atomless

are Boolean monomorphisms of finite B o o l e a n

The objects are

Boolean algebra which operations

between

(subalgebras).

objects and

o~

is

subalgebras.

Functors.

appropriate

the case.

'functor'

A functor

morphism

that all categories

p e C

and~ moreover,

a morphism if

condition means

in a more concrete way than is often

F: C ~ ~'

p:OZ~

: F(Ot) ~ F ( ~ )

~

~

is a function which assigns e ~'

such that

is the inclusion morphism.

F(p)

is an identity

if

p

of the continuous

topologize

~.

~

= {q ~ ~

functors

which are a

topology.

We

: pc_q}.

on

for all

By the compatibility

topology.

(Note that this

is.)

ones in the following

The weak topology

P ~ o@

then

Set u(p)

p e o~.

to each

F(pq) = F(p)F(q)

is an inclusion morphism,

We shall be concerned with combinatorial subclass

considered are

in the sense of section I.

We use the word

for

°V

(Boolean algebras).

From now on we shall assume

F(p)

The morphisms

objects.

are n o n - e m p t y and closed under the B o o l e a n

the full s u b c a t e g o r y

are

subspaces.

of a fixed countable

The morphisms

The objects

dimensional vector space

which are subspaces°

space linear transformations category

(vector spaces).

is the smallest

topology with

condition

(I.i0)

U(p) the

open U(p)

together with the empty set form a base for the weak We call this base the standard base.

D e f i n i t i o n 2.1. if

(i) F

F : ~ ~ ~'

is c o n t i n u o u s on morphisms,

F(p) = U F(q) ~,

A functor

c F(~)

: q ~ p ¢ q c o~] and for s o m e ~ t h e n

In this case we w r i t e 0 ~ : T h e o r e m 2.2.

is said to be c o m b i n a t o r i a l that is

(ii) for a n y ~ '

there e x i s t s ~ c

o~

c °C',

if

such that for all

F~(~').

A functor F : { ~ ~' c o n t i n u o u s on m o r p h i s m s la

c o m b i n a t o r i a l iff the inverse image of a standard base

set in Ob C' is

a standard base set in 0b C. C o r o l l a r y 2.3.

C o m b i n a t o r i a l functors are continuous,

preserve

inclusions and i n t e r s e c t i o n s and are closed u n d e r composition. Note that to e s t a b l i s h closure u n d e r c o m p o s i t i o n in the m a n y variables

case the c o m p a t i b i l i t y c o n d i t i o n

(i. i0) is required.

The next t h e o r e m is m o s t useful. T h e o r e m 2.4. is unique

Let F : o~ ~ ~, be a c o m b i n a t o r i a l functor then there

e x t e n s i o n G : @ ÷ ~' of

F

which is a c o m b i n a t o r i a l

functor. The e x t e n s i o n

G

is g i v e n by

a(~) = U{F(~)

: ~

2 ~

~

~ o~]

and s i m i l a r l y for morphisms. The f o l l o w i n g c h a r a c t e r i z a t i o n is also useful. T h e o r e m 2.5.

Suppose

o¢

is closed u n d e r a r b i t r a r y intersections~

Let F : { + ~' be a functor continuous on m o r p h i s m s then c o m b i n a t o r i a l iff T h e o r e m 2.6.

F

Suppose

preserves arbitrary intersections o{

F

is

of objects.

is closed u n d e r a r b i t r a r y intersections.

Then F : @ + {' is c o m b i n a t o r i a l iff

F

is continuous on m o r p h i s m s

S

Vp(f) = lim Spn(f) n..~¢~

= lim lim Sp,~(f) Sn +~ If(Yk) c~10 n-oo k=l le premier terme, not6

v~(f,F),

est appel6 p-variation fine de

est appel6 p-variation grossi6re de explicite de la p-variation fine :

DEFINITION 1.

f

sup

f

F.

f

sur

F ; le second

Nous allons donner ici une expression

v;(f,F).

Pour toute fonction

appelle p-variation fine de

f(x k) lp

sup

A

£ : T -~IR et toute pattie

A

de Ii~, on

le nombre n-1

(12)

v;(f,A) = lim

sup

7"~.(lffti+I) - f(ti) [P Aa) ,

~10 scAf)T s fini oh_h s = (t I, t 2 . . . .

, tn).

i=l

E n p a r t i c u l i e r on appelle p - v a r i a t i o n fine de

(13)

v;(£)

f

le nombre

= v;(f,T).

Le th6or6me suivant donne deux autres expressions de la p-variation fine.

THEOREME 6.

Soit une fonction

c anonique. Pour toute pattie

A

de

T,

f C 55p(T),

dont

Exk , yk ]

(kCN)

f' o g

est la factorisation

d6signant les intervalles

contigus ~ g(A), n

(t4)

v ; ( f , A ) = scAinf ~.= Vp(f, ] t i ,

(15)

v;(f,A} +

If'(yk) - f ' ( x k ) [ P = inf =

avec

s=(tl,

t2 . . . .

, tn),

ti+ 1 [C1A)

to = i n f A ,

s

cA

tn+l = s u p A "

Vp(f , ~ti,ti+l~(~A) ,

8

If x s domep x e fo(E) @p(x)

then

then

x e fo(E)

x = fo(Yo)

or

x e fl(E) but not both.

for some n a t u r a l number Yo"

= @p(fo(Yo) ) = fo(Po(Yo) )

If

Set

and similarly if x e fl(E)

and

x = fl(y ) set

~(x) Clearly,

when

morphism

=®P(fl(Yl)) p

= fl(Pl(Yl))"

is an i n c l u s i o n m o r p h i s m

is an inclusion

A ~ B, ~p

@A ~ ®B.

We verify that

@

Suppose p,q are morphisms

is a functor.

of $ x $ such that codom p = dom q.

Then for i = 0. i codom Pi = d°mqi

and

e(qop)fi(x) iff

iff

:

fi(y)

(qi o pi)(x ) : y

eq(fiPi(X))

iff (qop)i(x) = y

by definition

of

o in $ × S

= fiqiPi(X) = fi(y)

iff

eq(~(fi(x))

: fi(y).

Finally we check that object of

o$

®

is combinatorial.

If

A

is an

set

(®~i) i = {x : fi(x) c A]. (Recall that

(B)i

Then x e @i implies (®~ Ix]) i ~ A i Similarly with continuous Example 2.2.

is the i-th co-ordinate (®~-[x])i ~ i i

B

and ® < - A

e $ × $.)

(i = 0,1) and if

then from the definition Ix] replaced by a set

of

of

B.

@, x e ~(Ao,AI).

Since @ is clearly

on morphisms ® is now combinatorial. Cardinal m u l t i p l i c a t i o n

We define a c o m b i n a t o r i a l lies m u l t i p l i c a t i o n

(cf.

Sierpinski

functor ® : $ x S + $.

of cotmtable

cardinals.

[1958] p.135).

This functor under-

9

Let j : E x E + E be a b i j e c t i o n

(which m a y be asslmmed to be

recursive). For objects

A

of

®A : [j(x,y) This defines morphisms

S x S

set

: x [ A o ¢ y ~ AI].

the functor on objects

p

of

$ × $.

so now we must define it on

Set dom~p = dom Po®domPl ,

codom ® p ~ codom Po ® codom pl,el0 j(x,y) Po(X),

is to be defined when

pl(y ) are defined and then

~p(j(x,y))

= j(po(X),pl(y)).

We leave the reader to check that ® is c o m b i n a t o r i a l

noting only

that (®~A)o

: [x : J y(j(x,y)

~ A)]

and

( o ~ A ) 1 = {y : ~ x(j(x,y) Example

2.3.

~ A)}.

Ordinal addition.

®: ~ x ~ + ~

underlying

We define a c o m b i n a t o r i a l

the addition

: (Q, ~ ( l ~ ' I )

is said

to be a precombinatorial

operator if I) 0 ~ ~ 6~'

implies

2)

p: 0~ -~ ~ '

if some

card G ( ~ ) 3)

= card G ( ~ ' )

U[G(~) : ~ c

G(~) n G(~')

= ~

for

~,~'

is am isomorphism in

c o¢(~),

°C(~)

then

and

°C)

i s a s u b s e t of a b a s i s f o r ~ ' .

A combinatorial functor

F: C ( ~ )

~ ~(~')

strict if there is a precombinatorial operator

is said to be G

inducing

F

in the sense that

F(~)

= o~U[G(Of') : O f ' ~_OT ~ ~ '

~ o~}.

All 3-valued combinatorial operators are strict and

G(~)

= F(O"(') - U ~ : F ( O f ' )

" ~'

~ °~(t'~O '~ ~ '

is the precombinatorial operator inducing Theorem 4.1.

~Of'~

F.

Strict combinatorial functors are closed under

composition. A combinatorial functor finite implies funetor

F(~)

F : ~ ~ ~'

F

finite. where

¢,@'

is said to be finitary if Each finitary strict combinatorial are suitable and

dimension induces a number theoretic function

F # d i m ~ = dim F ( ~ )

for

~

~,@'

have

F#

given by

in

°C

]4 where

dim denotes

dimension

an induced number theoretic {'

combinatorial

function.

in the appropriate function If

category.

Such

is said to be a strict

C = C'

{

to

we call the function

strict

C-combinatorial. Corollary ~.2.

Strict combinatorial

functions

are closed under

composition. n [i]C

Let

denote the number of i-dimensional

an n - d i m e n s i o n a l

object in the category

have dimension).

We omit the subscript

Theorem 4. 3-

(Myhill normal form).

combinatorial

iff there is a function

n

in

C

subobjects

of

(which is assumed to

where there is no ambiguity.

A function

f :E ~ E

c: E ~ E

is strict

such that for all

E

f(n) : ~ c(i)[~]. The function

e

Stirlin~ coefficient

given by Theorem %.3 is said to be the f u n c t i o n for

the Stirling coefficient Example 4.1. Here any f(n)

In

S, [~] = (~) =

= ~ci(~)

and In

c i = Aif(i) ~,

of an n - d i m e n s i o n a l elements

[~]

It is easy to show that

is unique. nl i~ (n-i)~

f: E ~ E* : [0,±I,~2,...]

Example 4.2.

q

function

f.

has a unique

where

(Af)(i)

expansion

= f(i+l) -f(i).

is the number of i-dimensional

space and if the field,

K, of scalars has

then

k(n)+ ['~] -- k(i)+k(n-i)+ where k(n)

= qn _ 1

and k(O)+ = k(O),

subspaees

k(i+l)+ = k(i+l).(k(i)~).

15

Not every strict S-combinatorial function V-combinatorial since the

[~]

f :E ~ E

is strict

increase too rapidly.

The generalizations to several variables are straight-forward

(~l,j ~F~'~l = [~].[~],

etc.)

Example 4.3.

S, x + y = ([).(~) + (~).(~), and

In

are strict combinatorial and so are In

~, x + y

However,

(x+l) y

x.y = (~)(~)

(but not

x y)

and

not strict combinatorial and neither is the identity.

the closure functor

V: S ~ V, of Example 2.5, is strict

combinatorial. Theorem 4.4.

If

F : C I ~ @2

is strict combinatorial then there

exists a strict S-combinatorial functor cI

.....F

S

H

such that

> c2

vl~

/IV2 .........

H

>

S

commutes where

V i (i = 1,2) is the appropriate closure functor.

Corollary 4.5.

[n]@

is S-combinatorial (as a functor of

n)

@

Example. [~]v =

(q s = t~ FV(s)

f

g ~ resular,

if

k s = t .

27

W e n o w w i s h to s h o w that assignments

LEMMA

f ~ if

Val(x~f)

= f(x)

Proof:

Let

f = [g] .

12.

Val((st)~f)

Proof: BV(s')

Let

Val((st)~f) V a l (tj f))

=

13.

14.

Then .

= Ix(g)]

= [g(x)]

.

= [g(x)] = f(x)

.

.

= @ , for all

x E FV(s')

= A([s'(g)],[t'(g)])

N o t e that

ii)

[g] .

~

Let

U FV(t')

.

Then

= A(Val(s,f),

M0 ~ s = t . s u c h that =

k s = s'~ BV(s')

(~xs'(h))(t) =

.

= [(~xs')(g)] =

o

Let

N FV(g(y))

h = ~

(~,xs'(h))(g(x))

.

= ~ • for all

Then

(~xs')(g)

= s ' ( h ) g ( x ) = s'(g)

=

.

[s'(g)] = V a l ( s , ~ [ t ]) .

Val)

is a s t r u c t u r e .

11-13.

Let

s3t

for all s t r u c t u r e s

By T h e o r e m

Val(s,f)

= Val(s~[t])

val(O~xs)~f)

By lemmas

Proof:

f =

[g] ~ w h e r e

=

[t']

.

g

M0 N

s = t

implies

is the i d e n t i t y m a p ~

BV(s') Hence

are equivalent:

iii) M 0 ~ s = t .

s i m p l y s h o w that

k s = s'~ ~ t = t'~ and Is'], V a l ( t ~ f )

The f o l l o w i n g

M ~ M > s = t

i~ we m u s t Let

be terms.

N FV(s')

= BV(t')

Is'] = [t']

~ s = t .

and choose n FV(t')

~ and so

= ~ .

~ s' = t'

.

~ s = t ~ a n d we are done. Let

If }

f(x)

M0-

so that

[(s'(g)t'(g))]

f~t ] =

(Completeness).

i) k s = t

Hence

=

M 0 = ([D@},~A},"

2.

Suppose

s' t'

N FV(g(x))

and

f = [g] ~ for

x •

Val(x,f)

Choose

A(Val(O~xs)~f),[t])

THEOREM

Then

[(s't')(g)]

.

Proof:

s',t'

[g]

= BV(t')

Choose

(~xs'(h))

LEMMA

Then

A(Val(()~xs)~f),[t])

y E FV(s')

Hence

~ for v a r i a b l e s

Write

.

Proof:

=

substitution

= A(Val(s~f),Val(t~f))

f =

N FV(g(x))

is a s t r u c t u r e .

is a r e g u l a r

Ii.

LEMMA

LEMMA

g

M0

M =

({D@},[A

},Vall) , N =

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

is a p a r t i a l

surjective

map

from

D

({E~},{B from onto

}~Val 2) M

E

onto ii)

be s t r u c t u r e s . N

f

just in case (x)

A system i) e a c h

is the u n i q u e

f

element

28

of

E (~'v)

(if it exists)

y ~ Don(f) involve

LEMMA

.

Val.

15.

then

[f}

The following

If

assignment~

Note that

such that

{f}

h

is determined

is a partial

on

f(g(x))

s j where

f (Vall((st)~g))

(Vall(S~g))

Finally~ y E Dom(f)

= h(x)

We must

s

onto

} x

show

does not

(Vall((~xs)~g))

x

of type

o .

f(Vall(X~g))

By induction

U

=

hypothesis~

Hence

= Val 2((st),h)

= Val2((~xs),h)

f (A(Vall((kxs),g)~y))

is an M-

.

of type

= Val2(t~h)

g

~ o

are fixed,

= B(Val 2(s~h),Val 2(t,h)) f

N • and

for variables

of type

M,N~[f

= Val2(s,h ), f ( V a l l ( t , g ) )

we must show o

M

= f (A(Vall(S~g)~Vall(t~g)))

f (A(VaI l(s~g))~Val l(t~g))

, for all

fo ' and this definition

from

3 for variables

= f (g(x)) = h(x) = Val2(x~h)

Now

by

homomorphism

is an N-assignment~

By induction

= B(fcfv(x),f(y))

Lemma does.

f ( V a l l ( S 3 g ) ) = Val2(s~h ) ~ for terms

Proof:

f

f (A(x,y))

.

. To do this~

= B(Val2((Xxs)~h),f(y))

let .

Now

x

f.(A(Val l((~xs)jg),y))

= fT(Vail(s~g

)) = Val 2(s~hf

(y)) = B(Val 2 ( ( ~ x s ) , h ) , f ( y ) )

.

We are done.

LEMMA

16.

Suppose

M ~ s = t

implies

Proof:

Let

there is a partial

N ~ s = t ~ for any terms

{f}

be an N-assignment. variables Val2(t~h)

x

a

u o

B, let

Let

IBI

homomorphism

Proof:

We define

surjective

s~t

{f}

map from

},Val)

g

and assume

so that

TB

onto

onto

N o

Then

of

@ .

M ~ s = t .

h(x) = f ( g ( x ) ) ~

Let for

= f(Vall(t,g))

Hence

=

N ~ s = t .

B .

be a structure

IDOl ~ IBI .

Then there

M .

by induction B

onto

s~t .

of type

be the cardinal

from

M

Val2(s~h ) = f ( V a l l ( S ~ g ) )

M = ({Dff},[A

partial

partial

Then

from

homomorphism,

Choose an M-assignment

~ by Lermna 15j for terms

LEMMA 17. is

be a partial

of type

For sets

homomorphism

DO °

on the type symbol Suppose

f~f

~ .

Let fo

be any

have been defined~

29

surjectively~ f

(x)

to be the unique element of

= A(f Let

a c c o r d i n g to the clauses for being a partial homomorphism.

(x),f(y)), z E D(~)

.

for all

T H E O R E M 3.

y E Dom(f )

Choose

x(y) E f ~ l ( A ( z , f ( y ) ) )

D (~'~)

x E B (@~)

.

Then

f

such that

We must show that so that for all

f

f (x(y)) =

is surjective.

y E D o m ( f )~

(x) = z .

(Extended Completeness).

following are equivalent:

(if it exists)

Define

Let

i) ~ s = t

s~t be terms~ B

The

a n infinite set.

ii) for all structures

M ~ M ~ s = t

iii) T B ~ s = t .

Proof:

By T h e o r e m 2~ it suffices to show that

TB ~ s = t

M0 ~ s = t .

By Lemma 17~ there is a partial h o m o m o r p h i s m from

Lemma 16~

TB ~ s = t

if

LEMMA 18.

The r e l a t i o n

Proof:

then

implies TB

M .

onto

By

M ~ s = t .

~ s = t

is recursive.

This follows from the following k n o w n fact about the typed ~ - c a l c u l u s

(even w i t h r e c u r s i o n operators):

e v e r y term reduces to a unique i r r e d u c i b l e term~

up to changes in bound variables~

no m a t t e r h o w the reductions are performed

Sanchis

[2]~ Tait

COROLLARY.

If

B

[3]~ and B a r e n d r e g t

Let

B

[i] for elaboration).

is infinite then the r e l a t i o n

independent of the size of

TB ~ s = t

19.

be finitej g: B ~ B .

Fix

There are

x

i(g)~

is r e c u r s i v %

and is

B . Define

g

I

= g~ g

that the extended completeness t h e o r e m fails for

L~MA

(see

j(g) > 1

(0~0) .

= gog

k

We will show

B .

such that for

to be a v a r i a b l e of type

k+l

Let

n > i ~ we h a v e

x

1

= x~ x

k+l

g

= xox

n

k

= g

n+j

=

(Xy (x (xky)) ) .

T H E O R E M 4. with

For each n o n e m p t y finite

T B P s = t j such that not

Proof:

For each

g: B ~ B

B ~ there are terms

s~t

of type

(0~0)

> s = t .

define

i(g)~j(g)

as in Lemma 19.

Choose

i

30

i(g)

greater than each g

i

= g

i+j

Hence

N o t e that not f u n c t i o n on

j = H j(g) g

x i = x i+j

TB

~ .

T h e n for each

x

g: B 4 B

m e a n not

M = s # t[f]

~ s = t • consider

M ~ s = t[f])

TB ~ s # t ?

.

£

are w r i t t e n

calculus of the same type.

(with equality)~

~•&•Y .

£ ~ appropriate

~ for structures

Take

M ~ ~

A formula Let x~y

0

~

M ~ formulae ~v

to m e a n

of

£

LEMMA 20.

If

B

LEMMA 21.

If

s•u

variables~

such that

(~y(~xy))

(~y)

= )

0 °

x ~ FV(s) U FV(t)

iS existential~

~

be

# t)

= (0~i>) *~ (YX)(S ¢t)

so

~ where

f • is

For terms

(~x) (s = t) .

(~y(kxx))

s~t ~ let

<s~t>

<s,t> has type

~ where

be the term

0 .

TB ~ 0 ~ I o

have the same type• then

.

there is a n existential

(~x) (s = t) .

.

Thus

f .

the closed t e r m

, so that

B

~

with the same free

with at least two elements.

Note that b y Lemma 20~ T B ~

y ~ FV(s) U FV(t)• y ~ x .

( = ) -- s # t ,

- (Vx)(s

~ i

T B ~ 9 ~ ~ ~ ~ for all

= 0 & (yt) = i) - s ~ t ~ where TB ~

£ ~ and M - a s s i g n m e n t s

for all M - a s s i g n m e n t s

have the same type• t•v

If

Let

of

has at least two elements• then

LEMMA 22.

Proof:

are obtained from the a t o m i c formu-

is called existential if it is of the form

T B ~ (s•t> = -~ (S = U & t = v)

~

~

M ~ ~[f]

be the closed term

~ where

s•t ~ of the typed ~-

are introduced as a b b r e v i a t i o n s in the standard

are d i s t i n c t v a r i a b l e s of type

(~x((xs)t))

£

Specifically•

The Y - q u a n t i f i e r s q u a n t i f y over a g i v e n type onlyo

d e f i n e d in the obvious way. manner.

s = t ~ for terms

The formulae of

~4e

Does the C o r o l l a r y to

for the theory of functionals of finite type over a n o n e m p t y domain. the atomic formulae of

f .

Below~ we give a negative answer.

W e introduce a m a n y - s o r t e d p r e d i c a t e calculus

M ~ ~[f]

T

m a y be i n t e r p r e t e d as the successor

M ~ s = t[f] j for all M - a s s i g n m e n t s

for not

T h e o r e m 3 hold for the r e l a t i o n

lae by u s i n g

we have

We are done.

M ~ s # t

will o f t e n w r i t e

.

To see that not

x i = x i+j • since

T

N o w let

, and set

Hence

TB ~

By Lemma

(Yx)(~y)

TB ~ ( Z z ) ( V x ) ( < ( ( z x ) s ) , ( ( z x ) t ) > z ~ FV(s) U FV(t)• z # x~y .

(Zy)((ys) = 21~

( =

= Hence

TB ~

(Zz)

31

((Xx)

LEMMA 23.

If

=

~,~

(Xx))

-

are existential,

same free variables

as

(Vx)(s

# t)

.

then there is an existential

~ & ~ , such that

TB ~ P ~

p

with the

(~ & 4) ~ for all

B

with at

least two elements.

Proof:

Let

~

be

(Zx)(s = t ) ,

had their bound variable -

((Zx)(s

= t)

&

If

~

LEMMA 24.

free variables

(Zx)(u

changed = v))

to

be

x).

(~x)(u = v ) ~

Then

TB ~

(where

~,~

may have

(~x)(<sju> = )

.

is existential~

as

~

then there is a n existential

( ~ ) (~) , such that

T B ~ p ~ (~x)(~)

p

with the same B

~ for all

with at

least two elements.

Proof:

Let

bound variable

LEMMA

25.

~

be

changed) o

(~y)(s = t) ~ where Then

For each formula

TB ~

~

with the same free variables~

of

y # x ~ (where

~

may have had its

(~z) (S(zO) x (zl)Y = tx(zO) (~I)) "

£ ~ we can effectively

such that

(~x) (Zy) (s = t)

find a n existential

T B ~ ~ " 9 ~ for each

B

with at least

two elements.

Proof: LEMMA 26. formula ~ and

From Lemmas 22~ 23~ 24. There is a one-one

~

of

£ ~ f(~)

5.

B

function

f

such that for each

formula with the same free variables

of Lemmas

For each

one-one reducible Proof:

B 3 the set of sentences

to the relation

since otherwise

£ ~ TB ~ ~

if and only if

from corresponding

22~ 23~ 24.

We can assume that

is infinite),

as

with at least two elements.

This is an effective v e r s i o n of Lemma 25j obtained

effective versions

of

is a n existential

T B ~ ~ ~ ~ ~ for each

Proof:

THEOREM

total recursive

~

of

£

such that

TB ~ ~

, is

TB ~ s # t . B

has at least two elements

[~: T B ~ @}

if and only if not

T B ~ s # t , where

TB ~

f((~~))

is recursive. (~~)

(or for that matter~

Note that for sentences

if and only if not

= (~x)(s = t) .

T B ~ f((~~))

32

2.

The typed ~-calculus

with primitive

recursion.

We will refer to this extension of the typed ~-calculus The R-~-calculus .

has the additional

The variables

variables

of the R-~-calculus

s ~ their types,

sets of bound variables

BV(s)

= {X~n} ~ BV(X~n) = ¢

~ then = BV(s)

(st)

are given by

ii) if

is a term of type

U BV(t)

iii) if

s

s

• ~ FV((st))

is a term of type

0 ~ FV(0)

¢

= ¢

vi) R

(Sl,...,Sn+l)

Let

([D~},{A})

D~

are disjoint.

A system

is of type

ix) A ( V a I ( R

~

~ and their

a term of type

~ t

U FV(t)~

@

BV((st))

a variable

=

of type

@ ~ then

= BV(s)

v) N

is a term of type

((~(0~)),

(~,(0,~)))~

]~Val)

U {y} (0~0)

FV(R ) = ¢,

. It will be convenient

~ for appropriate of

A

denotes

is an R-structure

~ ~ D~

= z~ A ( V a l ( R u ~ f ) ~ y , z ~ k

= 0

that be

the appropriate

A

i)

({D~}~A

iv) Val((st,f)

A(Val((~xs)~f),~)

vii) Val(0~f)

to assume

Xl~...~Xn+ 1

just in case

iii) Val(x~n,f ) = f(x~)

v) for all

vi) D ° = ~

y ~ D(~'(O'~))~z

((SlS2),..o,Sn+l)

A(Xl~..o~Xn+l)

ii) D ° = ~

~f),y~z,O)

=

= Val(s~

viii) A(Val(N~f),k)

})

~ where = k + i

+ I) = A ( y , A ( V a l ( R , f ) ; y ; z , k ) ~ k )

for

6 D ~, k ~ ~ .

Note that if

({D~}~{A})

such that

({D~}~{A}~

structure

({D~}~{A

Obviously~

be

Let

({DO}~{A

= A(Val(s,f)~Val(t~f))

Val)

is an R-structure.

to mean

there is at most one

Val

Thus we m a y refer to the R-

}) .

we m a y view

in section

is a pre-structure~

T

as an R-structure

just as we viewed

T

as a

i.

As in section I~ we write M ~ s = t

(@~)

- {y}, BV((~ys))

= ¢

is a term of type

~ where each occurrence

is a pre-structure

structure

= ¢~ BV(0)

be a pre-structure.

A(A(Xl~X2)~...~Xn~l)

s

~ ~ y

= FV(s)

FV(s)

.

We l e t

the

of the

is a term of type

= FV(s)

is a term of type

iv) 0

= ¢, BV(N)

i) x~ n

is a term of type

(~,~), FV((~ys))

=

~ for each type symbol

their sets of free variables

is a term of type

BV(R )

R

are the same as the variables

(~ys)

FV(N)

0~N ~ and

of the typed L-calculus.

The terms

FV(x~)

symbols

as the R-~-calculus.

Val(s~f)

M ~ s = t[f]

= Val(t~f)

to m e a n

Val(s~f)

~ for all assignments

= Val(t~f) f .

~ and

In this section

33

we will show that the relation Let {f]

M = ([D ] , [ A

T

~ s = t

},Vall) , N = ([E },[B

is a partial homomorphism from

are viewed as structures

M

i ~i "

is complete

onto

],Val2) N

be R-structures.

A system

just in case it is one when

(the definition did not involve

Val)~ and

fo

M~N

is the

identity.

LEMMA i.

Suppose

Xl~...~x n Then


M

are respectively in Dom(f l)~...~Dom(f n)~ and

f(A(Xl~...~Xn)) Proof:

tion

If]

onto

N .

Suppose

A(Xl~.°.~x n) E D(y •

= B(f~l(Xl)~'''~f~n (xn)) "

By i n d u c t i o n

on

n

of partial homomorphism.

.

For

n = 2

~ this

is

Using this~ we have

straight

from

the

f(A(Xl~...~Xn+l))

defini-

=

= f(A(A(x l~x2),...~xn+l)) = B(f (A(xl~x2))~f~3 (x3) ~''" ~f~n (xn)) = B(B(f~l(Xl)~f~2(x2))~f~3(x3)''°''f~n+l(Xn+l)) appropriate

" B(f~l(Xl)~''''f~n(Xn+l))

for

'r .

The following Lemma is the analog to Lemma 16~ for the R-calculuso

LEMMA 2.

If

[f}

assignment~ h then


is an N-assignment~

f (g(x)) = h(x)

f (Vall(S;g)) = Val2(s~h) ~ for R-terms

Proof:

By induction on

M

s ~ where

s

M~N;[f }

onto

N ; and

for variables of type

x

g

is an M-

of type

~

~ .

are fixed.

The variable~ appli-

cation~ and X-abstraction cases of the induction are as in the proof of Lemma 15.

We have

fo(Vall(0~g)) = f0(O) = 0 = Val2(O~h ) .

We must show that all

f00(Vall(N~g)) = Val2(N~h)

y E ¢0 ~ fo(A(Vall(N~g)~y))

we have

f0(A(Vall(N,g)~y))

.

= B(Val2(N~h)~f0(y))

It suffices to show that for •

Since

f0

is the identity~

= A(Vall(N,g)~y) = y + i = B(Val2(N~h),y) =

B (Val 2 (N~h) j fo (y)) " Finally we must show It suffices to show that y E Dom(f ) . f (z))~ for all

f

(Vall(R(r~g)) = VaI2(R ~h) ~ where

~ = (~(0~@))

.

f (A(ValI(R ~g)~y)) = B(Val2(R(y~h) ~ f~(y)) ~ for all

It suffices to show

fo(r (A(ValI(R@~g)~Y~Z)) = B(VaI2(R~h)~f~(Y)~

y E Dom(f )~ z E Dom(f ) .

Again 3 it suffices to show

34

fcY(A(VaI l(R 3Y)'Y~z~k)) z E Dom(f)~

k E ~ .

f (A(ValI(R

,h)~f

= f(z)

f (A(ValI(R

= B(Valm(R

(y),f(z),k

A(ValI(R

~g)~y~z~k)

LEMMA 3o

Suppose

Let

T

M .

onto

Proof:

LEMMA 5.

M

to Lemma

.

T O~

T

Assume true for

,g),y,z,k),k))

=

(y) , f (z) ,k) ,k) =

y E Dom(f )

from

M

onto

N ~ where

N ~ s = t ~ for any R-terms

16~ using the previous

s = t

Note that

.

M~N

s~t .

Lemma.

Then there is a partial h o m o m o r p h i s m

case of (the proof of) Lemma

T

> s = t .

from

18.

i 91 .

is

~ s = t

if and only if for all countable R-

T

elementary

Then by Lemma 4~ all R-structures

~ s = t .

Assume

substructure

of

T T

will be a countable R - s t r u c t u r %

We now wish to complete 1 2/ NI "--

(y),f(z),O)

k .

M ~ M ~ s = t .

Suppose not

M

y E Dom(f )~

(y),B(Val2(R@,h),f

implies

be any R-structure.

We claim that

Suppose

countable

= B(f

~ k E Dora(f0)

M ~ s = t

The relation

structures

sense.

Then

A special

Proof:

for all

+ i) , by eermna I, since

E Dom~f)

Analogous

L E M M A 4.

,h),f

there is a partial h o m o m o r p h i s m

are R-structures.

Proof:

(z)~k)~

,g),y,z,k + I)) = f (A(y,A(ValI(R

(y),f(A(Vall(Rfy,g),y,z,k)),k)

= B(Val2(R@,h),f

(y),f

We show that this is true by induction on

,g),y,z,0))

k , and write = B(f

= B(VaI2(R

To this end~

let

P

~ s # t[f] containing and

.

M ~ s # t[f]

be the set of indices

have

Then let

Rng(f)

the proof that the relation

M

T

M

M ~ s = t . be a

~ in the appropriate .

So not

~ s = t

of primitive

M ~ s = t .

is complete

recursive well

~/This half of the proof was motivated by a proof by R. Gandy and G. Kreisel (Communicated to us by H. Barendregt) which showed that there are two unequal p.r. functionals which agree on all primitive recursive functional arguments.

35

orderings element

whose is

primitive is

field is

I .

~ • whose

We can arrange

recursive

linear

0 ~ and whose greatest Let

(ao•...~an~O)

least element

the indexing

ordering is

0 ~ and whose greatest

so that every

<e ~ whose

element

is

field is

i ~ and so that

be the function

f

e

~ • whose P

given by

is the index of a least element

is complete

f(i) = a i

F~ .

for

i ~ n ; 0

otherwise. Let

F: O ~

f(O) = I . 0

~ .

Let

otherwise.

We wish to define

f = ~ (F) E ~ ° ° e

f(n + i) = F((f(0),...,f(n),~)) Let

~e(F)

= g

be given by

if

Let

F((f(O)•...•f(n)•~))

g(n) = f(n)

if

<e f(n) ;

f(n + i) # 0 ;

0

otherwise.

LEMMA

6.

If

Proof: and

ao >e

7.

F(%(F))

Since

* ~e(F)

e E P , let

e ~ e 6 P

Assume

Clearly

~e(F)

For each ~e(F) = 0

LEMMA 8.

If

be

(ao~...~an,~)

n = 0 .

or

then

~(F)e

P(~(F))

, where

# 0

I

n ~ O, a 0 = i~

= ~ ~ and we are done.

an = F(%(F))

Hence

if and only if for all

e ~ P .

each

.

So

F(~(F))

F ~ F(~e(F))

= ~e(F)

F(~e(F))

For all

There

a closed R-term

Proof:

(ao~al~a2~...)

= 0 • and

~e

Hence

be the functional

= F(~)

or

F(

# 0 .

or

= F(~)

~e

~ ~

= an+ I

of type

ai+l <e ai ~ and such that

for

0 ~ n .

= 0 # F(~)

(((0•0)~0)•0)

given by

otherwise.

is constantly

0 .

7.

is a total recursive (((0~0)j0)~0)

This just says that

F: ~

F(~e(F))

(F)) # 0 ; i

if and only if

from Lemma

of type

be such that

F((ao•...~an~O))

= (a0•al•a2•. ..) .

e • e E P

Obvious

Let

a i # 0 • and so we may choose

e ~ we let

if

Proof:

LEMMA 9.

~e(F)

= F(O)

~ 0 .

