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,<) be the rational numbers. preserving
injections
and exhaust
Q
such that
of countable
order types.
Let fo, fl : Q ~ Q
fo(Q)
and every element of
and fl(Q)
functor Let
be order
are disjoint
fO(Q) precedes
every element
of fl(Q).
fo(Q)
These injections countable
exist because
dense u n b o r d e r e d
theorem of Cantor,
(order)
of the constructive
fl (Q)
J i
I
the ordering pictured above is a
linear ordering and therefore, isomorphic
character
to
Q.
of the proof
Further, fo, fl
by a
because
m a y be chosen to be
general recursive. We now identify dense ordering
~
Q
with
E
and endow
E
with the induced
Then if we use exactly the same functor as for
10
c a r d i n a l a d d i t i o n we obtain the required c o m b i n a t o r i a l functor.
Example 2.4.
O r d i n a l multiplication.
We define a c o m b i n a t o r i a l
functor ® : ~ × ~ -~ L w h i c h u n d e r l i e s ordinal multiplication. j : Q × Q ~ Q
be a b i j e c t i o n such that for all
Q × Q, J(ao, al) ( J ( b o , b l ) a ° (bo).
Again
j
Let
(ao, a l ) , ( b o , b I) in
if, and only if, a I ( b I or (a I = b I ¢
exists since
Q × Q w i t h the l e x i c o g r a p h i c
o r d e r i n g is a c o u n t a b l e dense u n b o r d e r e d linear o r d e r i n g and therefore i s o m o r p h i c to Q. recursive.
A g a i n also
j
m a y be chosen to be
Now use the same functor ® as for c a r d i n a l m u l t i p l i c a t i o n
and show that
®
is indeed a c o m b i n a t o r i a l functor.
In the same w a y we can define
(weak)
cardinal exponentiation
and o r d i n a l e x p o n e n t i a t i o n f o l l o w i n g Sierpinski
E x a m p l e 2.5.
Let V : S ~ V
[1958].
be defined as follows.
Identify
$
w i t h the c a t e g o r y of one-one maps b e t w e e n subsets of a c o u n t a b l y infinite
set
A.
Let
for a vector space
U
f
be a one-one map from
over a field
category of vector subspaces of ations b e t w e e n them. the object morphism
V(~) p
of
Then
V
U
V
be the
w i t h one-one linear transform-
~f(a)
: a c ~
~of
S
] and to each
the m o r p h i s m such that the f o l l o w i n g d i a g r a m
c or~u t e s
v(p)
V
and let
onto a basis
assigns to each object
g e n e r a t e d by $
K
A
is a c o m b i n a t o r i a l functor.
I] 3.
Model Theory.
certain
Now we turn to model theory for it is from
recursive models that we get out best examples
theory of recursive
combinatorial
for the
functors and the model theory
also enables us in some cases to relate combinatorial to number theoretic We ~onsider
functions.
a fixed
infinite model ~ equivalent
funetors
.
Xo-categorical
theory
T
which has an
We may assume that every formula is
to an atomic formula
(or else add new predicate
letters and axioms to make this so). For a formula
@
with
Vo,Vl,...,v n
free then if
T//I- Vn (Vo,'",Vn ) Eao,""%-l] ~ither,
for some positive natural number ~ b
(where
~ ~k
means
~ l k v n ~ ( V o , - - - , v n) "there are exactly
or there are infinitely many elements ~I--
k,
6[ao,...,an_l,b]-
k") b
in
?r/~l-- ~[ao,...,an_l,b ]
finite A ~
subset of
I~I
and if
over a set A.
A ~
l~I/~I
formula).
over~
relation
is a finite If
b
is algebraic
closure
, then
such that
T
cg(A)
such that
A ~
I~I
b ~ l~rl. if the
segment of the natural numbers and
"a satisfies
@
in ~
sequence of elements of
is a decidable
completeness
is
of a set
theory then
" I~I
T
proof.
is recursive and
@
a
has a recursively
presented model as can easily be seen by effeetivising Henkin-style
b
over some
is said to be recursively pyesented
the satisfaction a
such that the
~ao,...,an_l].
subsystem ~ y ~ 7 ~
universe of ~T~ is an initial
(where
if
The a!~ebraic
is algebraic
A model ~
b
is al~ebraic over
is the smallest b
such that
I~l
If there is some formula
former case holds we say that an element
algebraic
[ao,''',an_l]
a standard
12 We are now almost ready to define categories which are suitable for developing the theory of recursive equivalence and recursive combinatorial functors. A category
~ ~ @(~)
is said to be suitable if
C
is the
category of elementary monomorphisms between algebraically closed subsets of a recursively presented infinite m o d e l ~ o f X0-categorlcal
theory
T
which is decidable,
an
has decidable atoms
and is such that algebriac closure is fully effective,
o~(~)
is the full subcategory of finite objects and m o ~ h i s m s . any suitable category satisfies all the conditions To say
T
at most $
Note that
in section i.
has decidable atoms means that given a formula Vo,...,Vn_ I
is an atom of
@
with
free then we can effectively decide whether
Bn(T)
or not.
To say algebraic closure is fully
effective means that given an explicit index of a finite set A ~
!~I
we can uniformly effectively compute an explicit index
of the algebraic closure, X0-categorical means
cg(A), of
cg(A)
A.
Note that
is finite if
A
T
is finite.
The examples of Section i are (up to isomorphism) categories
(if the field
K
in 1.3 is finite).
theory of an infinite set with equality, unbordered
linear ordering,
vector space over
K
So we shall now use as
@(~)
theory
for ~
being
suitable
For i.i
T
is the
for 1.2 that of a dense
for 1.3 that of an infinite dimensional
and for 1.4 that of an atomless Boolean algebra. $, ~, V
and
~
for the categories which arise
the reeursively presented model of the appropriate
T.
In the case of
$
and
V
but not
L
or
B
we have the
notions of dimension and basis akin to those for vector spaces. This is also the case in many other suitable categories. case of other categories,
in particular
(In the
~, there is a notion of
dimension but this does not have sufficiently strong properties
13
for our purposes.)
We shall not however give the formal criteria
here but just treat dimension.
In
$
$
and
V
when considering categories with
dimension of
element is a basis element.
In
A = cardinal of V
A
and every
dimension is the usual vector
space dimension and basis has the usual meaning.
4.
Combinatorial Functions.
For categories with dimension we
define a subclass of combinatorial functors. A map
G: 0b(°~)
-> ~ ( 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
-- ~c .(n)
(qi-l)... (q-l) for some non-negative integers
5.
Recursive equivalence.
Two o b j e c t s ~ , ~
j $ J
cj.
Now we turn to recursive equivalence.
in a suitable category
$ = @(~)
are said to
be ~-recursively equivalent if there is an isomorphism q :O~%~in
xJ.
@(~)
recursive function.
which can be extended to a one-one partial The equivalence classes under this relation
are called @-recursive equivalence types (~-RETs, for short).
16
Finite ~-RETs may be identified with isomorphism types of objects in
~.
We write
A set
A
~
for the @-RET of an object
is said to be effectively Dedekind finite if there
is mo one-one partial recursive function which maps proper subset of itself.
An o b j e c t ~
Dedekind finite (Dedekind,
called Dedekind ~-types.
A
onto a
is said to be effectively
if its underlying set is
The @-RETs of Dedekind objects are
The set of Dedekind ~-types is denoted
For each (classical)
isomorphism type of infinite object in
there are continuum many This result is proved,
@
for short)
effectively Dedekind finite.
A(@).
~.
(c = 2X0)
distinct Dedekind ~-types.
as are most of the theorems below,
by a
Baire category argument and it turns out here that the representative of Dedekind C-types contitute a second category set in a suitable Cantor space. Recall that an explicit index of a finite set gives the cardinal of the set as well as its elements whereas a standard index merely allows a recursive enumeration of the set. A combinatorial functor recurslve
(partial recursive)
F: ~ ~ ~' if
F
is said to be finitary
maps
(the r.e.) algebraically closed objects of a recursive fumction
f o~
F0~
the map from
index
~' FF ~
and (ii) xq of
F ~x
@'
into the finite and (i) there is
which assigns to each explicit index
of a finite object in in
o~
an explicit
in
o~
(a standard)
x ~ '
index
r~
rF0~ 7
of
to the explicit
is partial recursive.
So the recursive requirements are just effectivizations of the conditions
in Section 2.
Clearly finltary recursive
combinatorial functors are partial recursive. Theorem 5.1.
(i)
functions
on
K
Partial recursive combinatorial functors ¢-RETs given by
K(A)
= (F(~))
where
F
induce
A =~i).
17
(ii)
Partial
composition.
recursive
combinatorial
functors
are closed u n d e r
Similarly for f i n i t a r y reeursive ones.
with the n o t a t i o n of part
(i), if
H = F oG
then
Moreover,
H = F ° G. - -
A pre-combinatorial
operator
G
w
w
is said to be recursive
if
there is a recursive
function which when applied to an explicit
index of ~
gives a standard index of
in
precombinatorial combinatorial recursive,
°C
operators
functors.
induce partial
A strict c o m b i n a t o r i a l If it induces a (recursive)
6.
C,C'
which
functors
Let
strict V
is finitary
functor is finitary recursive functor is f i n i t a r y number theoretic
recursive
if and only
function with a r~cursive
F i n a l l y we exhibit
stem from combinatorial
will be assumed
Theorem 6.1.
Recursive
function.
Transfer principles.
principles
recursive
Also the closure functor
if the u n d e r l y i n g
Stirling coefficient
G(~).
to be suitable
F,G: C ~ C'
inducing functions
functors.
Throughout
categories.
be finitary
F, G
some of the transfer
recursive
on C-RETs to
combinatorial
C'-RETs.
Then
the f o l l o w i n g are equivalent (i)
F(X)
= ~(~)
for all
C-RETs
(ii)
F(X)
= ~(~)
for all Dedekind
~
~-RETs ~.
It is not true in general that if q-isomorphism
types
and
(= finite C-RETs)
F(X) that
= G(X)
(1) and (ii) hold.
Consider for example addition of order types then for all finite order types
~,~
but
cannot hold for all infinite ~-RETs. other p r o p e r t i e s Theorem 6.2. Then (i), (iii)
~ + ~ = ~ +
w + i / I + w However,
so equality
in the dimension
case
do carry over.
Hypotheses
as for Theorem 6.1 plus
(ii) are equivalent
F(X)
for all finite
= ~(~)
to
for all finite
C-RETs
~.
C
has dimension.
18
As a corollary all the usual associative, distributive
laws concerning
laws for e x p o n e n t i a t i o n
addition and m u l t i p l i c a t i o n
for cardinals
have also shown that p r o v i d e d identities
commutative
and
and related
carry over to S-RETs.
the isomorphisms
We
are u n i f o r m then
do carry over from finite types to infinite ones.
full technical
details may be found in [Crossley and Nerode
For the case where
~(~)
= G(X)
holds only sometimes,
have to consider chains of objects.
The 1974]. we
A chain is a sequence
op> 1 Of I p2 >... where the
pi
are @-morphisms
and the ~ i
are objects of
The length of the chain is the number of morphisms. ~o
1 ~___> ...
n-equivalent
and
1 q > ...
~o
of length
if there are C-isomorphisms
P >
1
2 .P.... >
q
q
n
.
P
.
Two chains
are said to be
r°,r I y.. • , r n n
.
.....
@.
such that
> ~n
q
commutes. The equivalence
classes are called n - c h a i n types.
A frame is a subset of object
~
is attainable
B c~there
exists
For a frame ~= i)
CF(B )
F
if
~
~ F B ~
o@
closed under intersection.
from a frame such that for some
with this property.
F
ii) the n u m b e r - t h e o r e t i c defined]
(where
FBU~SCF(B)7
if for every finite
B c o~[ ~c
F
A frame
the set of indices of elements of
, is partial
then there is a smallest F
F
is recursive is r.e.
index of
recursive.
set
F.
f u n c t i o n with domain
~ B ~ is an explicit
An
If
and
[ r B n : CF(B ) B)
R
if
is
given by is a relation on
19
on chain types then a frame whose objects are in C×...×@,
m factors,
D e f l n i t i o n 6.3.
F
F
is an
is in
R.
instead of
representative~ We write
C(R)
isomorphism F say,
~
C(°R)
functors to
Let
and let
~(~)
C = C(~)
= G(C)
is an m - a r y relation.)
of a relation X
on chain
such that some
noting that O - c h a i n types are just
F
combinatorial
functors,
on chain types and we have:
F,G: C + @' nR
R
from a recursive nR-frame.
Now finitary recursive
induce functions
Theorem 6.4.
R
@-types
is attainable
for
types.
if
~(nR)
types is the set of all Dedekind
if every n - c h a i n
(Note that we are using
C
The extension
nR-frame
be finitary
recursive
combinatorial
be the set of finite n-chain type solutions then
@(nR) ~
has dimension and ~
~
~ A(@) : ~(X)
= ~(~)}.
If
has degree I then the other
inclusion also holds. ~
has de~ree i if whenever
formula
~
such that ~ I - -
~'/'(f t-- ~]. 7 Vn~( Vo, • • •, Vn )
a c cg~ao,...,an_ I]
@[a O .... ,an_l,a]
there
is a
and
[ a o .... , a n _ l ] .
Our final theorem holds when all the categories
have dimension.
We state it only for the case of relations on a single category for simplicity. numbers
Let
XwE
denote the set of those sequences of natural
such that the sequence
is zero for all but finitely many
places.
We identify an n-ary relation
factors,
with the relation
have n
x ~ R'
R'
if, and o n l y if
elements of the sequence
on
X E w
xln c R x.
R
on
such that for where
A relation
this way is said to be a finitary relation. ~(R)
extends n a t u r a l l y
to give
@(R').
XnE = E×...×E,
xln R'
n
x ~ X E w
is the first
w h i c h arises in
Then the n o t i o n of
we
20 Theorem 6.5. Let ~ finite
(Compactness
be a set of r.e.
is a sequence of Dedekind
on
C-types which simultaneously
@(R), of a relation
Monash University Melbourne, Australia Cornell University Ithaca., N~Y., U.S.A.
finitary relations
with dimension.)
X E. w is satisfied by some element of
subset of ~
every extension,
theorem for c ~ e g o r l e s
R
in
~.
If every X E
then there
satisfies
21 BIBLIOGRAPHY Aczel,
P.H.G.
Crossley,
[1966] D. Phil thesis
J.N.
Oxford (1966).
[1969] Constructive Order Types (monograph)
Amsterdam Crossley,
J.N.
[1970] Recursive Equivalence,
Bull. L.M.S.
(1970) 129-151. Crossley,
J.N. & A. Nerode
[1974] Combinatorial Functors
(Ergebnisse der Mathematik und ihrer Grenzgebiete
Bd. 81,
Springer). Dekker,
J.C.E.
[1966] Les functions combinatores
et les isols,
Paris. Dekker,
J.C.E. & J. Myhill
[1960] Recursive Equivalence Types,
University of California publications
im mathematics,
n.s. 3 (1960) 67-214. Ellemtuck,
E. [1965] The universal properties of Dedekind finite
cardinals,
Arm. of Math., 8~2 (1965) 225-248.
EQUALITY BETWEEN FUNCTIONALS
Harvey Friedman The ~-calculus,
in both its typed and untyped forms~ has primarily been
regarded as an attempt to formalize to provide a new foundation of the ~-ealculus
over
~,
in an embryonic
state of development,
another aspect of the typed ~-calculus~
(i.e., with the functionals
closed term in the typed ~-calculus
awaiting
of finite type).
which is readily
finite type structure
In an obvious way~ each
defines a functional
of finite type over
~ .
simple if it is given by some closed term in the typed ~-calculus.
Several definitions without
It seems fair to say that this aspect
This is its connection with the full classical
Call a functional
and ultimately
advances.