F(~) = I• F((ao•al•a2•...)) Clearly•

F ~ FC~CF~

= (a0~...~an.l~0)

For all

Proof: a0 = i .

then for all

"'° >e an ~ an # 0 o

Otherwisej

LEMMA

e E P

~e

function

~

such that for each

such that in

is a primitive

T ~ • Val(~(e))

recursive

e ~ ~(e)

is

= ~e "

functionalj

defined

36

effectively

from

T H E O R E M 6.

The relation

Proof: fact that

e .

T

~ s = t ~ for R-terms

By Lemma 53 the relation is P

is complete

State Hniversity of New York at Buffalo

i HI .

s3t 3 is complete

i HI .

By Lemmas 83 9~ together with the

H I ~ we see that the relation

is complete

37

REFERENCES

[I]

H. Barendregt~ Some extensional term models for combinatory ~-calculij Dissertation~ University of Amsterdam.

logics and

[2]

L. E. Sanehis~ Functionals defined by recursion~ Notre Dame J. Formal Logic~ vol. 8~ no. 3~ pp. 161-174.

[3]

W° Tait~ Intensional interpretations (1967)3 198-212.

of functionals of finite type~ JSL 32

TEE USE OF ABSTRACT LANGUAGE IN ELH~4ENTARY METAMATHEMATICS: SOME PEDAGOGIC EXAMPLES G. Kreisel,

Introduction.

i.

G. E. Mints and S. G. Si~0so n

The logical need for abstract language is well-known 3 where

by 'abstract language' we mean such things as the language of set theory or the many-sorted

languages

of arithmetic

of higher type]

or of concatenation

fact that there are(true)number

in contrast to the 'concrete'

theory .

This logical need consists in the

theoretic and metamathematical

which can be proved by use of evident (and familiar) properties notions 3 but not by use of evident number theoretic Naturallyj

languages

propositions of abstract

or syntactic properties.

one needs an abstract language in the sense above to express properties

of abstract notions. Besides this logical need 3 there is also a practical or mathematical need for abstract languages 3 which has long been taken for granted; that 3 in certain areas~ for example~

so much so

in current analytic number theory (where the

abstract languages can be~ demonstrablyj

eliminated

cians tended to confuse logical and mathematical

'in principle~ 3 mathemati-

needs;

specifically 3 many

thought that analytic notions- of the complex plane and its structure- were needed logically 3 for the existence of some proof of the theorems considered~ when in fact they were needed for intelligible

proofs~

of manageable

complexity.

Perhaps the most striking example of the distinction between logical and mathematical need is provided by the use of the axiomatic method applied~ number theory.

in~ say~

Here certain relations are explicitly defined and shown to

satisfy more or less familiar

'axioms' in the theory of finite fields or topology.

39

Then results from these axiomatic theories are applied to the explicitly defined relations

to yield number theoretic theorems .

Since formal number theory is

incomplete s there is of course the possibility that higher set theory is needed

logically somewhere to verify that the explicitly defined relations satisfy the axioms in question or that the results are in fact consequences of the axioms. (This last verification is elementary by completeness if all formulae are of first order and if logical consequence is meant.)

But these are only possibilities; in

fact s they are not realized in current practiee~ even when there is general agreement on some kind of need for abstract methods.

The obvious conclusion is

that the need is simply not logical; no strong (existential) axiomsfor the abstract concepts are needed.

For a very convincing analysis of the other kind of

need s in terms of the measures of complexity provided by length and genus of derivations~ the reader should consult Statman's dissertation

[St].

Over the last two decades it has become clear that even elementary metamathematics benefits from the use of abstract language.

As our title indi-

cates s we wish to illustrate this view by use of pedagogic examples.

But at

least the thoughtful reader will demand a little more discussion of this switch since it conflicts s prima f a c i %

with the principal aim for which Hilbert intro-

duced metamathematics s that is~ the aim of eliminating abstract language altogether.

So it would at least appear circular to use such language in metamathe-

matics itself.

What has changed since the turn of the century when Hilbert

formulated his program?

For one thing some of us have

come to doubt the parti-

cular hypotheses (about the nature of mathematical reasoning and hence of mathematical rigour) which suggested Hilbert's program - and s in fact~ suggest that it would be easy to carry out.

Those hypotheses do not appear even remotely plausi-

ble on the basis of intellectual experience; alias philosophieal~

only on the basis of ideological~

views of what 'ought' to be reliable knowledge.

But apart

from this so to speak negative development s this correction of false first impressions s we also have a positive development in the present century: the discovery of natural and manageable

'models' of abstract languages~ models to which

40

we refer constantly in this paper.

Here the reader should not forget that even

the notion of hereditarily finite set has become really familiar only in the second half of this century!

Otherwise

it is hardly likely that Ackernmr~wotuld

have gone into as much detail as he did in [A] to discuss the axioms of general set theory excluding the axiom of infinity,

or that Godel would have arithmetized

syntax instead of coding it in terms of hereditarily finite sets (where the representation of finite sequences of sets is quite straight-forward). reader to recognize the defects of the past.

There is quite enough of (logical)

value in the ideas of the pioneers to make pious reverence Beside~ any overestimate underestimate

of progress

of achievements

-- We invite the

quite inappropriate.

of the past necessarily

implies an

since then.

Returning no~ to our immediate purpose~ we naturally choose the most familiar metamathematical

results for our pedagogic

and so-called cut elimination theorems. of abstract language~ terms.

L

Completeness involves~

of a system

namely completeness

The main results and especially the r%le

and proofs~

More details are to be found in Sections 2.

language

in formulations

examples,

R

are described below in general I and II.

of rules w.r.t,

a formula

A

of a

obviously 3 the abstract notion of realization or model.

To be quite precise 3 a realization is given by a domain (of individuals

or

several domains in the case of many sorted languages) and its satisfaction tion defined on some subclass of formulae in L ular case of first order predicate function symbols than, possibly~

logic

As is well-known

in the partic-

(here taken to contain no other

constants for individuals)~

relation on atomic formulae determines

the satisfaction

the satisfaction

relation for all formulae~

and therefore the passage from the former to the latter is not analyzed. if -- as mentioned in para. i - one is interested

in 'restricted'

e.g. recursive ones~ the distinction becomes crucial: model for first order arithmetic faction relation metically definable

rela-

classes of sets,

for example~

the usual

in the language of rings has a recursive

when restricted to atomic fo~nulae, one for the whole language.

However~

satis-

but not even an arith-

As is to be expected from

41

experience in logic over the last decade, we shall consider different kinds of models, not only the usual binary ones9 to avoid conflict with ordinary (mathematical) usage we shall then speak of valuations

rather than 'models'.

we shall not have occasion to use the familiar boolean valuations~ valuations and partial valuations in connection with Takeutlfs The partial valuations

introduced by Schutte

conjecture

reformulated

higher order logic.

valuations were used quite recently

by Girard to say so to speak the last word on this c o n j e c t u r e ~ W e these valuations

for our extensions

but semi-

[Sch] some 15 years ago

for inpredicative

as t e r n a ~

Actually~

of the completeness

theorems

shall use for the usual

rules with and without cut. Summarizing: A

L'

A

is valid or

is true in all valuations,

depends on the class subclass

the notion

2

of sets considered~

the kind

of formulae on which the valuations

A

p

of valuations,

the 'proof theoretic'

is derivable by means of the rules

R. of

R~

we consider for-

mally (or 'pseudo') well founded trees (of formulae) where each node (literally) finite level and the formula at

N

N

N

R

from

in the tree ordering; we

shall also say that the trees are locally correct or locally regulated

(by

R).

is terminal 3 the formula there must be an 'axiom'. The formulation

of

very familiar f r o m ~ - l o g i c often);

is at a

is ~derived' by means of

the formulae at the nodes immediately preceding

N

side of

namely

Instead of restricting ourselves to finite iterations

If

and the

are given.

Now we use abstract language to reformulate completeness~

Val (g ,L',A) P

in symbols

'A

is derivable'

the class ~

trees is

(where rules are in any case iterated transfinitely

cf., for example, [KK] p.151.

finitely branchin$

in terms of well-founded

derivation

In the case of finitary rules, and thus of

trees, formally well founded trees are finite provided

of sets or paths ensured by our formal theory is 'sufficiently

It is not if both the trees and the paths considered

are recursive

rich'.

since there are

infinite recursive binary trees containing no infinite recursive path.

Evidently,

42

if ~

satisfies KSnig's lemma ~ (and the trees considered are in ~ )

then our trees

will be finite. The extensions of the usual completeness theorems which we provejall have the following form. R

and

For appropriate

~ (depending on

R ) and

L' (depending on

A) A

has a ~ -founded R-tree if and only if Val (~ ,L',A), 0

provided only ~

is closed under some simple primitive recursive operations.

Actually~ our results are a little sharper in the direction ~- : specific tree

TA~

there is a

determined primitive recursively from the for~ala

A~

with

the property: V a l (~ ,L~,A) -~ (TA

is locally regular and ~ -founded).

P TA

is sometimes called the universal ~search tree' for

A

though~ for our

exposition~ it is more appropriate to call it the universal refutation tree since it codes all infinite countable counter models to

A

(with domains consisting of

terms familiar from Henkin's complete and consistent extensions). Naturallyj by what was said in the last paragraph but one~ the extension is interesting or at least novel mainly if ~ For 3 if ~

does satisfy the l e ~ [A : A

does not satisfy Konig's lemma.

the set has a ~ -founded R-tree]

is obviously independent of ~ ; but so is Val (~ ,L',A) by a familiar use of the first 'basis' theorem in the literature (which concerns strict ~||-predicates; this does not conflict with the fact, noted above, that a given satisfaction relation on L' in such a ~ satisfaction relation in ~

cannot generally be extended to a

defined on the whole of L).

So our reformulation

follows from known, albeit useful facts about predicate logic. of the extensions when ~

As to the interest

does not satisfy Kbnig's lemma (for example, if

There are some technical distinctions concerning Konig's lemmm which it is useful to keep in mind here] they are summmrized in the Appendix.

43

consists of the recursive sets), there is nothing problematic if we are interested in recursion theoretic complexity of models;

in this case our extension has the

same kind of interest as does the usual completeness theorem for unrestricted models.

There is an extensive literature on the subject culminating in papers of

Vaught [ V ~ or Jockusch-Soare [J-S].

Note that our results do not conflict with ~V~

since Vaught uses the satisfaction relation on atomic formulae as data (for 'giving' a model) which we never do for logically compound A. However 3 it is possible to analyze the interest of our extension in a more sophisticated way relating it to general mathematical experienee~ expecially to the technique of studying distinct notions (here: rules) which are ~equivalent' in familiar domainsDby extending the domain of definition to

D+

where they are

not equivalent; or~ in medieval terminology 3 where the notions are extensionally different.

Here

D

refers to finite derivation trees,

D+

to ~ -founded ones.

We give such a sophisticated analysis at the end of Section I; not unexpectedly~ it leads to novel problems very much in the spirit of (other) modern mathematics~ for example geometry 3 if we compare rules to algebraic and ~ -founded trees to analytic structures. 3.

Rules with and without 'cut'.

Inspection of the usual completeness

proofs 3 at least when guided by the eonsideration~of para. 23 provides the data~ that is~ the kind of valuation ~ and the class

L~

(depending possibly on

A)

~here the satisfaction relation is given for which our extensions of the usual completeness theorems hold. Roughly speaking 3 for rules with cut~ say binar~ and out cut 3 say

L~ consists of all formulae of the language (of R23 P2

A);

R13 01

for rules with-

is a semivaluation (defined precisely in 1.2) and

sists of a sufficiently large class of subformulae of

is

A(and h e n c %

L~

con-

in effect~

on all formulae because a semivaluation o ~ that class of formulae is a semivaluation on the whole language). (*) if

~

[A:A

has a f -founded

Evidently Rl-tree ] = [A:A

has af-fou_naea

Re-tree }

satisfies Konig~s lemma~ simply because the cut free rules are complete perhaps in the usual sense and rules with cut are sound. (This provides the easy but~not

44

very well-known proof of 'cut elimination',

for f£rst order predicate

a finite subsystem of first order arithmetic; general structure of so-called syntactioproofs with some crude considerations

on ~-logic

rapid proof of (*) above [~isses the point~ holds for some classes well-founded

(that is, 2

iallyweaker' conditions

~

for which

cf. Appendix).

However,

both the

of cut elimination and comparison

in Section II (p. 41 Specifically,

) suggest that the

we conjecture that (*)

$ -founded trees are not automatically

need not satisfy Konig's lemma, but only 'substant-

closure conditions).

-- Naturally here we cannot expect the relevant

to be so strikingly weaker than (true) well-foundedness

in the case of ~-logic and other infinitary systems of rules: bound to magnify distinctions predicate

logic, in

as they are

the latter are

that are subtle in the case of ordinary first order

logic (a fact which, to one of us, is the main source of interest

in

infinitary languages). Turning now to 'negative'

results,

that is, failure of (*), it is quite

easy to see that (*) does not hold generally if ~ recursive formula

sets. A

language of

The easiest way to find a counter example is to look for a

which has no model with a recursive total valuation on the whole A

but does have a recursive semivaluation

After having found such an

A,

founded derivation with cut of manipulations, possible

formula

A

(in which

~ A.

~ A

is true).

Depending on one's practice with syntactic oneself)by looking at

does no_~t possess a Rec-founded derivation

J. Stavi has improved on our original result by constructing which is true in some (primitive)

recursive partial or 3 equivalently,

recursive

terna~j valuation

semivaluation,

of

His

proof operates directly on the valuations without using the corresponding plete sets of rules (without cut, rasp. with cuts limited to subformulae

introduction

to ternary valuations,

comof

interest as an

for example, as used by Girard

A~

Precise

of these valuations are given in Stavi's exposition in 1.2.

We believe that Stavi's exposition has additional pedagogic

a

but in no

(on the subformulae

and hence in no recursive binary valuation of the whole language). definitions

A

it is usually quite easy to write down a Rec-

one will or will not succeed in convincing

cut free proofs)that

without cut.

consists of the class of

[Gi].

A).

45

Specifically 3 though Girard is careful to use 'weak' metamathematical methods, they are not weak enough to force a clear se2aration of notions 3 for example by analyzing the passage between partial (ternary) and semivaluations. formulae considered in ~his system

AA

(The

~-

are more than sufficient to define

infinite paths of infinite recursive finitely branching trees). NB.

The notation in Stavi's subsection is different from the notation

here; for example, he uses Greek 1. c. letters for formulae while we use Roman capitals (for nearly everything). 4.

Transformation o f ~

-founded trees~ sharpening (*) in para. 3 by:(**)

imposing (additional) requirements on mappings to

R2-derivations (of

A) when (*) is true.

from

Rl-derivations (of A)

In the literature the distinc-

tion between (*) and (**) is sometimes expressed by nonual form and normalization theorems (when cut free rules are called 'normal').

For finite Rl-derivations

the distinction was subtle because the property of being an Rl-derivation is decidable end so on__eemapping~ 0~ is obtained trivially by simply running through all Q-derivations till one hits an R2-derivation with end formula A.

From the

start~ there was little doubt about the interest of something like the particular normalization procedures in the literature though the analysis of that interest was problematic.

It is fair to say that one of the most useful immediate conse-

quences of the shift from (i) sequent formulations preferred by Gentzen himself and by Takeuti te (ii) natural deduction formulations preferred by Prawitz was just this:

in

contrast to (i)~ in case (ii) a particular mapping suggested

itself; so there remained a well-defined normalization problem even after the corresponding normal form theorem had been established

by model theoretic methods

@hich ensured at least on__~enormalization procedure~ the trivial years later we cannot be satisfied with such virtues:

p

abov@.

Five

~uod decet bovem dedecet

Jovem. Once again we introduce abstract language, still

the main theme of our

*On the other hand AA is 'too ~eak s for the particular applications made in [Gi]. For his purposes it is not necessary that the metamathematical proofs be formalized in AA~ it ~ould be sufficient to have them formalized in full classical analysls.

46

paper (though it has not been mentioned for some time) where the literature avoids it; specifically~ we think

of the derivation trees in (**) as arbitrary (sets)~

not as inductively defined objectsp represented by natural numbers; in contrastj in the case of ~-logic, to F e f e r ~ n obvious additional requirement on continuity of

p~

[F] or Carstengerdes p,

[C].

The most

suggested by this step is

where a neighborhood of a derivation

is determined (as is

usual for trees) by a finite number of finite initial paths in the given derivation tree. Naturally, it remains to verify that this so to speak porte-manteau requirement, which makes sense throughout mathematics, sent context.

is also of specific interest in the pre-

This is plausible inasmuch as experience (of informal proofs) shows

that we can often read off desired information from the last few steps of a proof without going all the way back to the beginning; for example, in classical mathematics if we want to know whether or not a given proof ~ of an existential theorem ~ x B

(with logically compound B) provides a specific (or even any) realization

realization x ~ : B ~ x / x ~ ;

cf. p.65 for more detail.

Another question which suggests itself immediately is this:

can

f

be

extended to a continuous total mapping, defined on all trees, not only RIderivations?

There is a simple device for doing this, a kind of delay mechanism, I

familiar from recursion theory when one replaces El-well orderings by primitive recursive or even Kalmar elementary ones or partial recursive neighborhood functions by similarly elementary ones.

This device is here applied by (i) adding to

R2, if necessary, the rule Repeat which allows a string of repetitions of the same formula on a (non-branching) path of an R2-derivation and (ii) verifying that, if ~is

closed under recursive operations, ~A:A has an R2-derivation ~ = IA:A has an R~eP-derivation},

where 'R-derivation' means a tree which is G - f o u n d e d

and regulated by the rules R.

Though the questions raised here are meaningful for f - f o u n d e d

(finitely-

branching) derivations in ordinary predicate logic, we discuss them, in II, for ~-logic:

as was mentioned already, the distinctions are easier to see.

In

particular, at least as long as we consider G~-logic for finite formulae, the

47

normal form theorem (*) requires nC~ closure of ~

under all arithmetic opera-

tions~ not the much weaker closure condition~ of satisfying Konig's lemma, mentioned on p. 6. tains 3 with each definable from

Conversely 3 it is certainly sufficient for (*) that X~ the ~-th jump of

X

are recursive):

X

~

con-

(in which all sets arithmetically

in this case any semi-~-valuation~

can be extended to a total (binary)-~-valuation also in ~

.

X3

in~ ~

Here it is evident

that this closure condition is much weaker than the requirement that all ~ - f o u n d e d ~-derivations are well-founded. (This is the crude consideration on ~-logic mentioned in para. 3 above). To conclude this discussion of transformations of derivations~ let us note more generally that 'additional requirements' on normalization procedures P

need a genuine study of proofs (not only of the set of provable theorems) if

p

is required to preserve some (structural) relations between proofs~ for example~

if 3 for such relations

~

f

f'=>0f % p f

As mentioned in para. 1 above; Statman's dissertation contains studies of such relations 3 defined in terms of simple isomorphism types of derivations. Such requirements are to be distinguished from another t~pe of additional requirement or 'sharpening ~ of (*), which can be perfectly adequately expressed in terms of sets of derivable theorems without explicit reference to derivations and their transformations.

For example, our

p

(called

~I in Section II) has

the additional property that - except possibly for the rule Re,eat - every rule applied in the derivation

pf

is also applied (somewhere)

on

f ;

(applications of) rules are ~added'~ though some may be eliminated. contrast to requirements of preserving relations between

f

and

second sharpening can be expressed quite adequately as follows. range over trees of formulae and let

Deri(fi,A )

no However; in

f'; Let

this fl

and

f2

stand for the conjunction

*One of us used precisely this phrase 'preserving relations between proofs' [Sl~f II, p. i00] hut 5 pages later got involved in a discussion of the relation between (the proofs expressed by -- what corresponds to--)f and pf if p is the normalization operation for systems of natural deduction.

48

Lci(fi)^~(fi)^ ~fi : ~ where and

LC.

means local correctness for the rules

1

kf.

l

is the end formula of

so we need not quantify over

fi

means well-foundedness~

(which is of course determined by

below).

A

RiJ WF

So the present

fiz

and

'sharpening' of (*), in

particular 3 of

V fl {oerl(fl'A ) * 3 f2Der 2 (f2A )} is given by:

Vfl{Der!(fl,A)

~3f2[Der2(f2,A)^

R(fl, f2)] }

that (except possibly for Repeat) all rules used on on

fl"

subsets

f2

Clearly this is equivalent to the conjunction~ R/I of

R i (i = ij2)3

[A:A

where

R

expresses

are also used over all suitably matching

of

has an R[-derivation] S {A:A

has an R~-derivation]

This involves only sets of derivable theorems~ as stated earlier~ not the derivations themselves. Remark.

One of us has advocated~

starting in [SPT II] and continuing,

perhaps ad nauseam 3 since then 3 that proof theorists give up the literally unrewarding aim of establishing essentially model theoretic results (about the set of derivable theories) by proof theoretic manipulations Preservation results of the form: determined by

kf

and

kf'!)

f

~i f' -~0f

~2 of

(of derivations). (where

f

~ f'

is not

are good examples of genuinely proof theoretic

results while the second kind of 'sharpening'

is not.

The conviction behind this

propaganda over the last 5 years was that though traditional

proof theory looks

dull it was the statements of the results~ not the methods used that were dull. In other words~ the interest (of the methods) was real enough 3 but its analysis (expressed by the choice of theorems actually formulated) was defective. -Evidently those who pursue the subject for what were called ideological reasons cannot agree:

no amount of structural analysis can compare, in glamor~ with

replacing dubious rules

R1

by valid ones~

R2~

Perhaps it should be pointed out that there are also quite different

49

attempts (implicit) in the literature 3 of making existing proof theory more interesting; for example# Takahashi's imaginative idea [Ta2] that proof theory is concerned with the topological properties of the space of truth values.

(This

is certainly consistent with what most people know of the proof theory of classical logic with its discrete 2 element space of truth values).

Somewhat perversely

he chose continuous model theory [CK] to illustrate the general idea. 5.

The need for enrichments of derivation trees:

~hat i s ~

In view of the stress on continuity properties 3 of the mapping

0

proof tree? in para. 4 3

we must be prepared to pay attention to the exact choice of data (which constitute the arguments of

p~

that is 3 the derivation trees on which

0

operates).

It may fairly be said that a principal difference between the styles of modern mathematics and of a hundred years ago is just this:

we have learnt to become

aware of the effects of small differences in the choice of data and# consequently~ can use those effects to lead us to a correct choice of data.

In contrast~

earlier workers attempted to evade the problem of choosing among 'slightly' different data by establishing brutal equivalences. - According to taste the reader may think of examples from category theory and the reason for enriching (the graphs of) functions by adding a bound on their range as part of the data; or from the theory of continuous functions on the unit

interval and the reason

for enriching (or 'equipping') such functions with a modulus of uniform continuity (to determine upper and lower boundscontinuousl~ on the uniform convergence topoiogy)~ or from the theory of r.e. sets and the reason for equipping any such set with some enumeratio~to satisfy the reduction principle by mappings

~i ~ ~2

which are continuous for the product topology (on the enumerations~ say

ex~ ey,of

the sets

X

and

Y):

~l(ex, ey) c X ^ ~2(ex,ey) c Y ~ ~i ~ ~2 = ~ A ~i u ~2 =

X I ~ X 2. The principal criteria of choice of data will be to ensure that

O

can

be chosen to be recursive and continuous and to have the further structural properties analyzed in para. 4.

It will turn out that these criteria are only

peripherally corrected with the ~m~in traditional metamathematical requirements

50

which denmnded data permitting quantifier-free

proofs of (the usual) metamathe-

matical results about transformations. The data fall into two parts~

(i) the presentation

'underlying ~ trees and (ii) the information, which is put at each node.

'abstract'

besides the formulae

or

'proved',

As to (i)~ the basic choice is between a presentation

of the set of all initial paths (without~ if a node is terminal~

of the

for example,

determining explicitly

or more generally the exact number of its immediate pre-

decessors ) and one that includes the relation between nodes and explicit presentations illustrated recursive

of the set of immediate predecessors.

The difference

in the Appendix in the case of binary trees which are given by a

set of initial paths (where each node

predecessors)

'happens' to have

whether there are

2~ I, or none.

pre-

In short, at each

we are given complete information about the location of the nodes used

in the last inference which leads to (the formula at) N. local

indicating here

In Section II we use the latter kind of

sentation also in the case of infinitely branching trees. N

< 2

and those given by an additional recursive function defined on all

nodes and taking as values a (code for) the set of predecessors~

node

is very well

information

about the

on the level adjacent to

'underlying'

--We think of this as

tree because it refers only to nodes

N. ,

As to (ii)~ the traditional literature

on infinitary

introduces additional global information at each node lying tree~ n a m e l y ~ b o u n d nated by CN

N~

on

the ordinal length,

IN,

and partly about the formulae occurring in

on the logical complexity of either all formulae on

formulae

N;

(usually called degree).

plexity are required~

proof figures

partly ~ about the under-

of the tree TN~ TN

TN

domi-

namely~ a bound or at least of cut

Though not exact values of length and com-

the bounds chosen must be compatible with the tree ordering.

- The immediate purpose for this additional

information

is of course clear~

For finite proof figures this is rarely done because so to speak ever$~hing one wants to know can be read off from a finite figure. Realistically speakingjthis is of course not true; we can often operate on a finite figure if we are given (relevant) additional information~ but not if we have to search for it systematically.

51

especially of

~N:

modulo principles of transfinite induction formulated in the

language of arithmetic (to which Gentzen's work had drawn the attention of proof theorists)~ this additional information allows us to paraphrase the basic condition of well-foundedness (WF on p. ll).

O A ~ l - statement expresses the

campatibility of the ordering of the notations rSle of the bounds

CN

~N

with the tree ordering.

The

was also clear for certain specific proofs of cut-

elimination according to the lexieographic order It should perhaps be mentioned that this

(CN,~N). additional informationj though

perfectly adequate for formulating the metamathematicalarguments (especially of cut elimination) in the language of arithmetic, is not particularly convenient for a purely quantifier-free forn~lation:

0 one is left with ~ l - proof predicates

which appear as premisses in implications~ for example, of the so-called reflection principle~ even when applied to quantifier-free formulae.

For partial

results (possibly retaining quantifiers) a proper choice of extra information at each node is due to Lopez-Escobar [LE] who puts~ at derivation~ say

~3

of the formula 'proved' at

N,

a suitable formal

N (where

~

is in the system

in which the metamathematical result in question is to be formalized); more detail on this kind of enrichment will be found in Section II.2(c). Even if 3 subjeetively~ the additional information

~N' CN~ ~

was ad hoe~

that is~ was actually obtained by tinkering~ it is pretty clear that the information is pertinent to the aim of 'arithmetizing' metamathematics in an elementary way.

We do not question this here.

What we wish to study is this:

Is

this information on which we have stumbled in pursuing the quite different aim (of giving quantifier-free proofs of metamathematical results), also useful for our present more structural aims?

More precisely,

Must~ for example~ ~-derivations f (CN, IN)

be enriched by bounds

to solve the 'functional equation' (for

0) :

Derl(fl, A ) -~ Der2(Pfl,A ) ) by means of a continuous The answer~ given in Section iI~ is negative.

p?

Next we ask:

Must we make a fresh start to solve the functional equation

52

above when the

fl

c°rresp°ndinglY3

are enriched by Pfl

(CN, IN)?

(And

is to be enriched too. )

Again, the answer is negative2 since there,s a mapping

~

defined on the k~nder-

lying' (universal) trees which maps bounds into bounds, compatible with the tree ordering 3 and closed under of)

~

is independent of the formulae on

~0'

~(CN, IN).

that is, if

IN < sO

for all

N

fl"

As one might expect;~ is

so is (the second component

In other words, the business of bounds is not of the 'essence',

but can be separated from the structural analysis. 6.

Conclusion.

Evidently our analysis of the data (determining deriv-

ations) and of requirements on operations~

p~

is quite superficial compared to

the corresponding work in venerable parts of mathematics, for example; geometry. There the data, determining geometric figures(in terms of the 'structure' of the space: combinatorial, algebraic; differential; etc)are analyzed in an incomparably more sophisticated way, and the same applies to various kinds of 'equivalences' and'isomorphism types ~.

But it seems to us that our analysis has at least the

right smell.

I. GENERALIZATIONS OF T ~

C01{FLETENESS T H E O ~ 4

FOR FIRST ORDER PREDICATE LOGIC.

In subsection I we sketch what is 3 perhaps 3 the most basic case where (i) the models considered have as domains terms familiar from Henkin's construction of complete and consistent extensions, and a model (of A) is determined by the satisfaction relation for all formulae in the language of up from the relation symbols occurring in include cut.

A,

A,

that is~ built

and (ii) the rules of inference

The reader should have no difficulty in working out the next most

familiar case when (i') the models considered are determined by semi v a l u a t i o n s in thesense described precisely in Section 2, and (ii ~ ) the rules of inference exclude cut. In subsection 2~ most of which was written by J. Stavi, a formula

A

is constructed which is (i) true in every recursive partial or ternary valuation, and a f o r t i o r i

in every recursive complete and consistent extension (of the

53

language of

A)

hut (ii) false in some (primitive) recursive selmi-valuation.

Thus 3 by subsection 13

A

has a Rec-founded derivation with cut~ but none with-

out cut. In subsection 3~ there is a discussion of possible uses of our extensions or 'generalizations' of familiar completeness theorems~ with special stress on new problems formulated in terms of (the notions used in) the generalization. We can hardly hope that these new problems, th@product of research over the last 40 years 3 can have the same immediate appeal as the original completeness problem for logical validity which~ after all~ is intelligible without an_[y specialized knowledge!

So both for objective and for subjective pedagogic rea-

sons we go into the limitations of the original problem; so to speak its insensitivity to the class of models considered (which was a great advantage for preliminary orientation).

Naturally~ as is familiar from other parts of scienc%

what were mere auxiliaries in the solution of the original problem turn out to be rewarding principal topics of research~ and by no means of less permanent interest than the original problem. - As examples of interesting (and precise) problems we should mention here two kinds of 'basis' problems: (i) for a given kind

(p,L')

of valuation (cf. p. 4) to find

~f~,L' ) = > [ V a l o ( f ,L',A) < = > (Pl, Li),(P2,L~)

~(

p~L')

s.t. for all

A:

Valp( f(o,L , ),L',A)] and (ii) for a given pair

to find g(1,2) such that, for~satisfying conditions

~(1,2) ,

Va[val01(f,~,a) Val02(f,~,a)]. 1.

Universal Refutation Trees:

Model Theoretic Analysis and the Search for

Co~lete Sets of Rules. Historically 3 (Frege's) rules came first and their completeness validity) was proved some fifty years later (by Godel).

(for

One so to speak started

with the 'linguistic evidence'~ and slowly passed to the (model theoretic) 'semantic' notions consistent with this 'evidence'.

We reverse this

here~ starting with the semantic notions of model and validity.

procedure

To some of us~

those notions 3 once isolated~ are clearer and~ above allj more obviously significant than any particular set of rules can ever hope to be.

54

We begin with quite crude distinctions,

for example~

in terms of the

cardinality of models and only afterwards analyze more precisely the closure conditions on the class of models actually used. (a)

Cardinality:

validity for arbitrary models is equivalent

to validity for countable models.

(If

A

has an uncountable

a countable elementary submodel of

M

in which

model

A

has a countable model

is

MO

with domain

al,...,a~T ,

the union of copies

name of

M;

of

M0

and each

A

is true.

a (~) l

model

M

If

has a finite

A

there is

M~whose

domain

is regarded as a distinct

a.: i

R(ail ,...3aik )

The passage from not 'normal'

M0

is true in

to

M~

+ M0

iff

R(ail,...~aik)

is true in

MO.

uses the fact that we consider general models and

or 'equality t models.)

(b)

Termmodels,

also called canonical models.

Validity for countable

models is also equivalent to validity in all countable term models of the particular kind introduced by Henkin language

L~

variable

x

is described). and

only constants. is imposed on where AcL~.

3xA If

cA.

A

(for exposition~

Here, if

A

see [KK], p. 83~ where the

is a formula with the single

is true in the model so is

A[x/c A]

where

cA

are the

is false, e.g. of the form

B^~B,

then no requirement

The canonical models considered have as dommin the terms

cA

The reader will find relations to other kinds of canonical models

in Ex. 2 on p. 105 of [KK]). The role of these particular term models for a study of the recursion theoretic

complexity of models is clear.

consist of its domain and its satisfaction

Suppose our data~ determining a model, relation (or 3 equivalently~

definition for a language augmented by names for all elements of the model)~ and suppose the closed formula the satisfaction where A[x/c A]

c

relation.

in the domain

is true~ that is, belongs to

Then, in general, we need a search through

are names of the elements is true.

~xA

the truth

of the model to find one, say

A[x/c]

cA J such that

For our canonical models this operation is more primitive.

85

(c)

At this stage we set ourselves the problem of coding all count-

able canonical counter models to a given formula models of

~ A.

A~

or, equi~alently,

(The decision to look at counter models~

such

in the case of classical

logic 3 is based on the familiar fact that validity here has nothing to do with explicit constructions: ions).

one constructs counter examples to proposed refutat-

The strategy for this coding procedure is by now familiar, for example~

from so-called semantic tableaux (although,

in contrast to our aim~ these

tableaux provide only a certain class of semivaluations,o~ in which

A

is false).

recursively

on

A,

We describe a binary tree

at whose nodes

or; more oftenjP N ~ 3 ~If

of

~A

mM

;

to make them as elementary or explicit as possible~ specific construction of Corollary:

*

Let &~

T A.

one must pay attention to the

As a result we have this.

be any class of sets closed** under the operations

(*);

Or indeed~ in a (Kalnmr-) elementary way.

** As far as 'GSdel numberings' are concerned we naturally require them to satisfy the familiar conditions on canonical numberings(which are isomorphic by means of elementary operation@. Our # are to be closed under those too.

56

the paths which are in ~ given by data in ~ Corollary: Remark.

A

and go through

TA

correspond to models of

~ A

- Another

is true in all models in ~

if and only if

TA

is ~-founded.

The reader who can see in outline the (familiar) construction of

should go on immediately to (d) o ~ 0 ~ h e r e

TA

we pass from this construction to

the discovery of rules of inference. Construction of constants

~

TA

(for

fcrFeL~).

A

belonging to

true 3 rasp. A i

At stage

Ai

of

2i (i > O)

At stage

Ai

L~

in an

each possibility

false) will be 'envisaged' in a branching of

sure that each path contains at some node.

that is~ not containing the

We enumerate all closed formulae

order respecting subordination of terms. (A i

L~

TA

to make

either to the left or to the right of k

(2i+i)~

which is broken up into a finite number of

'logical' analyses, the internal (logical) structure of the fo~nulae in (N

at stage

2i) is 'considered'.