There is, however, understood.
for mathematics.
is currently
further conceptual
the concept of rule or process,
of convertibility
types, have been considered.
between terms in the ~-calculus~
The motivation
for introducing
with or
these defini-
tions has primarily been to analyze the notion of the identity between rules, processes.
The relation
these definitions
~ s = t defined
of convertibility.
terms define the same functional convertible
functionals
funetionals
of convertibility~
to one of
We show here that any two non-
of finite type. i/ This~ coupled with
tells us that "equality between simple
is recursive."
Let us call two functionals show that,
is equivalent
It is easy to see that any two convertible
of finite type.
terms define different
the known decidability
in the text~
or
in contrast
strongly unequal
to the above~
if they differ everywhere.
"strong inequality between
We
simple functionals
is as complicated as the set of true sentences of type theory over ~ ." Augment the typed ~-calculus by (primitive) recursion operators, and call the result the R-h-calculus. tive recursive
The functionals
functionals
of finite type.
that "equality between primitive recursive
This research was partially !/This result was obtained
denoted by closed R-terms are the primiWe conclude the paper by demonstrating functionals
supported by NSF PO38823.
in 1970.
is complete
~ii " 11
23
I.
Th e typed A-calculus. We first describe
the syntax of the typed l-calculus.
The type symbols are given by i) o symbol if
~,~
are.
The variables
their sets of free variables given by i) x~n term of type
(~•~) , t
= FV(s) U FV(t), BV((st))
, y
of type
by
((y,T)
The terms
is a type
s , their types,
, and their sets of bound variables
a term of type
= FV(s) - {y}• BV(O%ys))
x~n .
ii)
(y • FV(X~n) = {X~n}~ BV(X~n) = ~
FV((st))
a variable
are written
FV(s)
is a term of type
is a type symbol
~ , then
(st)
= BY(s) U BV(t)
~ , then
(lye)
= BV(s) U {y} •
BV(s)
ii) if
s
is a
is a term of type
iii)
if
s
is a term of type
are
~ ,
is a term of type (@~)
, FV(~ys))
=
The collection of all terms is denoted
Tm . We now describe the semantics we will use for the typed l-calculus. A pre-structure
each type symbol
is a system ( { ~ } ,
~ , and
= A(y•z))
An assisnment
is denoted Aug.
If
then
in the system
the set of all variables, y
where
~
is a nonempty set, for
A T: D (~'~) X D ~ ~ D ~ , for each type symbols
We require the following extensionality (•z ~ D (y) ( A ( x • z )
{Ag}),
condition:
if
x~y ~ D (~'~)
~,~ .
and
x = y .
([D@],[A
and such that
is a variable
})
is a function
f(X@n) 6 DC; .
then
fY
f
whose domain is
The set of all assigr~nents
is given by
fY(x) = f(x)
for
({D~],{%T})
is a
y # x ; fY(y) = (~ • A structure is a system pre-structure, i)
and
Val: T m X A s g
Val(x~n,f) = f(X~n)
(~,~)
, t
has type
where
s
is of type
Suppose function
Val
~
iii) for all • , x
such that
]• Val)
such that
-~ U D@ , such that the following
ii) Val((st),f)
= %T(Val(s,f),Val(t,f)) ~ 6 DO • A
is of type
({D(;],[A T})
refer to structures
({Df;},{A
clauses hold:
, where
(Val((lxs),f),o0
s
has type
= Val(s,~)
,
@ .
is a pre-structure.
([D~},[A T}IVal )
([D~],[A T]) • meaning
Then there is at most one
is a structure. that for some
Thus we sometimes Val,
([~}•[A
}~Val)
is a structure. For any structure case
Val(s,f)
= Val(t,f)
([D@]:[A T}) ~ we write , (where
([~},[%T})
([D~},{%~},Val)
~ s = t[f] ~ just in
is a structure).
Write
24
({Dg]3[A f .
~}) ~ s = t
if and only if
B
be a nonempty
T B 3 often referred where
set.
B 0 = B, B (g'~) = (B~) B"
decidable~
If
A
s
for all infinite
a variable~
the substitution
This m a y be inductively (rs)tX = (rtst)X x
A substitution
is a function x .
g
, if
Similarly~
2.
((~xs)t)
3.
(~x(sx))
= s , if
4.
s=t s=t~t=r s = s t t=--~ • .......s=r .....
5.
s=t (rs)=(rt) ,
x ~ FV(s)
LEMMA 2.
s=t (sr)=(tr) ~
LEMMA 3.
As an intermediate
is
stept we es-
x ~ then let
t
of
for each free occurrence
v)
let
s(g)
~ if
such that
denote the simultaneous y
in
in
s j by
g(y)
s .
y # x
, for variables
into terms~
y # x • g(x)
substitu-
.
= @ .
•
theorem.
~ if
, if
Fix a structure
BV(s)
N FV(t)
s .
y ~ FV(s)
U BV(s)
i.
M ~ (>~xs) = (~ySy)
for variables
(hys)~ = (~ys~)
t ~ and use induction on
From Lemma
x ii) Yt = y
x
to the typed h-calculus.
s=t (hxs)=()~xt)
Val(sy,f y) : V a l ( s , ~ )
Proof:
TB ~ s = t
.
Val(st, f) = V a l ( s ~ v a l ( t ~ f ) )
Fix
TB = ([Bg}'[A"T })u
U BV(s)
BV(s) n FV(t)
We first prove the soundness
Proof:
B .
of a structure~
a term of the same type as
of each variable
y ~ FV(s)
x = s t , if
LEMMA i.
B .
from all variables
We now introduce axioms and rules (~xs) = (~ys~)
over
example
B t the relation
i) x~ = t
(hxs)~ = (~xs)
tion of each free occurrence
i.
t
of the term
defined by
iv)
has the same type as
.
theorem for the typed h-calculus.
is a term~ x
denote
A
for all assignments
(x,y) = x(y)
and is the same for all infinite
tablish a completeness
of
We introduce the important
to as the full type structure
We wish to show that~
iii)
}) ~ s = t[f]
Below 3 we will often leave off the subscripts Let
x st
({Da}~[A
y ~ FV(s)
[j BV(s).
.
M = ([Dff},[A},Val)
= ~ .
•
25
Proof:
A(Val((>~xs)~f)~)
= Val(s~)
= V a l ( syX ~ f~X ) =
A ( V a l ( O,ys x)y j f ) ~ )
~ by
Lermna 2.
L E M M A 4.
Val(((Ixs)t),f)
Proof: x = Val(st~f)
LEMMA 5.
= V a l ( s t ~ f ) , if
Val(((Xxs)t),f)
n FV(t) =
= A(Val((}~xs)~f)~ Val(t~f))
¢
m
= V a l ( s , ~ a l ( t j f )) =
~ by Lemma i.
Val((},x(sx)),f) = Val(s~f)
Proof:
BV(s)
~ if
x ~ FV(s)
A(Val(O~x(sx)),f),(~) = V a l ( ( s x ) , ~ )
.
= A(Val(s,¢,
Val(x,~))
=
A (Val (s, f) 3~) .
L E M M A 6.
If
Proof:
M ~ s = t
Fix
f .
= A(Val((~xt),f),c~)
THEORk7~ i.
then
Then
A(Val(O~xs),f),~)
= Val(s,¢
= Val(t,~]
=
.
(Soundness).
Proof:
M ~ (~xs) = (>~xt) o
If
~ s = t ~ then every structure
By i n d u c t i o n on the proof of
M ~ s = t .
s = t , using Lemmas 1-6.
For the Proof of completeness~ we consider a p a r t i c u l a r structure d e f i n e d from the r e l a t i o n the
[s]
~ .
Let
[s]~ for terms
s , be
{t: ~ s = t} .
It is clear that
are the e q u i v a l e n c e classes of the e q u i v a l e n c e r e l a t i o n
~ s = t .
This
is because of the rules 4. We w i s h to define a specific type
~ }.
type
~ .
Define
A
([s]j[t]) = [(st)] , w h e r e
W e m u s t n o w check that s = t ~ then
s
~t = t' ~ then
LEMMA 7.
M0
Proof:
and
M 0 = ({D~},[A}).
t
M0
is well-defined.
are of the same type°
~ (st) = (s't')
s
Take
D@ = {[s]: s
is of type
(~TT) , t
Secondly, note that if
, by rules 5.
[s],[t] E D (C;'~)
is of
Firstly we remark that if
is a pre-structure.
Let
is of
Suppose that for all
r E D@
, we h a v e
S = S I,
26
[(sr)]
=
have
[(tr)]
[(sx)]
By axiom
=
3~ ~
.
Then
let
[tx]
.
Hence
~
= s~ >
(~x(tx))
O~x(sx))
x
be a v a r i a b l e (sx)
=
(tx)
not .
free
in e i t h e r
By r u l e
= t .
5, ~
By r u l e s
s
or
(~x(sx))
=
4~ e s = t .
t .
We
0~x(tx))
Hence
•
[s] =
It] Let
us c a l l a s u b s t i t u t i o n
FV(g(x))
LEMMA
n BV(g(y))
8.
such
Let
that
b
= @
be a f i n i t e
By s u c c e s s i v e
Let
be an assignment
Choose
t
let
g
such
x E FV(t)
, again
Val(s~f)
is w e l l - d e f i n e d .
LEMMA
Suppose
by Lemma
gl~g 2
for all v a r i a b l e s = @
, for all
Proof: k = 0
.
U BV(t)~
•
Let
Choose
w
~- ( X w ( s ( h i ) ) ~ )
(~.w(t(h2))X)g2(x)
Set
o n the
Val(s~f)
x3y
.
Then
k
type
as
~ for all
(~.wsX)(gl)
.
of
for
3 so t h a t
k (S(hl))Xl(x)o
Hence
now
U PV(t)
~ for a l l
show that
= g2(x)
i0.
Proof:
Val(s~f)
Obvious
is w e l l
defined~
from Lemma
9.
and
Val(s~f)
Let
The
=
Hence
case
x E FV(s) U FV(t) .
[J
U BV(s)
We have
h i = (gl)x~
= Val(t,f)
,
N FV(g2(y))
.
z E FV(t)
"
.
.
h 2 = (g2)xx .
= ~- s ( g l )
~ t ( g 2) .
2 LEMMA
t
.
x I- ( X w ( s ( h l ) ) w ) g l ( x ) (x)
[g(x)]
= ~
w ~ FV(s)
Let
= (t(h2))g
=
= BV(t)
n .
y E FV(s)~
Val(sjf)
~ gl(x)
= t(g2)
= (~.wtwx) (g2) .
that
FV(s)
true
f(x)
We must
N FV(gl(x))
k
x
to d e f i n e
N FV(f(x))
such
k s(gl)
is a t e r m
4.
each
BV(t)
It(g)]
k = n + i , and assume same
that
su#ostitutions
there
.
We wish
such
=
Then
1 and rules
a term.
cardinality
U FV(g2(z))
So
n b = @
~ s = t ~ BV(s)
= (Xw(t(h2))~) .
for a l l v a r i a b l e s
a term.
k s = t ~ and
y E FV(t)
o f the
hypothesis,
in c a s e
s
of a x i o m
M0 , s
that
8.
BV(t)
substitution
Suppose
By i n d u c t i o n
w ~ FV(gl(y))
By induction
for
are regular
x E FV(s)~
is t r i v i a l .
U FV(t)
Then
x
and
applications
be a regular
to be a t e r m
9.
set of v a r i a b l e s ~
= FV(t)~
Proof:
B y Lermna 8~
just
.
> 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)
= <0,1>)
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 .
(<(ys)~(yt)> = <0,i>) -- 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~
(<(ys)j(yt)> =
= Hence
TB ~
(Zz)
31
((Xx<((zx)s),((zx)t)>)
LEMMA 23.
If
=
~,~
(Xx<0,1>))
-
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
is a partial homomorphism from
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 a partial homomorphism from
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
<) is the root of
U {rN ~ ~ : N ~ ]
TA~ P<) TA
[F:Fg~N,N~]]
(FN,~)
= ~, A() = [A].
determines a countable canonical model ~
are given by
t~ms, for
(F:FCPN, NS~ }
N,N'g%, PN ~ ~ '
: ~
and
and
must be the whole language.
Each countable canonical model infinite paths
depending primitive
we have two sets of formulae
where the true formulae of
the false formulae by
A~
with the following properties.
Each infinite pathAthrough ~[~
N
TA,
subformulae of
~{
tkrough
TA
M
of ~ A
determines one or more
for all of which =M.
The extension or 'generalization'
stressed in this paper comes from
analyzing a little the operations (*)
~[:
~ ~->
m~
and:
M ~ - > 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, <
The sentence
~
will be the conjunction of the
L:
~l - ~ :
Reflexivity, symmetry and transitivity of
~4 - ~ :
Substitutivity of
=
= •
in the other predicates
Vxyzx'y'z' ~x = x'~ y = y,^ z = z'^
[e.g.
J6
is
sum(x,y,z)}~ s~m(x',y',z')).]
C~9 - C~12: single-value~ness axioms for zero, suc, sum, prod.
[e.g.
~i0
is
-
~yz ~suo(x,y)^ sue(x,z)]~y ~ z)~ ~13 - G16:
Existence of zero, successor, sum, product.
[e.g.
G15
is
Cxy~z~ sum (x,y,z)]. C~17 - ~25:
The axioms
Shoenfield
[Sh, p. 32] 3 with
prod.
~ x ( S x ~ O)
Thus
~xy (x'Sy = x'y+x)
NI - N9
O,S,+,"
becomes
of the weak number theory
eliminated in favour of
V x y ~ (zero(y) ~ suc(x,y))
becomes - g~13-~(16
N
given in
zero, suc, sum,
and
--
Vxyzuvw ([Suc(y,z)^ prod(x,z,w)A prod(x,y,u)~ sum(u,x,v)t~
w = v).
The exact choice of translation is somewhat arbitrary. the two examples given, the fact that each of i~ the form V Vl,...3vm~
where
~I
sentence.
~13"
~15
~i'
is a
~l
sentence (i.e.
q0 is a quantifier-free formula).
Thus ~ = ( . . . ( ( ~ l ~ ~2) ~ ) ^ Inspection shows that each
~l 7 - ~25
We use only, as in
"-')^
~25),
or %^ . . - ^ % 5
except the existence axioms
have the form
VVl...v2uq0
where
~13 - ~16' k = 0,i
or
for short. is a 2
and
is quantifier free. To show that model
~
has the desired properties we consider first the standard
65
< ; 0,i,2,3...)
= (~o~=,[O],[(m~n)In--In+l],{(Z,m,n)In=I~m],{(~,m~n)In=I.m],
for
L(C)
(where
cn
~ith this model.
is
Since
n).
Let
V~
be the total valuation function associated
~ ~ ~, V~(~) = 2.
To establish the positive part of theorem we show first that the restrictions of
V~
to certain classes of simple sentences are prim. rec.
use of well-known properties of Len~a l:
The restriction of
N
V~
Then we shall make
to prove the negative statement about
~.
to the set of all quantifier free sentences is
prim. rec. Lemma 2:
Suppose the
~Vl...Vk~
L(C)-sentence
or ~Vl...Vk~U~
where
is true in
~
and has either of the forms
is quantifier-free.
,
Then
V~ISub(~).
is prim.
rec. Lemma ! is routine.
As for lerm~a 2, if
~
is (say)~Vl...Vk3U~(Vl,...,Vk, U)
and true then all subsentences which are not quantifier-free are of the form VVi+l...Vk3U~(dl~...3di~Vi+lS...~Vk3U Lemma 5: Proof: ~i
V~ISub((~) Since
)
(0 < i < k; dl~...~di~C)~
and also true.
is prim. rec.
~ = ~l ~ ''" ^~25 ~ each subsentence of
or is of the form
~1^"'^~j
(2 < j < 25).
~
is a subsentence of some
Each
(zi
is of the form considered
in lemma 2~ which proves lemma 3. Thus
V~ISub(~)
Now let
V
is a prim. rec. full s-val, for
derive a contradiction from n < ~ ) let
is the n th successor of 3U(~n(U)^S~o(u,v))). particular sively
0 Since
= (C3 = D% 3 zero ~ ~...~ < V(~) = 2 ~n(V) (i.e.
(i.e.
~
v(~) = 2
do~dl~d2~...¢C dn+ I
L
is zero(v) and ~e have
true.
be the C-model
, c03c1~c2,...).
be the formu/a of ~o(V)
~
We shall
~).
V(~ i) = 2
V(~x zero(x)) = V(Vx3y suc(x,y)) = 2.
V(zero(c)) = 2;
which makes
be any recursive total valuation and let ~
determined by it 3 that is~ ~
For each
~
which expresses that ~n+l(V) for
is 1 < i < 2~.
:In
Therefore we can define recur-
as follows:
dO
is the first
is the first
caC
for ~hich
ceC
for which
V(s~C(dn, c) ) = 2.
v
66
Since induction on
V n,
is recursive 3 the sequence
(dnJn < ~ )
is recursive and, by
it is clear from the construction that
~
~
~n(dn)
We nOw make use of the special properties of the theory
N.
for each
n.
It is proved
in [Shj Sec. 6.7] that every recursive function is representable in
N
and it is
clear from the proof that the same applies (under a natural notion of representation) to every partial recursive function.
Now let
A~B
be two disjoint recursively
enumerable (r.e.) subsets of ~. Applying the representation theorem to the partial i x~A ftunction F(x) = 0 xeB we conclude that there is a formula q0(v) of the otherwise language of N such that for every n < ~ :
(where
n
is the numeral
neA ~ >
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
~
~
resp.) we have to know that ~.
and
n~A = >
Of course, to get from (V(~(dn)) = O
~n(dn)
A
V(~(dn)) = 2
<~0~ ~ ~ < d n ) ~(dn)
)
,
to
V<~
is in the domain of
That is why we put the disjunct
~x~(x) ^ ~ V x ~ ( x )
V,
i.e. -
in
~.
The proof of the statements in (b) concerning s-val.'s is thus complete. Now let V(~) ~ i ~3
V
be a 3-val. such that
for every atomic
hence
V JSub(~)
Since
~.
V(~) = 2.
This will imply that
We shall show that V(~) ~ i
for every sentence
cannot be recursive by what we have just proved.
V(~) = 2
and
V(Vx~(x)m~Vx~(x))
is (necessarily)
0
or
i,
68
v(~) :
it is clear that
hence
2,
(by looking at the equality axioms sentences have the value 2 under e =c,
[c = o ^
= o ^
~i-o~)
where
=d
zero(°)
Thus
Remark:
V(~) ~ i
= d ^c
< d ~c]<
suc(c,d),
satisfactory analogue
of the intended
~
~,
(a)
v(~)
= i
or
which completes the proof.
to the language
L = {=,~}, ~
and
if
~.
But
of lemma 4 since a rather of the hereditarily
(in the proof we know).
and arithmetic properties,
(set-theoretic)
In other words, not 'natural'
model.
As mentioned earlier,
our extension of the
theorems is of obvious interest if one wants to know about validity
in classes of models of restricted
(recursi~
and the improvement by Vaught IV].
results into proper perspective; not arithmetic
theoretic)
complexity;
cf. Mostows:ki
In a sense our extension puts their negative
specifically,
that the set of valid formulae is
if the models considered are determined by a satisfaction
which is recursive when restricted d_~o get an arithmetic satisfaction
sum( c, d, e ), prod( c, d, e )
referring directly to the canonical~numeration
Discussion and Open Problems.
to atomic formulae.
relation
By subsection 1 above, we
set if we consider instead those models which have a recursive
relation on the full language
founded tree is obviously arithmetic). Mostowski
d
for all atomic sentences
expresses a mixture of set-theoretic
completeness
s~(o,d,e)
it follows easily that we cannot have
finite sets, is needed as a conjunct in
[M~
su~(c,d,e~
finite sets are taken to be the intended model for
artificial axiom,
3.
,
c = d, zero(c),
we do not know an altogether
properties
the following
,
The proofs above apply, mutatis mutandis,
the hereditarily
c,d, eeC
sue(o,d~suc(c,d)
-~(i,i) = i f 2
is of the form
c
It follows easily
d = d ~ e = e ~ prod(e,d,e~prod(c,d,e)
[e = e ^ d
Noting that
that for all
e = ~ e
= e ^ d = d ~ e = e^ = e ^
i < i < 25.
V:
zero(e~
= c ^ d = d ^
for
v(~ i) = e
(since the property of being a Rec-
In other words, the negative results of
and Vought are sensitive to the precise measure of complexity used:
to
69
their particular analysis of what is meant by a 'recursive model'. (b)
As is implicit in subsection l(e) and at the beginning of subsection 2,
our extension is also obviously relevant to establish the stability of results in elementary metamathematics w.r.t, the exact meaning of the abstract notions involved. (i) 2-founded}
In one direction we have stability or 'invariance' of
{A:TA
since this is equivalent to ordinary logical validity for all
provided only that
G 0
is a 'basis' for strict ~ l
predicates.
is ~ ~
0
It is fair to say A
that this result is not a 'mere' refinement but quite essential for the role of Ilogical' reasoning in classical mathematics:
for most restricted domains of mathe-
matics 3 we get no more schemata (of the logical language) which are valid for the sets arising in that domain than those that are generally~ logically~ val[~ . is so because~in most branches~ all the sets
~G0
This
do arise. - Naturally, they don't
arise in so-called combinatorial ~mathematics about finite domains.
It is an empirical
fact that here there have been few applications of Clogica~ model theory~ and~ as a matter of logical t h e o r ~
validity in finite domains does not admit a complete
axiomatization. The reader familiar with intuit~0,istic mathematics should note that logical validity seems much less useful ther~ : different schemata are valid for branches which do and for those which do not use choice sequences. (ii)
In another direetion~ we consider stability of relations between
different rules (or~ equivalently 3 validity for different kinds of models ) such as the equivalence (**)
on page 23 in section 2.
Question:
~
For which
So we have the
do we have
[A:A has a f -founded derivation without cut} = [A:A has a ~ -founded derivation with cut} It seems plausible that such ~ afortiori
~ -founded
TA
or
need not satisfy the condition on ~ 0 TACF
need not be finite (well-founded).
thing~ as we shall see in II.1 o n ~ -derivations 3 the a n a l o g u %
say ( * * ) t
holds under quite weak closure conditions on (the c o r r e s p o n d i n g ) ~ ~ - f o u n d e d ~-derivation~need not be well-founded.
in (i) above. For one to (**)
in particular~
Also~ more generally~ cut-
elimination does not at all presuppose completeness ~.r.t. (usual) logical validity;
70
for example 3 it holds for the intuitionisticrules
(which are not complete for classi-
cal validity). The question above~ together with the exercise at the beginning of 1.2~ illustrates the kind of use that can be made of ~ -founded derivations between metamathematic~results
of different
'depth'.
The inversion theorems
1.2) are quite elementary 3 holding for a general c l a s s ~ class (as shown by Stavi's example);
to distinguish (in
; (**) holds for a smaller
and (i) above is even more special.
Some related questions will be put at the end of Section II. (c)
The use of our generalization~
from well-founded
to
~ -founded trees~
described in (b) should not be thought of as an isolated exercise 3 but as a typical example of modern mathematical practice. of given operations
To understand the mathematical
(here: rules of inference)
(here: finite iterations).
one extends the domain of definition
This can be rewarding(mathematically)
restricted domain remains the principal area of application course~ by (a) and (b) above~
operations
(here:
even if the original,
of those operations;
our extension is in fact of intrinsic
The general nature of such uses of generalizations different
behaviour
interest.
is clear enough.
If
sets of rules) yield the same results in a particular
domain s attention to these differences
is 3 by the nature of the case~ unrewarding;
as one saysj they are 'subtle' and (in the original domain of application) At the same time these boring differences must be considered theorems
of
in proofs~
boring.
thus the
stated cannot - as one says - do 'justice' to the methods of proof .
More prosaically 3 in such a situation there is a high chance of errors and omissions. This chance is reduced if one states theorems to the content of the proof; cf. the technique proofs of z:
=
~ a x
for real
!
in 'proper generality'~
more adequate
of avoiding errors in convergence
by checking the behaviour of
~ a~z ~
for complex
Ix .
Naturally 3 it is more difficult to be sure about the relevance of our specific choice of generalization~
that is~ of replacing finite (or~ equivalently
in the case of binary trees: well founded) by guiding idea is this.
~ -founded.
To master large finite configurations
For what it is worth our one simply must
71
not
use their build-up !- and if one doesn't~ the results will automatically apply
to suitable infinite configurations.
So at least one way of attaining that mastery
of large finite configurations is to make a guess at those 'suitable' infinite ones. In our particular extension 3 the (mathematical) analysis of 'not using the build-up' is: not using operations defined by transfinite recursions from the terminal nodes of the derivations (which are only f -founded)~ or again:
using operations which
are continuous and depend only on a finite Stump of the tree (starting at its root). It does not seem far~fetched to push the analogy with function theory a little farther 3 comparing G - f o u n d e d trees with analytic data and rules of inference (given by finite Post systems) with algebraic data.
Useful algebraic properties
can be inferred from analytic behaviour; for example~ analytic data about the poles of meromorphis functions in the complex place imply algebraic properties of being a rational function and so forth.
So it would be natural to look for ~
the 'analytic' equivalence:
has a f -founded R-derivation} = [A:A
-founded R'-derivation}
[A:A
such that has a
implies an 'algebraic' or 'intensional' equivalence
namely:
R
and
(by appropriate elementary conversions).
RI
are interconvertible
There seems to be a good deal of somewhat
inarticulate knowledge about rules in the proof theoretic literature which could ~enefit from this type of analysis~ for example~ the idea of (sets of) rules which are directly derivable from each other~ and thus in a much closer relation to each other than equivalence w.r.t, finite derivations.
At least for some relations of
this sort it may be easier to find an 'extensional' analysis in terms of a suitable choice of ~ themselves.
than to find appropriate conversions between the rules
R
and
R'
Be that as it may, merely remembering the possibility of choosing
between various analyses should have a salutary pedagogic effect: one will avoid that peculiar mixture of missionary zeal and hard-sell advertising which advocates of any one method of analysis employ and which repels the reader of logical sensibility.
despite the stress that ideological advocates of finitism put on the possibility of such a step-by-step build-up (e.g. of the power set of a finite set)~ on the fact that the configuration can be so built up (in principle).
72
II.
e-LOGIC: TRANSFORMATIONS O F ~ - F O U N D E D
DERIVATIONS AND 0Fe~-REFUTATION TREES
Some of the principal questions below are also of interest when restricted to the case of ordinary predicate logic discussed in I.
As already mentioned in I3(b),
some matters are easier to settle for ~-logic, in particular, the relations concerning
to closure conditions on ~
needed for two kinds of equivalences:
~'fotundedness and well-foundedness, or between validity in models g ~ validity,
(ii) between derivability ( o n ~ - f o u n d e d
out use of cut.
(i) between and logical
trees) by use of cuts and with-
However, the single most promising reason for concentrating on ~-
logic seems to us to be this: To state results which express adequately the mathematical content of Gentzen's work on formal arithmetic, one of the true highlights of proof theory.
In
accordance with i3(c), the extension to trees which are not well-founded, appears to provide concepts adequate for this aim. Evidently this aim assumes that the usual formulation of Gentzen's work (in particular,
his own as a consistency proof of formal arithmetic by means of
induction) is not adequate.
c O-
The authors of this article do not agree among them-
selves on the precise degree of inadequacy of the usual formulations.
The following
exposition represents the view of one of us. First of all, though realistically speaking Gentzen's metamathematical methods satisfy pretty well (Hilbert's) requirement$on mathematical principles that ought to be particularly reliable (on Hilbert's conception of mathematical rigour), the consistency proof simply does not add to our actual conviction of the truth of theorems proved in formal arithmetic.
From this angle, the value of Gentzen's proof is essen-
tially negative, in discrediting a certain conception of mathematical rigour. Secondly, as is well-known Gentzen's formulation is, on the surface, almost ludicrous.
The main principle of formal arithmetic is ordinary induction, on
and the metamathematical methods use induction on
~,
CO, which is 'bigger' than ~. So
clearly some more or less subtle distinction is involved here which is not conveyed by the usual formulation (and, being satisfactory). of the Appendix.
These matters ~ i i
'subtle' not likely to be foundationally really
be developed further below and in subsection 1
73
Lastly, quite pragmatically, the role of theoretic analysis is
mathematically marginal.
c0-induction within Gentzen's proof It is used in a proof of termination
or conver6ence of a certain transformation procedure of derivations.
Would it not be
strange to formulate a solution of, say, a differential equation merely by reference to the methods which are used to prove the convergence (of some procedure of approximation)?
At best one might hope to find that the method of proof (for example by
the convergence test: Ifm+l/fml < p < i)
could be developed to show something about
the nature of the solution, like analyticity or continuity in the parameters.
These
further properties are often of obvious significance in the contexts in which the differential equation arises, where the method used in the convergence proof is of no direct interest.
The discovery of such additional properties was then th__~esignifi-
cant step even if those gence proof.
properties
(What had to be added?
are easily derived by inspection of the converAnswer:
the ~ d e r s t a n d i n ~ o_~fth__~eproblem
which was needed to know what to derive, was not needed to find the convergence proof.) One of us has made several attempts to formulate results (derived from Gentzen's work) into which ordinals enter explicitly, in particular
e0
in the case
of arithmetic; now ordinals are mentioned in the results themselves, not only in the course of proofs.
Naturally, these results are best stated for abstract languages
(and axioms which are conservative over formal arithmetic) where ordinals arise naturally, which they do not in the context of first order arithmetic. such results are bounds on
Examples of
Z 1 .orderings that can be proved to be well-fottuded and
particularly, bounds ~stab]/shed by Friedman) on the lengths of certain hierarchies which can be proved to exist [Fr]. But it has to be admitted that these applications are somewhat marginal, and in any case they use little of the machinery of Gentzen's analysis; more formally, the applications use only facts about the set of theorems of formal arithmetic, very little about the derivations.
Another attempt to discover
significance in the ordinals used in Gentzen's work relied on a certain model of the development of mathematical knowledge, by use of socalled autonomous hierarchies [OL].
If indeed these models are faithful to traditional ideas (associated with
finitist or predicative doctrines) and if we look closely at those models we simply
74
become skeptical of the significances of those doctrines; cf. the 'negative' uses of Gentzen's consistency proof mentioned earlier.
In short, however brave or well-
intended these two attempts may have been, we cannot consider them successful. We are now persuaded to make a third attempt, by use of trees which are not well-fo~ded;
naturally, not the ordinals (such as
cO )
play a central role here,
but certain operations which are defined (not only on well-orderings but also) on ~-founded
trees, such as exponentiation to the base 2 or 3 or
on any binary relation; and generates well-orderings (< CO) finitely often to a well-ordering (< CO)).
~
(which is defined
when applied
To make this new attempt
plausible let us review quickly the developments which have occurred since the earlier attempts.
(i) We have become aware of the intrinsic, model theoretic signifi-
cance of trees which are ( g - f o u m d e d but) not well-founded, as explained in Section I.
At the same time, for such trees restrictions on the (logical complexity of)
predicates to which induction is applied are by no means arbitrary; roughly speaking, unless a predicate P.