FN,

Specifically, suppose our language is

~,V, ~

. J

and let

Nj (j ~ D(i))

be the nodes introduced at stage

likes number theoretic functions may compute

2i.

(The reader who

p; paying attention to the fact

that at each odd stage several formulae are analyzed and so the depth of say,

_2i+i TA

increases by

> 1

between

T~ i

and

_2i+i ±~ .)

TA

The propositional

analysis is standard: If at the node resp.

~ G

we have

is being analyzed then

~' k A u ~ 3 ,

resp. F u {G} k A'.

predecessor is

branching: If at 3xF

Nj

F ~ ~' u IF, G}.

{F}oP' Nj

~

and

we have

[~F} u F' * ~ N. J

ment that ~f formulae

3xF

F[X/ek]

for

{G}ur'

and ~ F,

[Fv

G} u P' ~ &

the i~mrlediate

there is a

~

[]xF} u P' ~ m

k ~ i

F ~ IF v G} u ~',

If we have

is true so is

F k [~G} u A',

has just one immediate predecessor:

If we have

is 'considered' and replaced by

or

then (sometime

F[x/~]~

F[X/eF].

during stage

2i+l)

in accordance with the require-

If ~e have

P ~[~F} u ~'

are added to the right of ~ ;

then the

put differently

57

at the terminal nodes the first JxFe~

i

N

where

_2i+l TA

of

constants

o n scmenode

U [~:N~]~

N

ck

dominated by

of the language

N

of the i th formula

A.

o~

The mapping ~ ~.

(in

M3

and ~

N2

then each

F[x/~ G] e

~

clearly determines a model

~A,

say

~:

the truth value

Secondly, TA

A.

For consistency,

at the node of

argue by induction

is quite explicit because we know where

a counter model

M

to

A

determines

one or

by telling us which branching to choose at each

are the immediate predecessors

are false then

PNi

of

must all be true,

N; ~i

and

FN

are all

must all be false

M), for i=l or i=Z or both. At stage

2i

('cut') there is no choice:

take that branch which has (2i+i)~ F ^

of

(completeness).

more infinite paths through

true in

through T A

First of all, each path

2i

o ~ logical complexity.

N1

This makes sure that if

being read off from the position of

introduced at stage

if

for

if

(that is 3 complete and consisteqtextension)

node:

D [F[X/Ck]:k i i]

G eL A.

What is there to prove?

Ai

L A.

of an infinite path

STOP the construction at node

to look for

Nj, ~

G

if

Fv

is true.

A. i

GePN. and so J Nevertheless

on the left of ~ , F v G

is true in

if

Ai

the choice

we must

At stage

and conversely. M~

M~

is true in

is open if

at the nodes introduced at even stages determine the

since the formulae of model. (d)

From refutation trees,

terminal nodes of

T.

TA,

to rules of inference

where the order of occurrences

cut:

from

2i 3

P u {Ai} ~ A

At stages

A).

At

we have

FeF ~ A

for ex.,

At stages

(for

of formulae in

P

or

A

is neglected.

the reversal of the construction of and

r ~ A u {Ai]

infer

TA

is the rule of

r ~A.

2 i + I we have the usual rules for ~ , v ;

the 'usual'

58

contraction rule for the existential quantifier ( ~ ) from

r + [Fix/el, 3~F} u n'

infer

P ~ {~xF} ~ a'

and the tunusual' rule (3~:

(*)

from

[F[x/~]] .J P' ~ A

infer

the 'usual' rulefeorresponding to where

c

[3xF] ,~ r' ~ a

(*)

does not occur anywhere in

;

requires, as premise , P' ~J h.

IF[x/o]} u F' ~ A

To show then that the usual rule

is sufficient to replace (*) we use the familiar substitution or Standardization Lemma.

Let ~

be the language of

(in place of the term~

eo~Cl,...

~F

of

5A) ,

A

and let

but not necessarily well-founded derivation tree for rules.

Then there is a standard tree

nodes

A,

T

be a locally correct,

regulated by the usual

such that, for any two inferences, at

N~ N', of

[]XFN] cN

T'

expanded by the terms

and

CN,

Corollary.

u ~

are distinct. If T l

~ %

from

(In T' some

[%[x/eN] } u P~ 'eigen'

variables

~ ~N of T may be renamed.)

is a derivation, by the usual rules, of a formula

containing constants, then

A

A

not

has a derivation by the unusual rules too.

The corollary follows if, throughout a standard form of

T 1,

cN

is

replaced by ~FN, since this replacement does not disturb the logical connections locally.

As to the proof of standardization,

the only point to notice is that

T is

n o t assumed to be well-founded, and so we do not use a construction by (transfinite) recursion, but proceed from the root of the tree (with even subsript) for the jth node from

[ F N [ X / C N ] ~ F ~ ~'~N

F~ U~N, and

N.

by

c2j

Then terms

on the path joLning node of

T

on

T'

on all nodes ci, N'

N

at which

and replace N|

at a node

'below' N'

to the root of

T.

of T

We reserve the terms

[~XFN} ~ ~

}-~

c2j

is inferred

CN, which does not occur in N T,

if

cN

is not bound between

which differ from all

are replaced by

c2i+l.

cN

N|

occuring

Since each

is, by definition, at a finite destance from the root, we do not run

out of constants.

59

Exercise.

The reader may wish to verify directly that the usual rules are sound

for validity on the term models described in (b) above; that is, if ~ - f o u n d e d tree regulated by the usual rules~ determined by data in ~ -Hint.

A

A

has a

is true in every term model

Some care is needed in the choice of data determ-

ining a tree; for more detail~ see II.l(a). (c)

Concerning closure conditions on ~

if the trees

T

considered

~ ~

and

sens%

discussed in the Appendix) then

of

is logically valid.

T

~

~

it is of course evident that 3

satisfies Konig's lemma (in the usual T

is f i n i t %

and hence the end formula

The significance of these facts for the (unexpected)

role of logical validity in mathematical reasoning is well-known$ we shall return to this matter in subsection (3) below, 2.

b(1)

on p.32.

i~-founded Derivations With and Without Cut:

A Comparison.

As mentioned in the suam~ry 3 at the beginning of the present Section I j A

is not false 3 that is~ true or undefined, in each semi-valuation

if and only if a certain 'refutation' tree is ~ - f o u n d e d .

CF T~ ,

e Z"

regulated by rules without cut,

The definition of 'semi-valuation' is given below and also in

[Sch] which contains a proof of completeness for semi-valuations (from which a proof for our generalization looks for it).

to ~ -founded derivations can be extracted if one

Before passing on to the main results the reader m~y ~ish to

review his knowledge of proof theory in terms of the generalization. Exercise.

Find weak closure conditions on ~

(for example~ closure under prim-

itive recursive operations) which are sufficient for so-called inversion theorems such as (for conjunction)i I P * A A Bs~ P ~ A~

and

P ~ B~

has a (cut free) ~ -founded derivation ~

have such derivations} or

(existential premis) if the variable has a cut-free

x

~ -founded derivation iff

does not occur in A~ P ~ ~

The corresponding results are patently true for ~ (for all

~

both

Pu &:3xA~

F ~

has one.

-founded derivations with cut;

considered in subsection 1); model theoretically because all valua-

tions are defined on all formulae of the language and the equivalences are true

60

for validity;

proof theoretically

can attach to a derivation of resp.

A A B ~ B

because,

e.g. in the case of conjunction,

P ~[A^~lu ~

and cut with

a (finite) derivation of

AA

we

B ~ A~

A ^ B. - In the case of cut-free derivations

a

little care is needed9 model theoretically 3 because we have to choose semivaluations with suitably nmtching domains of definition; because we cannot apply without restriction of induction on the length of derivations -founded.

proof theoretically~

(on the predicates used) the principle

since our derivations

are only

So we must either pay attention to the complexity of the predicate

to which the principle end formula~

is applied,

or simply use a proof which starts with the

not the terminal nodes of the derivation.

Returning now to our trees

TA

CF TA

of subsection 1 and

we have the

relation: If ~

(is closed under~

say~ primitive

recursive

operations and)

satisfies K$nig's lemma

~ACF

(**)

is ~ -founded if and only if

or~ equivalently,

TA

is ~

-foundedj

the same formulae have ~ -founded d e r i v a t i ~ s

~ith and without

cut. The proof of (**) is trivial modulo the well-kno~n theorems for finite derivati0~s. proof showing that if true in s o ~ * considered

A

(The reader may also give a new model theoretic

is true in some semi-valuation

total binarySvaluation

in subsection 3 below3

cut-elimination

in ~

of the whole language of

the~ A A).

For reasons

it is plausible that the equivalence

holds under weaker closure conditions;

in other words,

with the equivalence between ~ -foundedness

is also

(**) above

(**) has 'nothing' to do

(of binary trees) and ~ell-foundedness.

It will now be shown that (**) is false (for suitable A) if is the class of recursive by showing that recursive

A

sets

2

is valid in all recursive total valuations

semi-valuations.

Actually even more is true (for the

but the given semi-valuation ~hieh is also in ~ •

but not in all A

considered):

need not have an extension 3 to a total valuation,

61

A

is valid in all recursive valuations defined on all subformulae of

the recursive semi-valuations formulae of

in which

A

is not true cannot

A ; thus

be defined on all sub-

A.

The remainder of this section was written by J. Stavi; we have retained his notation which differs from ours.

An Example of Satisfiability by Recursive Seml-valuations A.

Basic Notions

Definitions:

(a)

We regard

mined and true resp.

-~# T 3~ ) T 3 = 23

~3(x)

as the three "truth Values" - f a l s %

For each (propositional)

ordinary truth function on , ^,v#

0,1,2

and

{0,2}

*3

operation

associated with

*

*) let

*2

undeter-

be the

(we take the operations to be

the natural extension of

*2

to

[0, i, 2]:

1 3 = O~

=2

- x,

A3(X,y ) = m i n ( x , y ) v3(x,y)

= max(x,y)

-~(x,y)

= v3(,3(x),y)

(b)

Let

L

= ma~(2-×,y)

be a set of relation symbols, C

Sent the set of sentences of the language above and of the quantifiers A total valuation,

f o r every

.

V

, ~ ).

for

L(C),

L(C)

a nonempty set of constants,

(built up by means of the operations

is a function

V : Sent -~ {0,2}

such that

~ ~ Sent: (i)

operation )then (ii)

v(~p) = ~ x

If

~ = *(qOl,...,~n)

(where

0 < n < 2

and

*

is an n-ary

V($) = *2(V(q01),...,V(~n)). If

q0 = V x @ ( x )

then

V(~)=

min V(@(c)), cgC

and if

~ =Sx~(x)

then

v(~(c)).

c~C A ternary valuation is a function (peSentj(i) and (ii) hold, wlth Remark:

*2

V : Sent -~ {0,132 }

replaced by

"3

such that)for every

in (i).

Total valuations and ternary valuations are uniquely determined by their

62

restrictions [0,2]

([0,i,2])

valuations L(C), itself.

to the atomic sentences,

can be extended to a total (ternary reap.) valuation.

correspond biuniquely to (two-valued and three-valued)

that is, a model with domain

C

into

Thus, these

C-models for

such that every constant in

C

denotes

Of course 3 we are talking about models in which the equality symbol (if

present in

L

Definitions

(cont.):

at all) is not assigned a special realization. (c)

such that the following D i = {~DIV(~)

(I)

A seml-valuation

is a partial function

"downward" conditions

hold; where

For any

paD 2

(i)

~ = T ;

(ii)

~ = ~ ~

D = domain of

V~

one of the following is the case:

and

~ED 0 ;

(ill)

~ = X^ ~

and

X£D 2

and

~eD 2 ;

(iv)

~ = Xv ~

and

X6D 2

o_~r

~eD 2 ;

(2)

V:Sent ~ [ 0 , 2 ]

: i] (i : 0,2).

(v)

~ = X ~ ~

For any

(i) (ii) (iii)

and

X6D 0

o__rr ~eD 2

one of the following is the case:

~eD 0

p : i

;

~ = ~ ~

and

%6D 2 ;

p = X ^~

and

XeD 0

o~r

(iv)

~ = Xv@

and

XeD 0

and

(v)

~ = X ~

(d)

The set

Sub(~)

= {~]

and Sub(p)

XeD 2

Sub(p) = [~] u

(e)

Sub(~).

~eD 0

;

9ED 0

;

an__~d ~gD 0 .

of subsentences

of

~

is defined recursively by:

~ ;

for atomle

Sub(~) = [~] u S u b ( P l ) u

cludes

and every function on the atomic sentences

"'" !~ Sub(~n)

U Sub(~(c)) c6C

A full semi-valuation

for

for

~

for

~ = Vx~(x)

~ = * (~l,...,~n) or

;

]x~(x)

is a semi-valuation whose domain in-

63

(f)

We abbreviate "semi-valuation" by "s-val." "ternary valuation" by J

"3-val.". Remark:

The important features of the definition of a s-val, are:

need not be closed under subsentences (if

Xv ~

must be true but the other may be undetermined (I) (eD2)

is true

X^ ~

then one of

X~

(~D)).

The requirements go only "downward".

then of course

(eD2)

(i) The domain

If

X

and

~

are both true

cannot be false but it need not belong to the do[m~in

D.

If we change the definition by adding the natural "Ul~,ard" conditions (X~eD 2 =>

X^ ~ D 2 ;

XeD O = >

X^ ~D 0

etc.)

~e obtain the notion of a partial

valuation; the reader may check that the partial valuations in this sense are just the functions of the form

VI[~IV(~) ~ l}

where

V

ranges over 3-val.'s.

Thus,

partial valuations correspond bituqiquely to 3-val.'s in a very simple way. B.

Statement of the Results Suppose now that

L

is finite and its relation symbols are listed in a

finite sequence~ and suppose stants.

C = [Cnln < ~ }

where

Co, el, e2,..,

are distinct con-

If we choose a (primitive) recursive G$del numbering of the constants~ and

distinct numbers for th~other s~2ools~ a Godel numbering for all formulas of

L(C)

is determined uniquely up to (primitive) recursive isomorphism by familiar conditions. Thas we can legitimately talk of (primitive) recursive valuation functions of all kinds. Theorem:

For a suitable choice of

L

the following hold:

(a)

There is a sentence

which is false under every recursive total valuation but is true under some prim. rec. full s-val, for (b)

~ (see def. (e)).

There is a sentence

~

which is true under some prim. rec. s-val.

but not true under any recursive full s-val, for

and not true under any 3-val.

(or partial valuation) whose restriction to

is recursive.

Remark:

We have also proved this for a langaage

predicates

(=~e);

probably one binary predicate

however 3 will take for that

Sub(B)

O~S~+~.

L

L

which contains two binary (e)

is enough.

The proof below,

the language of arithmetic (for ordered rings) except

are replaced by the relation symbols zero~ sue 3 sum~ prod.

For this

64

language the proof follows directly from some well-known properties of "Robinson's arithmetic" or its variants.

C.

Proof of Part (a) of the Theorem Let

(with

L

consist of the relation symbols

2,1,2,3,3,2 arguments resp. ).

following sentences of

=, zero, suc, sum, prod,

N F ~(~)

neB = >

N F~q0(n)

sno).

Transferring this result to our language Len~a 4:

For every two disjoint r.e. sets

such that for all

there is a formula

n~A = >

~ ~¥V(~n(~) ~ ( v ) )

n~B =>

a ~ VV(~n(V) ~ ~ ~(v))

every two disjoint r.e. sets

~(v)

of

L

in which

~

AIB

there is a formula

~

~ ( d n ) ~ = > V(q0(dn) ) = 2

is true• ~e see that for

~(v)

of

L

such that for

n~ neA = >

n eB ~--->~

Since set

we have

n:

Applying this lemma to our model ~ 3

all

A~B

L

V

~ ~(dn)~ ~--> V(~(dn) ) = 0

is recursive and the operations

[nJV(q~(dn) ) = 2}

tradiction if we take

is recursive and separates A~B

The contradiction shows that

,

n ~ dnp~q0(dn) A

from

B.

are recursive~ the

But this is a con-

to be two reeursively inseparable disjoint r.e. sets. V(~) ~ 2,

i.e. - ~

is false under

V.

This completes

the proof of (a).

D.

Proof of Part (b) Let

A3B

be two disjoint recursively inseparable r.e. sets and let

~

be a

67

formula as in lemma 4.

Let

~

be the following sentence

-

v (Vx~p(x) ^ -, W ~ o ( x )

We shall show that theorem.

~

has the properties stated in part (b) o f the

The positive statement is easy:

semi-valuation that makes

~

true.

Let

to the domain and letting

V~(B) = 2.

Let V'

Then

V = v~Jsub(~)

be the prim. rec.

be obtained from V'

V

by adding

is a prim. rec. s-val, making

true.

~ow let reeursive.

v

be ~_~_~ s-val, such that

and

VJSub(~)

We shall derive a contradiction from the assumption that

Without loss of generality Assume

V(~) = 2.

Clearly

for

i < i < 25.

sum(cl,c",c

TM)

is a subsentence of

atomic sentences, hence ISub(~)

where

is

V(~) = 2.

dom(V) = Sub(~).

V(~i) = 2

V = V

dom(V) 2 Sub(~)

V

V ( V x ~ ( x ) ^ ~ Vx~(x)) = O ~ so

Since every atomic sentence is ~xy3z sum(x,y,z)),

V(~) = 2,

Sub(~)

V

hence

(e.g. -

is defined o~ all

determines a (two-valued) C-model ~

such that

Vr#t is the total valuation function associated with ~9~ .

We can now get a contradiction exactly as in the proof of (a), by producing

a recursive separation between

doJdl3d2~..,

such that

~/~

a subformula of

B:

First construct

and then conclude that

~1 = ~(dn),=>

n~S = >

~I=: ~ ~(dn)~=> V(~(dn)) = 0

~

~

Of course, to get from (V(~(dn)) = O

~n(dn)

A

V(~(dn)) = 2

ai+l).

This is special, in t h a t

[ai:i C ~]

s

generally)~-logic is

applied to (single sorted) languages with a predicate symbol say always realized by

(where

~

where

~

is

(ai:i C ~)~ hut the full domain of the models is left open.

(A variant of the latter uses many sorted languages and fixes the domain of one of these sorts in all the models.)

Even though our results are more'refined' than

the traditional ones (and would therefore be expected to be more sensitive to 'small

F6

differences), the principal conclusions below happen to hold for each of the variants mentioned. (a)

Generalizations of ~-eompleteness theorems (for various kinds of c0-valu-

ations and various kinds of rules of inference).

As in Section I for ordinary

predicate logic, for a given formula

A, in the (same finite) language of

find (primitive recursive) trees

and

(i) g

T~

T~

A, we

AT~F with the properties:

is regulated by the usual rules for ~-consequence with cut and, for

satisfying a few recursive closure conditions, T~

is g

-founded if and only if

~-valuations in ~ ,

A

is true in all (countable) total

which are defined for all formulae in the language of

A.

An ~-valuation satisfies of course the 'adequacy conditions' in that it commutes with the propositional operations and B[x/an]

is valid for some

+

and

is valid if and only if

~.

Note that for every formula cursion equations of

3 xB

A

of the form

x, T~

is ~

of all arithmetically definable sets.

R - - > B~ where

-founded if ~

R

are the re-

is contained in the set

The reason is that, as is well-known, there

is no arithmetically definable truth definition for the (unique) ~ m o d e l of short, every arithmetic formula, true or false, has a ~

R.

In

-founded~-derivation

with cut.

(ll)

TA ~CF

is regulated by the usual cut free ru&es for ~validity,

and for

satisfying a few primitive recursive closure conditions: TA ~CF which

C ~

is

-founded if and only if

and which are defined on

A

is true in all ~-semi-valuations

A.

Clearly, in contrast to (i), there are plenty of formulae have arithmetic ~-semi-valuations of logical complexity on the complexity of

R--> B

which

0 depending essentially Zn,

B.

Strictly speaking, as far as completeness is concerned~ these results are best possible; in the sense that, for a given TA ~CF A

which are

founded c0-derivation trees with,

is valid for total ~-valuations,

validity of

A

A, we have specific trees

resp. without cuts provided

respectively ~-semivaluations C ~

ensures derivability of

A

T~, and

.

Thus

by a specifiq derivation obtained

77

primitive recursively from the formula tingent on validity in g - m o d e l s . ) because it asserts validity of specifie trees

TA

Exercise.

and

A

itself

(only ~ - f o u n d e d n e s s

is con-

But as a soundness theorem our result is weak

A

only for those

A

which are derivable on the

CF T~ .

Analyze the data determining trees C ~ ,

regulated by inference

rules for ~o-logie, with and without cut, which ensure soundness; that is, if and

T CF

are provided with these d a t ~ a n d

are true in all~-va!uations~

are~-founded

T

then their end formulae

resp. in all a~semivaluations C ~ .

Hint. The crucial point is this: when we are given a valuation

V

in which

the end formula is false3 the data occurring at the nodes of the trees must determine, say~ primitive recursively an infinite path N ~V

of

~V

all

FN

are trae (in

V)

and all

has been defined up to the node

derived by a cut.

To continue

~V

N

(in

are false.

where at the nodes Suppose now that

T) and that the sequent at

N

is

one step further it seems necessary to be given

the cut formula used:as part of the data at TA

~N

~C~

N.

(In the case of our canonical trees

there was no need to ad___ddsuch information explicitly because it was provided by

the specific construction rules for Remark.

TA. )

Experience in many branches of mathematics suggests that it is gen-

erallyworthprovin~ plausible impressions like the one above concerning the need for extra data at the nodes (besides the formula derived there).

Naturally, we can

see by inspection that the additional data are used for our ~articular construction of

~V"

The question is whether

the soundness theorem is t ~ e

under a few primitive recursive operations and for all ~

(for all ~

-founded trees of fo~nulae,

regulated by the rules considered)~ even if the ~-derivation trees (explicitly) enriched or

-closed

T

are not

'equipped' with the additional data. -- It goes without say-

ing that the distinction involved arises also in the case of finitary derivations, but the difference is less striking.

The reader should consider the data determin-

ing the trees themselves, whether a tree

T

or by a function defined on the nodes

of the universal tree (which is finitely-

N

is given by its set of initial segments

branching for ordinary logic and infinitely branching for ~-logic) with values determining whether or not

N

lies on

T

at all.

In the rest of this paper we

78

shall use the following data: A derivation tree is ~iven by a function with the root ..., m ~

)

labelled

with

f

(01, and nodes at level

m. E w.

This corresponds

defined on the universal tree n > 0

being labelled

(n, ml,

to the path choosing the node

(l,m I)

at level 2 etc.

f

i

at level i, its node

N

m2th immediate predecessor

consists either of a sequent

chosen) name of the rule by which premises at the level above all, f(N) = ~

is inferred,

and a location of the

or if our derivation tree does not contain N at

derivability

N

is empty.

in ~-logic with and without cut:

It follows from (i) and (ii) in (a) that if ~

class of all arithmetic a ~

FN ~ FN

sets, there are (false) arithmetic

-founded w-derivation,

formulae

of rules with and without cut for ~

of ordinary predicate

logic.

on E

'weak,' in a sense to be described in a moment, semi-~-valuation

C~

If

X C ~

closure conditions

corre-

operations):

so does the ~th jump, say

tively~ with the satisfaction

X (w).

of all the sets in the

for those formulae

of say analysis, w-model or, alterna-

relation for the atomic formulae

of the second sort, for sets).

if the semi-valuation

which are

is defined).

where one begins with an enumeration

valuations

on ~

of the formulae on

The reason is clear (and familiar from the theory of ~-models

are the constants,

are enough

can simply be extended to an ~-valu-

One such condition is this (besides the usualelementary sponding to syntactic

This should

and which ensure that

(preserving of course the truth valuations

which the semivaluation

which have

-founded derivations

On the other hand it is quite easy to write down conditions

E ~'S

A

with cut, e.g. T~, but none without cut.

to ensure equivalence

ation

model theoretic

is contained in the

be contrasted with l.l(e) where much weaker closure conditions

each (countable)

at a

together with some (conventionally

or some other symbol indicating that

(b) ~ - f o u n d e d comparison.

N

FN ~ FN

The value of

A

a. E c 1

A

c

To be precise we consider ~-semi-

which contain enough information

is defined on

where

to ensure that :

it is defined on all atomic formulae in the

79

language of guage of

A.

Given such a semi-valuation

A, the unique extension to

formulae is arithmetic in

X

B

X

and any formula

B

in the lan-

of the (total) valuation on the atomic

and so the satisfaction relation for the whole lan-

guage is certainly primitive recursive in The closure condition on E

X (~).

:X~--> X (~), is certainly weak in the sense that

it does not imply 'S-foundedness

=---~ w e l l - f o u n d e d n e s s ;

it obviously does not do so for arbitrary recursive trees; for e.g. ~

= H

satisfies the closure condltion~ but there are recursive trees which are not wellfounded, yet have no descending

hyperarithmetic path.

It remains to verify that

the same applies to our particular derivation trees (which are regulated by rules of ~-inference).

To do so~ consider

rec. ordering is well-founded

T~

where

A

expresses that a certain prinu

(which can be done in the language of rings with a

free function or relation variable) and choose an ordering which is HYP-founded but not well-founded.

Putting it differently, the condition:

X~--~ X (~)

is also

weak in the sense that

validity in ~-models C ~

Discussion.

does not imply logical validity in ~-model.

In 1.3(b) we referred to the results above in connection with a

rcorresponding' open problem for ordinary predicate logic.

It seems quite clear

that more refined analysis is needed for the open problem.

The weakness of the

closure condition: X ~'-~ X (~)

is patently connected with the restriction to finite

formulae in the language of ~-logic.

By now it is--or should be~-superfluous to

remind the reader that this is a mere fragment of the language, or~ more precisely, of the class of languages appropriate to e-logic.

There is an imbalance in the

syntax itself (where we have finite formulae but infinite well-founded derivation figures) and trivial counter examples (for example 3 to the interpolation theorem, , Cf. Stavi's use of equality axioms in 1.2 which ensure thatevery ternary valuation is binary on the atomic formulae.

8O

which holds of course for a sensible choice of language*).

It might he profitable

to refine first the results above on e-logic for suitable infinitely long formulae and infinite derivations,

perhaps in terms of the segment

metic hierarchy where

is the length of certain autonomous progressions

r

duced by one of us when infected by traditional and) studied by Sch~tte and Feferman° some

well-founded

~-derivation

founded ~-derivation well-founded

of

derivations

of

~

interest in

of the h~gperarith(intro-

'predicative' mathematics

Here we know that (*) if the formula

A by use of cut are in

A without cut.

Hr

A

and

so is some well-

True~ we want more; not a result about

but something about,

say, ~-fourlded ~-derivations.

inspection of one of the more careful metamathematical

proofs

But

(which does not apply

induction on the given derivation to unsuitably complicated formulae),

may solve

our problem too. (c)

To complete these preliminaries

we consider,

for illustration,

concerning ~-logic with and without cut,

a Img.ir of more simply related sets of rules;

'simply'

in the sense that it is easy to show that they generate the same sets of theorems on C - f o u n d e d

trees, for example if ~

is closed under recursive operations.

have chosen these rules because their so to speak model theoretic easily decided, illustrating

and so--it eeems to us--they are, pedagogically,

refinements

recursion theoretic properties of ~-derivations

model theoretic

questions of transformations inclusions

in particular,

of transformations

Realistically

(of derivations)

speaking the recur-

arise only when the

(between sets of theorems) are known to hold.

sense, model theory comes first and recursion theory afterwards. relationship

for

regulated by the rules considered.

Remark (concerning matters of pedagogy). sion theoretic

relations are appropriate

concerning non-model theoretic matters;

We

is opposite to the historical

working on cut elimination ~rocedures

In this

This lo~ical

sequence of events since people had been

for 25 years before semivaluations

were even

. The 'counterexample' is almost as trivial as that for ordinary predicate logic when the constants T or l are left out, A - - > B holds, but A and B have no symbols in common.

81

mentioned (in [Sch]).

The reader interested in 'sociological'

aspects of logical

research may wish to use these facts to reconsider the notorious lack of interest, of the silent majority (of logicians), in proof theory: or to an objective lack of interest?

Was it due to 'prejudice'

of the results actually stated in the (bul~

of) proof theoretic literature. Repetition. Given a set ~ by adding the rule: derived

F ~

from

P ~ ~

derive

F ~ ~

be the system obtained

with the stipulation that the

is equipped with (some name of) the rule above, for example, Rep.

Evidently

~

derivation trees.

and

~+

generate the same sets of theorems o ~ well- fo~anded

This (extensional) equivalence obviously holds for Z - f o u n d e d

trees too provided ~ For suppose an

of (formal) rules, let ~ +

contains all sets which are recursive in the trees considered.

~ + -derivation of, say, A

is ~ -founded.

It cannot contain an in-

finite (uninterrupted) sequence of application Re2. dominated by the node

NO

say

beeause, by the stipulation above, such a sequence has no branching and is patently + (primitive) recursive in the given derivation trees. So the ~ -derivation collapses to an ~-derivation

by olmitting the rule Rep altogether.

The collapsing process

is not total recursive, by pp.46-47; it is partial recursive by searching at each NO

for the last application of Re__p_pin the sequence beyond

NO

(and the search

terminates for our .~-founded trees).

One important use of R ~ ,

in work considered in 11.2, depends on the follow-

ing result where the trees considered are regulated by f

~+

and given by the data

at the end of II.l(a). There is a (Kalmar) elementary operation, call it

an ~+-derivation

then

~T = T

and, for all

H

such that if

T

is

T

+ H(T)

is an

~

-derivation (but not necessarily well-founded).

The definition of

H

is clear.

If

f((0))

is not well fondled, take

to be a derivation of, say, 0 = 0. If

(Hf)(N')

(n, m I . . . . .

m~)

has been defined for

on the path

n+l

DT: cuts are pushed

of

0T

Put more positively,

What

0T

Now

p

then there

pT

contained a cut of But this contra-

has a cut of degree n at ()

is

'down' the tree order-

with cut degree c. if

N.

and

N

N.

when considering the main step or, as we said~ more

when formulating various clauses in the functional equation for

in effect~ a uniform scheme for reducing

'simultaneously'

p--we

infinitely many

Of course when a cut, say a maximal one in the tree ordering of a given

derivation

d

latter are not tree

cut of degree

has a cut of degree n+l in the stump between

for~lly:

cuts.

and (the level of)

is finite and = n+l, and if

To avoid misunderstanding:

giv%

I}

there is no

c > n, there would be a node

dicts what we just said. then

between

> n in this stump of

If the cut degree of

roughly as follows.

0

is determined by what happens on a finite number of

in the stump of

is no cut of degree

(even) for deriva-

has been

'considered'

'reconsidered'

and replaced by cuts of lower degrees, the

in this simultaneous

reduction;

we go on

'down' the

d. The standard treatment applies

formula

C

predecessors the cut rule

at

N NI

straightforwardly

is the argument of both rules (r I and

N 2.

For example,

if

C

is

to the cases where the cut

and

IC' and

r2)

at the immediate

~¢is

the name of

93

dI

d2 t

P2 f-C', ~2; r2

Pl' C' ~ ~i; rl Pl "I C', ~I; ~

then

p(dN)

is given for suitable

rl, " r 2"

by

~(d1)

P(d2) T~

v~

r l, c'~" al; r I

r 2 ~c', a2; r2

% u ~ 2 , - - q _ u q ; ~ - c,

The new cut formula that is, ~ C'

C'

is patently of lower degree than Jhe cut formula

which is eliminated.

Remarks.

H

(i)

on the end piece of

d N.

(ii)

0

T

are either

T

rl, r 2

or Rep, depending

The use of Re~, in the last inference of

T, as required above.

modulus of continuity of of

H

The 'suitable' rl, r 2

required to make sure that any stum~ of stump of

pT

0(dN)

is

is determined by the corresponding

Put differently--in more civilized language--a

at the argument ((), N)

should be bounded by the level

N.--The reader should verify that, by help of this property, the action of

that is, the value of

(0T)(N)

is nicely expressed in terms of values

(a finite number of) nodes

N. l

positions of the nodes

depend of course not only on the position of

N. m

at levels between

the universal tree), but also on the formulae Ni, that is on the values If say

C,

V xA,

C

pI ~ Vx

and the level of

for

N.

The

N

(in

and names of rules occurring at the

T(Ni).

is a conjunction, we have two cuts. and

()

T(Ni)

P,

A, A I

If

has the derivation dI P I ~- A[X/aNl] , 2~; r I rl~

VxA,

~i; i-V

C

is a universal fomula~

94

we convert vatians A[x/t]

dI

into standard form,

dl[aNl/t]

for

PI ~ A [ x / t ] ,

is less than that of

suppose

P2' V x A ~

of) V x A

V x A

~

(however

V x A

V x A are suppressed altogether, A[x/t]

to

finite segment

V ~

(), N'

Whenever

that is, of

is applied in

at some node

in

~,

t.

this con-

These uses of

dl[aNl/t]

(the cut

N', inspection

to

pT

(p

N'.

of the

N 'above'

N'

with

Note that new deriva-

so to speak

and hence we have the bound on the modulus of continuity of

p

'stretches'

T)

required in Remark

(p.56),

Next we have the case where the cut formula at both rules at the predecessors principles

Now

V x A--as in the standard treatment).

~

T

may be).

(an ancestral occurrence

determines whether there is some node

tions are added in this passage from

the degree of

t

V~,

and replaced by cuts with

the cut formula V x A, and thus tells us what to do at

(ii) above

Hence we have deri-

V x A, for some term

having lower degree than

Thus, if the rule

too.

'complicated' d 2.

is the argument of an inference, A[x/t],

dI

where, by definition,

has the derivation

sists in contractin~

formula

and call it

N1

and

N2

are taken care of by (ii) above.

of

N.

N

is not the argument of

Standard uses of inversion

The (only) delicate situation arises

when we have that seesaw of cuts involved in what Gentzen called

'end piece

' or

the similar figure which,

in the related context in the style of natural deduction,

is called

In the case of finite derivations

'main branch.'

course finite; but also in the case of well-founded For general ~

these objects are of

(infinitary)

~-derivations.