P C g,
induction on a ~
-founded tree simply is not valid
This is p~omising because the 'subtle' difference, mentioned on p. 35
Gentzen's use of just this matter:
Co-induction and
for
between
e-induction applied in formal arithmetic concerns
the logical complexity of the predicates to which the two princi-
ples of induction are applied.
(ii) More generally, we have become sensitive
to the
basic difficulty involved in using proof theory sensibly for model theoretically meaningful results (which concern only the set of provable theorems):
there is a
high chance of wasted effort since the detailed operations on derivations studied in the (metamathematical) proofs get lost in the statement of the results.
This
realization was only a first step since the difficulty remained of discoverin~ properties of the derivations themselves, say, structural ones that we want to know about (and apply the methods of proof theory to such matters).
The first concrete
evidence in this direction, for a sensible use of Gentzen's methods~ was obtained
by considering (intuitionistic) derivations of existential theorems asking not merely for some term t for which for that term
t
A[x/t]
~xA
and
can be formally derived, but
which is provided by a given derivation
~
of
~xA
[M~.
75
More substantial
evidence is contained in Statman'~ dissertation on
[St~ partic-
ularly Chapter III. To combine (i) and (ii) it is natural to go to the extreme where the trees considered are re~utation trees (coding counter models C g )
and to extend Gentzen's
transformations so that they are well-defined (and of interestS) when applied to such trees.
Here the following problem suggests itself (which we have not solved~
neither for ordinary nor for o~logic): Is there a mapping
p
defined on all trees (regulated by the usual
rules) with cut and taking, as values, trees without cut such that, for suitable I , p
maps ~
-founded derivations i n t o ~ - f o u n d e d
derivations with the same
end formula~ and p
maps refutation trees coding a c l a s s ~
A
into a class ~ -
is 'contained' i n ~ ;
of counter valuations to
of semi valuations satisfying in other words, D
~ A
where
~-
eliminates not only cuts from
derivations but models from refutation trees. In subsection 1 we give some analogues for and in subsection 2 we state some results on
~-logic to the results in Section I~ transformations which were presented
by one of us at the S}unposium on the Theory of Logical Inferences at Moscow in March 1974.
i.
Preliminaries on ~-lo~ic.
We shall here confine ourselves to a single sorted
language including distinct constants with a 'successor symbol'~) is realized by:
ao, al~ a2, ... (for example, O, sO, ssO,...
and consider models with domain
a i ~--> 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
<0)
to
and = f(N') (~f)(N)
for
N'
N = (n+l, ml, ..., mn, mn+ l)
= f(N)
provided
f(N)
satisfies the data at
H(f)
82
f((n, ml, ..., m n )); otherwise change Rep beyond
f(N)
so as to satisfy the data and apply
N.
Naturally, in g~neral 3 Hf
will not be well-founded.
This simple construction indicates clearly how, with the help of (Re~
and)
K; one can extend operations defined on s a y ~ - f o u n d e d derivations to arbitrary trees. Note, for reference below, that the lo~ical complexity of
f that is~ f
is locally correct, LC(f) 0 HI"
is regulated by our rules, is
for short,
Naturally
LC(Hf) is identically true, but en revanche Hf E f is again
0 HI .
To get familiar with the properties of Re~ the reader may consider the following: Exercise.
Find rules
such that
A
has a primitive recursive ~ + - d e r i -
vation, but no primitive recursive ~-derivation. P(x,y~, in several variables
x, y, such that
primitive recursive function
@
allow the inference
from some
~ ~P
satisfying
Then, if
~ ~ P[~/~
sive ~+-derivation of max ~
is true, say
~ ~ ~ P[~/~
~
~ P[~/~
rules) not the multiple-succedent inference m.
~
Hint: ~ ~ P
Consider a polynomial is true, but there is no
P(~ ~x~.
Use rules ~
but (in contrast to the 'usual'
F ~ ~ y P, ~ P[~/~, ~ ,
from
P[~/~, ~ 7 ~ .
F ~ p ~[ y ~ ,
~ yP,
there is a primitive recur-
by repeating this sequent
m0, followed by a computation of
which
m0
times, where
The verification of this
state of affairs is primitive recursive. Put differently, the rule Rep acts as a dela7 d e ~ c e (in the sense of delaying decisions).
This kind of thing is very familiar in recursion theory when r.e.
or partial recursive objects are replaced by total or primitive recursive (or even
Kalmar elementary)ones ; cf. the theory of well-orderings or of neighbourhood functions of countable functionals.
The passage from partial to total operators raises
immediately another problem (which is familiar in the whole of functional analysis). Extensions of o~erations beyond their (principal) domain of definition. if we are principally interested only in the domain of ~ - f o u n d e d ~-derivations, and consider transformations, question arises whether is, not simply by giving requirements on
p
p
Even
derivations or
say p, defined on this domain, the
can be extended beyond this domain--not trivially, that p
a conventional value outside the domain.
may involve definability properties;
Non-trivial
for example, whether
is the restriction of some primitive recursive functional.
p
Clearly, this particular
requirement implies (the weaker requirement of) continuity since (primitive) recursive operations are continuous.
(Even the latter is not trivial since, for general
, the domain of ~ - f o u n d e d trees is not closed in the space of all trees).
In
view of the relationships mentioned, one will formulate-when possible--'positive' results asserting the possibility of extensions for (primitive) recursive operations, and 'negative' results for continuous operations. The matter of continuity or uniformity applies strikingly to the little argument earlier on showing the equivalence between ~
-founded
~
-derivations and
~-derivations whether the
(when ~ is sufficiently rich). There were 2 cases to consider: + ~ -derivation did or did not contain an infinite path consisting
only of repetitions(beyond a certain node).
Clearly, the decision between these 2
cases cannot be made continuously in the trees considered. Exercise.
Show that there is no recursive (and hence no recursively contin÷
uous) mapping from ~ -derivations to ~ derivations, for s~t~ble @ Consider ~ - d e r i v a t i o n s
of formulae
~ ~ P
for polynomials
coefficients and use the fact that the set of ~-derivable
P
~int:
with integral
formulae
~ ~ P
is
not recursive. Presumably, by use of formulae containing relation or function s~nnbols (in place of 3 ~ P) one will be able to show that there are no such continuous mappings, even if they are not required to be recursive.
84
Warning:
The problem is to show that there is no recursive mapping f r o m ~
derivations, to ~ - d e r i v a t i o n s ;
+-
it is evident that the Im~rticular 'collapsing'
mapping ia our little argument described above~ is not recursive. Remark on the transformation of refutation trees.
Though we do not go into
this matter in the present paper, it is perhaps worth mentioning that the extension of
familiar cut elimination procedures to trees which are not
~-founded
is
essential if they are to be applied to re~utation trees. We believe that these preliminaries pinpoint sufficiently the issues that need attention to make the formal details in subsection 2 below almost routine.
2.
Cut-elimination: anal~zin~ the issues involved.
The reader is assumed to know
one of the standard treatments~ at least for ordinary predicate calculus~ for example by Gentzen or in the books of Kleene, Sch~tte or Shoenfield. The statements of their main results do not mention explicitly any particular procedure for eliminating cuts, but only the possibility of doing so: the cut rule is said to be permissible (that is, conservative w.r.t, the remaining rules).
The standard treatments
proceed~ generally~ by induction on the socalled cut-degree~ that is~ the logical complexity of the cut formulae used, and on the 'size' of the derivations (which is a finite or countable ordinal in ordinary, resp. e-logic).
The logical complex-
ity of the predicate to which induction is applied, is not analyzed. it is clear that the results must hold for suitable r - f o u n d e d well-founded, further analysis is needed to find such ~
.
Thus, though
trees which are not
More to the point,
different ~arts (ler~aas) of the standard treatments require different closure conditions o n ~
.
Those auxiliary 'lemmas' are liable to have independent uses since
we are in fact interested in different ~ (which need not be ~ - - f o u n d e d
except for quite trivial ~
need concepts in terms of which these be made explicit.
--at one extreme, in re~utation trees ).
To analyze this we
uses' and the relevance of the lemmas can
As usual, this last step will need few formal changes in the
~roofs; it needs an understanding of
what we are talking about and what we want.
Below the following issues ~ill be treated. 'reduein$ ' operation
0
on trees
T
(a) The explicit definition of a cut-
in the sense that
(i) if
T
contains no
85
cuts, 0T = T
and
(ii)
if
~
is a bound on all the cut degrees of
is the degree of any cut formula of bN < c
if
~
to all trees
is finite). T
pT
at the node
N, then
By use of the operation
H
so as to satisfy (i) and (ii) if T
considered (including Re~).
The sizes of
in terms of ordinal f~anctions when
T
and
T
and pT
T
and
i + bN ~
in ll.l(c), p
c
bN
(and so
is extended
is regulated by the rules pT
are compared quantitatively
are well-founded, and generally by
means of appropriate operations such as exponentiation to the bases 2, 3 or (which are defined for arbitrary relations and satisfy the familiar laws).
(b) Given
D, one asks for closure conditions on ~
T
~-founded;
for trees
T
such that
pT
of bounded cut rank, say
if
is
c, these closure conditions in-
duce new~ possibly stricter conditions ensuring that is.
is~-fo~ude~
oCT
is
~-founded~
if
T
Lower bounds on these conditions are set by the 'model theoretic' requirement
that rules with and without cut generate the same set of theorems by use of ~ founded tress; cf. ll.lb (we do not discuss the case of infinite paper).
c
in the present
The two principal closure conditions so far considered are formulated in
terms of comprehension principles satisfied by ~
and in terms of socalled bar
(=tree) induction or of closure under the corresponding (generalized) ordinal functions.
The choice of these conditions is made for formal reasons concerned with the
metamathematical principles needed to prove that ~ (c)
preserves well-foundedness.
From a more explicit or 'mathe~tical' point of view, it is not satis-
factory to 'hide' the issues involved in restrictions on methods of proof: lets 'it all hang out'
Specifically, the trees
one
T, given in the first place by
the data mentioned at the end of II.l(a), are enriched by adding additional information at each node
N; for example by bounds on the 2 measures mentioned in (a),
that is, on the size, say
~N' of the tree
TN
cut degrees of all the inferences occurring on
dominated by T N.
N
and
CN, on the
Other enrichments, by adding
various (finite) formal derivations, are suggested by p. 122 of [SPT If] and [LE]. We give here just two samples of questions about such enrichments:
(i) How can
p
+
of (a) which is defined on the (fairly) bare trees, be modified to say operate o ~ enriched trees? +
T I and
p , to
Can this be done 'extensionally' in the sense that, if
+
T2
differ only in the enrichments (and so are the same tree
T
in the
86 ÷+
sense of the original data) then so do
++
p T , and
p T2~ and
pT
provides the bare @
data of
p +T i+ (i = i, 2)
Inspection of standard treatments provides a suitable p .
(ii) How exactly are enrichments related to restrictions on metamathematical principles?
Naturally, there are two kinds of restriction:
one on the expressive power
of the language used (in metamathematics), the other on the ~roof theoretic strength of the principles (stated in a given language).
As to the first restriction, in
particular in connection with (b), something like the enrichment by ordinal (notations for the) bounds
~N
above is needed to paraphrase the notion of well founded-
ness if the language does not contain functions or set variables, for example first order arithmetic.
Even if free function variables are included, (the hypothesis of)
well-foundedness,
in (b), cannot be expressed; but modulo a system of notations (de-
fined arithmetically),
the hypothesis is replaced by the--generally stronger--require-
ment that the tree orderin~ o_~fthe nodes of T +
be compatible with the given ++
ordering of the notations (at the nodes; and the conclusion, that fo~qded, must then be strengthened accordingly).
p T
is well-
If the notation system and its 0 H I.
ordering are defined recursively, the compatibility requirement is
This re-
duetion is, however, still inadequate if the metamathematical language is quantifierfree, in which case an implication with a ly.
0 HI
hypothesis cannot be expressed direct-
Additional enrichments, by use of suitable formal derivations, are used for a
further reduction because they permit a quite elementary kind of self correcting o~eration 8, to be compared to H in Ill.l(c): if then + + + + condition,~ST = T ~ and, even if T does not, 8T
T+
satisfies the c0~patibility +
does, where
T+
and
8T
have
the same end formula.--As to the use of enrichments for the second purpose, of reducing the proof theoretic strength of the metamathematical principles used, it appears
that, at least up to now, this matter always looks after itself: when we
demand explicit solutions stated in a language of weak expressive power, the natural metamathematical arguments tend to be quite elementary; cf. the possibility of 'algebraic' treatments even of ordinary number theory if the results are stated in 'suitably' explicit form. At the symposium on the theory of logical inference (March 1974 at Moscow), one of us presented a particular system of ~-logic, and a treatment of cut-elimination.
87
The latter contains most of the formal details needed to analyze, in that particular case, the issues (a)-(c) above. The first is pedagogic. some standard treatment.
We do not repeat the details here, for 2 reasons.
Realistically speaking, most readers will be familiar with So they will not only find a repetition of familiar details
boring, but are liable to miss the few novel points altogether.
Since the issues
in question are quite general, it should be possible to present them in general terms, with an occasional reference to some general features (illustrated by any) of the standard treatments.
This is precisely what we shall do.
quite simply, this:
The second reason is,
The details in question are inadequate; literally so for the
t.ype of applications of cut-elimination which seem most promising at the present time. Specifically, the choice of language is not properly consideredj more seriously, the choice of 'trivial' axioms for closed atomic formulae, that is, true numerical equations, excludes even a possibility of formulating some of these applications. shall go into these inadequacies in II.2(d) below.
We
Here it should be remarked that
these inadequacies do not touch the--unstated--aim of the symposium paper , which accepts the standards of traditional proof theory and simply tries to show how much can be done by primitlve recursive methods. phical, in the popular sense of the word~
The trouble with this aim is philosoThe whole project of going into the de-
tailed structure of cut elimination procedures is teratological, if those traditional aims are accepted because they concern only principles of proof and their consequences, not the derivations themselves (as we have said many times before). (a)
Let
R
be one of the familiar rules with cut, R-
supplemented (if necessary) by the rule Rep of ll.l(a). that the data rules
R
of
is
T
is locally correct for
P ~ m; in other words,
rule used in the last inference.
(*)
Let
f, in the sense of li.l(a), determine a tree
(or, equivalently, T
V f V F V~[Der(f,
sometimes also called
f(())
consists of
without cut, both Der(f, P ~ ~) T
mean
regulated by the
R) and that the end sequent F ~ m
and the (name of the)
We are concerned with the property
p ~ m ) -->
~ f' Der-(f', F ~ A)],
'permissibility' of cut.
We call
R-
closed under cut if
88
V f'l V f2' V F I V E2 V ~ I
V~
V C[[Der-(fl, D I ~ $1'C)ADer-(f2'~C~E2 ~ 2 ) }
f0 Der'(f0' Fl U P2 ~- ~
A functional
Pe
a
Remarks (i) Pe
.
is called a cut elimination procedure if Oe:f ~--> f'
and a functional
recursive
U %)]
is called a cut functional if
~:(fl' f
As is well-known, for finite trees
T
~--> fo
get a locally correct cut free tree with end formula
r ~ ~.
considered are well-founded, Pe
d
f), also called
are assumed to be only
for elementary classes ~
f'
rules, a till you
Similarly, if the trees
'bar recursion'.
If the trees
we do not get auto-
matically a 'trivial' solution of (*) by a recursive (functional) Pe trivially a solution of (*) from one of (**).
realizes (**).
by tree recursion (on the
ordering of the tree determined by ~-fo~dnded,
realizes (*)
with decidable
is ensured by the truth of (*); simply run through all
can be defined from
nor do we get
Since the trivial solutions above are
obviously unintended, our scheme of considering ~
-founded trees instead of well-
founded ones is at least a candidate for an analysis of our intentions here. the other hand, in the presence quirement that 0e(f)
0e
-->
(ii) Om
of Re~, (*) is empty without at least a tacit re-
preserves some kind of
to be the single line consisting of
~
-foundedness; otherwise we can take
(F ~ ~ j R e p )
repeated infinitely often.