3 they cannot be expected to be finite (they are if ~

satisfies

K$~ig's lemma for finitely branching trees in the sense of the Appendix). the

'main'

steps in the standard treatment involve

device is needed to ensure the continuity of

p.

the whole end piecej

Since some

An obvious step is to use

as follows: Starting at

N

and going down the tree ordering~

times to the end formula of

dN

we apply Rep

unless the end piece of

dN

k has

depth < k. Since the end piece is finitary,

one can decide

primitive recursively

(in the

95

data of

dN) whether or not the proviso is fulfilled.

Warning:

Even if

dN

contain one; in this case and hence an infinite path

does not contain an infinite recursive path~ P d N m a y ~

will contain an c ~

infinite recursive (finitary) tree,

(or even of degree < 0'--by familiar basis

results). Before describing the relevance of this warning to topic (b) of the introduction to 11.2, we suggest to the reader that he should 'compare'--as one says-(i) the functional equation for

p which is here extracted (in outline) from

standard proofs of the permissibility of cut, and the primitive recursive solution of this equation with

(ii) the functional equations without the use of R e p a n d

their solution by means of the recursion theorem, e.g. in [C].

The suggestion

would be quite silly (if not empty) without some indication of the features which are significant for

the comparison.

First of all, if ~

-founded trees

are considered at all, the simple look of uses of the recursion theorem is deceptive since one must check if the proofs use induction of suitably limited complexity.

More importantly~ the recursion theorem operate~on

notations, and so

will often yield unique solutions to a functional equation even if the latter possesses infinitely many 'extensionally' different solutions]

Thus the work

with the recursion theorem is incomplete, unless the structure of those 'unique' solutions is further analyzed. Digression on applications to infinitary derivations of infinite formulae (not:to the

~-logic of finite formulas), where the recursion theorem was first

used in the way mentioned above.

We are well aware of the fact that formulae in

the finite language of e.g. first order arithmetic can be 'replaced' by infinite propositional formulae which express the same proposition. In particular, as for Tait has stressed3~(hackneyed ) questions of consistency or underivability of some schemes of transfinite induction, the answers for arithmetic can be read off from the corresponding work for infinitary propositional logic.

But this procedure

turns out to be a hit of a detour when examined more closely.

Given the mapping

from (formulae of) ordinary first order arithmetic to infinite propositional

96

formulae, one translates finite derivations with induction into infinite derivations of propositional logic~ and then applies cut elimination for infinitary propositional calculus.

But to use this work for less hackneyed results, for

example for the less obvious metamathematical properties of arithmetic itself (such as delicate reflection principles discussed further in II.2(c) below), the cut-elin~ination procedure for the infinitary language has to be formalized; this needs finite codes for the infinite

formulae.

Inspection shows that, when

one starts with a finite (quantified) formula (of arithmetic) each infinite formula that occurs in the transformation is e~uivalent to some finite arithmetic formula~ So the most natural codes for these infinite formulae are precisely their finite equivalents.

This reservation about the use of infinite formulae for the present

quite specific purpose is of course consistent with the Discussion in ll.l(b) on the use of infinite formulae for solving other questions. (b)

For which ~

does

p

of (a) preserve

~

-foundedness?

If the answer is to be formulated in terms of instances of the comprehension principle satisfied b y ~ obviously, D

, pretty sharp results are known.

does not preserve

derivation of cut degree theoretic) information.

n+l,

~-foundedness

if some

but not of cut degree

As to lower bounds~

F ~ A ~;

has a ~

°founded

cf. ll.l(b) for (model

As to upper bounds, by the Warmin~ at the end of II.2(a), 0 ~-comprehension

it is sufficient that ~ - s a t s i f y 0 lently Hn-COmprehension (for

n = i, ..., ~), often called: arithmetic compre-

hension principle with parameters.* bounded cut degree, say c, our

(with parameters) or equiva-

p

e

Provided the derivations considered are of preserves

is a cut elimination procedure, called

Oe

~-fo~udedness

for such ~

and

at the beginning of ll.2(a).

For further progress one has to pay attention to the choice of concepts in terms of which sharper results can be stated.

But we have not considered the

Amusingly, by an observation of Friedman mentioned in the Appendix, this is equivalent to the requirement that ~ satisfy K~nig's lemma for semi-infinitely branching trees. But we do not know a proof of the preservation of ~ -foundedness~ in which this alternative is used.

97

matter s ~ f f i c i e n t l y t o

Natu~llythe

literature has

considered many of the obvious questions and obvious parameters

in these areas;

for example,

have any serious comments.

extensions

to systems with infinite formulae and p a ~ m e t e r s

of ordi-

nal length of formulae and ordinal bounds on cut degrees. It is perfectly possible these that results in terms of~quite simple minded syntactic parameters will turn out to be useful.

But it seems at least equally likely that better results will in-

volve more delicate restrictions (mathematical)

on the class of formulae used, involving their

content and not only their (logical) form.--We have in mind the

following analogue in the area of ordinary predicate decidability,

logic, in particular,

where there are striking results for mathematical

for classes of formulae of the form: formula in the language of

A).

A --~ F (A

being the axioms and

What can we expect from enrichments? to 11.2.

functional equations of fumctionals

any

those

content.

of the kind described in the

can often be solved by operations

for solving our particular

siveness

F

The first possible use that comes to mind is quite general

if the arguments

as the restrictions

that is

e.g. by means of logical complexity;

classes had to be discovered by reference to their mathematical

introduction

theories,

These classes are quite artificial for any of

the socalled logical classifications,

(c)

on

(functions)

equation (for

in a more restricted class

are suitably enriched.

p), no enrichments

By II.2(a),

are needed as long

considered involve only continuity or (even) primibive recur-

(except that, perhaps~

suitable enrichments).

the effects of the rule Rep can be achieved by

As mentioned in the introduction

to 11.2, even more is

true: not only are the enrichments not needed, they do not even 'affect' the solu+ tion, in the sense that p of ll.2(a) can be enriched to p , in such a way that ÷ the relevant retract of Exercise. language,

p

is

p

itself.

The reader familiar with

'ordinal assignments'

ordinal bounds for the length of)well founded

to (or, in civilized

w-derivations

the following hints sufficient for working out the details (assuming~ latively,

that he wants to do this).

the increase in the ordinal

may find

more specu-

On the one hand the standard estimates,

length of a derivation by reducing the cut degree

for

98

('by i), ar~independent the properties

of the particular

rules used in the derivation.

of the ordinal functions used in these estimates,

etc., are simply their

'recursion equations'

2a~ 3 ~ or

is needed because of the (novel) use of Rep.

since each application bounds for

~

to ~

of Rep introduces at most an ~-sequence, ; so if, for e x a m p l %

have an easy extension to ~ - f o u n d e d the familiar formulation

example,if the formulation

vant.

defined for

Natturally~

then ~

derivations.

c O J is defective

~ ~ ~0

This is manor

and hence changes

= ~.

In short we

trees particularly when re-

(and possibly misleading; ~

~--~

but looks as if well-fo~Jndedmess

of

the reader will not forget here that a correction

in short~ a significant

tablishing the significance The significance

of ~

but also a selection restatement.

of a restatement

trees.

But it may fairly be said that

introduces the binary function:

requires not only some formal restatement reformulations),

~ = ~

for the class of well-founded

stricted to trees of ordinal ~

of course,

the corresponding

on trees also apply generally and not only to well-founded

A final modification

~; ~

for

this is~ were rele-

of such defects

(among all correct

Part I was devoted to es-

in terms of

~-founded

trees.

-founded trees may add some interest to a

pursued by one of us--of axiomatizing,

~

and continuity at limits~ which are

valid also when applied to order types that are not well-founded; operations

Furthermore

for a given measure of complexity

project ~, the

class of theorems of a theory, which are of complexity ~ c~ by means of axioms of complexity ~ c; for example,

in the case of arithmetic

restricted to predicatesof

by means of ~-induction

(for

~ ~ ~0)

complexity ~ c ; cf. p. 331 of [SPT] or

[P].

Other variants of this project establish reflection ~rinciples

tic) applied to formulae of complexity ~ c using as metamathematical principle of ~o-induction applied to predicates quantifier-free

of complexity

c -induction for the case of quantifier-free

(for arithmemeans the

c"; in particular~ formulae.

In this

O

connection,

enrichments

of (infinite)

derivations were envisaged,

~-derivations

in particular,

by adding (finite) formal

on p. 122 of [SPT If] and used in [LE],

but with different answers to the obviously crucial questions: What are these formal derivations~

say

DN~ at the node

N

to prove?

99

From which formal system should we take

DN?

Neither [SPT II] nor [LE] seems to us to face these issues squarely. former chooses quantifier-free systems and has nated by and

DN

prove that the subtree domi~

N, assumed to be enriched by ordinal notations, is

(ii) compatible with the tree ordering.

can be expressed in free-variable systems.)

The

(i) locally correct

(Since both properties are

~i0

they

The purpose of the choice was evidently

'reductive', avoiding the use of logically complex operations in the exploitation of the

meaning

of logically compound formulae.*

But this conflicts with the evidence,

collected in the present paper, in favor of using abstract language in metamathematics; the choice of [SPT II] is bad since a formal derivation anything about an abstract tree!

DN

can't possibly prove

The data determining the trees involved must at

least be defined in the language of

DN.

Of course, the choice 'works' for the

specific project mentioned since the particular trees that turn up, are (primitive) recursively defined. project.

But there is nothing terribly exciting about this particular

One of us would go so far as to say that we probably have a good chance of

finding a better project simply by requiring that the choice above be inadequate for solving the problem! In [LE] (or, more precisely, in a preliminary draft, since we have not seen the final text) there is a different choice:

DN

simply proves the sequent

at

N, and the formal system used is first-order arithmetic itself (with intuitionistic logic, the corresponding results for classical logic having been proved earlier by an ad hoc trick~ using properties of the socalled no-coumter example-interpretation [KL]).

This choice is perfectly meaningful for abstract trees, though this

--for us~ crucial--fact was of course not used, and not even mentioned in [LE].

This was needed to avoid the defects of socalled operational semantics, discussed loc. cit. This interpretation explicitly aims to be reductive, yet uses logically compound expressions in its explanations~ This pointless enterprise is to be contrasted with familiar explanations of the(classica~ meaning of logical operations since these explanations are not intended to be reductive.

1O0

However,

it is not clear to us that this choice is adequate for the project of

getting sharp results,

for example,

tion principle mentioned above, (d)

of proving the sharp versions of the reflec-

for formulae of bounded complexity.

Second thoughts on the Paper presented at the Symposium on the Theory

of Logical Inference,

Moscow 3 1974.

As mentioned in the introduction

we are not altogether

satisfied with

(i) the choice of language for ~-logic nor

(ii) the choice of axioms (and rules) for atomic formulae. discuss these defects in some detail ditional proof theoretic)

to II.2,

It seems necessary to

just because they do not affect the (tra-

aims of that paper;

such defects do not spring to the

eye of the reader who is prepared to accept an author's aims (and is therefore particularly (i)

The language considered

(besides + and ×)with axioms to be discussed further in (ii) below and

free (monadic) function variables. A

is quite familiar from the proof theoretic

u such as Schutte's book [S:i~ : first order arithmetic with function

literature constants

in need of outside help before he questions traditions).

being defined as

(x ~ 0 --> B)]. are decidable, theorems.

A-->

The logical operations

0 = I

and

A,t B

as

(~ , ^,

V)

^

is embedded in the set of ~-models as far as

(~ , ^, ~0 is concerned.

To the student of ~-logic the language is artifically socalled second order arithmetic no function

A)

thus since atomic formulae

The infinitary rules make the system complete* for

the fragment

(--> , ^, V, ~),

3 x[(x = 0 - - >

The rules of logic are intuitionistic; the classical fragment

are

quamtifiers.

not essential.

Not even

can be expressed in this language since there are

For the preoccupation

of traditional

For on the one hand, validity in countable

to validity in all ~a~models;

restricted.

proof theory this is

~-models is equivalent

and on the other, the notion of countable ~-model (of

a formula of second order arithmetic)

is perfectly nat~arally expressed in the re-

stricted language as follows. Since the matter of completeness of the (intuitionistic) system is not directly connected with the present paper we do not go into it here; except to remark that the familiar incompleteness results, for example, on the assumption of Church's thesis, apply to the case of ~ l o g i c too.

101

A function variable is used to 'code' or enumerate all sets of an arbitrary countable ~-model and ~uantifieation over sets in that model is then expressed by numerical quantification (over the codes of the nth set).

In this

way, the restricted language is quite suitable for standard questions concerning the set of ~ e o n s e q u e n c e s

of given axioms in the unrestricted language.

However, for the kind of structural properties that are of interest to us, the 'reductions' or 'embeddings' just described are quite inadequate, and even deceptive.

To get an idea of the de~th of the deception involved the reader is

recommended to look at Chapter ! of Statman's dissertation [St], concerning explicit definitions (and the choice of derived rules or 'axioms ~ which are demonstrably satisfied by the defined expressions).

A ~rofound misconception is involved whenever

two objects differ trivially in terms of that conception, but differ strikingly in structure and general behaviour (when viewed by the light of nature). To be specific, it is simply not true that the structure of derivations in classical logic formulated in the 'full' language

( ~, ^, v,~,

faithfully in the fragment ( I , A , ~0 in -which

(v,

Nor does the intuitionistic theory for the fragment

-~, 3)

(l , ~ ,~0

3)

is represented

cau be defined. reflect the classical

theory ~ h o u g h it contains the same class of theorems) since, obviously, some classical derivations in that fragment are not intuitionistically valid. More positively, we shall now £ormulate a classical analogue to the problem solved by the socalled E-theorems by one of us [Mi] for intuitionistic systems, where attention to structural properties is essential. to each derivation that

A[X/td]

~ of an existential theorem, say,

is (for~ally) derivable.

And

press the definition which the proof (say 'unwind'

d).

application

td

The E-theorems associate ~ x A

a term

td

such

is seen, by inspection, to ex-

d, expressed by) ~

provides (when we

As mentioned already, the E-theorems constitute the first genuine of cut-ellmination or normalization:

in the normal form

Idl

of

formula

Idl

is inferred.

3 x A

of

d, in particular, in

the term A[X/td]

td

actually occurs

from which the end

Now, in the case of a classical derivation

I02

dc

of

A[x/t]

~ x A

there is of course no guarantee that there is any

is known to he true,

let alone formally derivable

t

for which

(in the system considered);

-C

and a f o r t i o r i

there is no guarantee that

Problem: or not: or not

~c

To determine

~c

which realizes

provides a formal derivation,

sidered.--0f

course~

provides

such a

for any given derivation tc d

provides a term

d

say

it is not assumed that

questions of explicit realizations

~ x A

d' d'

dc

of

of

t. 3 x A

(i) whether

and if it does, (ii) whether A[X/td]

in the system con-

is intuitionistic

are quite independent

since, after all~

of matters of constructive

provabi lity. A simple minded candidate for a solution is just this : of normalization A[x/t]

or cut elimination:

dc

actually occurs at a node of

of systematic

Clearly,

and s~ if a theorem of the form

(People not disturbed by the possibility

lead to the same

Do different

t?)

it would be premature to assume that the relevant information con-

tained in

~c

where

is the(canonical)

(ii)

IdCl.

IdCl

error may be interested in the purely formal problem:

current rules of normalization

A-

--~

find suitable rules

is preserved when

dc

is 'translated'

equivalent

of

As observed at the beginning

A

in

into a derivation of

~Vx~A-

( S, ~,V).

of (i), the symposium paper takes as

axioms a set of atomic formulae closed under cut or under the rule of equating equals to equals.

(Here there is simply no problem of proving closure under cut

for atomic formulae.)

The function constants considered are taken t o be comput-

able; at least tacitly, we are assumed to have a valuation up out of those constants.

Certainly~

function for terms built

the choice of such sets of axioms is per-

fectly natural for traditional proof theory which assumes that we are really in doubt about the validity of the formal principles the validity of the metamathematical proof by means of such methods, of

H 0I

sentences

methods to be used.

V X[fo(X ) = O]

0 HI

Then a consistency

answers a basic question about formal derivations where

fo

sistency proof would tell us that, for each (in our set) if the

of proof studied, but not about

is some function constant. n, f0(sno) = 0

sentence is formally derivable.

The con-

is one of the axioms

103

But if we have no doubt about the validity 0 HI

know that those

sentences are true) we cannot avoid the question:

What more does a formal proof of VX[fo(X) = O]

(Thiz

VX[fo(X) = O]

provide than that

is true?

is to be contrasted to the ease of say a

formal proof provides a function f

of the formal principles (and

f:x --> y

~

statement

such that

Vx ~yB

Vx B[y/f(x)]

where a is true and

lies in a restricted class depending on the formal principles considered).

More specifically~ What more does a con@tructive proof of

Vx [fo(X) = O]

provide than

can be expected from a non-constructive one? One of us is attracted by the Conjecture: computations of

A constructive proof

~

provides a specific sequence

dn

of

fo(sno) = O, and, in general, a non-constructive proof does not.

~YB. Contrary to an almost universal (and almost amazingly thoughtless) misconception, a constructive proof does not generally provide a particularly efficient means of computation~

After all, when we have a proof of

very efficient (and sound) rule of computation is simply this: to O.

Now~ suppose

tive proof of set theory.

f0

Vx [f0(x) = 0], one put

is such that we have a non-constructiv%

f0(sno)

equal

but no construc-

k/x [fo(X) = 0], e.g. if this formula expresses the consistency of Then constructive rules are bound to be less efficient computation-

ally. Clearly, we cannot hope even to begin a study of the Conjecture (by means of cut-elimination) if we use the fact that all true formulae the axioms:

fo(sno) = 0

are among

we thereby avoid any analysis of (numerical) computations or of

their relation to proofs of identities~ that is~ formulae of the form V x [fo(X) = 0]. tion~ d

n

-NB.

For e x a m p l %

The conjecture requires a proper choice of normal deriva-

it is clear from inspection that many

!

provide computations

which use recursion equations for functions that do not occur in the defini-

tion of

f .

Thus a choice which makes all normal derivations of

computations from the defining equations for appropriate here.

fo

fo(sno)

simply

(as in [ML] ) would be quite in-

104

9-

Reappraisals.

(a)

We hope that our (combined) knowledge of proof theory and

other logical literature informed

judgment.

is wide enough for sound judgment;

Granted this we need have no doubt that our abstract formula-

tions provide a much better basis, proof theory, conditions

for analyzing

than what is done elsewhere

on ~

In particular,

, for

~-fo~ded

existing methods and notions in

(we mean analysis in terms of closure

trees and various kinds of valuations

in ~

).

we believe that we have tightened the standards of scientific

rigour used in current (proof theoretic) those of an immature

practice~

its standards are more like

science than what could r e a s o n a b ~

than 50 years(of proof theory). (i)

in other words, for

be expected after more

We mention ~ examples.

Systems and methods are accepted as

attempt to disting~ish between a meaningful

'natural' without any determined

sense of 'natural'

(that is, modes of

thought to which we return even after trying out alternatives)

and mere familiar-

ity (to a particular author often ignorant of existing alternatives). (ii)

Mere coherence

is accepted as evidence for an adequate analysis,

example~ an analysis of the significance (and, as happens sometimes,

of formal discoveries

of Gentzen's

no other test than that of coherence

semantics which,

or normalization was in turn, was put to

For a long time~ no attempt was made to find other properties

Gentzen's discovery. dissertation (iii)

(of the pro-

for the obvious appeal

of

(This has changed, e specially with Chapter Ill of Statraan's

[St]). Isolated observations,

often ponderously,

curiosities

the

(which we regard at best as a useful negative

cedures above) which could account more convincingly

ated~

Specifically~

formal discovery of cut elimination

sought in its role for socalled operational

test).

of obvious interest

such an obvious appeal can be obviously reliable even

if it is difficult to analyze the nature of the interest). significance

for

admittedly of some logical charm, are elabor-

without any attempt at distinguishing

and useful scientific

tools.

This happened,

for example,

proof theory of socalled subsystems where some (unexpected) between restrictions

between mere in the

relations turned up

on methods of proof and the content of the theorems proved.

But these relations were not tested for their interest,

as if people were paralyzed

by the surprise of finding such relations at all. We are quite aware that judgments on the points raised in (i)-(iii) need not be easy.

But similar questions arise in (the development

too, and they have been satisfactorily accumulation

settled; often,

the various

by the

of proof theory should be mentioned,

a

results (which we should like to put through tests)

were accepted simply for their claimed meant--and we may agree--was

'philosophical

interest.'

that many of them had philosophical

see no guarantee that they are therefore (b)

quite prosaically,

of knowledge and pointed critical comparison of alternatives.--For

reference in (c) below, a 'peculiarity' kind of pun:

of) other sciences

(even: philosophically)

But what was character;

we

interesting.

We certainly do not wish to overstate the interest of the present

paper (whatever its virtues compared to current practice may be). particularly

critical attitude is appropriate--as

ditional proof theory as its starting point. has its roots in Hilbert's program, ~otheses.

We must not forget that this theory

which arises from

untenable epist~molo~ical

but the brutal fact remains that there are

between (the tendencies

ence there are correlations done on them.

with any work that takes tra-

We may not like Jehovah's threats to punish the sins of the fathers

on the third or fourth generation; correlations

In fact, a

of) related generations.

between the

qualities

Similarly,

of hypotheses

in sci-

and of the work

We do not expect that milch, let alone the bulk of work concerned

with the properties

of the (hypothetical)

ether has a permanent place in science

--however natur~,l or coherent the idea of the ether may have been to those who dabbled in the subject. Conversely,

we do expect that some facts which people have stumbled on in

the pursuit of a false

or even absurd

hypothesis will be of use provided the

106

interpretation

of these facts is rigorously disentangled

from the hypothesis.

readers may (not) like the comparison with the la~d Columbus followed his false* hypothesis

Some

stumbled on when he

about a sea route to India along the latitude of

Spain (or thereabouts). As a practical conc~sion, expect the most interesting

to which we return at the end of (d) below, we

developments

of existing proof theory to concern

topics which are far removed from its original aim, which was to formulate and support a particular (c)

(untenable) hypothesis.

To avoid (what would be) a basic ~sunderstanding,

that the views expressed in (a) and (b) are by no means

it should be stressed

'anti-philosophical';

fact, they are not even directed against traditional philosophy--neither academic

sense of traditional

of philosophical

contemplation.

As far as the heuristic philosophical different

questions

We merely advocate certain distinctions.

tradition is concerned,

genetically and phylogenetically

able doctrines~

themselves

~lue

of the

At one stage (which certainly occurs, both onto-speaking) we know very little, and have only first fact, the traditional philosophical

to us when we know little.

Contrary to fashion-

it may safely be assumed that the questions are meaningful at

that stage.

We should go f u r t h e ~ W h e n

(meaningful)

questions to ask~

phenomenon,

epistemological

it seems necessary to distinguish between

As a matter of historical

questions present(ed)

in its

'puzzles') nor in its popular sense

or, more generally,

stages of knowledge.

impressions.

(or

in

for example,

about its velocity,

we know very little~ they may be the only

Suppose we know very little about some physical

light; not enough to ask s~ecific questions

(for example,

at a stage when we do not even know if it has a well-defined

velocity or how to measure it).

Even so~ we can always ask if it is real.

Or,

to return to our present paper, when Hilbert introduced his consistency problem~ We have not checked whether Columbus' hypothesis was not only false, but even absurd (when he reached the West Indies) in the sense of being easily refuted f r o m ~ a l l ) the evidence available to him about (i) the distance he himself had travelled, (ii) the length of the known sea route to India and (ill) known estimates at his time for the diameter of the earth.

107

what could we have asked about proofs or principles structural

questions,

natural deduction:

Surely, no Specific

say, about the genus of formal derivations

But questions about validity,

principles presuppose

of proof?

little knowledge

in particular,

(of the principles

in systems of consistency of

one is talking about).

--We certainly do not assume that questions which strike us when we know little, never continue to be rewarding when we know a lot. reason why this should happen often. that they lose their meaning,

our

when they have merely lost their interest.)

is the much-touted

'basic' philosophical

'dangerous'

dramatic

conceptions

tradition which seems to us to require

claim about far-reaching

of

particularly

The matter is relevant to our own subject

In fact it seems plausible that the claim above is behind the

lax standards of rigour criticized in (a), at least~ claim,

consequences

(the latter being, allegedly,

if they are not analyzed).

in several ways.

we see no

(As was said already, we also cannot accept

The second aspect of the philosophical attention,

But, eonversely~

in this sense: granted the

it would indeed be unlikely that the (pedestrian)

tests which we want to

apply, could be of much use; the battle is lost before we even start, because (according to the claim) those

'basic' conceptions will have

whatever word is currently in fashion)~

'bewitched'

Be that as it may, the empirical evidence

of the development of the sciences--and of some proof theorists, theory--does

us (or

not seem to support the claim.

(So experience

be useful for examining the general merits of the claim.)

if not of proof

in our subject may We leave it to

the reader to go over the various hackneyed instances usually cited in support of the claim, and also to look for various banal counter-examples. (attractive and) instructive so needs some analysis;

We find it more

to go here into a case which is not clear-cut and

specifically

the work of Herbrand

(which one of us cites

frequently in this connection). The case is interesting because,

at least during his very short (logical)

career, there were some pretty evident connections between Herbrand's philosophical conception of mathematical his specific

reasoning and--both positive and negative

logical interests.

We describe these connections

sides to--

before taking up

108

the quite separate matter of farreachin~ fruits of a different,

model theoretic

consequences

conception).

By now it is a common place that Herbrand needed to prove the completeness cause (ideological)

(or Skolem) had all the formal tools

theorem but stopped short of even formulating

finitist doctrines

other side of the ledger,

(if he had lived to see the

led him to reject it as senseless.

On the

it should be said that without those doctrines~

a model

theoretic proof of the socalled uniformity theorem (if A, is valid so is some disjunction for terms

t

in the language

of

AI V .-. V A n A)

~ ~ A

there are terms

A.l

is of the form

if

~ = ~(d) for some primitive recursive

ation are also related to the doctrine;

on the hypothesis

where

theorem (which states:

elaborate version for more complex formulae than

(by one of us) that even Herbrand's

~ ~ A, for quantifier-free A[~/~

would be the natural stopping place, and there

would be no reason for going on to Herbrand's vation of

it be-

is a deri-

~; with an

~ x A; the reasons for the elabor-

cf. SPT I I p .

errors

d

130).

It has been pointed out

[DA~may be related to the finitist doctrine,

that he, in fact, knew the (easy) model theoretic proof.

believed that all finitistically meaningful

Since he

statements have finitistic proofs a n ~ a y ~

there was no reason to check his own proof very carefully.

(This hypothesis may of

course reveal more about its proposer than its subject.)--Granted conception may be said to have prevented him from discovering

all this, Herbrand's

the more important com-

pleteness theorem (which, after all, was discovered independently,

perhaps just be-

cause its interest is so clear), but to have led him to his own less important~ also more recondite theorem.

but

This state of affairs is certainly consistent with what

was said earlier about the heuristic value of philosophical

conceptions at an earl~

stage of a subject. On the other hand we see no reason to suppose that Herbrand's would have had far reaching consequences~

that it would have prevented him from

learning fro___mex~erience and from chang in ~ his views accordingly. and as an example of the

'~ureasonable

conception

effectiveness'

In a sense--

of the test of formal rigour

(of which we shall find further instances in (d) below>-the

very errors in his

work would have helped him see the defects of the conception;

particularly,

if our

109

hypothesis

concerning these errors is sound.

We could well believe that some slow-

witted or basically tired individual, terrified (consciously or unconsciously)

of

never having another idea in his life, might be reluctant to give second thoughts to his conceptions

or to his convictionsjfrom

without finding anything to replace them. the case of Herbrand.--In conception

But we find this quite implausible

is this)principally,

a property of the individual or of the conception? of the type described that,

about the effects of philosophical

first impressions,

in

any case, when an individual is reluctant to change his

Are there so many individuals claims

fear of seeing them fall to pieces

conceptions

statistically,

the dramatic

and, more generally,

of

are sound?

As so often with glib claims of powerful psychological (which are so fashionable

in this century),

the dramatic

'influences'

claims above acquire a

certain air of unreality when we try to apply them to ourselves.

We certainly

were not born with the conviction that abstract language was suitable for elementary metamathematics!

we learnt this from experience.

ethereal as to be immune to the temptations (d)

To continue

(and conclude)

positive practical consequences consequences pressions,

Or are we so exceptionally

of false conceptions?

on an up-beat note we shall now formulate

of the reappraisals

in (a)-(c).

The

'negative'

consisted in some kind of debunking of problems which, on first im-

seemed plausible and/or fundamental~

but are not.

So the

'positive'

consequences

may be expected to consist of problems which seemed marginal or mere

refinements,

but in fact are not.

many aspects): (i)

Loos_~al and mathematical

reasoning.

Some problems on cut-free axiomiatizations.

of logic as fundamental, some

We consider just one topic (and only 2 of its

'additional'

axioms,

and of mathematics

as

like an after thought.

Most logicians tend to think

some kind of a p p e D d a g % Accordingly,

they look for a

(very) few fundamental systems such as (finitary or infinita~y) icate logic, and impredicative how

'look after itself'.

test, namely the

second or higher order logic.

This impression

'reduction'

of arithmetic

given by

first order pred-

The rest would some-

seems to be supported by the first crude to logic.

The test is crude inasmuch

110

as a mere embedding of theorems is considered. both in ~-logic, in (finitary)

Formal arithmetic

of II.l (a special case of infinitary predicate

second order logic to which Peano's

the set of numbers

is

can be embedded logic), and also

successor axioms are added (and

defined in the style of Dedekind).

It seems to us significant,

that this crude test was not followed up by

more searching tests, not even by the most obvious ones; obvious the moment we look for tests at all (provided only we are familiar with cut-elimination proof t h e o r i s t s

are).

Specifically,

d

suppose the derivation

as many

of formal arithme-

v

tic is translated d

of

d2

of second order logic with normal form

How are Id$1?

into

d

and

~-logic with normal form

d2

In other words,

or only of arithmetic To be precise, distinct

(terminating)

related?

Id I*

For example,

do we really have a

if

Id I = Id'I

'reduction'

do we have

of arithmetic

the notation above hides a problem~

Not only are there

result.

properties

of the successor

This last fact is obvious be-

cause cut free rules are complete for validity , not for consequence

set of theorems

reasonin~

cut elimination procedures with different normal forms

rules or axioms--affect~the

cannot be derived from

Id21 =

results?

Id I, but also the precise choice of defining function--by

I~I*.

A ^ A), and so two axiomatizations

(since

A

which generate the same

(by additional use of cut) need not do so if only cut-free rules

are used. Even when mathematical

theories were formulated in a cut-free

as to be closed under cut), as has happened occasionally, discussed as an issue. of logic as

style (so

this matter was not

This omission is consistent with the idea mentioned above,

'fundamental';

the idea could hardly be convincing

if the exact choice

for the canonical normalization or cut-elimination procedures where, as we shall see in a moment, the word 'canonical' is a little deceptive.

111

of axiomatlzations

for mathematical

solved by the allegedly conspiracy of silence, experience

theories were a major problemJ

'fundamental'

work on logic). Without

what, no doubt~appeared

the matter does deserve discussion.

[Sch 2] gave

to him ~ (if not the~) natural axiomatization

STE of simple

and solemnly went on to show that this STE does

not admit a normal form theoremgwithout different axiomatization, Takahashi

lating the issue. East.

reaction to actual

Schfitte

type theory with extensionality,

result.

suspecting here some

let us only note that the straightforward

is different:

(which is not

even mentioning the possibility that a

also equivalent

to STE, might admit the normal form

[Tal] gave such an axiomatlzation

but once again without formu-

We see no reason to doubt generally the wordless wisdom of the

But in the ~resent instance there seems to be a case for a bit of explicit

analysis in the Western style, to help us avoid the repetition of errors. Uesu [D-l], with Takahashi's tended to axiomatize to him by Pohlers~

acknowledged help~put

Zermelo's

When

down some (cut free) rules in-

set theory, he made a formal error,* pointed out

a student of Sch~tte,

and Uesu [U2] gave another axiomatization

-still without a word on the issue involved. Now we come to our first problem. scheme for discovering axiomatizations

Without asking for some grand general

(of a mathematical

theory given by its set

of theorems) which are complete for cut-free logical inference, see this done for a few specific theories; zation involved the mathematical Remark. appropriate

if the choice of axiomati-

content of the theory.

We no more expect

axiomatizations

preferably,

we should like to

'general' rules here than for

in ordinary mathematics.

the discovery of

The reader may compare

(i) the present switch from one or two ~fundamenta~ logical systems to several (but not to___~omany) specific ones to quite useless formalization

of

(ii) the passage from the (mathematically)

'all' of mathematics

in a universal

system (of say

This is the example of the 'unreasonable effectiveness'of the test by formal rigour alluded to in (c). Uesu 'might have'-~ne says-found the axiomatization [U2] by mere tinkering~ and we might still be in awe of the w.w. of the East.

112

set theory) to the discovery of relatively few systems (for Bourbaki's structures\ mere) which have greatly increased the intelligibility of mathematical reasoning. (ii)

_Logic and combinatorial mathematics.

As was noted in 1.3(~g), logic

has not found many applications in combinatorial and constructive mathematics. One theoretical point was mentioned, namely that in these domains of mathematics the set of valid axiomatizable. classes~,

formulae (in the usual logical language) is not reeursively In the language of ~ - f o u n d e d

the property of being ~

(binary) trees:

for the relevant

-founded is not recursively enumerable.

differently, these domains of mathematics obey more are universally (logically) valid..

Put

'logical' laws than those which

As far as ~roofs (of theorems in the logical

language of first order) are concerned, we do better, in these domains, to use ad ho___~cmethods than rules of inference (in the language of first order).--Realistically speaking, the same applies also to those domains of mathematics where we do have a complete formalization : modern mathematics uses--constantly and efficiently--set theoretic arguments to prove theorems (that can be) formulated in first order language, for such structures as p-adic or real closed fields.

In fact the same

applies to propositional logicJ as suggested by examples used in the theory of infinitary propositional calculus~ particularly by Stavi. 0 < i ~ N+I, 0 < j ~ N+I

(*)

[~ n

and

i ~ j.

~ ~(Pni i,j

Suppose

0 < n < N~

Consider the propositional formula

^ Pnj )] ~

~

i

~

n

Psi "

We should not dream of refuting (*) by a propositional deduction, but instead we note that (*) implies the existence of a l - i [i:O < i ~ N+I)

mapping of [n:O < n ~ N] onto

and that this second order statement is false.

However, our formulations (of validity for models restricted to ~ in terms o f ~ - f o u n d e d

or

trees) open up the possibility of applying metam~thematics

to those restricted domains of matherr~tics~ as illustrated by the following example (to use material mentioned already in II.3(c) above). For general ~ , AIv

we cannot expect that a formula -.. ~

A

n

(for some finite

n)

113

is~-~alid where ti) , whenever

A i is

3~A

A[~

• ] iand

is ~ - v a l i d .

A

is quantifier-free (for finite terms

But it does not seem implausible that, by

examining elementary metamathematical proofs (of the uniformity theorem just mentioned)~we can find some useful class language of

A) such that, for each

~ 3~A

of possibly infinite terms (in the which is ~ - v a l i d ,

disjunction A[x~i] : t i C

is~-valid.

some infinite

114

APPENDIX:

KONIG 'S LEMM_& FOR FINITELY AND S E ~ F I N I T E L Y BPANCHIg[G TREES

This topic is related to the subject of the present paper in 2 ways. Quite specifically, K~nig's lemma is mentioned repeatedly in Section !; in this connection, the results in subsection 2 below permit the formalization of various model theoretic proofs in a conservative extension of arithmetic (obtained by adding a suitable formulation of K~nig's lemma to socalled elementary analysis). More generally, K6nig's lei~mm illustrates veT/ well a phenomenon which pervades all uses of abstract language (and not only ours for metamathematics) and which has repeatedly led to a simple minded confusion between (mathematical) content of abstract principles and

(i) the corabinatorial

(ii) their proof theoretic

strength which depends on the logical complexity (of the definitions) of the structures to which the principle is applied.

We discuss the general phenomenon

first because--in point of fact--it has led (one of us) to the specific work on K~nig's lemma in subsection 2. i.

However the latter can be read independently.

The (logical) complexity of sets to which mathematical principles are

applied. (a) The case of induction. pay attention to logical complexity.

It is a commonplace that mathematicians rarely A memorable illustration of this fact is

provided by the apparent absurdity of the ~news' that Gentzen used

Co-induction

to prove the consistency of arithmetic the main (mathematical) principle of which is ordinary, that is,

~induction.

The fact that Gentzen applies ~0-induction

only to elementary (primitive recumsive) predicates, while, in arithmetic,

~-in-

duction is applied to all first order predicates (in the language of rings) cannot provide a convincin~ distinction unless one has doubts about the sense of logically compound expressions.

An equivalent way of putting the matter uses a two sorted

language, for numbers and predicates or sets of numbers.

~he abstract induction

principle VX[[O

~ X-->

Vx~C

X-->

sx C X)] --> V x ( x

C X)]

is supplemented by independent set existence axioms expressing closure under say primitive recursion and the 'logical' operations (that is; projection since ~nion

115

and complementation happen to be primitive recursive). to Gentzen's use of

The system corresponding

~o-induction supplements the abstract principle (of ~o-induc-

tion) only by the non-logical set existence axioms.

But why should one 'leave

out' the logical ones? A good reason would seem to be provided by (manageable) models which satisfy the non-logical axioms but are not closed under projection--and the collection of (primitive) recursive sets is certainly such a model (when of the individual variables and

s

the suacessor).

~

is the domain

But the 'reason' is deceptive

because this model is an y-model and so automatically satisfies induction without restriction on the

X.*

It must be admitted that we do not know 3 at present,

any models that are not standard w.r.t, the integers (that is, not ~-models), satisfy induction for predicates of limited complexity~ and are sufficiently transparent to be of intrinsic interest.

In other words~ we do not have here a

clear parallel to the typical situation of ordinary axiomatic mathematics where the study of restrictions on the 'usual' axioms (for example, in the passage from Dedekind's second order axioms for

E

to the theory of real closed fields) pro-

vides obviously interesting structures which satisfy the restricted but not the usual axioms. As already mentioned in il, we regard the subject o f ~ - f o u n d e d

derivations

in the present paper as providing a 'home' for Gentzen's mathematical ideas. Here the restiction on the logical complexity of the predicates to which (bar) induction on G - f o u n d e d trees can be applied, is certainly not arbitrary~

The prin-

ciple is simply not valid unless, roughly speaking, the predicates C ~ . that remains is to make clear the interest of the ~

So all

-founded derivations them-

selves, which was done in i.3(a) without any logical sophistry. To avoid misunderstanding it is perhaps worth making two remarks.

First

of all, even though we do not have any really manageable non-standard models, . This is by no means an isolated phenomenon. For example, ~-models automatically satisfy bar induction without any restriction on the predicates to whichthis principle is applied; only the (partial) ordering along which the induction proceeds is required to be a set in the domain of the ~-model.

116

there are a few ve~ 7 elegant uses of (the language of) such models anu some superficial properties

(common to all such models),

for example,

the indefinability

of

the set of standard integers of the model: we know this much, without knowing anything about its genuinely mathematical

properties.

is described in subsection 2: it is particularly purpose is to define a good

One such use 3 due to Scott

[Sc~,

appropriate because its principal

a~model (for elementary analysis to which K~nig's

lemma is added). The second remark concerns the study of restrictions which abstract principles

such as induction are applied.

there is a logical view which requires the restrictions,

on the predicates to

As already mentioned for example because of

(genuine or ethereal) doubts about the existence of sets having certain formal properties. sensibleJ

But this view is not at all necessary to make the study in question Whether we like it or not, any given application

ciple can only involve a finite number of predicates. predicates will be limited by the mathematicians'

Moreover,

in practice the

view of the problem which he

is trying to solve by means of the abstract principle. theoreticall~the

of an abstract prin-

If he is thinking number

predicates will be expressed in the language of number theory;

if he is thinking in a broader context,

for example of the integers embedded in

the complex plane, the predicates will be in the language of function theory, the additional

set existence axioms will be those familiar in the given context.

it certainly can do no harm to have some idea of the consequences for example,

and

for the class of, say, number theoretic

of the abstract principle.

of a given

'view',

theorems provable by means

This so to speak negative interest of non-standard

models seems to us quite real.* (b)

Trees and branchings.

To fix ideas let us consider trees

T

whose

nodes which are all at a finite distance from the root, are labelled by integers. K~nig's lemma, in ordinary mathematical

language,

asserts this:

* if this is accepted, one can at the same time (i) accept the intuitive impression that non-standard models are of interest and (ii) dismiss the contention that non-standard models discredit our usual conception of the (standard) integers.

117

If

T

is unbounded and each node of

number of immediate descendants then

T

For example,

if

T

is either terminal or has a finite

(= predecessors

in the tree ordering)

has an infinite path. T

is binary~ that is~ each node has at most 2 predecessors,

and labelled 0 or i, then any infinite path

~

can be coded by a set

X

c

as follows: n ~ X

the label at the nth node of

~

is

i

n ~ X

the label at the nth node of

~

is

0.

To formulate the principle in set theoretic language of sets of natural n~nbers,

language,

we shall regard

sequences of O~ i, closed under initial segments. (i)

specifically in the usual

the abstract principles

T

as a predicate of finite

We then have a choice between

together with appropriate

existential

set

theoretic axioms or (ii)

a schema where the language~

Evidently icate path

T

is defined by some given class of formulae in

in the style of the comprehension

(ii) is sensitive to the class of formulae chosen~for an arbitrary pred-

P, we can obviously arrange ~

schema.

s.t.

X

T

to define a tree consisting

~--~ P; in short, the comprehension

of a single

principle follows from the

schema (ii) (by use of quite elementary axioms). As to (i) it has long been known that the abstract principle by arithmetically

definable

sets in the sense that~ if

T C ~0

and the first basis theorem in the literature asserts that if some

X

C ~.

Also it has long been known that if

need be no recursive

T

is satisfied

so does some T

X ;

is recursive

is recursive then there

X ~'

(recursive)

Both the positive and negative results apply equally to the two types of~ trees mentioned abovej namely those labelled by 0 or i and those labelled by integers without any bounds on the labelling. ence between these two t ~ e s .

If

T

If~ in contrast,

there is an obvious differ-

is recursive and the labels are O or i, the

decision whether a given node is te~ninal, recursive.

However~

has on__~eor two descendants

is also

the labels are not bounded and even if we are given

118

that the tree is binary (in the sense that each node has at most 2 descendants) there is no obvious reason why the decision above should be recursive: tion between an (arbitrary) facie, only r.e.

node and its immediate descendants

Put differently,

the rela-

is, at least prima

we have no reason to expect recursive isomor-

phisms between the two classes of recursive binary trees which have botmded~ unbounded labels.

resp.

Some years ago one of us presented this state of affairs to H.

Friedman who gave a very satisfactory answer.

~he abstract version of K~nig's

lemma for binary trees with unbounded labels together with closure under a few primitive recursive operations principle

(with parameters),

arithmetic.--For though Friedman's

is equivalent

to the full arithmetic

and thus not a conservative

reference in (d) of subsection 2 below, result is very satisfactory

comprehension

extension of first order it is worth noting that

for a good choice between bounded and

unbounded labels on finitely branching trees, it also casts doubt on the mathematical interest of such equivalence

results

clear that qua combinatorial

principle,

arithmetic

(w.r.t. to elementary analysis):

(or any other) comprehension

Exercise.

K~nig's l e m ~

it is quite

has nothing to do with the

principle.

The reader may wish to derive formally,

from K~nig's lemma for

unboundedly labelled finitely branching trees, that for any ordering 2R where

2R

R, R

is well-founded

if and only if

is well-founded. has as domain finite descending

eographically

(w.r.t.

R).

sequences

This observation,

(in order

R) ordered lexi-

which we learnt from W. A. Howard,

also shows that K~nig's lemma for unboundedly labelled finitely branching trees (together with elementary analysis) is not conservative over first, order arithmetic. Here it seems quite clear--and, perhaps, worth making precise--that, qua combinatorial principle, KSnig's lemma does have something to do with the step from 2R

R

to

above. In contrast to Friedman's result, as mentioned in subsection 2 the prin-

ciple for bounded labels etc. is conservative

over arithmetic.

Naturally~

the

119

restriction to the particular have a recursive bo~ud~

say

labels O, I is immaterial:

~(n), for the labels at the nodes at distance

from the root of the tree; since our tree is recn~sive recursive determination

such a bound allows the

of all the labels.

Remark on an alternative

analysis by Troelstra

language of natural n~mbers and functions, logic.

it is sufficient to

[Tr], for systems in the

but using intuitionistic

Leaving aside some technical refinements,

rules of

the most striking difference

is this. At least formally 3 the intuitionistic

version* is insensitive

to the logical complexity of the definition of

T~ and to the

distinction between the two types of labelling. the case of bounded labels: finite branching,

(We shall call the other: semi-

finite branching.) Moreover--and length--this

this is of course the reason for going into the matter at some phenomenon

the proof theoretic complexity

is by no means isolated.

For example,

in classical logic

strength of choice schemata depends sensitively on the logical

of the relation~

say

R~ in

Vx 3yR--~

3fVxR[y/f(x)]

or in bar induction etc.; in intuitionistic generally speaking the opposite is true. tion treats f~uctions~

,

logic of functions

The intended intuitionistic

including the (logical) operations

tial quantifier or in disjunction~

interpreta-

implicit in the existen-

in a quite narrow sense~ and a s y ~ e t n c a l l y

from species (which need not have characteristic formal systems for intuitionistic

(not of species~)

mathematics

which the functions are recursive**).

f~ctions--and

permit

Consequently,

the~sual

an interpretation even if the tree

in T

is

of: if every path through T is fimite then T itself is fimite~ where a ~ath is given by a function (of ~) naming the nth label on the path. at least as long as the logical operations of realizability.

are reinterpreted

in the sense

120

defined by a logically complicated expression, this is so to speak nullified

by

th___~erequirement of finite branchin~ (in the sense that, for each node, there is either no descendent or one or tw0).--The reader may wish to verify, that, for logically complex

T, a mere bound on the number of descendents would alter the

situation completely. 2.

~models

of K~ni~'s ! e ~ a

(for finitely branching trees) derived from

non-standard models of first order arithmetic.

The models considered below were

introduced by Scott [Sc~, also in connection with K~nig's lermma (for binary trees with labels 0 or i).

Naturally, 15 years later we pay attention to aspects of

these models, and consider variants, not treated in the original publication. We begin with a description of the models~ then look at some properties or socalled 'axioms' which they do or do not satisfy 3 and finally nmke some relevant applications. (a) arithmetic

We start with any complete and consistent extension of first order P~ in short, a model of

P in the sense of Section I. i . As is

well-known, there are such models which are arithmetically definable, even in in such a way that the nth natural number of the model, say primitive recursive function of

n.

~,

i(n), is given by a

Furthermore, by the formalized version of

the completeness theorem (adapted to Henkin's proof, from the presentation in Hilbert-Bernays for G~del's proof which gives a on the atomic formulae, not on all formulae), the obtained can be formally proved in P of

is consistent, Con(P) P

for short.

P

~

satisfaction predicate only ~

to be a model of

satisfaction relation so P

from the assumption that

Naturally since the property of being a model

is a single sentence, the proof is given in a subsystem of

interested in such matters should note that 'being a model of

P'

P.

(The reader

is expressed by

infinitely many sentences if a model is given by its satisfaction relation for atomic formulae.) The class of sets (of natural numbers) or relations (between natural numbers) to be considered are what Scott called

'binumerable' in the given extension;

we prefer to call them more simply: the sets and relations

121

definable

that is, explicitly definable by a formula of (n:i(n)

satisfies

P

~

F,

sets are definable

(and, indeed, all those sets are definable

subsystems of on

P, say

'without parameters':

F].

As is well-known all recursive of

P) o__nn ~j

(in the given model of

P, for example,

Q).

on

~

in all models

in all models of quite small

Not all r.e. sets are necessarily definable

in each such model since there are models of degree < O' The class of sets considered contains not only all recursive

closed under reeursive operations;

0

where

it satisfies

- CA:V RS[Vn[3 pR(n,p) < - - ~ 3 qS(n,q)] I-> 3 X Vn[X(n) 3 pR(n,p)])

R, S

are variables over binary relations and

To see this, and

more formally,

sets but is

FS

suppose

R

and

with the free variables

S

X

are defined on

x, y; resp. x, z.

Then

is a variable over sets. ~, by the formulae X

FR

is defined on

by

3y[F R^ (V~Sy) I F s]

Here again one does not use the full ensure (by Rosser's Exercise.

condition)

'force' of

P, but just enough axioms to

that all the models are end extensions of

The reader should write down variants of K~nig's

~.

lemma (for

finitely branching trees) which are satisfied in the model, and consider their logical relations;

any such tree being given by a set of (formally finite) paths

closed for initial segments together with a bounding function for the labels at level n.

In particular,

consider the variants obtained by taken for the trees

(i) sets and functions definable on

~

in the complete extension of

P

considered,

(ii) sets and functions which include all those that are recursive in parameters which are (b)

definable outright. Defining trees (and a reminder of t~e definition of flJnctions on

in contrast to: definitions of their graphs).

Both naively and for the specific

122

purpose of satisfying KSnig's lemma, it is natural to use definitions, say trees ~

which define, in our (nonstandard) model of

restriction of ~ t o

P,

T, of

end extensions of the

¢J, that is

any initial path of %

of length n

where

n

is a natural number

has labels which are also natural (= not: nonstandard) numbers. (Here, as in subsection i above, a tree is given by its formally finite initial paths.)

Clearly, a standard tree is recursive in the sense that the set of finite

initial sequences (coded canonically) is recursive and if there is a recursive bound for the labels at level (non-standard) model of

n C ~, there is a formula

T

which defines~ in each

P~ an end extension of the given tree.

In particular,

this is clear for trees whith labels 0 or i only. Exercise a tree

~

(for a reader

fond of 'omitting type theorems').

Show that if

(is reeursive and, say, binary but) does not have a recursive bound

for the labels at levels

n C

defines an end extension

of

~, there is a model of

P

in which no formula

~.

Thus our requirement on finitely and not merely semi-finitely branching trees makes good sense in terms of (definitions applied in) non-standard models; even if we do not mind what the definition

'does' to sequences of non-standard

length~ we do not want non-standard labels at a standard, genuinely finite distance from the root of the tree.

(The 'extension' of definitions

T

to non-standard

models illustrates modern mathematical practice discussed in 1.3(c).) The 'reminder' referred to above~ concerning functions involves -~iqueness

properties

f

on

~

of definitions in the whole model not only on J

The ~raph, say

Gf of

provided only evidently, since

Gf

f, is said to be defined o__nn ~

(n,m) C Gf satisfies is functional o_~n ~, for each

will do, but the e may be non-standard

< i(n),x > A function

f

by

satisfies

is said to be defined by

F F;

~ C ~, only

m = f(n)

x s.t.

F. F

if, in addition, for each

n C

~.

t23

V x[ ~i(n), x ) satisfies F] x : i(fn)} is true in the model. (c)

K~nig's lemma holds in the models considered in (a)~ for trees

defined by formulae Suppose ~

T

is unbounded (in

non-standard model (of length.

Thus

described in (b).

~

P), T

e).

Since

P

contains induction~ in any

must be satisfied by some path

has an infinite path definable on

e

parameters (in the model), since the proper segments themselves standard and therefore in ~ goes through ~ .

~

of non-standard

by a formula with ~

of standard length are

, and so the restriction of

~

to

Here we have used our choice of definitions in (b) for finitely

branching trees which excludes non-standard labels at a finite distance from the root o f ~ . To get a definition without parameters one distinguishes between two cases: (i)

The (nonstandard) tree defined by

T

has a necessarily nonstandard

bound~ that is, it is finite in the sense of the nonstandard model.

Then, since

induction holds, there is a highest level~ and we take the node say

N) with the

smallest label at that highest level.

Having just given an explicit definition

(without parameters)~ we also have a definition of the path joining

N

to the

roots of the tree. (ii)

The tree is infinite in the sense of the (nonstandard) model (and~

formallyj finitely branching). tree defined by

Now we take the leftmost infinite branch of the

T.

So in each of the eases there exists a formula which defines~ on infinite path through ~

.

T

an

(The reader who does not like undecided disjunctions,

will easily find a single formula to define a path; let the tree defined by

~

is finite 3 and let

p(i)

definitions of paths given in (i) and (ii) resp.

and

FT p(ii)

Then take

(F~ ^ F (i)) v (p(ii) ^ ~F#

•)

express formally that be the explicit

124

Actually, for the particular applications mentioned below it would be sufficient to consider the class of sets definable by use of (nonstandard) parameters, or to consider models in which each element is definable outright . (d)

Applications and references to the literature.

with a comic side to it, is this.

Let

~-CA

An obvious application,

be the subsystem of analysis (also

called second order arithmetic) in which induction is applied to arbitrary formulae~ but comprehension is restricted to

~ - f o r m u l a e as in (a), and let KL be K~nig's

lemma for finitely branching trees in abstract form, as described in l(a) above. Then we have a proof in primitive rectu~sive arithmetic of

Con(P) --> CO~(K~ + ~

Since

KL + ~

proved in

- CA

KL + ~

literally contains

- CA) .

P, this means that

Con(P)

cannot be

- CA.

As one of us reported in a review of [G], Gentzen's original version of his consistency proof for (P) was criticized for having used, allegedly, KL.

The

criticism overlooked (among other things) the crucial point, discussed in i, that the strength of K L is very sensitive to the complexity of the trees to which the abstract principle is applied; for example by l(b) one gets a trivial proof of Con(P)

by applying KL to e.g. arithmetic trees, and by Friedman's observation

mentioned there 3 also by applying KL to recursive but semifinitely branching trees. By the remark above, Con(P)cannot be proved by use of KL applied to finitely branching recursive trees. More amusingly~* when Spector applied the abstract principle of bar induction (for infinitely branehing trees) to trees labelled by objects of hi~her t ~ e ) f i r s t reactions essentially repeated the oversights involved in the earlier criticism of Gentzen's work.

It was

'argued' that it can ~ k e

no difference whether the prin-

ciple is applied to trees of natural n~mbers or to trees of objects of higher type at least according to Marx's sense of htm~our applied to patterns in history: the first instance (of a pattern) may be tragic, any repetition is comic. cf. also Walpole whose man of feeling apparently does not recognize a repetition as such,

125

(where, incidentally, it was far from clear just which operations of higher type were to be meant; especially since, demonstrably, no_~tall such operations were admissible~ for example, the operations had to be extensional and even continuous --for the product topology at types (0 w ~

~) --~ 0). -The reader may also wish to

pursue the parallel in subsection 2 where invariance properties of definitions of trees (at finite levels) were found to be important: in the case of trees of higher type objects, the same formula will define different trees ~ccording to the class of such objects present in the model considered (even if only ~ -models are allowed). However inappropriate these first reactions may have been in the logical contexts (of consistency proofs for arithmetic and analysis respectively) in which they occurred, they are obviously symptoms of the conflict between mathematical interest and

(traditional~ logical analysis which is part of the theme of the

present paper.

After all, if naively we wish to convey the idea of a proof, we

do so by mentioning the abstract combinatorial principle used and, perhaps, the 'kind' of predicate to which it is applied; it is only the particular classification~ by means of logical complexity, that is rarely convincing. sionally~ it is~ for exam@le, when

ZI

or

SI

(And, if~ occa-

is connected with recursion theory

or invariance~ the discovery of the particular connection constitutes the principal interest; not merely the fact that

ZI

is a stage in the hierarchy

Zn

for

n = i, ~, ...*). As a practical consequence, for further research in the present area~ we are led to be more--or~ perhaps, too-discrlminating w.r.t refinements which we consider to be rewarding.

As an example, we consider the improvement of the rela-

rive consistency results (of KL + ~

KL + ~

- CA

- CA

over

P)

to:

is conservative over

P,

also established by primitive recursive (metamathematical) methods; this is an improvement since relative consistency asserts that

KL + ~

- CA

Put differently, we do not think of the 'simple minded' step from as a particularly significant direction for generalizations.

is conservative ZI

to

Zn

126

over

P

w.r.t, the single formula 0 = i.

this subject, reviewed in [Tr].

There is a considerable literature on

Also the work in (a)-(c) above can be modified to

yield the conservation result by consideringjin place of and models of

P

n

established in

P

itself.

P, finite subsystems

Pn

(One has to keep count of the eom-

plexity of the formulae in (c) to which induction is applied.) While, from the point of view of logical complexity, P

is as natural a

system as one can reasonably expect (in view of the incompleteness theorems), we reamin skeptical: What does

P

have to do with KL?

One has the impression that certain

'optimal' subsystems

PO

of

P ajre much more

relevant; not because they are 'weaker' or 'smaller' but simply because the proof of the

result for

P

itself aboundswith formulae in the 'neighbourhood~of

~.

So we look for a statement (of another resultS) which is both memorable and sums up this feature of the analysis.

Roughly speaking, we expect to find 'subsystems'

P'~ not necessarily in the language of

P

itself, such that the 'corresponding'

systems EL' of elementary analysis together with KL are conservative over

P'.

We should not be surprised to find that some of the 'subtler' hierarchies of could be used to formulate sharp results here. As always when

'subtle' or pedantic distinctions appear to be necessary

for a proper statement of the facts~ it may be time to look at bigger things which magnify the distinctions (and thus remove the horrors of pedantry).

In the partic-

ular case of KL, we should pass from the language of number theory to that of set theory, or better still (since we have found ~ - f o ~ n d e d

derivations useful) to the

language of trees which are--unlike sets--not well-founded by definition. last passage is also suggested by work on infinitary language~ ~ A sets

This

for admissible

A: the analysis of the roles of admissibility, for example by Barwise

and Stavi~ uses closure conditions on

A

which are extremely natural in the language

of sets; but we hardly ever use the corresponding

tree-theoretic operations.

127

REFERENCES

[A]

Ackerman~W., Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Math. Ann. 114 (1937) 305-315.

[C]

Carstengerdes, W.~ Mehrsortige logische Systeme mit unendlichlangen Formeln, Arehiv Math. Logik Grundlagenforsch 14 (1971) 38-53 and 108-126.

[~]

Chang, C.C. and Keisler, J.H., Continuous model theory, Princeton, 1966.

[nAA]

Dreben, B., Andrews, P., and Aandera, S., False lemmas in Herbrand, Bull. Ag~ 69 (1963) 699-706.

[F]

Feferman, S., Lectures on proof theory, Springer Lecture Notes 70 (1968) i-i08.

[Fr]

Friedman, H., Iterated inductive definitions and 4-AC, pp. 435-442 of: Intuitionism and Proof Theory (ed. ~iyhill et al. ) North Holland Publ. Co., 1970.

[0]

Gentzen, G., The collected papers of Gerhard Gentzen, ed. M.E. Szabo, Amsterdam 1969; rev. J. of Philosophy 68 (1971) 238-265.

[ai]

Girard, J.-Y., Three-valued logic and cut-elimination: the actual meaning of Takeuti's conjecture, Fund. Math. (to appear).

[J-S]

0 Jockasch, C.G. and Soare, R.I., H.-Classes and degrees of theories. A.M.S. 173 (1972) 33-56; rev. Zbl+ 262 (1974) 19; no. 02041.

[~]

Kreisel, G. and Krivine, J.-L., Elements of mathematical logic, second revised printing, North Holland Publ. Co., 1971.

[KL]

Kreisel, G. and Levy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift math. Logik Grundlagen 14 (1968) 97-142.

[~]

Lopez-Escobar, E.G.K., On an extremely restricted (to appear).

[Mi]

Mints, G.E., On E-theorems, Zapiski 40 (1974) 120-118.

[~]

MartinlZf, P., Hauptsatz for the intuitionistic theory of iterated inductive definitions, pp. 179-216 of Proc. Second Scand. Logic Symposium (ed. Fenstad) North Hollard Publ. Co., 1971.

[Mo]

Mostowski, A.~ On recursive models of formalized arithmetic, Bull. Acad. Pol. Sc., cl. III, 5 (1957) 705-710.

[OL]

Kreisel, G., Ordinal logics and the characterization of informal concepts of proof, pp. 289-290 in Proc. ICM Edinburgh 1968.

[P]

Parsons, C.~ Transfinite induction in subsystems of number theory (abstract), JSL 38 (1973) 544.

[Seh]

Sch~tte~ K., Syntactical and semantical properties of simple type theory, JSL 25 (1960) 305-326.

[Sch i]

......, Beweistheorie, Berlin, 1960.

Trans.

o~rule, Fund. Math.

128 [Sch 2]

, On simple type theory with extensionality, pp. 179-184 in: Logic~ Methodology and Philosophy of Science III~ North Holland Publ. Co., 1968.

[Sco]

Scott, D.S., Algebras of sets binumerable in complete extensions of arithmetic, pp. 117-121 of: Proc. Syrup. Pure Math. 5, AMB, 1962.

[Sh]

Shoenfield, J.R., Mathematical logic, Addison-Wesley, 1967.

[SPT]

Kreisel, C., A survey of proof theory,

JSL 33 (1968) 321-388.

[SPT II] , A survey of proof theory II, pp. 109-170 of: Proc. Second Scand. Logic Symp. (ed. Fenstad), North Holland Publ. Co., 1971. [ S t]

Statman, R., Structural Complexity of proofs, Dissertation, Stanford, 1974.

[Ta i]

Takahashi~ M., Simple t ~ e theory of Gentzen style with the inference of extensionality, Proc. Jap. Acad. 44 (1968) 43-45.

[Ta 2]

~28

, Many valued logics of extended Gentzen style II, JSL 35 (1970)

[Tr]

Troelstra, A.S., Note on the fan theorem, JSL (to appear).

[U !]

Uesu~ T., Zermelo's set theory aad Gx%C, Co~ent. M~th. Univ. St. Pauli 16 (1967) 69-88.

[U 2]

IV]

, Correction to 'Zermelo's set theory and G*LC', ibid. 19 (1970)

Vaught, R.L., Sentences true in all constructive models, JSL 25 (1960)

39-53. PS (suggested by last-minute correspondence between two of the authors) concerning mainly the roles of continuity and~'-foundedness for analyzing cut-elimination procedures, especially in Part I I .

In ~4 of the Introduction (pp. 8-9) and

elsewhere (e.g.p. 51), continuity is considered as an additional requirement (p. 9, I . 5) to be s a t i s f i e d by 'standard procedures' (p. 51, I . - 4 ) ; the ultimate aim being to use the refined cut elimination procedures for e f f i c i e n t solutions (p. 9, I . 12) of the kind of open problems stated near the end of I I . 2 ( d ) (cf. p.54, I . - 3 ) . I t is quite clear that these problems are not solved by what were called ' t r i v i a l ' procedures (in the f i r s t paragraph of §4 on p. 8), namely procedures obtained automatically from normal form theorems(or, equivalently, the completeness of the cut free rules for the class of models considered).

Through an oversight we f a i l e d

to discuss e x p l i c i t l y the obvious pedagogic question whether continuity and/or ~-foundedness considerations alone are sufficient to distinguish between t r i v i a l and standard procedures. negative.

As w i l l be seen in (1) and (2) below, the answer is

129

Correction (of a second pedagogically serious omission at the beginning of §4 on p. 8).

There are two familiar t r i v i a l procedures. The f i r s t , described loc. c i t . ,

involves running through a l l cut-free derivations; this type of procedure is patently discontinuous in the case of i n f i n i t e derivations.

The second t r i v i a l procedure

involves simply the canonical 'refutation' trees (for a formula A) called in the f i n i t e case (§2 on p. 22) and

TAcF

T~cF A

in the i n f i n i t e case (p. 39); given any

derivation tree, T ' , with the end formula E(T') one has the procedures: 2:

T/~-->TE(T')

the i n f i n i t e

in the f i n i t e case and

[~ :

ECT') in

case.

We failed to mention the second procedure where i t should have been stressed (~4 on p. 8).

I t is plausible that, at least s t a t i s t i c a l l y , ~2 is quite e f f i c i e n t in

the f i n i t e case in this sense: for constructing a cut free derivation of A, i t CF may be s t a t i s t i c a l l y e f f i c i e n t to make a fresh start, to construct T A once one knows that A is valid, instead of trying to convert a given derivation with cut into one without cut by means of a 'standard' procedure. LL)~~,~- c ~ s , ~ - y ~ I.

Evidently,

I~

is continuous.

2.

Also (as w i l l be shown elsewhere) ~ is essentially optimal as far as

~-foundedness is concerned. Roughly speaking, for any A,

T~C~s optimal among all A cut free derivations of A in the sense that, f o r ~r" satisfying a few closure con~CF ditions, A w i l l be ~'-founded provided there is any cut free ~-founded derivation of A at a l l . In short, the Pedagogic Remark (bottom of p. 54) which refers to differences between the authors' views, understates the extent to which the value of our analysis depends objectively on the applications alluded to there.

For i f

continuity and ~-foundedness were the only conditions imposed on cut-elimination procedures, the t r i v i a l 'procedure'

~z

would do, despite the fact that i t is

completely independent of the structure of the derivation trees to which i t is applied (except for the end formula). Concerning applications, i t is evident that ~z is inadequate for anything like E - theorems of [Mi].

(Here i t should perhaps be mentioned that the last 8 lines

130 of p. 64 mention only a small f r a c t i o n of what is proved in [ M i ] , since the same t

is also given by various functional i n t e r p r e t a t i o n s of ~_, and so the i n t e r -

pretations of A [ t ~ ] are derivable too. t h i s 'small f r a c t i o n ' ) .

Clearly

~

is inadequate even w . r . t .

In addition i t seems plausible that

F~

is also inadequate

f o r ensuring the separation of ordinal assignments from the cut e l i m i n a t i o n procedure itself.in

other words, f o r establishing that d e r i v a t i o n trees need not be 'enriched'

by such assignments (for a s t r u c t u r a l theory of proofs); ef. bottom of p. 14 and p. 60(c).

The matter i s , at least prima f a c i e , of obvious pedagogic i n t e r e s t since,

by p. 87, Gentzen introduced the whole business of ordinal notations as a r e s u l t of a misunderstanding and so a certain skepticism concerning the s i g n i f i c a n c e of such notations is j u s t i f i e d .

At the same time the separation would support the

impression that Gentzen's work in toto is s i g n i f i c a n t , only the roles of ( i ) the well-known business of ordinal notations and ( i i )

of the almost anonymous d e t a i l s of

the s t r u c t u r a l transformations themselves would be reversed.

(Projected back to the

f i n i t e case, ordinal assignments involve nothing more than the length of the longest paths of the derivations considered). Some remarks (to avoid possible misunderstandings).

(a) Mathematically, we do

not wish to claim that ours is the only reasonable s t y l e of analyzing differences between t r i v i a l

and standard cut e l i m i n a t i o n procedures; f o r the reasons given at

the bottom of p. 33 and e x p e c i a l l y on top of p. 34, some of us f i n d our use of trees that are not necessarily well founded promising for getting e f f i c i e n t procedures for operating on larpe f i n i t e trees.

At least f o r one of us t h i s constitutes a change

of an e a r l i e r view (in [SPT]) stressing generalizations to uncountable formulae, but with well founded proof trees.

As Jo Stavi has pointed out to us,

here ~he

obvious generalization of) Pz cannot be used, simply because 7 cFwould not be A determined for uncountable A (nor for an A for which there is no enumeration of i t s subformulae in the, say, admissible set from which the syntactic objects of the language are taken).

(b) Bibliographically, we do not claim that pp. 55-57 give

an even remotely adequate summary of the Moscow Symposium paper loosely discussed in II.2(d) (or, more precisely, of the detailed analysis by one of us of a specific formal system, which is behind that paper).

I t may be superfluous to add that,

131 apart from the author who actually wrote l l . 2 ( d ) or, for that matter, Section I I I , we have stronger reservations about his c r i t i c a l discussion than about the paper under discussion (which, after a l l , establishes precisely what i t says i t does).

PPS.

Though, of course,

differences and ~z or ~

the l i s t

between ' s t a n d a r d ' (p.

particularly

on p. 38 (and as a c o u n t e r p a r t trees

cut-elimination

procedures

91a) i s almost e n d l e s s , the f o l l o w i n g

perhaps be s t r e s s e d ;

canonical

of p o s s i b l y s i g n i f i c a n t

,TCF

,T~c t"

(p.51)

should

in view of the basic problem

to

(2) on p. 91a).

Since the

and code a l l c o u n t a b l e counter models to A A A up to isomorphism, in general f~ , resp. ~ / w i l l code counter models of

founded,

E('T~

not coded by ~ ' .

not n e c e s s a r i l y w e l l - f o u n d e d d e r i v a t i o n

may have some counter models). 'like'

(Since we c o n s i d e r ~ trees ~',

E(T')

We should expect some procedure

the standard procedures to s o l v e the problem on p. 38

positively.