(iii) Recall from ll.l(c) that, in the presence of Re~, we can extend a definition of
Pe
on [f:~ F, ~ Der(f, F ~ ~)] ,
to all
f
by use of the operation
H.
So below we shall tacitly assume that
f
is
locally correct. We wish to indicate in general terms how to extract cut elimination procedures from standard (syntactic) ~roofs of the permissibilit~~ of cut, (*), usually stated only for finite or well-fo~uded derivation trees.
As mentioned earlier, we have in
mind the older literature which did not derive (*) from (**) nor did it use the recursion theorem for defining
Pe
from a suitable functional equation for
89
Pe"
As so often, the early work lends itself to an abstract generalization because,
at an early stage, people rarely understand very well the particular objects they (talk about and) intend to study.
$o they seldom spot special properties of these
objects which can be used to shorten proofs, cf. the return to some early 'syntqctic' work on formal systems (with of
L
ZI
after model theoretic
sets of axioms) in later work o ~ predicate logic 'shortcuts' had been found for
L
.--In fact, even
a casual look at the proof theoretic literature suggests that the arguments apply to ~-founded
trees.
Ever since Gentzen, this literature shows only the end formula and
a few of its predecessors, apart from a lot of dots: evidently an argument cannot possibly depend on what is not shown~ such as the terminal points of the trees. formal language~ the older arguments begin with the end formula and proceed
!n
'down-
wards' in the tree ordering--in contrast to beginning with the axioms and proceeding 'upwards' (where inverted commas are used because, on this terminology, the root
()
of a tree is at the top). Proofs of (*) involve generally 3 elements; two of them are, obviously, preliminary, the third is the main (induction) step 'reducing' the degrees of cuts, but not necessarily their number. 'preliminary' and '~in' closure conditions on (i)
~
We shall see that this informal difference, between
step, corresponds to a precise distinction in terms of when~-founded
Derivations can be converted into socalled standard form~ that is, any
variable-bindin~ operation, say at a node aN,
are ~
P ~ Vx
trees are considered.
fo_r distinct
A, ~
from
N, binds a variable
t
fo__Kr a N
P ~A[X/aN] , ~
where
aN
does not occur in
aN
PUA.
and
N~ of
A conse-
t, any standard derivation of
can be converted into a derivation of
throughout.
where
N, N'; for example, in the inference, at
quence of this requirement is that, for any term P ~ A[X/aN] , A
aN
P ~ A[x/t], ~
by substitutin~
Clearly, this (substitution) operation is defined generally
, The standard form of the recursion theorem would apply only to notations for recursire f, not to abstract f, for which one uses the socalled relativized recursion theorem. For more essential defects of this kind of solution, see the discussion at the end of ll.2(a), p.58.
9O
(roughly, on all trees with countably many nodes) provided only the language contains infinitely many variables.
As a psychological experiment the reader should give a
proof of this convertibility by induction on the length of derivations, and see how stilted and ritualistic the proof sounds~ (ii) For the usual classical inference rules (apart of course from cut or modus ~onens) all the premises are valid if the inferred sequent is valid; for example, as in the Exercise on p. 22, if P ~A
^ B, ~
is valid so are both P ~ A, ~
and
P ~ B, ~.
(of course, as shown by cut, this invertibility property is not needed for the soundness of an inference schema).
For proof
~rocedure~ is of principal interest.
~
theory, something stronger, an inversion
For given rules ~ t h e
derivation of the conclusion of a rule into -NB.
Even if ~
procedure converts an
g-derivations
of its premises.
is sound~ such a procedure is not ensured by the inverti-
bility~which refers only to validity; trivially, if the rules are not (known to be) complete for validity, but also because the reference to 'convertibility' presupposes a mapping in some restricted class of operations (from ~-derivations
to ~-deriva-
tions). For rules
~with
cut (with or without Rep) it is clear that we have (elemen-
tary) inversion procedures; in the case above, just cut an P ~A
^ B, A
with some finite ............. derivation of
sulting derivation is ~ - f o u n d e d out cut, any ( ~ t ~ ) formula
A ^ B
derivation
is replaced by
A ^ B~A,
for the same class d-
of
~
P ~ A ^ B, ~
A~ resp. B at each node
the ar~mment of any inference rule applied at the nodes to dN,
N.
If
A ^ B
(dominated by
is the argument of the inference at N')
~-deri~tion
~
resp. A ^ B ~ B . as
of
The re-
A.--For rules ~ -
with-
is treated as follows: N, provided N'
A ^ B
on the path from
The
is not (0)
N', then the subderivation
has the form
We use the following terminology which is justified by the structure of the rules considered. There is a single formula, say C, either in the left sequent (P',C ~&') or the right (P~ ~ ~',C), such that the premises of the rule have the form P', Pi(C) ~ ~', ~i(C) where Pi(C), ~ ( C ) depend only on the argument C, not on the parameters
P', ~'; arguments are sometimes called
'principal' or 'active' formulae.
91
P' ~ A ,
~'; r I
F' ~ B, ~'; r 2
P'~ (where
rI
and
r2
applied at the node
A ^ B, ~'; ~^
are the nsmes of the rules loc. cit., and '~^' of the rule N'; the names form part of the data determining
'inverted' derivation, dominated at
4,).
Our
N', is then
F'~
A, A'; r I
P' ~ A, £'; Rep . This new derivation is locally correct for is a ~arameter of the inferences between
~( }
since by hypothesis the formula A ^ B and
N'.--Our
'treatment' of
d-
is
evidently (primitive recursively) continuous since, roughly speaking, we need look only at the finite segment
(),
N
of
d-
to determine the data at the node
N
the 'inverted' derivation.
(We say 'roughly speaking'; because we have left open
of
just which data are used to determine the trees themselves, in the sense of the Exercise in ll.l(a) above.) course, we need not decide at of
~^
anywhere in
~
Once a g a i n , ~ - f o ~ i d e d n e s s N
(unless
whether or not '~^'
occurs at
is clearly preserved.
A A B N
Of
will 'ever' be the argument
itself).
If such a decision
were necessary, the operation of inversion could not be expected to be continuous. Pedagogic Remark:
The discussion above shows--beyond any shadow of doubt--
that it is possible to present familiar proof theoretic arguments in relaxed detail, an obvious prerequisite for inspiring some degree of confidence in the reader; after all, if the author is not interested enough in a topic to give it detailed attention, what reason is there to suppose that the matter is, objectively, worth attention? In connection with our present topic of cut elimination , the views of the authors vary between two extremes; one finds that the introduction to II.2 is ample and that II.2(a) - II.2(c) are not justified without some striking application, such as solving the open problems mentioned near the end of ll.2(d); the other extreme would have us give a detailed treatment of the main induction step too (perhaps for some specific system of rules).
Reluctantly, we decided on the compromise below.
92
(iii)
After these preliminaries,
proofs of (*), the permissibility e~uation
(for
O )
we
of cut.
'consider' the main step in the standard More formally,
we set up a functional
which is evidently satisfied by a primitive recursive operation.
At one step the rule Rep is used--apparently
in an essential way.
Here the reader
should recall the Exercise in l.l(c) on the role of Rep in replacing partial recursire by primitive
rec~rsive operations.
The aim is to make by
~ I.
p
cut
'reducing'~
that is~ the maximum degree is reduced
This is prima faeie a global property of a derivation
tions of finitely bounded cut degree.
However,
the continuity properties of
ensure that the property follows from local conditions, happens at the node segments of
T
N
of
pT
T
such that if in this stump (of
ing. degree
T)
T
T
N
> 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)*) = ~<0n'll Z*) u ~(~ l) u(~ ~ 0 r~n-l(rr0) ~) u(z* 1 z~-l(z-1) ~), and so, noting that ~ = -~(~.~ ), we have (0 n'l I)* = -~(~. U m((O n-I I)*)) U >~, 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) <
144
Definition: S.
Let ~ be any Turing machine with tape symbols T and states
Assume b E T
instantaneous
where "b" designates
description
a blank tape square.
An
(i.d.) of ~ is a word in (T U (S × T)) t which
contains exactly one symbol in S X T.
Given any i.d. x = y. (s,t).z
for y, z E T , s E S, t E T, the next i.d.,
Nex~j~(x) is defined as follows:
if when ~ is in state s with its read-write head scanning symbol t, ~ enters state s' and writes symbol t' E T, then N e x t ( x )
y.(s',
t').z
y.t'(s',
u).w
is
if ~ does not shift its head, if ~ shifts its head right and z = uw for u E T, w E T ,
w.(s',
u).t'.z
if ~ shifts its head left and y = wu for u E T, w E T .
Next(x)
is undefined
the rightmost Next(x,0)
(leftmost)
symbol of x and ~ shifts right
= x if x is an i.d., undefined otherwise;
Next(Next(x, Finally, Comp(~,x)
if (s,t) is a halting condition,
or if (s,t) is (left).
Let
N e x ~ ( x , n+l) =
n) ). let # be a symbol not in T U (S X T).
The computation
of ~ from x is singleton set consisting of the followi~g word
in ({#] U T U(S x T))*: Comp~l~,x)
= (#.Nex~(x,O)'#,Nex~(x,l)'#
S × T = [(S,t)
I s E s and t E T}.
..... #.Nex~(x,n)'#}
We assume T n
(S × T) =
145
where n is the least integer such that (qa,t) occurs in Nex~(x,n) some t E T and designated halting state qa"
for
C°mpCS~'x) = ~ if there is no
such n. Remark:
Note that our definition of computation differs from the one
commonly in the literature. i.d.'s
x, not input words x.
exactly the same length. Fact 7:
The computation CompC~,x) is defined for Moreover, all i.d.'s in C o m p S , x) have
A key property of Comp(~,x) is given next.
Given ~ as in the preceding definition,
let ~ = (~} U T U (S X T).
Then for any i.d. y E ~ * , the n-I st, nth and n+l st symbols of y uniquely determine the n
th
symbol of N e x t ( y )
for I < n < ~(y) providing
Hence, there is a partial function ~ : ~
4 ~ such that if ~i,~2,~3
Next(y)
is defined.
are the n-I st , n th , n+ist symbols of C o m p S , x), then ~(~i,~2,~3)
is the
n+£(x)+l st symbol of Comp~f~,x) for I < n < ~(Comp(~,x))-~(x) and any i.d. x such that C o m p S , x) ~ ¢.
Also, ~(sI,~2,$3)
= ¢, if ~2 E (S × T) and
$2 is a halting condition of ~.
Lemma 3:
(Simulation) Let ~ be a Turing machine with states S, symbols T,
and designated halting state qa E S.
Let ~ = (#} U T U(S X T).
There is
a Turing machine ~(!D]) which, started with any word y,#.~ on its tape where y is an i.d. of ~ and ~ is a y-expression over D such that L(~) = ~n for some n > 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.
The
variable
qu~ variable,
excellence,
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sentence
values. truth
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on the part Logic,
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thought
letters
and q u a n t i f i c a t i o n
and corners.
their s c h e m a t i c they w o u l d
iThis
device
variable
work was
Science
status
was
to be
They
logic
felt
of truth
to no objects
they occur
any sense
as
nor to
and the predias values,
are not
obJec-
in schemata.
I
of creativity,
off basic m i s u n d e r s t a n d i n g s
to court
In that book wholly
in general
seldom
and in as I still
of the schematic
nor to classes
The s c h e m a t i c
been
or
in M a t h e m a t i c a l
I presented
logic
of
notation
of
sentence
and predi-
but
the
by
in a m e t a l o g i c a l
use,
clearly
earlier,
such m i s u n d e r s t a n d i n g s
letters
the trouble
appreciated.
was
that
Now and again
even get quantified.
A further objectual
at all.
had already
refer
five years
it unwise
Logic,
to p r o p o s i t i o n s
not with
Just
functions letters
that
now seems
to sentences,
of w a r d i n g
schematic
letters
of
of variable
as 1945,
are not bindable,
truth
cate
less
occur in sentences.
using
Greek
a variety
fixed numbers
as used in the
letters
to p r o p e r t i e s They
only
the status
much
a lively
of most
of
it takes
and to explain
of S y m b o l i c
values,
refer n e i t h e r
the n o t i o n
As recently
refer n e i t h e r
less to predicates.
quite
dissociation
letters
much tual,
This
to e x p l a i n i n g
and p r e d i c a t e
letters
But
to others.
letter.
pages
cate
idiom.
with
objects,
admitting
as the Journal
and q u a n t i f i c a t i o n . The sentence
against
variable
a notation,
is the schematic
I had to devote
to warn
as its values.
understood,
sophisticated
affinities
and par
It is the essence
of the r e f e r e n t i a l
strong
quantities,
is purely
an und f~r slch
variable.
and devices.
It used to be n e c e s s a r y numbers,
the variable
objectual
the essence
associated
Quine
that we must
is the b i n d a b l e
supported
Foundation.
in part
take
pains
to dissociate
substitutional
by grant
GS-2615
variable.
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
of q u a n t i f i c a t i o n .
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
of q u a n t i f i c a t i o n .
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
(<~m)Xo...x
then:
all work of bindable
~l'"~m
'
'
~ F~ x 1 . . . x
,
Ixy ~. ~ = ~. The n o t a t i o n of this version of q u a n t i f i c a t i o n theory of schematic p r e d i c a t e functors.
letters
The schematic
ber of argument places,
and the predicate
letters
carry exponents
since those places
consists
solely
'~' and the five to indicate the num-
are never visibly occupied.
166
Truth
functions
tion where George
come
Myro
got
last
able,
three
calculus
functors
transcending
teresting
thus
cases
of complement
two p e r m u t a t i o n
of p r e d i c a t e s
that bring
any g e n e r a l
to know w h e t h e r
can be e x t e n d e d functors,
as the
I used to have
and i n t e r s e c -
functors,
but
them down to one. 5
The Boolean the
through
m = n = 0.
is decidable; full
decision
the decision
to cover one further
the
the
of these
source
so it is
'I' and
of the b o u n d
procedures. procedure
or two also
isolating
force
It w o u l d
varibe in-
for the Boolean three
part
further
of the undecidable.
Havard University
5The pp.
system 8-25.
is p r e s e n t e d Myro's
disquotation."
in
"Algebraic
improvement
appears
logic
and p r e d i c a t e
in footnote
5 of my
functors," "Truth
and
167
REFERENCES Benmann, Heinrich, Mathematik und Losik , Leipzig, 1927. Bernays• Paul• "A system of axiomatic set theory•" Journal of Symbo!i¢ Lo$ic 2, 6-8, 13, 19 (1937, 1941-43, 1948, 1954).
~ h ~ n ~..........~........i '
, "Ueber elne nat~rliche Erweiterung in A. Heyting, ed., Constructivity in North-Holland, 1959)• pp.l-14.
, and A.A. Fraenkel, Axiomatic Set Theory, Amsterdam: Holland, 1958. Church, Alonzo, The Calculi of Lambda Conversion, Princeton,
North-
1941.
Frege• Gottlob, Grund~esetze der Arithmetik, Vol.l, Jena, 1893. G~del, Kurt, The Consistency of the Continuum Hypothesis, 1940.
Princeton•
Hilbert• David and Paul Bernays, Grundlagen der Mathematik, Vol.l• Berlin, 1934. Marcus, Ruth B., "Modalities and intensional languages," Synthese 13 (1961), pp.303-322. Nolin, L.• "Sur l'alg@bre des pr@dicats•" in the colloquium volume Le Raisonnement en Math~matiques et en Sciences Exp@rimentales, Paris: Centre National de la Recherche Scientifique, 1958, pp.33-37. Parsons, Charles, "A plea for substitutional quantification," Journal of Philosophy 68 (1971), pp.231-237. Peano• Giuseppe, Formulaire de Math~matiques, 1901. Peirce, C.S., C011ected Papers, Cambridge:
Turin, 1894-99; Paris,
Harvard, 1931-35.