132

WEAK MONADIC SECOND ORDER THEORY OF SUCCESOR IS NOT ELEMENTARY-RECURSIVE t

Albert R. Meyer

Let LSI S be the set of formulas expressible order logic using only the predicates Elgot[3,4]

in a weak monadic

[x = y+l] and

Ix E X].

B~chi and

have shown that the truth of sentences in LSI S (under the standard

interpretation < N, successor > with second order variables ranging over finite sets) is decidable. LSI S as WSIS.

Theorem I:

as

We refer to the true sentences

in

There is a constant E > 0 such that i f ~

accepting

all sufficiently

in the

In fact, we claim a stronger result:

started with any sentence

a designated

interpreted

We shall prove that WSIS is not elementary-recursive

sense of Kalmar.

which,

second

is a Turing machine

in LSI S on its tape, eventually halts in

state if and only if the sentence is true,

then for

large n, there is a sentence of length n #' for which ~'s

computation requires

'2 2 • "

2°I

LE' l o g 2 ~

steps and tape squares. t'By the length of a sentence we mean the number of characters in it including parentheses, digits in subscripts, etc. Any of the standard conventions for punctuating well-formed formulas may be used, except that in some cases conventions for matching parentheses may imply that for infinitely many n, there cannot be any wff's ~f length n. In this case, we assume that wff's may be lengthened by concatenating a finite sequence of "blank" symbols which leave the meaning of the wff unchanged, so that sentences of length n can be constructed for all sufficiently large n. tWork reported here was supported in part by Project MAc, an M.I.To research program sponsored by the Advanced Research Projects Agency, Department of Defense, under Office of Naval Research Contract Number N00014-70-A-03620006 and the National Science Foundation under contract number GJ34671 Reproduction in whole or in part is permitted for any purpose of the United States Goverr~nent.

133

Let t0(n) = n, tk+l(n) = 2 tk(n).

A well-known characterization of

the elementary-recursive functions by R.W. Ritchie [14] shows that a set of sentences is elementary-recursive iff it is recognizable in space bounded by

tk(n) =

.2n~ 2 2.

k

for some fixed k and all inputs of length n ~ 0.

Hence, WSIS is not

elementary-recursive. In these notes we prove a somewhat less powerful version of Theorem I, which by Ritchie's result is still sufficient to establish the truth of our title.

Theorem 2:

Let ~ b e

a Turing machine which, started with any sentence in

LSI S on its tape, eventually halts in a designated accepting state iff the sentence is true.

Then for any k ~ 0, there are infinitely many n for

which ~'s computation requires

222"

2°I

k

steps and tape squares for some sentence of length n.

The idea behind our proof will be to show that there are sentences in LSI S of length n which describe the computation of Turing machines, provided the space required by the computation is not greater than tk(n).

Since

a Turing machine using a given amount of space can simulate and differ from

184

all machines using less space, we will deduce that small sentences in LS! S can describe inherently long computations,

and hence LSI S must itself be

difficult to decide. Actually it will be more convenient to develop an intermediate notation called y-expressions

for sets of finite sequences.

We will show that

y-expressions can, in an appropriate sense, describe Turing machine computations,

Definition:

and that LSI S can describe properties of y-expressions.

Let ~ be a finite set whose elements are called symbols.

is the set of all finite sequences of symbols from ~.

For x, y C ~ ,

the concatenation of x and y, written x,y or xy, is the sequence * consisting of the symbols of x followed by those of y. called a word, and the length of x is written ~(x). the vacuous sequence of length zero in ~

A, B c ~

We use k to designate

which by convention has the

property that x.k = k.x = x for any x E ~ . identity k generated by ~.)

An element x E ~

(~

is the free monoid with

Concatenation is extended to subsets

by the rule

A.B = AB = [xy I x E A, y 6 B}. For any A c ~ , we define =0 A 0 = {~}, A n+l = An,A, A * = n~ These operations are familiar in automata theory.

A n" We introduce one

further mapping.

Definition: t

For any ~, the function ¥~: P(~ ) 4 P(~ ) is defined by the

rules

t P(S) = [A I A C S} = the power set of S.

is

135

y~({x]> = [y E ~'~ I ~(x) = £(x)} = ~ ( x )

for x E ~*,

~(A)

for A c ~'~.

=

U

y([x} )

xEA We omit the subscript on y ~ w h e n

~ is clear from context.

y-expressions over ~ are certain words in (~ U [~, ~, ~, ~, i, )~) are symbols not in ~.

where

Any y-expression ~ defines a

set L(~) c ~ . Definition:

For any ~, ~-expressions over ~ and the function

L:[y-expressions over ~

4 p(~*) are defined inductively as follows:

I)

~ is a y-expression

2)

if ~, ~ are y-expressions over ~, then 2~

over ~

for any & ~ ~, and L(~) = {&}; ( ~ • ~ ), ( ~ U ~ ~,

~ ~' and y ( ~ ), are y-expressions over ~, and

L(( ~ • fi )) = L(~).L(fi), L(( = U fl)) = L(~) U L(~), L( ~ ( ~ )) = ~t -e(~), and L(~ ! ~ ) 3)

= y(L(~)),

That's all.

Having thus made clear the distinction between a y-expression and the set L(~) it defines, we will frequently ignore the distinction when there can be no confusion. = L( ( ~ U ~ ( ~ ~ ~ ).

Thus we write ~

Similarly,

= c U ~(~) instead of

for any set of letters V c ~,

v*= ~(~. (~v>.~) since V

consists precisely of those words in ~

symbol not in V.

which do not contain a

Thus there is a y-expression & over ~ such that L(~) =

136

V

.

DeMorgan's

law gives us intersection,

V n = ~no

and then the identities

V , and

~n = y(v n) imply that from a y-expression of length s for ~ n w e y-expression

of length s + c for V n, and conversely

some constant c and all s, n E N.

can obtain a from V n to ~ ,

for

We shall show below that in general

s may be much smaller than n. Definition:

Empty

(~) = (~ I ~ is a y-expression

Since the regular closed u n d e r . ,

(finite automaton recognizable)

U, ~, and y, it follows that Empty(~)

in fact primitive recursive. for L(~) and tests whether well-known procedures however

indicates

One simply constructs

to do this.

automata

would have to apply the "subset construction"

can exponentially

concatenations

for ~(~.~)

are

is recursive and

a finite automaton there are

~ priori analysis of this procedure

that from deterministic

obtain an automaton

subsets of ~

the automaton accepts some word;

one would obtain a no__~n-deterministic automaton

y-expressions

over ~ and L(~) = ¢}.

or -~(~).

for y-expressions

~,

for ~.~ or y(~), and then

of Rabin-Scott

[l~ to

Since the subset construction

increase the number of states in the automaton,

in which k eomplementations

alternated with y's and

can lead to an automaton with tk(2) states.

space required by a Turing machine which recognizes procedure outlined above ca D be hounded above by

The time and

Empty (~) by the

137

tn(C ) =

.2c~ •

2

n

2"

for some constant c and all y-expressions of length n ~ O.

It will follow

from results below that such absurd inefficiency is inevitable. , Definition:

A Turing machine ~ r e c o g n i z e s

a set A c ~

if, when started

with any word x E ~* on its tape, ~ halts in a designated accepting state iff x E A• Let f: N ~ N .

The space complexity of a set A c ~ ~ is at most

f almost everywhere, written SPACE (A) ~

f (a.e.)

iff there is a Turing machine which recognizes A and which, for all but finitely many x E A, uses at most f(~(x)) tape squares in its computation on input x.

The space complexity of A exceeds f infinitely often written

SPACE (A) > f (i.o.) iff it is not true that SPACE (A) ~ f (a.e.). We shall use Turing's original one tape, one read-write head model of Turing machine, and define the number of tape squares used during the computation on input x to be the larger of ~(x) and the number of tape squares visited by the read-write head.

Then by convention at least

max{ ~(x), I} tape squares are used in a computation on any input word x. We briefly review some well-known facts, first established by Stearns, Hartmanis, and Lewis

[15],

about space-bounded Turing machine computations•

138

Definition:

A function f: N ~ N is tape constructible

Turing machine which,

iff there is a

started with any input word of length n ~ 0, halts

having used exactly f(n) tape squares. Fact I:

t o + 1 = kn[n+l]

is tape constructible.

For any k > O, tk is

tape constructible. Fact 2;

If f: N ~ N i s

tape constructible,

and SPACE (A) K f (a.e.) for

some A c ~ , then there is a Turing machine which recognizes A which halts on every input x E f= using at most f(%(x)) Hence,

tape squares.

SPACE (A) ~ f ~ SPACE (~'-A) ~ f. *

Fact 3:

If f: N ~

Nis

tape constructible~then

there is an A c G0,I}

such that

for any g: N 4

N

SPACE (A) ~ f

and

SPACE (A) > g

(i.o.).

such that g(n) 0. lim f(n) = n-~

Our proof consistsof a sequence of reductions recognition problem to another. recursive

In contrast

of one decision or

to the usual reductions

function theory, our reductions must be computationally

We introduce a particular

of

efficient.

notion of efficient reduction which is sufficient

for our purposes. Definition:

Let ~I' ~2 be finite sets of symbols,

A 1 is efficiently reducible

A I eff A 2

to A2, written

and A I c ~i , A 2 c ~'~ .

139

providing

there is a polynomial

p and a Turing machine which,

started with

any word x E E l on its tape, eventually halts with a word y E ~ 2 on its tape such that I)

x E A I = y E A 2, and

2)

the number of tape squares used in the computation on input x is at most p(~(x))

(and a f o r t i o r i

~(y) K p(~(x))).

We remark that all the reductions which are described below require only a linear polynomial steps, but to minimize

number of tape squares and a polynomial

the demands on the readers intuition

actually give a flow-chart

or table of quadruples

we describe) we allow polynomials restricted

than is necessary

number of

(since we never

for the Turing machines

of any degree.

Even so, eff is much more

to prove Theorem 2.

Fact 4.

eff is a transitive relation on sets of words.

Fact 5.

If A I eff A 2 and SPACE (A2) ~ f (a.e.),

then there is a polynomial

p such that SPACE (AI) ~ ~n[ max[f(m) Fact 6.

I m s p(n)] + p(n)]

If A I eff A 2 and SPACE (AI) > tk+ I (i.o.), then SPACE

Proof.

Immediate

from Fact 5 and that observation

polynomial p, tk(P(n)) + p(n) K tk+l(n)

(a.e.) (A2) > tk (i.o.)-

that for any

for all sufficiently

large n.

The proof of Theorem 2 can now be summarized.

Proof of Theorem 2:

We will establish below that Empty ([0,I}) eff WSIS Empty (~) eff Empty (C0,1))

for any finite ~,

140

and finally that for any k and for any set A c [0,I~

such that

SPACE (A) ~ tk (a.e.) there ks a finite ~ such that A eff Empty (~) From fact 4, we have A eff WSIS for any A and k such that SPACE (A) ~ tk (a.e.)Then from facts I, 3 and 6 we conclude that SPACE

(WSIS) > tk_l(i.o.)

for any k.

Q.E.D.

It remains only to establish the required reductions. Lemma I:

Proof:

Empty(J0,1})

eff WSIS

For any y-expression ~ over [0,i] we shall show how to construct

a formula F~ E LSI S with two free integer variables variable.

For any set M c N, let CM: N ~ [0,I} be the characteristic

function of M that is, CM(n ) = I = n E M. constructed

and one free set

The formula F

will be

so that for n, m E N, M c N, M finite:

F (n,m,M) is true =

[[n < m and CM(n)-CM(n+I)

... CM(m-I) E L(~)]

or In = m and k E L(~)]]. F

is constructed by induction on the definition of y-expressions.

If

is 0 or I, then Fo(x,y,X)

is [y = x+l an___dd~ (x E X)],

Fl(X,y,X)

is [y = x+l and x 6 X].

If ~ is ( ~ " 6 ~, then

F (x,y,X)

is (Zz)[x ~ z and z ~ y and F~(x,z,X) and F6(z,y,X)].

141

If Ot is ~ ( ~ %, then F (x,y,X) is (~X0)[F~(x,y,Xo)].

If ~ is ( ~ U 6% or ~ i

~ 2' then F

is [F~ or F6] or [x ~ y and -~ F~(x,y,X)],

respectively. It is clear that there is a Turing machine which, given an input E [0, !, ( , ), U, ~, ', -~ } , can test whether ~ is a well-formed y-expression and, if so, print out the sentence (~x)(~y)(~X)[F

(x,y,X)],

never using more space than some fixed polynomial in ~(~). well-formed,

the machine prints out (~x)[x = x+l].)

(If ~ is not

Hence, Empty((0,1})

elf WSIS •

Q.E.D.

It will be convenient to work with larger symbol sets than [0,I}, but a trivial coding will demonstrate

that this involves no loss of generality.

Let ~ be any finite set of symbols with I~ I >- 2. 0 ~ I.

Say 0, i E ~,

Then for any n e i, there is a y-expression over ~ for (~n)*.

To see this, consider a word in ~'~ not in (0 n'l i)*.

Such a word either

fails to begin with 0 n-I I, fails to end with I, or contains a subword in 0 ~n'l(F~-0) or I ~n-l(~j-l).

Hence

u ~((0n-ll)*) = ~~, and (En) * = y((O n-I I)*).

142

Now given any finite set E 1 choose n sufficiently large that I~nl ~ I~iI and let h: ~I ~ ~n be any one-one function.

Extend h to a one-one

map from P(~I* ) into p((~n).) by the obvious rules h(k) = k, h(X~l) = h(x).h(~l)

for x E D 1 , ~I E E l, and h(A) =

~ {h(x)} for A c E 1 . xEA

There is then a y-expression over ~ for h(~ 1 ), because a word fails to he in h(~ 1 ) either because its length is not a multiple of n, or else because it contains a subword of length n net in h(~l) which begins at a position congruent to one modulo n:

~ _ h ( ~ l * ) = ~((~n).) U (~n)*.(~n-h(~l))'(Dn)*. Lemma 2:,

(Coding)

Let ~I' ~ be finite sets of symbols with I~I ~ 2.

Let h: P(~I ) ~ p((~n).) be the extension of a one-one function from E 1 to ~n for some n ~ I. a

There is a Turing machine which, started with

y-expression ~ over ~I' halts with a y-expression ~ over ~ on its tape

such that h(L(~)) = L(9). Moreover the space used during the computation with input ~ is bounded by a polynomial in %(~).

Proof.

The transformation of ~ to ~ operates by applying the following

rules recursively. If ~ E El, 9 is set equal to an expression for h(L(~)). If ~ is ~ ~I ~ ~

or ~ ~i ~ ~2 ~ ' then 9 is ~ 91 ~ 92~ or

i 61 ~ ~2 ~ ' respectively, where 61 , 92 are the transforms of ~I' ~2" If ~ is ~ i

~I ~ ' then ~ is

143

where ~l is the transform of e I and ~'I is a y-expression over ~ for

D -h(D 1 ).

(Note that h(y~(A))

= yD(h(A)) n h(D 1 ) for A c D1 , which

justifies this rule.) Finally,

if ~ is ~ ( ~I ) ' then ~ is ~ (~(~I ~ ~ ) ~ ~) " since

h(D 1 -A) = h(~ 1 ) -h(A) = D

-(h(A) U (D -h(~ 1 )) for A c ~I "

It is clear that a Turing machine can carry out this recursive transformation within the required space bound.

Corollary: Proof:

Empty (~) eff Empty [0,I}

Q.E.D.

for any finite ~.

Code ~ into [0,I} via h as in Lenmaa 2.

Then ~ 6 Empty (~) ~

L(~) = @ ~ h(L(~)) = @ = L(~) = @ ~ ~ E Empty [0,I}. We now show how, given a y-expression

Q.E.D.

for ~n, one can construct

a y-expression of about the same size describing any desired computation of a Turing machine, providing the states and symbols of the Turing machine can be represented in ~ and the computation only requires n tape squares.

This construction will be applied recursively to obtain

y-expressions

of size n for ~ tk(n), and will then finally be used to

conclude that A eff Empty (~) for any A c [0,I} tk (a. e.).

such that SPACE (A)
0, halts with a y-expression ~ over ~ such that

L(~) = Comp~,bn.y.bn).

146

Moreover, there is a polynomial p such that ~(1~) never uses more than p(~(y.#-~)) tape squares in its computation.

Proof:

We shall describe how to construct the y-expression ~ for

Comp~bnyb

n) from y'#°@ where L(~) = ~ n

We begin by noting that the

words in ~* no_~t equal to Comp(~,h n yb n ), i.e. , m(Comp(~,bnybn)),

can be

characterized as follows: I)

words that do not begin with #bnybn#, or

2)

words that do not contain qa' or

3)

words that do not end with #, or

4)

words that violate the functional condition determined by

~

in Fact 7.

These four sets of words can also be described by the formulas I')

m(#.(e(~) N b ).y'(e(~) n b )'#'E ),

2')

m(~*,([qa}XT ) " ~*) ,

3')

~(~ .#),

4' )

U ~i,@2,~3 ~

[~*.~i~2~3 •L(~) .~(Y)-I.L(~). (~-~(~i,~2,~3)) "~e]

But it is easy to see how to construct y-expressions directly from (1')-(4'), and therefore ~ is simply the complement of the union of these four expressions.

Note that ~(~) ~ c.~(y#~) for some constant c which

depends only o n e ,

and not on y or ~.

Moreover a Turing machine

~I~) which constructs ~ from y#~ need never use more tape squares than ~(~), and so certainly runs within a polynomial space bound.

Q.E.D.

147

Definition:

k ~-tk-TM is a Turing machine such that for some polynomial

p, some function fk m tk' and all n > 0, when the Turing machine is started with O n on its tape, it halts with a word ~ on its tape such that I)

~ is a y-expression over ~ and L(~) = ~ fk(n),

2)

the number of tape squares used in the computation is at most p(n).

Lemma 4:

If there is a ~'-tk-TM for any finite ~', then there is a

~-tk-TM for any ~ such that I~ I ~ 2. Proof:

Code ~' into ~ as in Lemma 2.

Details are left to the reader. Q.E.D.

Lemma 5: Proof:

For any k m 0 and any D w i t h

A ~-t0-TM simply prints an expression for y(c n) from input O n,

where ~ E ~ is any symbol. ~-tk-TM.

IDI ~ 2, there is a rrtk-TM.

Let %

Proceeding by induction, assume there is a

be a Turing machine which, started with O n on its tape

for any n > 0, lays out tk(n) tape squares on its tape and then uses these tape squares to cycle through some number fk+l(n) ~ 2 tk(n) = tk+l(n) steps before finally halting. to o b t a i n %

as described.

Since tk is tape-eonstructible,

it is easy

Choose ~ as in the simulation lermna applied

to % . The ~-tk+l-TM operates as follows: obtain ~ such that L(~) = ~fk(n).

Given O n, use the ~,-tk-TM to

Apply N ~ )

of the simulation lem~na

to (q0,0)0n'lo#.~ where q0 is the start state of % .

This yields a

148

y-expression ~ such that L(~) =

Comp~,~where

x = b fk(n) .(q0,0)0n-I bfk (n)

But C o m p S , x )

is defined since ~

tape squares.

Moreover, ~(Comp(~,x)) e tk+l(n) since . ~ runs for at least

tk+l(n) steps.

halts on input O n within tk(n) ~ fk(n)

Hence, the output of the ~-tk+l-TM is simply y(~).

Since by hypothesis ~ is obtainable in space Pl(n) for some polynomial PI' and similarly ~ is obtainable in space P2(n+l + Pl(n)) for some polynomial P2' the entire process requires only polynomial space.

Lemma 6:

For any set A c [0,I]

Q.E.D.

, if Comp(A) ~ tk (a.e.) for some k ~ 0,

then there is a finite ~ such that A eff Empty(~).

Proof:

Let ~ be a Turing machine which recognizes [0,i] -A and for

every x E [0,I]

, ~halts

using at most tk(~(x)) tape squares.

By

Fact 2, there is such an 3. Choose ~ as in the simulation lemma applied t o N . The Turing machine which efficiently reduces A to ~ as follows:

(~) operates

given x E [0, I} , use a r~tk-TM to obtain a ~-expression

such that L(~) = ~ fk(n) for n = ~(x).

Apply N ~ )

of the simulation

lermna to (q0' u)°w.#,~ where q0 is the start state of ~, and x = uw for u E {0,I}, w 6 [0,I]

(We ignore the case x = k.)

This

yields a 7-expression ~ which we claim is the desired output. Since ~ requires space at most tk(n) , we conclude that Comp(~,y) where y = bfk(n).(q0,u).w.bfk(n) x E A ~xiSnot

is nonempty iff x is accepted by ~Tg

accepted b y e =

Comp(~y)

Hence

= # = L(~) = @ ~ ~ E Empty(~).

This verifies our claim that ~ is a correct output.

149

As in the preceding requires

lemma, the Turing machine transforming

space at most a polynomial

This completes

in ~(x).

the lemmas required

Q.E.D.

for Theorem 2.

It is not hard to extend this argument to obtain Theorem i. use a stronger

x to

We

form of Fact 3 due to Blum [i ] to obtain from the proof

of Theorem 2 more information about the frequency of the (i.o.) condition in the statement

Theorem 3:

that Comp(WSIS)

The following decidable

are not elementary-recursive: countable well-order, under ~.

> tk (i.o.).

Also,

full and weak second order theories

two successors,

countable

linear order,

unary function with countable domain, unit interval

first order theory of two successors with length and

prefix predicates,

and the first order theory of , where P(x,y)

[x is a power of two and x divides y], are decidable but not elementary~ These results

follow by reasonably straightforward

efficient reductions

of WSIS to each of these theories. y-expressions

are themselves

of interest as a decidable but non-

elementary word problem. Corollary:

Empty((0,1})

is not elementary-recursive.

Further remarks: (i)

The results and methods described here were developed in May, 1972. [9] This paper is a revised version of a preliminary report with the same title written at that time.

Since then, in collaboration with

t The decidability

of these theories

is shown in [6,12].

150

M.J. Fischer, M.O. Rabin, and L. Stockmeyer, J. Ferrante and C. Rackoff, close upper and lower bounds on space or time have been obtained for most of the classical decidable theories in logic as well as for various notations related to y-expressions. Some of the more interesting results to appear in forthcoming papers are (i)

(Meyer) The satisfiability problem for sentences in the first order theory of linear order is not elementary; in fact space tE.n(n) is required for some E > 0. much space.

An upper bound tc.n(n) follows from Rabin's

proof that S 2 S is decidable (ii)

WSIS also requires this

[12]~

(Stockmeyer) The emptiness problem for expressions involving only the operation of U, ., ~ is not elementary, that is, the y-operation is unnecessary.

The simulation lemma and its proof

become considerably more subtle. (iii)

(Fischer-Rabin) Any decision procedure for the first-order theory of iN,+>, that is, Presburger's arithmetic, requires t2(E.n) steps even on nondeterministic Turing machines. Ferrante and Rackoff[7],

following Cooper[5] and Oppen[ll],

have established an upper bound of space t2(E,n). (iv)

(Fischer-Rabin) Any decision procedure for the first order theory of requires time t3(E.n) even on nondeterministic Turing machines.

Rackoff has shown that space t3(c.n) is

sufficient.

In [12], Rabin inaccurately claims his decision procedure is elementary. In a personal communication, he has informed me that he was aware that his procedure required space tc.n(n), but that he misunderstood the definition of elementary.

151

(v)

(Fischer) Let g be any class of structures with a birmry associative

operator * and the property

that for arbitrarily

large n there exists s E S E g such that sn ~ sm for I ~ m < n, where sm = s , s * ... * s.

satisfiability of * requires

Then any decision procedure

m over g of sentences tl(E.n ) steps.

(vi)

in the first order language

This general result applies to

nearly all the familiar decidable the propositional

for

theories

in logic, except for

calculus and pure equality.

(Meyer) The decision problem for satisfiability

of sentences

in

monadic predicate calculus with only seven (approximately) quantifiers

requires

Turing machines;

time tl(E.n ) even on nondeterministic

time tl(c'n ) is achievable

on nondeterministic

Turing machines. (vii)

(Fischer-Meyer) sentences

The decision problem for satisfiability

of

in the first order language of a single monadic

function is not elementary. (2)

Abstract complexity theory has been open to the criticism of being ur~ble to exhibit "natural" decision problems such as speed-up appeared.

in which phenomena

Applying Blum's results

[2] on effective

speed-up to our simulation of Turing machines via WSIS, we can show that given any decision procedure construct a new decision procedure

for WSIS,

one can effectively

for WSIS which is much faster

(faster by t k for any k) than the given procedure on at least one

152

sentence of length n for all sufficiently large integers n.

Similar results

apply to the other decision procedures mentioned above. (3)

The relation elf can be characterized in a manner similar to

the definition of the elementary functions or the primitive recursive functions,

2.5 e , so called because it lies properly between the Grzegorczyk

2 ¢3, classes e and is defined inductively as follows: I.

x"

y, x+y, x.y, xLl°g2 yj E e2"5,

2.

e2"5 is closed under explicit transformation (substituting constants and renaming or identifying variables),

3.

e2"5 is closed under composition of functions, and

4.

e2"5 is closed under limited recursion, limited sum and limited minimization.

5.

That's all. *

If we identify words in ~ notation, and for any set A c ~

with the integers they represent in I~I-adic let CA: N -~ [0,I] be the characteristic

function of the set of integers identified with A, then B eff A if and only if CA(X) = CB(f(x)) for some f E e2"5 and all x E N. 2.5 Essentially g provides a high-level progran~ning language in which one can formally express the procedures we informally claimed could be carried out by polynomial space-bounded Turing machines.

In this manner

our proof could be presented in a completely formal fashion without appeal to intuition about the space requirements of computations.

We prefer the

latter approach.

tSee Grzegorczyk's paper for definitions. [8]. Closure under limited recursion actually implies closure under limited sum and limited minimization°

153

Acknowledgments:

Larry Stockmeyer's proof, that any problem decidable

in nondeterministic polynomial time is deterministic polynomial time reducible to the regular expressions not equal to ~ , provided the key idea of the simulation lermna. Jeanne Ferrante and Charles Rackoff worked out the efficient reductions of WSIS mentioned in Theorem 3.

Patrick Fischer

correctly suggested that the use of * in my original proof was inessential. My colleague Michael ~ Fischer's suggestions and attention were extremely helpful, as they invariably have been in the past.

October, 1973 Cambridge, Mass.

154

REFERENCES

I.

Blum, M. A machine-independent theory of the complexity of recursive functions, Jour. Assoc. Comp. Mach., 1_4, 2 (April, 1967), 322-336.

2.

Blum, M. On effective procedures for speeding up algorithms, Jour. Assoc. Comp. Mach., 1_8, 2 (April, 1971), 290-305.

3.

Buehi, J.R. and C.C. Elgot, Decision problems of weak second order arithmetics and finite automata, Part I, (abstract), AMS Notices, (1959), 834.

4.

Buchl, J.R. Weak second order arithmetic and finite automata, Zeit. f. Math. L0$.and Grund. der Math., ~ (1960), 66-92.

5.

Cooper, D.C. Theorem-proving in arithmetic without multiplication, Computer and Logic Group Memo. No. 16, U.C. of Swansea, April, 1972, t_o_oappear in Machine Intellisence ~.

6.

Elgot, C.C. and M.O. Rabin, Decidability and undecidability of extensions of second (first) order theory of (generalized) successor, Jour. Symb. Logic, 31, 2 (June, 1966), 169-181.

7~

Ferrante, J. and C. Rackoff, A decision procedure for the first order theory of real addition with order, Pro~ect MAC Tech. Memo 33, Mass. Inst. of Technology (May, 1973), 16pp., to appear SlAM Jour. Comp.

8.

Grzegorczyk, A. Some classes of recursive functions, Rozprawy Mat ematyczne, ~ (1953), Warsaw, 1-45.

9.

Meyer, A.R. Weak SIS cannot be decided (abstract 72T-E67), AMS Notices, 19, 5 (August, 1972), p. A-598.

I0.

Meyer, A.R. and L.J. Stockmeyer, The equivalence problem for regular expressions with squaring requires exponential space, 13 th Switchin$ and Automata Theory Symp. (Oct. 1972), IEEE, 125-129.

I I.

Oppen, D.C. Elementary bounds for Presburger arithmetic, 5 th ACM Symp. Theory of Computir~ (April, 1973), 34-37.

12.

Rabin, M.O. Decidability of second-order theories and automata on infinite trees, Trans. AMS, 14___1(July, 1969), 1-35.

13.

Rabin, M.O. and D. Scott, Finite automata and their decision problems, IBM Jour. Research and Development, ~ (1959), 115-125.

14.

Ritchie, R.W. Classes of predictably computable functions, Trans. AMS, 106 (1963), 139-173.

15.

Stearns, R.E., J. Hartmanis, and P.M. Lewis, III, Hierarchies of memory-limited computations, 6 th Switching Theory and Lo$ical Desisn Symp. (1965), IEEE, 179-190.

16.

Stockmeyer, L.J. and A.R. Meyer,

Word problems requiring exponential

time, __5 th __ACM8ymp. Theory __°fComputing (April, 1973), 1-9.

THE V A R I A B L E I W.V.

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from the The

of the N a t i o n a l

156

schematic it does priate whi c h

letter

not

itself

refer

expressions are not

of course

to any objects

as substitutes.

bindable,

there

substitutional

variables

in sentences.

These

way,

as m e a n i n g

gory

is thus

But

under the q u a n t i f i e r app r o p r i a t e tential

comes

grammatical

objectual.

To take

our langauge

every

admits

schematic

still

they

be read

x of the

letters,

in the o b J e c t u a l appropriate

can be e x p l a i n e d

out true

all s u b s t i t u t i o n s

and

cate-

by clear truth

is true

category;

of

and e m b e d d e d

quantification u n de r

appro-

and others

by q u a n t i f i e r s

cannot

object

if the sentence

correspondingly

over classes,

stitutional? can write

in the

for the exis-

But what

language

that

of any individuals, another

uniquely. a uniquely

if that were

assuming

in

does

come

law: of W as sole member)).

It may be called the 'W' and

of every

class

closed

if its m e m b e r s h i p obscure

singles

term that we

and remote,

out one

for every

variables

law of

'Z' are sub-

condition

as a s s u m i n g

expression

our i n d i v i d u a l

and

to quan-

look tempting;

it come to when

that

for

obscure

When we

a member

is as u n w e l c o m e

designating

welcome,

as

expression,

however

version the

naturally

require

at best.

however

condition

This

would

designating

set theory.

does

It then requires,

to write

is seen most

or particle,

.D (3Z)(Z has

in our language,

to be true

viduals

creature

for classical

unit subclasses.

uniquely

the s u b s t i t u t i o n a l 2 arises. Consider

a problem

is gospel

over i n d i v i d u a l s

be a r t i f i c i a l

(W)(W has members

able

or some

or some

and this w o u l d

however,

This

can be b o u n d

it as s u b s t i t u t i o n a l

a name,

individual,

remote; tify

substitutional:

quantification.

Our q u a n t i f i c a t i o n

every

But besides

quantifications

The u n i v e r s a l

purely

but merely

is the use by Ruth Marcus

that

that every

and so.

conditions.

is already as values,

happens we must

of those outright

be

indiin our

individual;

and

could go s u b s t i t u t i o n a l

too. Parsons

has p r o p o s e d

quantification

that

fication

counts

stance

still

that

contains

some values

of them.

averts

a modified this

as true free

condition

variables

standard

for s u b s t i t u t i o n a l

For him an e x i s t e n t i a l

as long as it has

objectual

By this

truth

effect.

the

a substitution

and comes

law of unit

quantiin-

out true subclasses

for is

true. I said that

21 am i n d e b t e d me on the

the b i n d a b l e

to Oswaldo

following

obJectual

Chateaubriand

line of thought.

variable

has

and Gilbert

close

associations

Harman

for s t a r t i n g

157

with

quite

a variety

it from variable these no place schematic mat i c

letter,

I have

dissociate

thus been

by those

We tend

into

it was

benefits 'Dx)Fx'

'{x:

familiar

Fx}'

'~

or class

can be defined

in his

clusion tion

lambda

ductions,

because

compact,

then,

quantification fixed

lexicon

subject range

of values

The

readings

The

are

under

'x' o f

the

class

when

~ y.~ F x)'

similarly.

The

So does

calculus,

'x'

the bound

for it is ulti-

directions

can be taken

of e i t h e r I the

is w a n t e d

However, to these not

and q u a n t i f i -

logic b a s e d

quantification, line.

Functional

Church took the one

other in my

to be p r e f e r r e d

too.

as basic,

of them.