Quine, W.V., A System of Logistic• Cambridge:
Harvard,
1934.
• "Logic based on inclusion and abstraction•" Journal of Symbolic Logic 2 (1937), pp.145-152. Reprinted in Selected Losic Papers. , Mathematical Logic, New York, 1940. , Elementary Logic, Boston, 1941. , 0 Sentido da Nova Losica, Sao Paulo, 1944. "On the logic of quantification," Journal of Symbolic Logic I$ (1945), pp.l-12. Reprinted in Selected Logic Papers. • Methods of Logic, New York:
Holt• 1950, 1959, 1972.
, Set Theory and Its Logic, Cambridge: • Selected Logic Papers, New York:
Harvard, 1963• 1969.
Random House, 1966.
168
, "On the limits of decision," Akten des XIV. Internationalen Kongresses f~r Philosophie, Wien• 1968, Vol.3, pp.57-62. • "Algebraic logic and predicate Bobbs-Merrill, 1971 (25pp.).
functors,"
Indianapolis:
, "Truth and disquotation," in L. Henkin, ed., Tarski Symposium, Providence: American Matehmatical Society, at press. • The Roots of Reference, Russell•
Bertrand, 1903.
The Principles
Evanston:
Open Court,
of Mathematics,
at press.
Cambridge•
SchSnfinkel, Moses• "Ueber die Bausteine der mathematischen Mathematische Annalen 92 (1924), pp.305-316. Tarski,
Alfred and F.B. Thompson, abstracts in Bulletin Mathematical Society 58 (1952), pp. 65f.
Whitehead• A.N. and Bertrand Russell, Cambridge, England• 1910.
England• Logik,"
of American
Principia Mathematica,
Vol.l,
ABOLITION
OF THE F R E G E A N Roman
This paper on Lewis'
is p a r t l y
systems
A n y endeavour,
stimulated
truth-valuations.
and
relations
Here,
are,
I want
S4
plenty
quieting
fact that the strong modal
systems)
are theories
logic, met i c
labelled
NFL,
based
(hence
exactly
matical
logic].
called modal com m e n t s
labelled
FL)
I essentially
construction
of NFL and its b a s i c
serve as a f r a m e w o r k
we will
arrive
logic
in Scott's
to arrive
(b)
mind, (a)
eventu-
to the s o m e w h a t
and
logically
however,
another
The m a i n s u b j e c t Also,
The general
truth-valuations,
dis-
all modal two-valued
that a x i o m a t i c
from t e x t - b o o k s
for three m e t h o d s
at NFL using
and
arith-
[created e s s e n t i a l l y
properties.
tween NFL and FL will be discussed. will
(a) the general
agree by the way,
[2] on that kind of logic.
following.
(but by no means
in the same sense
is not,
it
to o b t a i n m o d a l - s y s t e m s .
systems
(pure!)
or to d e f e n d
S 5.
in p a r t i c u l a r
and w e l l - k n o w n
This p a p e r
logic.
and
[i].
him to use both
visible,
on an e x t e n s i o n a l
is said to be b a s e d on
Frege,
doubt"
of ways
to call your a t t e n t i o n
of
Scott
1971,
operations)
induces
inferences
systems,
of course,
is the
on b e h a l f
semantics
(b) and a trick of m a k i n g
here
"a n a g g i n g
December
program
(or c o n s e q u e n c e
Furthermore,
ally at the strong modal
Lewis'
What m a t t e r s
with possible-world
There
on Entailment,
a great deal of p r o p a g a n d a
theory of e n t a i l m e n t
concerned
by a talk given by Dana
to r e c o n s t r u c t
is far b e y o n d my intention. Scott m a k e s
Suszko
in the S y m p o s i u m
however,
AXIOM
by
of mathe-
exercise
in so-
with Quine's here
is the
the r e l a t i o n
be-
theory of e n t a i l m e n t
of b u i l d i n g models
NFL.
In fact,
and logical
170
axioms
and rules
of inference.
As an i n t e l l i g e n t
reader
must be some h o c u s - p o c u s sists
essentially
to c e r t a i n notation
mulas But,
that all true
(names)
all
a common
it,
if you still
The trick
underlying
It is true that I even reject so far.
referent,
describe
axiom, the same
it, please;
I am sure,
is and you'll
you will
be free
is fairly
easy and also quite and
I am now a d d i c t e d
But try NFL cautiously.
I assure
you that NFL offers
all
unexpected
"impossible
when entering
into FL.
worlds".
you will
into the rough,
Surprisingly
of e x p e r i e n c e
lieve me.
Being
even
and this
else.
the p o w e r s e t
can,
I admit,
21
work
to use
you an
and beauty,
in your
it at once,
far
by my pocket when
you will be c o l l a p s i n g
be e x p e l l i n g
yourself
NFL
from a logical
world.
logicians
so close
something
Formally,
necessary
enough,
five years
to w o r k with
axiom hidden
to it.
that you go
But I am frank and fair,
the Fregean
headache.
Informally,
after
in its s i m p l i c i t y
the gate of NFL and be ready
you feel a c o n f u s i n g
to
innocent.
I do not insist
experience,
at
it ~o much.
NFL
So I tell you keep
They even
axiom
for-
categories.
the F r e g e a n
sentences
time,
to keep
syntactic
just forget
At that
syntactic
However,
surpassing
paradise
It con-
logic but only
inclined
called here
it seduced me s u c c e s f u l l y
the F r e g e a n
pure
is.
axiom.
intellectual
nature.
like
there
avoid his
as d i s j o i n t
false)
the F r e g e a n
that there
you need not use his archaic
you may easily
until NFL is constructed.
accept
Indeed,
you are n a t u r a l l y
similarly,
what
conjecture
in b u i l d i n g
to his assumption,
that is, have
understand
strive
Frege
Also,
and terms
(and,
better
NFL.
Of course,
For example,
w h e n you come
least
point.
or terminology.
(sentences)
thing,
underlying
in f o l l o w i n g
decisive
shortcomings.
you i n s t a n t l y
do not want
I know
it from
is the right w a y of p u t t i n g
it, be-
to NFL When not
sometimes, satisfied
as e x p o s e d
NFL.
Logicians with
convincingly
stubbornly
FL they choose by Scott
on it as hard as in a sweatshop.
[3]. So
171
m i g h t y is, Gottlob,
the m a g i c of your axiom!
Whatever
one calls elements of the index set I, the p o w e r s e t
21
(cheatingly) remains a
d i s t i n c t shadow of the F r e g e a n axiom. If we w a n t to follow Frege we must consent to his basic ideals of u n a b i g u i t y and extensionality.
To stress this point we start w i t h
his famous s e m a n t i c a l scheme of Sinn und Bedeutung.
It is obvious to-
day that the abyss of t h i n k i n g in a natural language does not fit into the Fregean scheme.
But this is another story.
to notice that we all live
ambiguity,
vague flexibility,
really enjoy them.
it must suffice
(and cannot c o m p l e t e l y get out of) that
messy abyss w i t h all its diffuse ghosts log~of
Here,
But not always.
(in Hermann Weyl's
[4] phraseo-
i n t e n s i o n a l i t y and modality.
We
When forced to c o n s t r u c t a theory,
we try to make our ideas precise and climb to the heights of e x t e n s i o n ality.
Then,
the s t r u c t u r e of our t h e o r e t i c a l thought c o r r e s p o n d s
s u f f i c i e n t l y well to the syntax of the B e g r i f f s c h r i f t ,
i.e., a forma-
lized language w h i c h does fit into the F r e g e a n s e m a n t i c a l scheme.
1.
Reference,
Sense and Logical Values.
We assume the p r i n c i p l e of logical t w o - v a l u e d n e s s and choose 1 and 0 to r e p r e s e n t the truth and falsity of sentences, Then,
respectively.
the Fregean scheme can be p r e s e n t e d by the following diagram:
S(~)
~
~
--f~
-- - - - - ~
i ///
/ g / ;f
t(~) = 1 or 0 if
~
is a sentence
r(~) Here, ~, ~(~)
~
is e i t h e r a name
i.e., what is given by is given by
logical
(or truth)
~.
(term) or sentence, ~,
Moreover,
value of
~.
~(~) if
~(~)
= the sense of ~
= the r e f e r e n t of ~,
i.e., the way
is a sentence then
The a s s i g n m e n t s
r, s
and
~(~) ~
= the are
172
related
as follows:
(I.i)
and,
s(~)
for sentences
(1.2)
# s(~)
axiom.
r(~)
~ r(~)
whenever
~(~)
~ ~(~)
~,~:
r(~) ~ r(~) Note
whenever
that the c o n v e r s e
On the o t h e r hand,
of
(1.2)
is a v e r s i o n
the c o n v e r s e
of
m a y say that we just need the a s s i g n m e n t ence of e x p r e s s i o n s the same,
i.e.,
Frege ([5], p° there
~,~
~(~)
put much
which
= ~(~). stress
9) that r e f e r e n t s
is a f u n c t i o n
f
is, however, (1.2)
that for any s e n t e n c e
= the s i t u a t i o n
the p r o p o s i t i o n not e d by names analogy
~,
as w e l l between
axiom.
~
but refer
set:
= ~(~).
~(~)
restated
to
by C h u r c h
on senses.
That
is,
~:
= r(~)
equivalent
to
(i.i).
of a f u n c t i o n
~
such
If
by
scheme
~
~.
as sentences. (I)
Is e s s e n t i a l l y to f o r m u l a t e
is a s e n t e n c e
b__y ~
is a name.
the pairs:
= t(e
terminology
described
expressed
if
in m e a n i n g
to the e x i s t e n c e
semantical
some t e c h n i c a l
lent of the F r e g e a n
the exist-
for any name or sentence,
_gCr(~))
we i n t r o d u c e
we simply
One
:
(1.4)
Now the F r e g e a n
somehow
seen to be v i r t u a l l y
is e q u i v a l e n t
is forbidden.
to cope with
functionally
such that
easily
~
on the assumption,
f(s(~))
Similarly,
~(~)
Then,
depend
(1.3)
This
differ
(i.i)
of the F r e g e a n
Therefore,
(a) names
once
~(~)
Assignments
r
another
equiva-
for all,
~(~)=
= the objec t de-
and
s
apply
one may expect
(terms)
and
then we say that
and we set,
Similarly,
complete
and s e n t e n c e s
to
a strong (formulas)
173
and,
(b) s i t u a t i o n s
and objects.
one m a y also e x p e c t tain d i f f e r e n c e s ences
in
cerned with
difference
(a) and
gean a x i o m has the
described
not a c c e p t
the F r e g e a n
Thus,
(b).
Now,
fer that
Under
the t o t a l i t y
in
at least
infer-
to names.
Thus,
of a n a l o g y
and
we b a l d l y
we do not w a n t
two d i s t i n c t
on the and,
situations
is d i v i d e d
to impose
by
any
or objects.
and objects.
assumptions
than two
say that we do
situations
at least one o b j e c t
of all s i t u a t i o n s
Cer-
are con-
there are no more
situations
obvious
(b).
it is easy to see that the Fre-
from above on e i t h e r
rather
(a) or
listed notions
interpiay
Hence,
a x i o m since
(i) there exists
(2) there exist
of the a s s i g n m e n t
w i t h assertion,
The
equivalent:
by sentences.
limitation
also.
concerned
familiar.
we have a a n a l o g y b e t w e e n
exists
differences
an i n t e r e s t i n g
following
situations
quantitative
because
and do not apply m e a n i n g f u l l y
to o b s e r v e
within
those
are quite
sentences
we are p r e p a r e d
some e s s e n t i a l
(a), e.g.,
and e n t a i l m e n t
However,
A difference
language (1.2)
or, more
we in-
or
(1.4),
explicitely,
into two n o n - e m p t y
collec-
tions. (II) The F r e g e a n summary
semantical
of the semantics
scheme
of the B e g r i f f s c h r i f t
and its full value
can be r e v e a l e d
of the e x p r e s s i o n s
involved.
and
E
can be e l a b o r a t e d
models,
respectively,
If one accpets
used
in s t u d y i n g
the F r e g e a n
continue
Frege's
Euclid's
program without
program without
at so c a l l e d
his
absolute
the w a y of E u c l i d
(formalized
a language,
logic
at FL,
geometry
~
can be e l a b o r a -
Frege
the F r e g e a n
fifth postulate.
and
logic may be im-
constructed.
follows
his axiom.
the syntax
of t r u t h - v a l u a t i o n s
the a s s i g n m e n t the
language)
the a s s i g n m e n t s
extensional
a x i o m and
arrive
and c o n d e n s e d
if c o m b i n e d w i t h
the m a c h i n e r y
Thereafter,
logic then one will
bilities:
Given
with
ing pure
arrives
only
and a t w o - v a l u e d
pos e d on the language. ted and e f f e c t i v e l y
is a s i m p l i f i e d
(III)
in c o n s t r u c t -
logic.
We will
It is like r e a l i s i n g In that case,
and there
one
are just two possi-
or that of L o b a c h e v s k i
and Bolyai.
Here,
174
we get NFL, opening
that is, the a b s o l u t e
a Pandora's
uncountable logics
Certainly, This
FL may have
it is not
NFL
the crystal
we have an e m b a r a s s e m e n t fact is very
so i m p r u d e n t l y
important
precious
items,
We are now ready (or perhaps,
simply
to c o n s t r u c t
A logic tautologies),
is m e a n t
which
generated
as an e n t a i l m e n t sense,
1930),
language. relation e,~,y,...,
This ~
relation
subject
and Transitivity,
we a l l o w ingly,
rules
the a n t e c e d e n t
an a r b i t r a r y
[]], p.
~
or not,
X
formal
that
we need we
tools.
(logical
operation"
Scott.
theorems,
in A. T a r s k i ' s
(sentences)
of some
Any entailment
X,Y,Z,...
is a
and f o r m u l a s Monotonicity
If we say
i.e.,
X ~ e
and the s u c c e d e n t
set of formulas
I mean
NFL b e c a u s e
laws of Reflexivity,
796.
our
from some set of axioms but
(or " c o n s e q u e n c e
to three b a s i c
us.
except
In particular,
in the set of formulas
as in
Therefore,
Identity.
sets of formulas
X entails
could p r e v e n t
in c o n s t r u c i n g
is a c o m m o n p o i n t w i t h
between
includes
into the chaos we
is n o t h i n g
not a set of formulas
via c e r t a i n
operating
There
problem.
Entailment,
here
totality
insight
and use the standard
2.
This
FL h e l p e d me to find there
NFL.
how to p r o c e e d
follow Frege
of
than NFL and FL.
we did not leave open any e s s e n t i a l
will
face is
chaotic.
since we k n o w FL so well.
some cataclysm)
not ask our i n t u i t i o n
we are
of NFL lies at the bottom.
of riches.
In fact,
other
we
at first glance,
and help us get some
created.
However,
is so w e a k that the t o t a l i t y
as NFL appears,
since
logic.
the r e a l m of p o s s i b i l i t i e s
(as w i l l be shown).
serve as a guide
certain
will
since
at least as strong
We may h o p e
FL.
box,
non-Fregean
or non
~
and any single
X ~
to be c o r r e s p o n d formula. (IV)
If
175
X
is finite and,
instead of ively.
X ~ e
If
of all
~ ~
F
X =
{~l,...,~n}
we may w r i t e then
~
or
X = ~ = the e m p t y set then
~i ..... ~n
k
e
and
is said to be a t a u t o l o g y of
t a u t o l o g i e s is an example of a theory of
formulas is said to be a theory of that is,
X
~ e , respect-
F
if
x
.