Peano had a f u l l - b l o w n

only

class

there

logic

of q u a n t i f i c a t i o n ,

on in-

abstrac-

though

his

reduction

to

alternative in set

where

theory,

re-

theory

is no call for classes

in Tarski's

or the p r e d i c a t e

unlike

phrase;

with

but

or

set

theory,

thus

particular

standardized,

of q u a n t i f i c a t i o n ; And of course

quantification

it is simply

one or a n o t h e r

to one or a n o t h e r

of a theory,

of q u a n t i f i c a t i o n .

of o b j e c t u a l

calculus,

appropriate

ontology

of the variables

are the variables the

there

'x' of the de-

and convenient.

of p r e d i c a t e s

matter.

moreover,

vari-

abstraction.

and

is a s t a n d a r d

logic,

of bound

The

'(~)(x)(x

fares

took another

the

to

its nature

'x' of q u a n t i f i c a t i o n

existential

theories

variable

- -

quantification

and m o r e o v e r

is complete, Such,

and

in q u a n t i f i c a t i o n .

uses

so.

in a l t e r n a t i v e

Even

is g e n e r a l l y

other

and i n t e g ra l

calculus,

quantification

also in e l e m e n t a r y functions;

substi-

I want next

'x' of q u a n t i f i c a t i o n

an

fx'

in terms

and abstraction.

quantification

an

abstraction

on w h i c h he based his

universal

primarily

of description.

of functional

can be made

objectual

and because,

as a d e s c r i p t i o n

of the d i f f e r e n t i a l

abstraction

line

as clear

so as to reveal

by doing

over into

abstract

a variable

use,

over into

goes

is defined

Reductions

cation

and the sche-

contexts.

definition

X

mately

the b i n d a b l e

variables

to be g a i n e d

goes

functional

variable

variable

apt to be confused.

we know how to p a r a p h r a s e

contextual

Fx}'

the v a r i a b l e

for

from the

this

dissociating

two

scription

'{x:

being

recognizing

from q u a n t i f i c a t i o n ,

Russell's

of the

there it also

it now from the b i n d a b l e

the q u a n t i f i c a t i o n a l

algorithmic

abstract

I am of both

to think of b o u n d

is because

ables

dissociated

dissociated

with which

it even

uncolored

though

I have

I have now d i s s o c i a t e d objects,

as far as it goes.

from associates

This

fond though

And I have

variable,

legitimate

and devices.

and other variable

in the universe.

letter.

tutional

of n o t i o n s

numbers

in ordinary

is the

for the variables it is clear language

from

that the

158

ontology thing

consists

of those

is such that' Very

I want 'all'

well,

to make and

variable

and

then;

needs

categoricals the b o u n d

variable

is fully

impose

reductive

Minutes more

basic

tion.

of

still

'all'

of

ago I e x t o l l e d

'some'

up with

and

'some',

that

shows

of the bound

variable

without

admixture.

clause,

mathematically

neither term,

a singular

want

tence.

some

Where

object

object

the

construction

sentence

is the relative a bound

and as

it is a g e n e r a l

a complex

perhaps

variable

original

of s c h e m a t i c a l l y of the

simplified to avert

and we

that may be simply

as the

segregation

term,

sentence

repeatedly,

noun

effect

clause

basic

regimented

abstract;

we have

is thought

is the explicit

This

the d i s t i n c -

a singular

or common

the same

is more

It is not

midway,

adjective

a with

order and fitted with

Fx'.

use where

a perhaps

original

clause

that

nor a class

its

a complex

of the

the relative

word

It has

to segregate

predicated

that'

description

a predicate.

that m e n t i o n s

'x such

abstrac-

no c o n n o t a t i o n

rather

idiom is the relative idiom:

as in a sense

It carries

tive work

'such that'

other

as we

or f u n c t i o n a l

variable

but

inte-

and yet these

of course

of reduction,

a use of the b o u n d

work when

abstraction,

neglected the

Conversely


abstraction

or function,

The q u a n t i t a t i v e

are wise'

except

direction

the

of the bound

its d i s t i n c t i v e

use in q u a n t i f i c a t i o n .

or class

The point

in the t r a d i t i o n a l

functional

in the d i r e c t i o n that

in?

work

function.

present

'Some Greeks

to class

is, however,

come

used in q u a n t i f i c a t i o n ;

definitions

than its

or

taken

'every-

of the q u a n t i f i e r ,

to the d i s t i n c t i v e

abstraction,

'all'

than r e d u c t i o n

There

basic

class

nothing

force

it is fully

mean

that'

the d i s s o c i a t i o n

are mortal',

no less than when

connote

is such

to its r e f e r e n t i a l

no variable;

used in description, gration,

does

is irrelevant

'All men

for the q u a n t i f i e r s

the q u a n t i t a t i v e

and i r r e l e v a n t

component

uses

where

is that

'some',

values;

'something

'F'.

senas

The

in respect

'F_~a',

'such of

ambiguities

of

cro s s - r e f e r e n c e . Other uses sitic upon this that'

and

ator is is

this

use.

'everything

'the

(thing)

'the class

thought

of the bound

of

functor

x such that';

be thought

of a functor

are

x such

x such

the pure

Similarly

that'

x such

of class

oper-

abstraction can be

'V' to a predicate;

quantitative

description

What brings

as para-

the d e s c r i p t i o n

Quantification

'S' or

'~' to a predicate;

of c o r r e s p o n d i n g l y .

represented

'there is s o m e t h i n g that';

the o p e r a t o r

of a functor

carries

of variables.

application

(a thing)

(the things)

is what

are readily

The q u a n t i f i e r s is

of as a p p l i c a t i o n

intrusion

variable

import,

with no

can be thought

and class

of as

abstraction

the variable,

and

if any,

can is

the p r e d i c a t e simple

itself,

adjective

Peano

saw this,

the i n v e r t e d

epsilon

words

three

in his

,3, and

,7, just

functor

for class

class

it is a relative

some

but then

slipped

term,

them.

a

or for the e q u i v a l e n t the

functors

But he i n t r o d u c e d inverted

confusion.

or predicate,

than

He i n t r o d u c e d

and he i n t r o d u c e d

He saw his

here was his

rather

a confusion.

'such that',

described

abstraction.

clause

compound.

into

languages,

as I have

the general

Boolean

for the words

romance

abstraction;

between

in case

or perhaps

epsilon

He did not

and the

class

no

as already

distinguish name,

a singular

term. The same the epsilon he a d o p t e d r~at.

conflation

that

is now

it as his

He i n v e r t e d

of predication; tion between

predication

variable

as relative

And,

to

of the

conflation,

whereas

conceived

is r a t h e r

for an u n b i n d a b l e context

was

clause

a predicate,

schematic

predicate

variable.)

With

to the

Happily

Peano's

contexts,

or

distinction

on this.

epsilon

as a sign purely

for

of the be

~) ~

clause

Cls'

bound

properly

of s u b s t i t u t i o n

(In this h i s t o r i c a l of a b i n d a b l e

sensitivity general

import

has

x such

explicitly

'(x~

over them with

letter.

between

to the rela-

But he must

He p r o v i d e s

alternative all his

is the inverse

to the role

at best

For and

of the Greek

'a is a thing

'such that'

to the o n t o l o g i c a l inverted

this

classes;

susceptible

predicate

insensitive

explicit

he quantifies

further

insensitive

ematical

a relative

epsilon.

from Peano;

sensitive

sensitive

designate

over the

singular;

thus

on two counts.

expressions

we may pass

tutional

because

clauses:

he was

is more to the point,

variables;

that'

upright comes

the initial

Peano was

He was indeed

pronoun;

'such that'

what

'such

and relative

'Fa'.

theory

of p r e d i c a t i o n ,

it for his

reduces

convicted

copula

in Peano's

in set

the two cancel.

that Fx'

that his

may be seen standard

he

and abstract

of values

caught

substi-

to g r a m m a r

of v a r i a b l e a

on,

in i n f o r m a l

math-

'such that';

and I shall

so

use it. Russell Mathematics function same,

but m u d d i e d

whose

matters.

indeed be seen

clauses.

designate;

so-called

were not really

I say

and this

functions any

over into The

'fictitious' with

because

of the

sentence, from Frege,

designata

Frege's

meaning

more

open

adapted

as fictitious

as u n s e s ~ t t i ~ t ,

such things.

was

of

propositional

is largely

property,

function'

accords

Principles

so-called

Mathematica

confusion:

'propositional

might

'such that'

term does not of his

'such that'

now by triple

The term

functions or

Peano's improving

that ended up in P r i n c i p i a

predicate.

tive

carried without

of rela-

a general

characterization

in a way

that

there

160

One senses general cause,

term,

in the m o d e r n

or predicate.

I expect,

of p r e d i c a t e

terms or

of our slowness

letters.

tual variables,

Peano.

Or,

variables

fleeing

to conflate

singular

general

class

abstract

terms

have

terms

been

on the

letters

status objec-

singular

the relative and here

one presses

duty

an uneasy

the

with

as abstract

Thereby

ontology~

for the

and partly

the schematic

such

abstraction;

quantificational

Predicates

a distaste

the effect

or classes.

becomes

this

of logic

to appreciate

tends

properties

clause,

into purely

sentences. stract

One

and so reconstrue

designating

'such that'

history

It is partly

and thus

clause,

we have

the bound

can operate

intermediate

one hand and o u t - a n d - o u t

between

in ab-

sentences

on the

other. We have isolate

to appreciate

and appreciate

in r e p r e s e n t i n g icate

letters,

can witness cumbered

the work

cate

of the bound

became

among

tended

under

that we may most

clauses

auxiliary

clauses

Bernays,

1934,

through

Open

sentences

for

status.

free merely

generally

thronghout

logicians

theories

the

predicate

of whole

substitution

substituted rules

were

them Nennformen. 'Fzw'

'x' and

the

for

class names.

are 'Fx'

complex, and

devised,

Hilbert

for

'Fz',

the purpose

Two sentences

of

they 'such

and

may be substi-

can both be got

'Z' on the one hand

I proceeded

only

open sentences

at length

scenes.

if they

to introduce

as o n t o l o g i c a l l y

to s u b s t i t u t i o n

that s e r v e d much

By coincidence

The predifor properties

it in retrospect.

behind

and

ele-

schematic

it is as a s c h e m a t i c

'F' as subject

such

though

'Fxy'

though

semantic

of freeing

for such

When

by s u b s t i t u t i n g

'w' on the other.

unen-

It had be-

by 1930,

substitution

'Fzw'

called

between

theories,

them as if they were

the sentences

after all,

tut e d r e s p e c t i v e l y some N e n n f o r m

pred-

that we

pronoun

indispensible.

first-order

letter

The rules

coordinate and

view

they avoided

the schematic

'Fxy'

invoked

clearly

their

'Fx~'

or for

as a relative

and

Still

into

for

or

contexts

the effect

therefore,

'Fx'

of its

have

predicates;

they must

It is

schematic

rather as a free variable

in w i d e r

indirectly,

that'

virtually

investigation.

But it did not

treated

itself.

in d i s t i n g u i s h i n g

logicians

to be seen

bindable

innocent

variable

in order to

into quantifiers,

and h i g h e r - o r d e r

became

Continental

sub c a l c u l u s

'such that'

variable

by s t e a d f a s t l y

letters

interested

taeories

letters

or classes,

They

these

to no clear a p p r e c i a t i o n

letter

letter

of the bound

letters

abstraction.

logicians

common

subject

schematic

clauses

letting

or f i r s t - o r d e r

use of p r e d i c a t e come

the work

'such that'

and not

by class

When ~entary

the

and exploit

similarly

and

in the

from 'z' and same

161

year, more

1934,

in my first

graphic

by use

stencils

in E l e m e n t a r y

blanks.

Expressions

C.S.

Peirce

Logic,

though

Also

in one sense

1945 p a p e r

that

the

at last

'such that'

They were

the r e s u l t i n g But quite

theory, Thus

in e x p l i c i t

on me to call playing

where

sentences

connection

with

term.

It remains

class

them p r e d i c a t e s ,

the role

I kept

of the

thus

can

abstraction

. F~

turgid

is to free

F"u =

--

'Fxz',

complex

certain

(u)[Func the

definitions

explicit

formalism,

even

is already

[

of

to set

available

in the

too.

Zermelo-

purposes

of

Such

the

the

of class

above

not

in his calling

.~

e

predicate

~))]

•

letters

of

thus

to

'such that',

from such embedding.

to complex

are already

. Fyz

predicates

familiar

We can

as fol-

for classes.

~

~

pause

.~

existence

(F"u)x)]

Theory over

and

Its

of

Losic

it

boils

down y e t

them.

not

the

grasped:

.

and predicate

name while for

.D. x = Z).

is then easily

predicates

case;

by assuming

GSdel

not

~

The effect

predicate

o f my S e t

I shall

.

z E u) --

P ~w)(x)(x

conventions but

Fyz'.

that

(x)(y)(z)(Fxz

The notation

though

clause,

the scenes.

(~w)(x)(x ~ ~

functors

axiom schema of r e p l a c e m e n t

defeated

recog-

no part

to the

of r e p l a c e m e n t

.3. ~ : ~)

x 9 (~z)(Fxz df --

Func F =dr

was

thus

and formed

contribute

as long as we confine

the

and apply

paralleling

further,

until my

relative

them b e h i n d

much

They

schemata

of p r e d i c a t i o n

then define

Under

by

system.

positions

The

indexed rhemes

a problem

It was not

(~z)(Fxz

lows,

with

called

or schemata.

the axiom

(u) (x) (y) (z) (Fxz

however,

formulas

had b e e n

substitution,

can c o n t r i b u t e

genuine

the N e n n f o r m e n

be called p r o p o s i t i o n a l

resilient

from substitution.

consider

Fraenkel

they

They were

could vaguely

And still

sentences

I made

I called these

same effect

for c a l c u l a t i n g

in fact

apart

not

they were

clause.

devices

1941.

of that

it dawned that

numerals.

the

they

functions,

nizing

In later books

to much

in 1892,

of substitution.

book.

of circled

requiring

purpose the

functors

of

class

the the

serves

class

axiom

{x_~_:

to

schema

some exist. would

be

Fxy~.

1940 m o n o g r a p h

availed

himself

of this

them relative

clauses

or general

convenience,

terms.

He intro-

162

duced

the e x p r e s s i o n s

which

he called notions.

plicit

way

I did much

Portuguese

lectures

for real tion

classes.

for notions

names

when

be seen

ultimate values

of b o u n d

hand,

are only

being

values

icates, What

variables,

a manner

are r e p r e s e n t e d

ues of bound

of s p e a k i n g

variables;

of 1937-54

Cantor

and K6nig were

passages

around

but

the

of a calculus

on the one hand

the point

of s i m u l a t i n g

long been

visible

tude

toward

the p e r v e r s e l y

utility

as an a l g e b r a to functors

coextensiveness, Three

of these

sentences pounding

3See

Set Theory

there.

of predicates,

and Peano's functors

sentences

This B o o l e a n

elementary

of union,

into

calculus

and Its

further

p.

whether

directly

other,

toward even

This bias

to

has

in the

atti-

of classes.

of logic has

its whole

by schematic

complement,

letters

inclusion,

and its dual

from p r e d i c a t e s , functions

'~'.

and four form for com-

ones.

of p r e d i c a t e s

Logic,

to think

calculus

Then we add truth

val-

in his

question

indeed,

'3' of n o n - e m p t i n e s s

sys-

are not

classes

on the

represented

intersection,

pred-

in Bernays's

classes.

bit

not

letters.

They g r a v i t a t e

level,

This

at all,

are really

otner in some b r i e f

been

names

Boolean

form p r e d i c a t e s

from predicates. these

class

so-called

is no call for classes

have

with

classe~

on the other

for they

"proper"

predicates.

classes

predicate

or the

of imaginary

at the most

subject

logicians

or toward

names

there

It is an open

the one century. 3

of complex

sentences

There

thing.

or

is

in not being mem-

names

classes,

it

are real

classes,

or " p r o p e r ' c l a s s e s

anticipating

turn of the

only

are schematic

the ultimate

Still

classes.

latter

really

terms,

simplicity

or virtual

seeming

the nota-

of class

general

and real

from sets

or virtual

how unready

some

or virtual

and not

general

the f o r m a l i s m

of classes.

These

Their

variables

are the real

We are n o t i n g in terms

classes.

variables.

with

clauses,

notions

ex-

in my

In Set T~eory

in p r e s e n t i n g

of virtual

and differ

notions

system

thought

classes

classes

as a s i m u l a t i o n

and Its L o g i c

to confuse

as ultimate

tem of 1958 are mere

integrated

thus

The notions

and their seeming

of virtual

up a s t r e a m l i n e d

of relative

integration

"proper"

of imaginary

level but in a more

out in 1944.

of pathos

to any

be taken not

of b o u n d

came

classes

a matter

classes.

the head

closely

in Set Theory

or s o - c a l l e d

of further

which

is a note

as p r i o r

from the close

Care must

under

classes,

There

elementary

later I w o r k e d

or virtual

should be said that

bers

years

it is really

and should

gained

the same

for virtual

q u as i - n a m e s

At a more

of 1942,

and Its Loci c twenty formalism

as eliminable

212n.

and p r e d i c a t e

functors

is the

163

easy

version

tifiers

in the new this

of m o n a d i c

any thought

notation

lief that

Boolean

has

it calls

in e l e m e n t a r y bit

for

logic

of Lo$ic

presentation

'F' and

for classes.

schools

logic

set

logic.

at this

convenience

level predi-

of

of the m i s t a k e n there

theory

that

and should not be seen

to

I avoid

the s c h e m a t i c

The

because

It is only

I have s w i t c h e d

to e n c o u r a g e

And c o n v e r s e l y

of a s o - c a l l e d

of m o n a d i c

that

'G' are still

foregone

are no quan-

letters.

of m o n a d i c


often been

and there

schematic

'F = G' in order not

of classes; of the

theory;

only the

of Methods

for the basic notation

letters

Boolean

but

third e d i t i o n

calculus

the identity

cate

quantification

and no variables,

be-

is the new irony is Just

this

as set theory

at

all. Behmann Boolean ever,

should be m e n t i o n e d

functors

in very much

he then promptly

tively

according

It is the

his

familiar

My theme was am p e r s u a d e d

style

predicates

that the e m b r y o

forth e x p l i c i t l y

We may

when we regiment clauses

How is a l o g i c i a n

to frame

a calculus

of p r e d i c a t e s ?

by

as dummies.

ician,

alive

predicate

and so his

He will

of exposition. letters

relative becomes

But he

embedded

free

all.

some

clauses

floating

schematic

pure

'such that' complex

letters.

little

their

old B o o l e a n

with

a true

that

so

of abstract singu-

A later

calculus

log-

of

in its usual on keeping

fearing

that

be a class name

as an o n t o l o g i c a l l y

logic to his

the

letters,

he insists

a

again.

innocent

representable

incidentally,

is,

of such ex-

abstract

logic

arguments,

would

p r ed i c a t e s ,

And so,

sort

become

I

using

terms,

abstraction.

overreacts:

with

it.

need to represent

with

class

and u n g e s ~ t t i g t

to appreciate

clause

He is dim on s c h e m a t i c

reacts

his

after

variable

calculus

quantification

for i s o l a t i n g

by free-

he has n e e d l e s s l y

unregenerate

prede-

set theorist.

Its r e s t o r a t i o n

4See The

a formal

be variables

'such that'

operator

cessor the

effec-

psychogeneti-

general

letters,

failed

names

relative

familiar

floating

surrendered

must

the

to s c h e m a t i c

style

How-

and thus

as b i n d a b l e

and it is the

his p r e d i c a t e

He has

letters

and his

predicates; modern

letters

as values;

functors.

variable,

are adjectives,

predicates

lar terms

the

do well now to retrace

of the b i n d a b l e 4 Its status

pressions,

objects

of class

pronoun.

Relative

these

of p r e d i c a t e

treated

by q u a n t i f y i n g

the status

the variable.

stands

he thinks

in 1927,

pitfall.

is the relative

predicates.

the

spoiled matters

cally,

'SUCh that'.

as one who,

Roots

involves

of Reference.

curious

ironies.

Its r e s t o r a t i o n

de-

164 pends, we saw, on a better a p p r e c i a t i o n of the bindable variable as an appurtenance of the relative the schematic predicate variable arguments,

and so the variables

Bound variables lus of predicates, programming,

clause, not of the class abstract.

letters then become detachable themselves

But

from their

disappear.

vanish thus from the scene, in our Boolean calcu-

but they lurk in the wings.

They

to switch to a computer metaphor.

figure in the

For when we apply this

calculus to verbal examples, we shall want usually to interpret

'F',

'G', etc. not just by s u b s t i t u t i o n of pat words or phrases such as 'man'

or 'Greek'

tive clauses as

or

'white whale',

'~ ~

~)(~

but by s u b s t i t u t i o n of such rela-

is son of ~)'

or '~ ~

(3~ 2 ~

here we have the b o u n d v a r i a b l e at its p r o p e r work. still use this variable in hidden of predicates,

foundations

2~)';

and

Also we might

of our Boolean calculus

thus:

--F ~df _x ~ ~Fx__ , FnG =df ~ ~

(Fx . G_~x) ,

F~G =df ~ ~ (FX V G_~x) , ~ ~ =df W{x ~

(F__xxD G__xx))

We well know that q u a n t i f i c a t i o n theory, which is so much more complex than the Boolean predicate tion in polyadic predicates.

calculus,

has its serious motiva-

When we move to p o l y a d i c

~ound variable quits the wings

and gets into the act.

of the bound variable is c r o s s - r e f e r e n c e tence where objective reference occurs;

to various places

for it also within the ongoing algorithm, and i n d e n t i ~ c a t i o n s

It is in such permutations

that d e c i s i o n procedures There is evidence

of arguments

able,

like monadic

ment places.

logic,

that the bound variable

and here it is, by the way, available.

Polyadic

tional schema that is fluted,

covering every q u a n t i f i c a -

as we might say, in the following sense.

letter has the same variable

'~' as its first argumen~

though this r e p e a t e d letter may in its different by different has one a ~

occurrences

logic remains decid-

as long as there is no crossing up of argu-

There is a decision procedure

Every predicate

logic

logic calls

of p o l y a d i c predicates.

cease to be generally

of a connection.

polyadic

in order to keep track of

and i d e n t i f i c a t i o n s

enters e s s e n t i a l l y into the algorithm,

in a sen-

and whereas monadic

calls for this service only in the p r e p a r a t i o n s ,

permutations

logic, the The basic job

of '(~)' or

the same letter

'(~)'

occurrences be bound

Every p r e d i c a t e

'~' as its second argument,

letter

if any; and so

165

on.

And, a final requirement,

stands in the scope of some

each occurrence of a '~' q u a n t i f i e r

'x' quantifier;

q u a n t i f i e r stands in the scope of some gave a d e c i s i o n procedure

each occurrence

'~' quantifier;

for such formulas

The variable,

then,

does not, however, Boolean predicate if we wish, predicate

but this appears

it seems,

and so on.

I

at the Congress of Vienna.

(A further proviso was that all the predicate n u m b e r of argument places;

of a 'z'

letters have the same

superfluous.)

is the focus of indecision.

set bounds to algebrization.

By s u p p l e m e n t i n g the

functors with a few more p r e d i c a t e

still banish the bound variable

functors we can,

for good•

functors that will do all necessary

It

For there are

linking and p e r m u t i n g of

argument places• The predicate

functors that I have in m i n d are somewhat reminis-

cent of S c h 6 n f i n k e l ' s

combinators,

but with a deep difference:

m e t h o d had the full strength of set theory, whereas what m i n d is equivalent logic•

to q u a n t i f i c a t i o n

Mine is closer to Tarski's

m o d i f i e d by Bernays, functors as before; score• cates

theory,

but complement

or f i r s t - o r d e r predicate

cylindrical

and to work of Nolin.

his

I have in

algebra,

especially

as

There are the Boolean

and i n t e r s e c t i o n

suffice on that

I construe them in a p p l i c a t i o n to ~ - p l a c e and ~-place predi'F~'

and

'G~' generally,

as follows.

F~ Xl...X_m~ ~F~ X l...~m (F~G~)xI" The variables

•

"~max(m,n)

~

"

F ~

X l .. "~m

G~ -

~l'''~n

'~I' etc. have no place in the system, but serve only in

my present e x p l a n a t i o n

of the functors.

To continue,

ther devices, which t o g e t h e r accomplish are a cropping functor, a constant

.

a p a d d i n $ functor,

identity predicate.

a permutation

~ m F

(pFm-)xl~3...x~2

the furvariable~

functor,

and

They are e x p l a i n e d as follows:

C [ [ ) ~ 2 "''~m m (]~l)F-m- ~ l ' ' ' ~ m

(

is v e r y board.

algebras

-a = 1 = the u n i t ; a n d

It i n c l u d e s ,

(with an a d d i t i o n a l

the o p e r a t i o n s a~

b = -a ~ b a ~

a

ideas

models.

relation

favorite

as

to the b a s i c

Then

and,

scheme

conventional

of y o u r

with

of c o n v e n -

of the F r e g e a n

structures

[ii].

truth

framework

a distinct

concerned

and

the

it b u t

A l l we h a v e

to the e n t a i l m e n t

A non-empty five

the

like

with

and associated

of s a t i s f a c t i o n

leads

algebra

as usual,

structure

not

independent

it in 1969,

notions

ally

is,

within

component

or m a y

of c a t e g o r i e s .

right

algebraic

to e l a b o r a t e

binary

of c o m p l e m e n t , and

a+

-a = 0 = the

b = zero

in paropera-

meet

and

(a~b)~(b~a). of the B o o -

182

lean algebra Given non-empty related

A.

Also,

we write

an a l g e b r a

subsets.

~,

(4.2) (4.3)

imagine

Denote

to the operations

a

/- [ (~.-'- t~'~ ->

C Y("/,,,'~ ~,~--> %~' "/t~) ") 1. %

We also have

the s y m m e t r y

[~-0 ~ * ( ~ )

~

I~-~], ~d

The e n t a i l m e n t of theories. LA duce

~

A set

and c o n t a i n s

,

X

~

tions.

Given

mulas

entailed

over

X,

w h i c h comes

i.e.,

X.

induces

a theory

~

also apply

and

X,

let

In other words,

TH(X) TH(X)

mulas.

consistent

Properties

TH(X)

are c a l l e d

theories

information are p r e c i s e l y

complete

theories

consistent

theories

which

satisfy:

either

~

~

~

q ~

T

Finally,

collection

(5.10)

includes

of all t h e o r i e s

if

~

either

the e n t a i l m e n t

is not

theory w h i c h obviously•

has the

in TH(X) does

~

following

then

5 resulted But,

very d i s t i n c t in d e f i n i t i o n s

the truth

one may prove

~

~

they all

on the those

is in

whenever

T

T

includes

in the sense

exists

that the

property:

a complete

but c o n t a i n s

sketched

of the e n t a i l m e n t s

the equalities:

for-

TH(X) ;

(XXIII)

and adequacy.

constructions

is that w e were

or

separation

there

we

also holds.

6. C o m p l e t e n e s s Three

or

is regular

not include

the c o n v e r s e

and since

store m u c h

Here,

(equivalently,)

Furthermore•

theories.

entailment.

or

X.

complete

underlying

T

we

theory

~ F M = the set of all

we may talk of c o m p l e t e

of c o m p l e t e

First•

= the s m a l l e s t

if

as a rule,

We intro-

for-

X

are m t h e o r i e s

•

includes

= the set of all

say that a set

sets

w~ff

X

opera-

on the a x i o m set

Maximal

iff

of c o n s e q u e n c e

the t h e o r y b a s e d is c o n s i s t e n t

a collection

to any entailment.

from the theory

a set of formulas by

is here

it contains

which

[~~(~,~)j

~

as e v e r y e nt a i l m e n t ,

whenever

a notation

laws:

~ (~ ~ )

of formulas

some o t h e r n o t i o n s

introduce

and t r a n s i t i v i t y

dealing with

in sections ~)

~)

the same thing.

3,4 and and Indeed,

188

Hence,

the c o r r e s p o n d i n g t a u t o l o g i e s c o n s t i t u t e e x a c t l y one set:

~($) t a u t o l o g i e s =

(6.2)

Two e q u a t i o n s in

~

(6.2) are said,

sent two c o m p l e t e n e s s theorems i.e.,

tautologies =

set of formulas

On the other hand,

~)

tautologies

in the current terminology,

to pre-

for SCI, c o n s i d e r e d as a formal system,

( ~ tautologies)

g e n e r a t e d from LA by MP.

(XXV)

(6.1) is said to p r e s e n t two so called g e n e r a l i z e d

or strong c o m p l e t e n e s s theorems for SCI (xxvI) It is not my intention to combat d e f i n i t i o n s or change terminology for my own amusement. emphasize,

Yet t e r m i n o l o g i c a l d i f f e r e n c e s

clear cut d i f f e r e n c e s in m e t h o d o l o g i c a l approach.

the case here and we decide to focus on equalities = SCI.

~)

in

~

(6.1) and to call them c o m p l e t e n e s s

The e q u a t i o n s i n v o l v e d in

theorems

sometimes

for the SCI.

=

This is ~)

theorems

and for the

(6.2) may be called w e a k c o m p l e t e n e s s

We thus stress the fact that we are not think-

ing w i t h i n the f r a m e w o r k of the theory of formal systems and we do not c o n c e n t r a t e on the set of theorems, axioms only in formal systems.

i.e.,

formulas d e r i v a b l e from the

Our t h e o r e t i c a l

framework is the gen-

eral theory of e n t a i l m e n t r e l a t i o n s and, therefore, w h e n facing a logical calculus we ask first for the e n t a i l m e n t e m b o d i e d in it and, w h a t amounts to the same thing,

for the a s s o c i a t e d t o t a l i t y of all

theories b a s e d on that entailment. (xxVII) Thus, .

in

(6.1) we have two d i s t i n c t c o m p l e t e n e s s theorems for

Two d i s t i n c t frameworks of semantical i n t e r p r e t a t i o n s

for SCI,

the theory of t r u t h - v a l u a t i o n s and the theory of models, provide entailments

~ (~)

and

w h i l e on that fact.

~(~)

which equal

~

Let us comment for e

You will fool y o u r s e l f and the public if you

c l a i m any p h i l o s o p h i c a l depth in p r e f e r r i n g

~(~)

over

~(~).

To

189

be sure, ferent

the m a c h i n e r y

of t r u t h - v a l u a t i o n s

and may p r e s e n t

different

But they are e q u i v a l e n t concerned. sive

In fact,

and that of models

difficulties

as far as t w o - v a l u e d

(6.1) h o l d s

even

in o p e r a t i n g

are dif-

with

and e x t e n s i o n a l

them.

logic

is

for the full N F L in c o m p r e h e n -

languages. (xXVIiI) Instead

of doing

(6.1).

We have

tions

entailment.

functions)

relativity tivity

completeness

theorems

of c o m p l e t e n e s s

to some

strongly

entailment.

semantical

theorems

ponder

a little

but e q u i v a l e n t suggests

that we are a c t u a l l y

and e x t e n s i o n a l

two d i s t i n c t

philosophy,

three d i s t i n c t

This c i r c u m s t a n c e

computable valued

falacious

definitions

of an

(as in the theory of

concerned with

Subsequently,

THE two-

reflect

for the SCI.

Thus,

is revealed,

interpretation

the equa-

that we have

the well

known

(I mean

rela-

again.

for the s y n t a c t i c a l l y

given

logic.) However, But equally tactically

human yearning

eternal

is human

constructed

You have

axioms

AX

theorems,

too.

Then,

interpretation weak

completeness

think But,

SI

that

SI

theorem

another

unless

you choose (e.g.,

framework logic.

"intended"

clear w h a t

the i n t e n d e d

elementary

school.

world to

little,

Hence,

ceremony

established.

But,

and you p r o v e

the

and

of the logic you have built. you.

To be sure, weak

there

completeness

it well but you d i s q u a l i t y begins.

First,

For example,

of a r i t h m e t i c

SI*

one c a n n o t

of a logic or p a r t i c u l a r

theory

it is p e r f e c t l y

is b e c a u s e we use it in

You k n o w that the i n t e n d e d m o d e l s

some doubts.

you have

(say m o d e l - t h e o r e t i c )

and a c o r r e s p o n d i n g

interpretations

model

RL.

syn-

logic! xxIx)

You are s a t i s f i e d

I assure

You even k n o w

you have

sentential

semantics)

SI.

an u n d e r s t a n d i n g

the c h e a t i n g

its use is w e l l

m a y evoke

a deviant

a semantical

possible

SI*

cannot be extinguished. Suppose,

and rules of i n f e r e n c e

have very

as~unintendedqand speak about

in some sense,

provides

for your

self-deception.

theorem relative

you a c t u a l l y

exists

for the a b s o l u t e

of s e t - t h e o r y

if you tell me that SI c o n s t i t u t e s

the

190

intended i n t e r p r e t a t i o n s sed,

for the logic you have c o n s t r u c t e d and analy-

I w i l l not b e l i e v e you.

I bet you are c o m p l e t e l y unable to use

the c o n s t r u c t e d logic and you simply do not k n o w w h a t you intended with it.

But,

I may try to help you a little to better u n d e r s t a n d

your own construction. Let

~(Y)

"y" means for

be the e n t a i l m e n t g e n e r a t e d by "you".

~(Y) .

i e.

Lindenbaum's

Furthermore,

AX

and

Anyway,

RL; the letter

try to prove the c o m p l e t e n e s s t h e o r e m

You may d i s c o v e r that your favorite

not w o r k at all. ~ (Y)

Look closer at what you have a c t u a l l y done.

SI

works or does

look at the c o l l e c t i o n of all theories of

b a s e d on your logic. ideas by R. W o j c i c k i

Then, an ingenious e l a b o r a t i o n of

[29] tells you that the language to-

gether with all your theories c o n s t i t u t e a m o d e l - t h e o r e t i c SI*

with a c o m p l e t e n e s s t h e o r e m for

ever,

~(Y) ; (also see

I am almost sure you are not s a t i s f i e d w i t h

you must be nonplussed,

some e x t e n s i o n a l p r o p e r t i e s congruences) SI*

~(Y)

in general.

So,

SI*

presents

or s o m e t h i n g very close

But,

if your logic has

(e.g., the e x i s t e n c e of so called logical

then you can factor

SI*

to

SI**

w h i c h is much simpler

and w h i c h also p r o v i d e s you w i t h a c o m p l e t e n e s s t h e o r e m for

(see[25]).

instead of

SI**

Moreover,

if in addition,

~(Y)

is regular,

you may take only a special part of it

still have a completeness

t h e o r e m for

~(Y)

(see

way one gets L i n d e n b a u m - T a r s k i q u o t i e n t models. only way to prove the completeness t h e o r e m logic.

Nevertheless,

It is a fact that the c o l l e c t i o n of all theories of a given

e n t a i l m e n t looks very chaotic,

(Y)

How-

since any e n t a i l m e n t and the c o l l e c t i o n of its

exactly w h a t you a c t u a l l y i n t e n d e d by

than

[30]).

SI*.

theories are like head and tail of the same coin.

to it.

framework

But, you are still u n s a t i s f i e d with

[25]).

Moreover,

(Post, Goedel) SI o.

ly explain to me that certain rules in

FL

AX

and

This is the it is the for Fregean

A f t e r a while,

s u d d e n l y declare that I m i s u n d e r s t o o d you; the r e l a t i o n the e n t a i l m e n t you i n t e n d e d w h e n s p e c i f y i n g

SI o

and

RL.

then

~(Y)

you

is not

You p a t i e n t -

(e.g., the Goedel rule

19I

~/ ~

of necessitation)

are a d m i s s i b l e w i t h i n the set of theorems

and are not valid in any other sense.

The axioms

define a formal s y s t e m w h o s e theorems are ~(Y)

AX

~(Y)

is not the e n t a i l m e n t you intended.

and rules

RL

t a u t o l o g i e s but

Eventually,

syntactic d e f i n i t i o n of w h a t you mean by entailment.

you give a

However,

you

still w o n d e r what use could be m a d e of the new e n t a i l m e n t since its t a u t o l o g i e s are the same as those of So it is.

~(Y).

To be sure, the Lord God will forgive you your attempt

to fool me with the e n t a i l m e n t

~(Y) .

However,

from c h e a t i n g y o u r s e l f if you w a n t to.

no one can save you

So, listen.

Use the W o j c i c k i

m e t h o d and c a r e f u l l y analyse the c o l l e c t i o n of all theories of your logic t i.e.,

the e n t a i l m e n t you truly intend.