The set A set
is c l o s e d under
c o n t a i n s every formula e n t a i l e d by
theories of the given entailment,
~
~
in general.
X.
X of ~
,
There are m a n y
Since the intersection
of any c o l l e c t i o n of theories is a theory again there always exists a least
(smallest)
theory:
the set of all tautologies. (v)
We are i n t e r e s t e d in a specific e n t a i l m e n t relation, NFL entailment,
imposed on a f o r m a l i z e d language.
that is,
First, we define a
class of f o r m a l i z e d languages w h i c h I p r e f e r to call "languages of kind W".
V o c a b u l a r i e s of languages of kind W involve:
tial variables, variables,
of d i f f e r e n t sorts,
tives, e.g., 4 junction,
(2) nominal variables,
~, ~
~>,
like functors,
~>
versal
V, e s i s t e n t i a l
3
unifier
(description operator).
nominal v a r i a b l e s and names,
i.e., q u a n t i f i e r s
one may b u i l d terms,
however,
connective,
and the
including
i n c l u d i n g s e n t e n t i a l vari-
The c o r n e r s t o n e of W - l a n g u a g e s
tion of identity which o c c u p i e d Frege's m i n d so much.
must,
(uni-
Starting w i t h a given v o c a b u l a r y and
and formulas,
formator falls into the group
dis-
(formators bind-
and even n u m e r i c a l quatifiers)
using familiar rules of formation,
ables and sentences. (Iv)
conjunction,
respectively),
(sentential and/or nominal),
not b i n d i n g
p r e d i c a t e s and connec-
of negation,
i m p l i c a t i o n and equivalence,
ing v a r i a b l e s
(3) formators,
(i) senten-
is the no-
The identity
(3) above as a binary formator w h i c h
split into an i d e n t i t y p r e d i c a t e and an i d e n t i t y
for obvious s y n t a c t i c a l reasons.
To e m p h a s i z e the ana-
logy, both versions of identity will be r e n d e r e d by the same symbol Thus, w h e n e v e r both is a formula,
~,~
are terms or formulas then
called an equation.
p r e d i c a t e or i d e n t i t y connective,
The symbol
~
~ ~
is then an identity
respectively. (vII)
176
The NFL e n t a i l m e n t r e l a t i o n in c o m p r e h e n s i v e W - l a n g u a g e s has b e e n defined in a syntactic way, by logical axioms and rules of inference in [9] and
[i0] and, a c o n v e n t i o n a l c o m p l e t e n e s s t h e o r e m for it has been
proved by S. L. B l o o m in for identity:
[ii].
NFL involves p r e c i s e l y the logical laws
trivial e q u a t i o n s
~
~ ~
are t a u t o l o g i e s and the law
of r e p l a c e m e n t of equals by equals salva identitate is g u a r a n t e e d by NFL entailment. If you do not have a clear idea of identity one-ness)
(equality,
then you had better stop reading right now.
sameness,
As an explana-
tion of this notion
(2.1)
t(~
=_ 9) = 1
iff
r(~)
= r(9)
is e s s e n t i a l l y c o r r e c t but somewhat c i r c u l a r and, hence, not very helpful.
We must b e g i n with the u n q u e s t i o n a b l e
Leibniz'
ideas.
It is, as Quine
only m e a s u r e of e x t e n s i o n a l i t y and contexts. opaque)
[12] made clear beyond any doubt,
(referential transparency)
A context is i n t e n s i o n a l
just if it v i o l a t e s
logic of identity along
(non-extensional,
some logical law of identity.
the
of notions referentially Since NFL
e n t a i l m e n t involves the logic of identity g o v e r n i n g all terms and formulas of the c o r r e s p o n d i n g language, we may say that NFL does not tolerate any intesional context or notion.
All notions encoded in a W-
language s u p p l i e d w i t h NFL e n t a i l m e n t m u s t be extensional. ticular,
equations
~ ~ ~
of b o t h kinds are extensional.
is the p a r a d i g m of an e x t e n s i o n a l
(and two-valued)
In parIn sum, NFL
logic.
In this article, we will not deal with the c o m p r e h e n s i v e uages of kind W.
More modest,
open W - l a n g u a g e s w i t h o u t b o u n d v a r i a b l e s
and w i t h c o r r e s p o n d i n g l y r e s t r i c t e d NFL entailment, s u f f i c i e n t l y interesting.
lang-
as in
[9], appear
They contain equations of b o t h kinds and,
hence, on that s y n t a c t i c level one may study the i n t e r p l a y of analogy and d i f f e r e n c e noted previously;
compare
[13].
But even open W-
177
languages cate
seem too c o m p l i c a t e d
is more
connective identity
than well
which
the s e n t e n t i a l which
k n o w n and quite
carries
predicate
a devil's
and make
calculus
is the crucial
for this article.
with
place
ables ~.
language
p, q, r ....
the v o l a b u l a r y
and,
connectives
of SCI may be found
connectives
to c o m m i t
~, note
As far as this
the c h i l d i s h
~ F ~-
if and only
if the F r e g e a n
relation
between
tog e t h e r
with
Although
(1.2).
~
set
[14]
and
Thus,
{0,i}
assignments
explained
t:FM +
of truth values. to formulas.
with
formulas e
connective
coincides
~,~
list of
Truth-functional the i d e n t i t y
entails
~
the
~ =
formula
to anyone
to
is not t r u t h - f u n c -
You r i g h t l y
with material
answer:
equivalence,
In the case of NFL the only and
and
~ ~ ~
is given by
(2.1)
that s i m u l t a n e o u s l y
t(&)
by Scott,
~ t(~).
connective
works ~.
the m a c h i n e r y
as well with Clearly,
we do not need
from the set FM of all S C I - f o r m u l a s We will
not
of the form
it is no d i s c o v e r y
as w i t h {0,i}
A short
one has to be w a r n e d
we only e x c l u d e
~ 9) = 1
~----~, ~
functions
of
variand
[15].
in c o m p a r i s o n
a x i o m holds.
of the
of s e n t e n t i a l
[17].
is a n o n - t r u t h - f u n c t i o n a l
of t r u t h - v a l u a t i o n s , V)
then,
truth-values
t(~
all
in
One m a y ask h o w it c o u l d be o t h e r w i s e ? and,
SCI,
~j ~ - ~ ~=m~
that a formula
of the i d e n t i t y
is t r u t h - f u n c t i o n a l
"~ j A;
A,
On the other hand,
that the nature
tional.
~
of c o n f u s i n g
saying
the so c a l l e d
consisting
[16] and
toys
the
to r e a c h the level of
i m p o s e d on formulas
toy is c o n c e r n e d
mistake
the m e t a - s t a t e m e n t s i.e.,
in
seem to be c h i l d i s h
connective.
with
studied
let us abandon
for NFL.
relation
with
predi-
It is the i d e n t i t y
connective,
of the battle
SCI has been e x t e n s i v e l y
properties
So,
step down
identity
SCI is the NFL e n t a i l m e n t sentential
innocent.
brand.
one more
The i d e n t i t y
only use c e r t a i n
These may be c a l l e d
shortly
to the
truth-value truth-valuations
178
(for SCI). as the
They
intended
characterize
"I ,
truth-functional
A ,
V,
~,
connectives
and
~ E
adequately as the
identity
connective.
3. T r u t h - v a l u a t i o n s . A function all
~,~,y,8
~:FM
+
{0,1}
~(~e)
(3.2)
t(~
similarly,
for
v,=,=
t([e ~ y]
where
~
and,
stands
not
m a)
= t(~)
b y the
usual
= 1
truth
if
t((~ ~ ~)
= 1
~
= 1
if
t(~
= t(y
[~ ~ ~]
of b i n a r y
tables
and,
= 1
= i
~(~
~
depend
~ ~)
= 0
a truth-valuation
depending
(3.7)
~(e)
_ ~)
connectives
A
~ 8)
= 1
=
V
=
and
finally
Given
every
iff
t(1(l m - ~
for one
(3.6)
~t'
if for
~(e)
as r e q u i r e d
t(e
(3.4)
~
A ~) = 1
(3.3)
(3.5)
to be a t r u t h - v a u l a t i o n
in FM:
(3.1)
and,
is said
in
on
X.
t: But,
X
whenever
~,
~ te
we are
iff
we
~(~)
6 ~(~).
immediately !(~ ) = 1
interested
have
an e n t a i l m e n t
whenever
in an e n t a i l m e n t
~(~ ) = 1 which
on a n y p a r t i c u l a r
truth-valuation.
So we
X
F
truth-valuation
h(t)~
iff
X
t~
for e v e r y
for
does
set: _t"
179
In this
section,
SCI-entailment, bitrary,
defined
possibly
Finiteness
if
X
such Obviously:
infinite
(3.8)
(3.10)
and
X
~
X
s.th.
a set of rule.
means
following:
the
of T A U T
X.
effective
~
in T A U T or not. finiteness
of
of
~(t)
(3.7),
X
However,
for the is an ar-
~
has
the
connectives
subset
Y
of
X
we
state
formulas
then
~ a
an)
in
for d e c i d i n g Note
that
v
a I ..... a n
to the
set of its
of this
in
of a set
X
in
a
is i n v a r i a n t .
the
of any
given
formula Another
formula
a
sub-
of f o r m u l a s
truth-valuations
for this
the
a[v/~]
by any
taut-
set.
under
the r e s u l t
Indeed
the r e a s o n
exist
if it is c l o s e d
X,
set T A U T
~ (a O = 9).
following:
properties
invariance
a
property:
= a).
in a sense,
of any v a r i a b l e the
the
or t h e r e
certain
words,
the D e d u c t i o n
a I ..... a n
into
invariant
for a n y
has
of the ~
property
provide
whether
is s i m p l y
is
a
an is
the o b v i o u s
formulas.
Functions called
9
reduces,
As e x p e c t e d ,
method
F
amalgamate
is its d e c i d a b i l i t y .
that
In
is a f i n i t e
~ ((a I A...A
In o t h e r
replacement
~
iff e i t h e r
Hence,
stitution
such
(3.9)
a
there
Moreover,
ao,a I ..... a n
TAUT.
in
instead
~ a. (VIII)
the e n t a i l m e n t
Call
are
Y
a,a = ~ ~ 9.
Properties
also
~
set of f o r m u l a s .
then
that
if
uniform
write
by t r u t h - v a l u a t i o n s .
~ a
(3.9)
ologies,
simply
property :
(3.8)
Thus,
we
t:FM
general but ~(a)
+
with
only
truth-valuations.
do n o t = 1
{0,i}
respect
for e a c h
the
the p r o p e r t i e s
They identity
general
respect
all
(3.1)
(3.2)
truth-functional
connective.
truth-valuation
and
Formulas t
m a y be
a
180
collected It a l s o and
into
the
set TFT,
is an i n v a r i a n t
-1(pA1p).
in TFT.
(p -= q) = Another
Here
are
(q --- p),
point
some
~ ~ ~
TAUT behave
that
theorems
(~p
every
is an e s s e n t i a l
level
predicate
for e q u a t i o n s
with
property
as well.
hand,
inequalities
ity i n v o l v i n g the i d e n t i t y
kinds. (Ix)
a difference ~ i 4-
the
Then
Again, also
not
(i.e.
identity
we m a y
~ ~)).
popular and
view
restate
Indeed
(3.11)
to the
connective holds
see t h a t
and
uniformly
as an a n a l o g y .
It is c o n c e r n e d , h o w e v e r ,
offer But,
in T F T
~ ~ ~ ).
postulate
appears.
predicate.
behaves
the
n(~
here.
can e x t e n d
the n a t u r a l
N F L does
identity
connective
with
(p_=r),
us a n y n o n - t r i v i a l
is t r i v i a l
One
is,
empty"
teach
are
trivialities.
the v e r y
is s a t i s f i e d
of NFL.
(q --_ r) =
no f o r m u l a
(that
Recall
cannot
in T A U T
W-languages
of b o t h
On the o t h e r
logic
A
Clearly,
"cognitively
postulate
equation
of c o m p r e h e n s i v e
identity
that
are
in T A U T w h i c h
are p u r e
~ I ~
p = p
of TAUT.
(p _= q)
They
respect?
(tautologies)
the n a t u r a l
This
p _= p,
tautologies.
p v Ip,
formulas
important.
in this
equationl.;
subset
the
or i n e q u a l i t y
postulate
(3.11
for e x a m p l e
~ ~q).
and
it as the n a t u r a l First,
are
examples:
is i n t e r e s t i n g
truth-functional
is a p r o p e r
to k n o w w h a t
But h o w does logical
contains
TFT
(p ~ q) ~
can be an e q u a t i o n
set of all
set a n d
Obviously,
Y o u m a y be e a g e r not
the
any
tautological
already
differently.
on the
In fact,
inequal-
level
p i,p
of SCI, is in
TAUT. A subtle hive
is t a u t o l o g i c a l l y
TAUT but The
circumstance
not
identity
plenty
(p A p) ~ p, TAUT,
since
symmetric,
in TFT.
However,
connective
of e x a m p l e s
they
is not
of this
(p V q)
follows
~
already
from
that
is,
(p ~ q)
~
are
~ q),
~(p
in TFT.
The
(p ~ q ) ~ (q ~ p)
tautologically
phenomenon.
( ~p
(3.11).
But,
by
(q ~ p)
is not
(n~p)
the
is in
There
(7 p v ~ q) (3.11)
connec-
in TAUT.
commutative.
Equivalencies A q) ~
identity
are
~ p, are
in
corresponding
181
equations
(~I p) ~
p,
are n o t
( n p ~ ~ q)
(P A P) ~ P,
in TAUT.
(P ~ q) ~
O n e has
(Ip
to p o n d e r
=~q),
~ (P A
the n a t u r a l
q)
postulate,
again.
4. M o d e l s . We are tional
going
semantics
applied of
not
to SCI.
formalized
firm part of the
the
referential
You may
languages
of u n i v e r s a l
theory
guess
the
Bloom
actually
class did
t l•c a lly-% b u s i n e s s
insight
into w h a t
portant
of a l g e b r a i c
SCI-language
link
semantics
philosophy,
to do at the b e g i n n i n g
is to
for SCI and
everything the
else
language
provides
is about.
free"
relativized us w i t h
Furthermore,
and e l u c i d a t e s
L.
is m a t h e m a -
as a
It i n v o l v e s
hence,
Stephen
some
some
it n a t u r of its
im-
properties.
binary
set
A
operations
together ~,
U,
with ~
an u n a r y
~ ~
operation
-
is h e r e
called
o
and an
algebra:
A =
(4.1)
The
class
ticular,
of all familiar
o)
join,
correspondingly, a V
(4.1)
Boolean
tion
Here,
where
algebras
-, ~ , U
are
and
",'-~, o >
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 <
one in
b
of t h e m b y ~
as
F
is in
F
iff
a
b
is in
F
iff
both
a,b
a u b
is n o t
in
F
iff
neither
(4.5)
a~b
is n o t
in
F
iff
is in
F
iff
b
is in
a ~. b
(4.7)
Then, pair
a
call of
F
~
o
(neutrally)
and
F,
that
an
F
the
are
are
in
that
two
F
elements
is
of
A) :
F
nor
b
both
a,-b
are
in
both
a~b,
b~a
iff
into
F
a
"f-set
is,
in
divided
suppose
(a,b
is n o t
= i.
A
and
follows
(4.4)
(4.6)
a ~-b
its u n i v e r s e
-a a ~
for
is in
F
F are
in
F
a = b
of
~".
Subsequently,
form the
structure
1481
and call the
it a m o d e l ,
special
(4.6)
force
condition
tain
any
any
f-set also
f-set
of
the Boolean
(4.9)
Thus,
of
algebra
This
is t h e
has
a o b = 1
if
algebras
~
with
~
the c o n d i t i o n s
the
in
~,
force.