This m e t h o d always ends

w i t h a c o m p l e t e n e s s t h e o r e m and is an adequate c r i t e r i o n of how good your logic is.

In short, your logic is good only if W o j c i c k i ' s m e t h o d

leads to nice models. gives you a mess,

If your logic is bad then W o j c i c k i ' s m e t h o d

naturally.

The r e l a t i v i t y of c o m p l e t e n e s s theorems is not o n l y an o p p o r t u n ity for fruitless d i s c u s s i o n s of intended interpretations.

In fact,

the said r e l a t i v i t y allows us to sharpen the c o m p l e t e n e s s p r o b l e m for a logic to w h a t may be called the a d e q u a c y p r o b l e m SCI p r o v i d e s a good illustration. p l e t e n e s s theorem:

~

=

~).

[8].

All models are i n v o l v e d in the comThis t h e o r e m remains true if we re-

strict the class of models i n v o l v e d to c o u n t a b l e ones. hand,

In fact, the

the class of finite models is too small,

But, w h a t about a class w h i c h contains

On the other

as p r e v i o u s l y noted.

just one, c o u n t a b l e or uncount-

able model? If Obviously,

~

=

a model

w h i c h means that for

~

~ - (TAUT)

t h e o r e m for SCI.

~

then the model

~

is called a d e q u a t e for

a d e q u a t e for

~

is also a d e q u a t e for TAUT

TAUT = TR(M).

The e x i s t e n c e of a model,

is a very strong form of the Now, we k n o w

[14] that

adequate

(weak) c o m p l e t e n e s s

192 (6.3)

has a d e q u a t e models of the c o n t i n u u m power,

(6.4)

TAUT

has c o u n t a b l e a d e q u a t e models and, every such

model is infinite.

Thus, one may n a t u r a l l y ask for c o u n t a b l e a d e q u a t e models The answer is

(6.5)

for

[31]:

any model,

adequate

for ~ ,

is uncountable.

This m u s t be seen as an extreme w e a k n e s s of the SCI.

7. N o n - F r e g e a n logics. The e n t a i l m e n t several facts: quate models,

is very weak.

Indeed,

for a stronger entailment,

of

~

is e x p r e s s e d by

the u n c o u n t a b i l i t y of ade-

v a l i d i t y of the natural p o s t u l a t e

I feel, however,

Again,

Its w e a k n e s s

the d i v e r s i t y of models,

variety of theories. possible.

~

(3.11) and the great

genuine logic should be as w e a k as

that we m i g h t have some reasons to seek i.e.,

the class of all

an e x t e n s i o n of

~

(XXX)

(even finite and structural)

in the same language is e n o r m o u s l y

large.

extensions

On the other hand,

each such e x t e n s i o n g e n e r a t e s a c o l l e c t i o n of its own theories w h i c h is, in general, riches.

very large,

also.

Thus, we face an e m b a r r a s s e m e n t of

Of course, one can think of e x t e n s i o n s of

ly defined and, then divide them into two classes: n o n - e l e m e n t a r y e x t e n s i o n s of an a d d i t i o n a l

(invariant)

be d e f i n e d as

~

besides

as s y n t a c t i c a l -

e l e m e n t a r y and

The former are d e f i n e d as

set of axioms added to

w i t h some a d d i t i o n a l rules

LA.

~

with

The later can

(schemes) of i n f e r e n c e

MP.

Consider, ference

~

~

for example,

the "G-rule",

given by the scheme of in-

193

It is not mentary

~

valid

extension

cal l e d ~ G - r u l e ~ p V

in v i e w of

~p

F G

Let

!

and w r i t e

observe,

(7.2)

However, in

T

iff

we call

MP

theory.

They

T.

of

T

T

X

extensions

where

~T

FT

T

it is e a s i l y (7.1).

This

an i l l u s i o n

that:

with

that

so. (xxxI)

~

is in

the property:

the t h e o r i e s

which

we will,

result

are p r e c i s e l y

~

which

naturally

invariant

0~

of

are concerned, are g e n e r a t e d

We w r i t e

choose

theories then

from formulas

those

of

is any theory(i).

is a set of formulas

is

~ G

and

Sb(~)

theories

T

X

T = TAUT T =

to be an i n v a r i a n t

= the

set of all those

by the s u b s t i t u t i o n

instead T

inductively

for a while.

Sb(x)

in

Since

we may on-

of

Sb(~).

such that

t h e o r y b a s e d on the a x i o m set

X,

of for-

Invariant

T = TH(Sb(X))

for some subset

= X

(XXXIII)

For example,

consider

the i n v a r i a n t

theory

le formula:

(7.3)

e.g.,

TAUT.

are p r e c i s e l y

entailments

for variables.

FM.

Then

do not think

such

is in

is in

LA U

the i n v a r i a n t of

T

of t h e o r i e s

So we discuss

If

theories

~

|

TFT,

that

no SCI theory if

in

creates

Please,

are p l e n t y

those

and

tautologies

mulas

logic.

rule will be also

to the G - r u l e

perhaps,

connection,

far as e l e m e n t a r y

ly c o n s i d e r

formulas

and,

intensional

if and only

~

another

formula

is e q u i v a l e n t

a proper n o n - e l e -

them G-theorieS. (XxxII)

As

by

~/ ~

in this

there

Sometimes

for the e q u a t i o n ~ ~

there exists T

and d e t e r m i n e s

for a fixed

our t e r m i n o l o g y

we aim at modal, Rather

~ .

stand

~

seen that the rule fact e x p l a i n s

of

(3.11)

(p~]

v

( ~

v

(%~5.

WF

based

on a sing-

194

It is e s s e n t i a l l y well known that c o n s i s t e n t i n v a r i a n t theory of Hence,

~WF

WF,

the F r e g e a n logic,

~WF'

i.e.,

WF

is an e l e m e n t a r y e x t e n s i o n of

~

is the only

is "Post-complete". with a certain de-

finite p r o p e r t y of maximality. N o n - e l e m e n t a r y e x t e n s i o n s of sense of B l o o m and Brown,

[32].

~

are not "classical"

Consequently,

rather strange and hard to develop.

their semantics

To be sure,

to have c o r r e s p o n d i n g c o m p l e t e n e s s theorems.

M

such that

On the other hand,

~G =

However,

A model

M

q u e s t i o n s con-

For example,

there

~M"

all e l e m e n t a r y e x t e n s i o n s of

is a d e q u a t e for

T

an i n v a r i a n t theory if, c o r r e s p o n d i n g l y ,

~

are classi-

I only m e n t i o n the a d e q u a c y

cal and have nice semantical properties. problem.

is

it is not d i f f i c u l t

cerned w i t h a d e q u a c y have m o s t l y n e g a t i v e answers. is no m o d e l

in the

(or for FT

=

~M

T)

where

(or

T

is

T = TR(M)).

Then, we have the following:

(7.4)

~T

has an a d e q u a t e model iff

T

has an adequate

model.(xxxIv)

Thus,

given an invariant theory

T,

we are i n t e r e s t e d in a n e c e s s a r y

and s u f f i c i e n t c o n d i t i o n for the e x i s t e n c e of adequate models or, equivalently,

for

T.

see

[33] and

[34].

sistent i n v a r i a n t theory

in

T

(7.5)

T

(or r e a s o n a b l e n e s s ) ,

T

~

or

[35].

A con-

is called q u a s i - c o m p l e t e iff the follow-

if no v a r i a b l e occurs both in

then either

a very old notion of

It is better known in the theory of modal

systems as H a l l d e n - c o m p l e t e n e s s

ing holds:

~T

It is very p l e a s a n t to k n o w that the de-

sired c o n d i t i o n is so called q u a s i - c o m p l e t e n e s s , ~os;

for

~

in

T.

~

and

~

and

~ V~

is

Then, we have the ~os theorem:

has an adequate model iff

T

is quasi-complete.

195

This theorem is also true for NFL in open W - l a n g u a g e s w i t h s e n t e n t i a l and nominal variables;

see

[13].

Again, we have the analogy as men-

tioned in section i. (xxxV) The e n t a i l m e n t W-language tion.

is called the Fregean logic in the given

(here, SCI-language)

All other

logic.

~WF

(nontrivial)

and will be d i s c u s s e d in the next sec-

e x t e n s i o n s of

~

are called n o n - F r e ~ e a n

We may also talk of F r e g e a n and n o n - F r e g e a n theories.

is called F r e g e a n if it contains WF or, equivalently, wise,

A theory

Sb( ~ F ) .

Other-

the theory is non-Fregean. As n o t e d previously,

the t o t a l i t y of all n o n - F r e g e a n

e n o r m o u s l y large and diverse.

Hence,

logics is

to n a r r o w the field of investi-

gation by m a k i n g certain choices seems to be an absolute necessity. Of course, we may d i s r e g a r d all n o n - e l e m e n t a r y e x t e n s i o n s of

~

subsequently,

of

where

T

c o n s i d e r only those e l e m e n t a r y e x t e n s i o n s

is an invariant theory.

ly large.

There exist,

w h o s e tautologies WF. (xxXvI)

T

But, even this class is t r e m e n d o u s -

for example,

i n f i n i t e l y many e n t a i l m e n t s

c o n s t i t u t e a P o s t - c o m p l e t e theory,

~

and, we should make use of it

Thus, as the first move, we decide to only c o n s i d e r entail-

ments w e a k e r than

~ WF"

the interval b e t w e e n

~

In other words, we focus our a t t e n t i o n upon and

~WF"

Thus,

c o n s i d e r from now on are e n t a i l m e n t s theory c o n t a i n e d in

Since

d i s t i n c t from

We feel strongly that the Fregean logic is a par-

t i c u l a r l y d i s t i n g u i s h e d e x t e n s i o n of

not contain

~T

T h e r e f o r e we want to r e s t r i c t the class of n o n - F r e g e a n

logics once again.

some way.

~T

and,

WF, ~

WF.

Then,

that is, is the pure

that each n o n - F r e g e a n logic

T ~ WF

~T

~T

the logics we are going to where

and,

(absolute)

~T = ~ non-Fregean

(of the r e s t r i c t e d kind)

iff

T

does

T = TAUT.

logic,

it follows

d i s t i n c t from

that is, some n o n - t a u t o l o -

gical a s s u m p t i o n s w h i c h we call ontological. PWF

is any invariant

is n o n - F r e g e a n w h e n e v e r

involves certain n o n - l o g i c a l content,

to the Fregean logic

T

Clearly,

the same applies

w h i c h is the g r e a t e s t one in our interval

196

of entailments. WF

Hence, by s t u d y i n g n o n - F r e g e a n logics, w e a k e r than

we may hope to reveal and analyse the o n t o l o g i c a l content of

the Fregean logic. The F r e g e a n logic is d e t e r m i n e d by the i n v a r i a n t theory b a s e d on single a x i o m

O~F,

(7.6)

i.e., the formula

WF

(7.3):

WF = TH(Sb( ~ F ))

In the next section we find other formulas w h i c h may serve,

like

~{F"

as single axioms of

WF.

Later, we will select three p a r t i c u l a r

mulas

~H

and, discuss three invarient theories b a s e d

~B'

~T

and

on them as single axioms:

WB,

WT

and

of t a u t o l o g i e s of three n o n - F r e g e a n Each logic hence, F WZ Now,

~WZ

where

Z

WH.

logics:

is

These theories are sets ~WB'

B, T, H or

~WT F

and

WZ

in a sense,

whole set

Sb( ( 0 ~ ' ~ )

[8]

=> ((,~ =~ (3"~ =.-, (,.,~ g')")

(3)

If

we m o d i f y

tained which

(4)

axioms might

necessarily XX.

for

have

and

(7)

identity its

own,

to

suit

intuitionistic

connective possibly

b a s e d on p o s s i b l e - w o r l d s

we w o u l d

interesting,

requirements define

and,

re-

a w e a k e n e d SCI

model theory

not

semantics.

C o n s i d e r all e q u a t i o n s as a d d i t i o n a l v a r i a b l e s and d i r e c t l y

apply the well known a r g u m e n t by L. Kalmer.

XIX. rules of

By schemes of inference or formulas we mean the sequential [8].

227

XXI.

This is a fundamental

NFL in c o m p r e h e n s i v e

languages.

or not, a c o r r e s p o n d i n g identity

(connective

XXII. (5.2), special

equivalent

complete

to ~ -compactness

extensions

by negativity).

XXIV. that:

if X

X ~ ~

~

,

t(~)

Put

non X

~ (~)~

axioms

axioms and the

~

argument,

finiteness

X

= 1

~t

~,

a valuation over

sets and regularity

non X

.

satisfied by

is in

Y

by

~y

M(Y)

and

in t(~)

~(~)~

.

in a complete

(see III)

w i t h respect to ~

is

set contains theorem on (again

~(Y).

Y

Y.

non X

X M

~(~) ~ and

X

is.

M

is a complete

= 0,

Observe

otherwise.

To show that: If non X ~ theory

Then,

is precisely

Hence,

shows

Y.

Fac-

to get the Lindenbaum-

into the abstraction

such that in

r

we argue as follows.

is contained

sending

Y non X

X

~(~) ~ .

~

if

There exists

satisfied by

~

X ~ ~

[

and ~

follows

To show that:

~(~)~.

and, hence,

~(Y)

of

(every inconsistent

~ (~) ~

if

~

then

X

using length of derivations,

of all formulas

Tarski quotient model

non X

of

Hence,

iff

This implies the L i n d e n b a u m

suppose

then, by regularity, both torAthe language and

cal m o r p h i s m

set).

(X ~ (

is not satisfied but every formula in

theory.

fore,

instead of the invariance

[25].

then

Y

mulas

with respect to

must be laid down.

is negative

of consistent

An inductive

that the set

Clearly,

set).

Compare

~(~) ~

such that

~

an inconsistent

a finite inconsistent

X

binding variables

(5.9) instead of the invariance

The e n t a i l m e n t

form together

if

(5.8)

(scheme)

of

identity axiom.

XXIII.

then

axiom

and/or predicate)

(5.4) or

of syntactic d e f i n i t i o n

For each formator,

invariance

One might use either

(5.2),

requirement

the canoni-

class of

~

,

is

the set of all for-

~(y)

~

and,

there-

228

XXV.

Completeness

Therefore,

theorems are seen here as suitable equations.

the often made d i s t i n c t i o n b e t w e e n soundness and proper

c o m p l e t e n e s s is overlooked.

It only stresses the d i f f e r e n c e in the

nature of proofs of these parts. induction.

S o u n d n e s s is usually p r o v a b l e by

Proper c o m p l e t e n e s s n e c e s s a r i l y calls for a c o n s t r u c t i o n

and is, as a rule, m u c h h a r d e r to prove.

XXVI.

Mostly, o t h e r e q u i v a l e n t

formulations are used:

sistent set of formulas has a model or, is v e r i f i a b l e

XXVII.

every con-

(and conversely).

W e a k c o m p l e t e n e s s of a logic does not say much.

S~upecki's

"widerspruchsvertragendes Aussagenkalkul"

[26] and Hiz's

s e n t a n t i a l calculus a d m i t t i n g e x t e n s i o n s "

[27] are examples of w e a k l y

complete truth-functional

logics.

"complete

The strangeness of these c o n s t r u c -

tions can be e x p l a i n e d only if we ask for the e n t a i l m e n t s Indeed,

they are not c o m p l e t e in our sense;

XXVIII. we get all

Here,

XXIX.

[28], pp.

203-204.

the role of c o m p l e t e theories is again crucial.

(up to isomorphism, countable)

(one-one)

First,

models as L i n d e n b a u m - T a r s k i

q u o t i e n t models m o d u l o complete theories. correspond

compare

involved.

Secondly,

c o m p l e t e theories

to truth valuations.

We assume s t r u c t u r a l i t y also,

in the sense of

[8], w h i c h is

almost no r e s t r i c t i o n at all.

XXX.

We say that

is an e x t e n s i o n of ly, every

~2

~i ~i

is w e a k e r

iff

theory is a

X ~i

~2 ~

(or equal) whenever

than X

~2

~i ~ .

and,

~

Then,

clear-

theory but not n e c e s s a r i l y conver-

sely. XXXII.

The fact that

~G

is not an e l e m e n t a r y e x t e n s i o n of

means that the G-rule cannot be, al set of axioms. o b v i o u s l y equals to

in general,

r e p l a c e d by any a d d i t i o n -

On the other hand, each p a r t i c u l a r G - t h e o r y TH(X)

for some

X.

T

One may ask for a nice and,

229

perhaps,

independent

XXXI.

In 1969,

X

such that

in particular,

axiom and convince me that instead of NFL

SCI something else should be constructed.

Let

~*

be the entailment

and the s u b s t i t u t i o n

extension of Since,

~

rule

Sb(TH(X))

is contained

and, a m o d i f i c a t i o n

•; for

[30], ~

[25].

and offers

that instead of each model

M,

the w~ole set XXXIV. for for

If

T.

showed that if adequate

for

TH(Sb(X))

if

M

~ ~ ~ in

for

is finite

M

is adequate

of theories

The essential

~

F~,

is in

of

point

is

defined

for

TR(M)

whenever

M

is adequate

TR(M). ~T

the~a obviously, for

(1-compact). for

T

The ultraproduct

T

then

But,

M

is adequate

Stephen L. Bloom

then some ultrapower construction

into the semantics

of

M

is

of ~os and his main

of NFL and SCI,

in particu-

One may dare to say that only the theory of models of NFL re-

veals the real nature and value of the ultraproduct XXXV.

NFL in open W - l a n g u a g e s

analogy. collection [13].

nice properties.

is a special case of that

iff

is adequate

~M

LA,

method works perfectly with

we must use the entailment X

by

we infer that the in-

the collection

facts.

is contained

~ T"

in

some interesting

theorem fit p e r f e c t l y lar.

Nat-

is a n o n - e l e m e n t a r y

has e x c e p t i o n a l l y

of Wojcicki's

is adequate

iff

~ *

Sb

precisely

as follows:

M

Clearly,

~*

~ M

X

defined syntactically

The semantics of

Conversely,

~T

Sb.

but the rule

variant theories constitute ~ *

He

if you do not want NFL then you will get something else.

XXXIII. MP

36 pages to

logic is rooted in the logic of modality.

tried to prove the Fregean

urally,

= the smallest G-theory.

a pupil of the late R. Montague wrote

show that n o n - F r e g e a n

and,

TH(X)

also provides

~os theorem on regularity of of q u a s i - c o m p l e t e

Moreover,

theories

construction.

other cases of that

~ *

with respect to the

holds

in open W-languages;

the theorem on common extensions

of models

see

(like in

230

[34])

is a l s o

XXXVI.

valid

If

no p r o p e r

M

for m o d e l s

is any

subalgebras

of o p e n

finite

then

the

model

W-languages

such

theory

that

TR(M)

(unpublished).

the a l g e b r a

of

it has

is P o s t - c o m p l e t e

(S. L.

Bloom).

XXXVII. algebras

Theorem with

(9.8)

circle

operation

property

(4.7)).

The

follows.

First,

write

They

are

equations

(x n

Y)U

z

x~

m

is c o n c e r n e d

class

(yv z ) ~

(y -~ x).

Subsequently,

following

(*)

if

axioms

either

add

of

following

x ~

WB

XXXVIII.

a model

infinite

(xu y ) ~ x,

list

Xl ~

-x.

then

x

0 and XlP ... x n

then T n = ~Xl••.XnA

is called an n-ary predicate or n-term.

We shall write

T n t l . • . t n = A[tl/Xl~

...~ tn/Xn]

where the right hand side denotes the usual substitution rences of i-variables•

of i-terms for free occur-

Thus~ Tntl...t n is a formula.

~[T"I xnl denotes

the result of replacing each part Xntl...tn of Bj such that the given occur-

rence of X n is free in B~ by Tntl...t n. first to rename bound variables

In all of these substitutionsj

in order to avoid confusing

we may have

them with free ones.

We

shall assume that this is done in some unique way. Formal deductions

In'are

in the style of natural deduction (Oentzen 1934).

They are finite trees which are built up from certain initial trees by means of rules of inference•

The initial trees are the premises n

A consisting of a formula A with an index n > 0

over it•

There are four rules of in-

ference~ an introduction rule and an elimination rule for each of D a n d S .

:

:

•

•

•

B

A~B

A

ADB

: B

A

',¢' i %

T'~

Thus~ in each of these rules~ we take given deductions j eg of B or of A ~ B and A~ and combine

them to obtain new deductions~

a restriction on V l ;

eg of A ~ B or of B, resp.

We must make

but flrst~ we must say when an occurrence of a premise in a n n deduction is discharged~ A is undischarged in A• In each of ~ Ej V I and E~ an

242

occurrence of a premise is discharged iff it is discharged in

one of

the given

deductions.

In ~ l~ an occurrence of a premise is discharged iff it is discharged n in the given deduction of B or else the premise is A.

Restrig..tion on ~ I ,

X a must not occur free in any undischarged premise in the given

deduction of A. A is deducible in ~(deducible

in ~ f r o m

B0, .oo, B n) Iff there is a deduction

in ~ ending with A and with no undischarged premises (all of whose undischarged k premises are of the form B i for some i _~ n),

2.2.

Prawltz 1968 showed that the logical constants .~(absurdity),'l, ~', /land

are definable in I by

± =Vx°x --~A = A ~ . AvB

= V X 0 ( ( A = X) = ((B ~ X) = X))

A /%B = V x O ( ( A = (B = X)) = X)

~z% = % { x ° C ~ Z ~ ( B

= x) = X)

With these definitions~ all of the intuitionistlc laws of logic are deducible in ~. Set A~

B = (A ~ B ) ~

(B=A)

E = kxy zl(zx n Zy) Then the theory of identity of individuals is deducible i n ' I n Assume that ~ c o n t a l n s constant '.

terms of E.

the individual constant o and the unary function

Set N

= ),xy~l[(zO ~ V y ( Z y D gy')) D Zx]

~+ = Xxyz za[6fuZuou^Vu~ 0,

0_< c < +0%

> 0

E > O,

~th(K ) et

choisissons

st(K) sont finis. Etant donn~s deux nom-

K

(commeil vient d ' 6 t r e dit) et un nombre

tels que

(23)

c
b

and a > ~ b

b[a/x]

and the notions of normality~ normalizability and well-

foundedness are defined just as before in 2.4o

Note that~ u n l l k e ~ , £ h a s

non nor-

mallzable terms, eg s = ((~X(XX)) (%X(XX))); and also has normalizable non wellfounded terms, eg (()0~K)s).

We shall write

ab = (ab), a b c =

((ab)c), abed = ((abc)d), etc. M

will denote the set of closed well-founded terms o f ~ .

If a ~ M

and a ~ b~ then b G M

a~b means that for some c, a n c and b>~.c,

The Church-Rosser Theorem (1936) asserts

that ~ is an equivalence relation on the terms o f ~ . We shall need Lemma

3.2.

the following result, which we prove in 4.

If b[a/x]Cl°..cn and a are in M, then so is ()u~b)aCl..°cn ( n ~ O ) .

Let I i denote the set of all closed i-terms o f ~ . A Droposition o v e r ~

is a species (i.e. property) R of elements of M

such that

247

i)

If a ~

ii)

R and a ~ b, then b ~ R.

If b[alX]Cl.o.Cn~ R, n ~

ifi)

0 and a ~ M, then (%Xb)aCl..°CnG R.

If Kal..oa n is in M (Joe. el, ..°, a n are in M), n ~ 0 ,

then Kal.o.anG

R.

An n-ary propositiqnal function overt, n ~ 0~ is a species R of n + l-tuples tlJ ..., tn, a

.

with tl, ..., tn ~ I i and a G M such that for all tl~

Ii ,

.., tn ~

the species Rtl...tn = { a : < is a proposition over r ( i . e , We e x t e n d ~ t o

satisfies i) - iii)).

the s y s t e m ~ b y

adding an n-ary relation constant for each

n-ary propositional function o v e r ' a n d over C .

t I ..... tn, a ) E R ~

a proposition constant for each proposition

If pn is such a constant~ then pn is the proposition or propositional func-

tion it denotes (n ~ 0)° I n denotes the set of all pn i n ~ . For each sentence (i.e. closed formula) A o f ~ , we define the species 5 of terms of C by induction on the number of occurrences of ~ o r ~ i n

A:

pntl...t n = pntl...t n

ce. A ~ B ~ V a l 5

cG %~XC6A

,oposition

(ca

~)

~-VPa F., Ia(cK ~ A[P/'xC~])

each s e n t e n c e , of

,

is a proposition o erC.

This is clear if A is atomic~ i.e. = pntl...t n. propositions o v e r ~ and c ~

Let c~d.

A~

B.

K ~ 5 and so c K C ~.

Then for all a e ~, ca ~ da and so d a G ~.

c[d/X]el...en E A = B and d ~ M. (kXc)del...ena ~ ~ by ii).

Assume that ~ and ~ are Hence, c K E M and so c C Hence, d ~ A = B.

M.

Let

For all a ~ 5, c[d/X]el...en a ~ ~ and so

Hence (kXc)del...en~

for all a ~ 5, Kal...ana is in M and so in~.

A ~ B.

Let Kal...anG M.

Hence, Kal...anE

A~

B.

Then

Thus, A ~ B

is a proposition overC. ............. is a proposition over C if A similar argument proves thatVX(~A

for each P a ~

A[Pa/

X c~] is

la° ~Qed.

248

We define the n-ary propositional

Let T be a closed n-term of J, n ~ 0 . function ~ o v e r ~ ( a

proposition if n = 0) by

~tl,,,t n

=

Ttl,o,t n

Thus~ there is a relation constant PT in I n with ~T = ~,

By induction on A, it

easily follows that

(l)

=

A[T/x n]

From (i) and the definition o f ~ X a A ,

(2)

A [ PT/xn

we obtain

c¢~XaA

----~c K E

A[T/x a]

for each closed Q-term T o f ~ . When a G ~

we say that a realizes A.

This is closely related to Kleene's

1945 recurslve realizabillty interpretatlon~ except that~ instead of coding functions by their Godel numberss we use the corresponding term o f f .

3.3. ~that

It will be convenient to make the purely notational restriction on A-terms of a variable occurs in one of them with at most one superscript.

and X B occur in c ~ C ~

then A = B; if X a and X B occur in c~ then a = ~

XA and X a occur in c. With each A-term a we associate a t e r m ~

xA

=

of~as

x

m

- -

ca

w - -

== c a

cT~ = "~K

follows:

I.e. if X A and not both

249

Let a ~ A form y a

and let i~iI, ..., Y~nn include all the free variables in a of the

If T i is a closed ~i-term o f ~ ,

a0

= a[TllYl , ° . . , TnlYn]

A0

= A[TI/YI , ,.., Tn/y n] Bm

B1

Then A 0 is a sentence°

Let g I , ° . ° ~ g

..., B n are sentences o f ~ .

set

Let b i ~

be all the free variables In a 0.

m

Then A 0 is called an Instance of A and

~i°

~0[bllzl,

b/zm]

...,

aI

=

Then BI,

is called an instance of a for A0°

Realizability Theorem. A0, then a l e

If a ~ = A ,

A 0 is an instance of A and a I an instance of a for

~0 °

We shall prove this in the next section. obtain the Well-Foundedness closed.

First, let a ~ A

Then a is an instance of a for A.

ability Theorem.

~m

B1

X1 .

D

.

(B n

Proof of the Realizability Theorem.

First~ we shall prove the Lermma. Proof.

by the Realiz-

Let

and let a be open then

n

founded°

4o1°

Let a ~ A

yBn n are its free variables,

A) ... )).

But, this term is closed~ and hence~ well-foundedo

4.

and let a (and so A) be

Hence~ a is well-founded~

and so, a is well-founded,

If X 1 , ..., X m~m ' Y1BI ' ° " ~ ~I

we

But~ an infinite reduction a>> b > > c ~> ,,. would yield an infinite

reduction a>> b ~ > c > > , . . ; now°

Theorem f o r ~ .

As an in~edlate consequence,

But~ that implies that a is well-

250

(kXb)aCl,..Cn>>d0>>dl)>... be a reduction.

We must show that it is finite.

Note that b is well-foundedt since

an infinite reduction of b would yield an infinite reduction of b[e/X]Cl...Cn . Likewise~ each c i is well-founded.

If each d i is of the form (kXb')a'c~...c'Ln where

h~/ b'~ a)/ a' and ci~/ c~, the reduction is finite because b, a, Cl~ ...~ c n are well-founded.

If some of the d i are not of this form, then for some J, dj = b'[a'/x]

Cl...C n ' ' where b ~

b', a ~ a ' ,

reduction d j pW d j + l ~

and c i ~ c

i.' But then b[a/x] Cl..oC n ~ dj, and so the

.., must be finite, since b[a/x]cl...Cn is well-founded. Oed.

4.2.

We prove the Realizability Theorem by induction on a.

Case i.

a = XA.

Then by definition, a I ~ ~0'

Case 2.

a = kxCb, where b ~ B

and so A = C D B.

Let c ~ ~0"

Then bl[C/x ] is an

instance of b for B0; and so by the induction hypothesis, bl[C/x]G ~0" clause ii) in the definition of a proposition over~, alc ~ ~0" Case 3,

a = kX~b where b ~ B and so A =~X~B.

Let P ~

I~.

Hence, by

Thus, al~ ~0"

Then bI[K/x] is an

instance of h for B 0 (since by the restriction on kX e, X ~ is not free in any C when yC is free in b).

al" Case

So, bI[K/x]G ~0 by the induction hypothesis; and so

I.e. 4,

a =

and c I ~ T 0. Case 5.

be, where h ~

B ~ A and c ~ Bo

By the induction hypothesis, bl~

So a I = blCl~ ~0

a = BT, where b ~ V X ~ B

a I = bIKE BIT/ X ~] by (2).

University of Chicago

and A = BIT/ Xe].

DIG

~XeB)0 and so,

B~0

251

References

Church, A. 1932,

A set of postulates for the foundations of mathematics.

Ann. of

Math. (2), 33: 346-366. Ghurch~ A. & Ross,r, J.B. 1936.

Some properties of conversion.

Trans, Amer, Math.

Soc. 39s 472-482 Curry~ H. & Feys, R. 1958. Dedeklnd, R. 1888, Gentzen~ G, 1934o

Comblnatory Logic.

North Holland, Amsterdam,

Was sind und was sollen die Zahlen?

Brunswick.

Untersuchungen uber das loglsche Schlussen, Mathematlsche

Zeitschrift 39: 176-210, 405-431. Girard, J.-Y. 1970.

Une extension de l'interpretatlon de Godel a l'analyse et la

the,tie des types. J.E. Fenstad. God,l, K. 1932-3.

Proc of the Second Scandanavlan Logic Symposlum~ ed.

North Holland, Amsterdam. 1971. 63-92.

Zur intuitionistlschen Arithmetik und Zahlentheorle.

Ergebnisse

eines math. Koll. Heft 4. 34-38. Martln-Lof, P. 1970.

Hauptsatz for the theory of species.

Scandanavlan Logic Symposium, ed. J.E. Fenstad.

Proc of the Second

North Holland~ Amsterdam.

1971. 217-233. Prawitz, D, 1965,

Natural Deduction.

.........

Some results for intuitionistlc logic with second order quantifi-

....

. 1968.

Almqvist and Wiksell.

cation rules.

Intultionism and Proof Theory.

R.E. Vesley.

North Holland, Amsterdan. 1970,

. 1970,

Ideas and results of proof theory.

vian Logic Symposium~ ed, J.E. Fenstad. 235-307.

Stockholm.

eds. A, Kin,, J. Myhill,

Proc of the Second Scandina-

North Holland, Amsterdam. 1971.

Logic Colloquium: symposium on logic held at Boston, 1972-73

Logic colloquium. Symposium on logic, Boston, 1972-73

Logic Colloquium 1981: Herbrand Symposium Proceedings

Logic Colloquium '73

Logic Colloquium 1984: Proceedings

Logic Colloquium '69

Logic colloquium '88

Logic Colloquium '88: Proceedings

Logic colloquium 2007

Logic Colloquium 1976: Proceedings

Logic Colloquium 2005

Logic: Colloquium Proceedings, 1977

Logic Colloquium 1982: Proceedings

Logic 1985: Colloquium Proceedings

Logic Colloquium 2006 (Lecture Notes in Logic)

Patras Logic Symposium: Proceedings

Logic colloquium '80 (no TOC)

Logic, Logic, and Logic

Logic, logic, and logic

Logic, Logic, and Logic

Logic Colloquium 2006 (Lecture Notes in Logic, 32)

Logic Colloquium '87: Proceedings of the Colloquium Held in Granada, Spain July 20-25, 1987

Logic at Botik '89

Logic Colloquium '87: Proceedings of the Colloquium Held in Granada, Spain July 20-25, 1987

Proc. 2nd Scandinavian Logic Symposium

Logic colloquium '78. Proceedings of the colloquium held in Mons, August 1978

Logic colloquium '69: Proceedings Manchester, 1969

Logic colloquium '78: Proceedings Mons, 1978

Logic colloquium '87: Proceedings Granada, 1987

Logic colloquium '77: Proceedings Wroclaw, 1977

Logic

Logic Colloquium: symposium on logic held at Boston, 1972-73

Logic colloquium. Symposium on logic, Boston, 1972-73

Logic Colloquium 1981: Herbrand Symposium Proceedings

Logic Colloquium '73

Logic Colloquium 1984: Proceedings

Logic Colloquium '69

Logic colloquium '88

Logic Colloquium '88: Proceedings

Logic colloquium 2007

Logic Colloquium 1976: Proceedings

Logic Colloquium 2005

Logic: Colloquium Proceedings, 1977

Logic Colloquium 1982: Proceedings

Logic 1985: Colloquium Proceedings

Logic Colloquium 2006 (Lecture Notes in Logic)

Patras Logic Symposium: Proceedings

Logic colloquium '80 (no TOC)

Logic, Logic, and Logic

Logic, logic, and logic

Logic, Logic, and Logic

Logic Colloquium 2006 (Lecture Notes in Logic, 32)

Logic Colloquium '87: Proceedings of the Colloquium Held in Granada, Spain July 20-25, 1987

Logic at Botik '89

Logic Colloquium '87: Proceedings of the Colloquium Held in Granada, Spain July 20-25, 1987

Proc. 2nd Scandinavian Logic Symposium

Logic colloquium '78. Proceedings of the colloquium held in Mons, August 1978

Logic colloquium '69: Proceedings Manchester, 1969

Logic colloquium '78: Proceedings Mons, 1978

Logic colloquium '87: Proceedings Granada, 1987

Logic colloquium '77: Proceedings Wroclaw, 1977

Logic

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