Hence,
algebra
with
following
and,
a circle
~,
of
if then
A.
In
(4.2),...,
~.
o
however,
a circle
case precisely
a = b
with
the u n i v e r s e
to b e a n u l t r a f i l t e r
It a l s o m a y h a p p e n ,
algebra
Boolean
algebra
in a B o o l e a n
a t all.
~.
~
remains
operation
filter of a Boolean an f-set
on algebra
case of a Boolean
(4.7)
additional
based
But,
the
is an a r b i t r a r y , ~
may
not
that every
operation,
if t h e c i r c l e
conultra-
actually
is
operation
property:
a o b = 0
operation
if
such
a ~ b.
that
(4.9)
holds
of
183
are the m o s t natural @ebras. (x) models.
examples
A general
lesson
The e x i s t e n c e
operations
the first
cases,
defined. (XII)
models,
is, m o d e l s
F
that
is any u l t r a - f i l t e r
two elements. sma l l e s t
Since
8"
such that Minimal
where
algebra. (XIII)
The
be r e d u c e d
important
logical
to mere
of special
possibly
with
of models
algebra
and,
call
two e l e m e n t
is, however, kind and,
and
exactly
speak about
F = {i}
the w e l l - k n o w n
circumstance
matrices
algebras,
and
Anyway,
contain
we may
it.
are just Henle
is a Henle
Henle models
A = {0,i}
It is e s s e n t i a l l y
so c a l l e d
~
in
on
forget
the class
examples
they all are i s o m o r p h i c
Henle model,
imposed
simple. (xI)
is done;
al-
can appear
One m u s t never
The most natural
F r e g e a n model.
are
theory
algebra
condition
to be r e l a t i v e l y
<~,F> of
not every
models.
into the S C I - m o d e l
is p r e c i s e l y
and will be c a l l e d Henle
is a specific
underlying
in c e r t a i n
step
follows:
of f-sets
in a l g e b r a s
It appears,
of algebras
the it the
Boolean
that all m o d e l s
in general,
one d i s t i n g u i s h e d
cannot
element.
(XIV) Given a model ning of v a r i a b l e s valuations
over
p,q,r, . . . . long to r.
as in
over ~,
If
A
~
the u n i v e r s e
i.e.,
r
The v a l u a t i o n
r
A
of
assignments
is a v a l u a t i o n
and are values
- -
(4.8), we d e s c r i b e ~".
what
is c a l l e d
This
is done by using
of e l e m e n t s
then
or r e f e r e n t s
~(p),
of
[(q),
of v a r i a b l e s
may be seen as e x t e n d e d
over
A
"run-
to v a r i a b l e s
~(r),...
be-
p,q,r,..,
under
the set of all
for-
i
mulas, .
Then,
is,
We simply
whe r e and
that
~ *
as an a s s i g n m e n t set:
~ ( n ~) = -[(~)
represents is,
one of b i n a r y
correspondingly,
clearly,
every
and we can i n t r o d u c e
tion.
We say that
model
~.
~
~
Subsequently,
by
other
~(~%
r if
operations
satisfies
semantical
falls
formula
* ~(B)
A , V ~ ~2, ~=~, n, ~; -~ ~'. , o
[(~)
semantical
~(~)
to every
B) = £(~)
connectives
the basic
~
[(a)
has a r e f e r e n t
the v a l u a t i o n in
and
one of b i n a r y
formula
r (xv)
is s a t i s f i e d
of e l e m e n t s
in
notion
a formula
~
under
of satisfac~
in
~
or,
into the f-set of the
notions
can be easily
de-
184 fined. Given a model true in
~
~,
a formula
if c o r r e s p o n d i n g l y
uation over
M.
~
e
is said to be satisfiable or
is s a t i s f i e d by some or every val-
We also define an e n t a i l m e n t
~M'
d e p e n d i n g on
=
X
F~
iff
~
is s a t i s f i e d by every v a l u a t i o n
satisfies all formulas in
(over
TR(M)
contains p r e c i s e l y all tautologies of
of all formulas true in F~.
However,
the m a i n m e r i t of the class of e n t a i l m e n t relations,
entailment
which
X.
It is easily seen that the set
single models.
~)
this is not
g e n e r a t e d by
The point is that we may use them all to introduce the ~(~) ,
w h i c h is r e f e r e n t i a l l y defined and is i n d e p e n d e n t
of any p a r t i c u l a r model:
(4.10)
X
~(~)~
iff
X
~e
,
for every model
~.
In c o n n e c t i o n with these notions we want to discuss the role of finite models,
i.e., those whose universes are finite.
First, we have
the finite model property:
(4.11)
If a formula is satisfiable in some model then it is satisfiable in some finite model.
It follows that
(4.12)
the set of all formulas true in all finite models equals to the set of all t a u t o l o g i e s of
~)
i.e.
formulas
true in all models.
On the other hand,
the class of all finite models is i n s u f f i c i e n t to
c h a r a c t e r i z e the e n t a i l m e n t
~(r) =.
to finite models is false. (xvI)
In fact, the r e s t r i c t i o n of
(4.10)
185
SO far, we have d e f i n e d two e n t a i l m e n t s are n a t u r a l l y eager to compare them.
However,
~t) ,
~(r)
and you
the s t r a t e g y is here to
first define a third e n t a i l m e n t and only then discuss how all these three e n t a i l m e n t s are related.
5.
S y n t a c t i c entailment.
E n t a i l m e n t s are very o f t e n defined i n d u c t i v e l y as g e n e r a t e d from some set of axioms via certain rules of inference. (xII) this method.
The
MP-rule,
~,
4~/
We also use
~ is the only rule of inference.
Select an a d e q u a t e set TFA (xVIII) of t r u t h - f u n c t i o n a l axioms and collect into the set IDA of axioms for the identity connective,
all for-
mulas of the f o l l o w i n g form:
(5.1) (5.2) (5.3) where
stands for one of b i n a r y T F - c o n n e c t i v e s
h, ~ ;-----~,~
(5.4)
(5.5) Denote the union
TFA L) IDA
define the e n t a i l m e n t (5.6)
X ~ ~
if
from Since are,
~
by
LA = the set of logical axioms and
as follows:
0( is d e r i v a b l e by MP in finite number of steps
X %2 LA. ~
is defined as d e r i v a b i l i t y r e l a t i o n
to some extent,
(by LA and MP) we
told how to use it in the language,
that is, how
to p e r f o r m r e a s o n i n g s i n v o l v i n g formulas of the S C I - l a n g u a g e such that conclusions
follow premises.
ticular d e f i n i t i o n
(5.6)
of
In this we do not feel bound by the parF
It is, obviously,
e x t r e m e l y im-
186
p o r t a n t to prove theorems about
~
The d e c i s i v e fact is that
LA
is d e f i n e d by finite n u m b e r of schemes of formulas and only a finite n u m b e r of rules
(schemes) of inference are used.
However, once
has been d e f i n e d w h a t e s s e n t i a l l y matters here is the e n t a i l m e n t itself,
i.e.,
the logic and,
axioms in LA) and all
c o n s e q u e n t l y all its t a u t o l o g i e s
F " valid rules of inference
those in w h i c h p r e m i s s e s always
~-entail
It is very easy to show that
~
0(i)...
;~,
(besides MP),
i.e.,
conclusions.
has the F i n i t e n e s s and Deduc-
tion p r o p e r t y p r e c i s e l y in the sense of
(5.7)
(besides
(3.8) and
(3.9).
Therefore,
~
Thus, v a l i d schemes of inference c o r r e s p o n d i m p l i c a t i o n a l schemes of formulas. (xIx) inference of t r u t h f u n c t i o n a i
logic
t r a p o s i t i o and so on) are valid.
(one-one)
to t a u t o l o g i c a l
In particular,
all schemes of
(modus ponens, modus tollens,
con-
This is g u a r a n t e e d by the a d e q u a c y
of TFA:
(5.8)
is in TFT iff
~.
is d e r i v a b l e by MP
in
finite number of steps from TFA. (Xx)
Hence, every t r u t h - f u n c t i o n a l We call
t a u t o l o g y is a
(5.5) the special i d e n t i t y axiom.
(5.4) c o n s t i t u t e s the i n v a r i a n c e axioms. cerned with each quently,
~
The group
(5.2),
(5.3),
O b s e r v e that they are con-
(!) formator we have in the language.
(XXI)
Conse-
deals w i t h identity c o n n e c t i v e as genuine identity.
r e p l a c e m e n t laws, cal:
(SCI) tautology.
salva identitate and salva veritate,
Two
are t a u t o l o g i -
187
<s.9>
/- [ (~.-'- 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(<)
is c o n t a i n e d in
o n t o l o g i c a l assumptions i n v o l v e d in universal statements
Therefore,
to the set of its own tautologies,
is an i n v a r i a n t theory.
For each t h e o r e m WZ. WZ
~WH"
is c l a s s i c a l and,
has both the finiteness and d e d u c t i o n properties. reduces,
for-
~
in
WZ
WZ. the
So, we are allowed to say that and u n d e r l y i n g
FW z
are
c o n c e r n e d with the totality of situations and
o p e r a t i o n s on them.
8. Fre@ean logic. Since
WF
is an invarient theory we may use the s u b s t i t u t i o n
rule w i t h i n it.
Hence,
ing theorems of
WF:
formal d e r i v a t i o n s e a s i l y lead to the follow-
(8.1)
(p~q) .~ (p~_q)
(8.2)
(p(~q) ~
(8.3)
(p%~q) ~_. (p~ q)
(p=_ q)
197
We will refer to all four formulas ously as
~F
Sb( ~ F )
a x i o m scheme,
(8.1),
since they are all e q u i v a l e n t
may serve as single axioms for set
(7.3),
WF.
(8.3)
ambigu-
in the sense that they
We call the formula
the o n t o l o g i c a l Fregean a x i o m and respectively,
(8.2),
m( F
(ontological)
and the Fregean
in c o n t r a s t to the s e m a n t i c a l or meta-
theoretical v e r s i o n of Fregean a x i o m stated at the b e g i n n i n g of this paper. Clearly
~F
is neither in TAUT nor in TFT and you may contem-
plate the superficial m e a n i n g s of d i f f e r e n t versions of the F r e g e a n axiom MF
~F"
The e s s e n t i a l point is, however,
is the only one in w h i c h
~F
is true.
that the Fregean model Moreover,
since
WF
is
post-complete
(8.4)
WF = T R ( M F)
The model
MF
is a d e q u a t e not only for the theory
the Fregean entailment,
~wF
=
FM F
It seems to be i n s t r u c t i v e to compare
(8.5)
and
(6.5).
(8.3) of the Fregean axiom identifies the identity
c o n n e c t i v e with the m a t e r i a l e q u i v a l e n c e connective. completely indistinguishable theory,
~
in the Fregean logic.
is t r u t h - f u n c t i o n a l and <-~
tity connective.
Hence they are In every Fregean
has all p r o p e r t i e s of iden-
Therefore, we do not lose a n y t h i n g if we throw one
of them out of the language.
(Which one?)
the language we o b t a i n the usual Fregean language,
but also for
i.e.,
(8.5)
The variant
WF
Anyway,
logic
if we so restrict
(FL) in sentential
that is, the w e l l - k n o w n t r u t h - f u n c t i o n a l
9. Three n o n - F r e g e a n theories.
logic.
198
The three previously invariant
theories
where
stands
Z
defined
announced
theories
b a s e d on s i n g l e a x i oms. B,T
or
H
and,
WB, WT
Thus,
the a x i o m s
and
WH
WZ = T H ( S b ( ~ z ) )
~B'
~T
and
~B
is the c o n j u n c t i o n
of the
following
are
equations:
(9.1)
[(p A
q) q
r] ~
[(q v r) A
(p q r)],
[p V
(9.2)
[(P ~
q) A
r] -
[(q A r) ~
(p A r)],
[p ~ ( q Y 1 q ) ]
(9.3)
(P => q) ~
( -P
This
is a s h o r t a x i o m
with
sentential <~
~ q) ,
system
variables
as r e p r e s e n t i n g
difference
Sb( ~ B )
theory
WB.
First,
commutative,
(P V ~ P ) ~
(q ~
sententia!
we can p r o v e
~q)
constants
~
p,
[36]), w r i t t e n
meet,
~ , A,
join,
V ,
co-
all t h e o r i e s
of the s m a l l e s t equational
laws,
~
Boolean Boolean
De M o r g a n
s i n ce we have
(P ^ n P ) ~
p,
(q~--~2P)].
connectives
the f a m i l i a r
Furthermore,
and
=_
Boolean.
and a b s o r p t i o n
and
A
(see
Accordingly,
r e f l e c t on t h e o r e m s
law.
algebras
of c o m p l e m e n t ,
co-difference.
(q^ nq)]
[(P ~ q )
and truth-functional
distributive
negation
for B o o l e a n
are c a l l e d
Let us b r i e f l y
(P~=~ q) =_
operations
and s y m m e t r i c
containing
double
~H
in turn below.
The f o r m u l a
~,
are
laws:
laws and the
the t h e o r e m s
(q A ~ q )
in two l e g i t i m a t e
we m a y i n t r o d u c e equational
two
defini-
tions:
(9.4)
Then,
1
1
and
situations.
0
--- (p ~ ~ p )
This
~w~ (P ~
-
(pA
~p)
are the u n i t a n d zero of the B o o l e a n fact can a l s o be e x p r e s s e d
w h i c h we define, (9.6)
0
q)
together with
=~2 ( ~ P
~
~q),
[~ ~Q
,
algebra
of
in terms of the o r d e r i n g
as follows:
(P A q ) ~
[ Op
rIp~l~ll~] ~<~ A £]q]
199
(9-7) ~w~__ (P ~--- q ) ~
0 (PL~q),
~
(p ~
q) ~--2 [(p ~ q)
~
p].
w~ If you are t h i n k i n g bra of situations are p r o b a b l y
trying
the c o n c e p t u a l tions.
then
WB
is the theory
I am sure you have m i x e d
to u n d e r s t a n d
framework
I advise
that
the t h e o r e m s
of i n t e n s i o n a l
you simply
to forget
of the B o o l e a n
feelings. of
WB
necessity
it.
First you
according
and strict
You may,
alge-
to
implica-
of course,
read
%%
formulas
of the form
necessarily sional
purify
shadows.
finition, Boolean
exactly
the m e a n i n g
that
of the w o r d
theories,
the
(the s i t u a t i o n
that ~( #
"necessity" formula
that) ~
but you must from any inten-
[7c(
means,
by de-
is the unit of the
of situations.
Obviously, in SCI
the c o n n e c t i v e (and NFL,
that the a m b i g u o u s an e x t e n s i o n a l
as it is n e c e s s a r y
In B o o l e a n
algebra
formator
r1~t
r]
is here
in general)
subject
as is every
and you may rightly
n o t i o n of necessity,
dimension,
extensional,
used
to the
in n a t u r a l
conclude
language, has
laws of B o o l e a n
theories
in
SCI. But, theory you.
even
WB
if you get rid e v e n t u a l l y
as a theory of c e r t a i n
No wonder!
You have
you are e x p e r i e n c i n g
theorems
of
algebra.
WB
This
any model(i)
M iff
Thus,
WB
is also
said
algebra
to another new.
that we deal
in the
seems
logical
Remember
and difference.
agree
following
ghosts
strange country
what
WB
with
semantical
to and
I promised
If you examine
in
the
some
some B o o l e a n
theorem:
for
M = ~A,F~ 1
is a model A
of
WB,
is a B o o l e a n
actually
i.e., algebra
WB
again.
since
with
is w r i t t e n
However,
sentential
is c o n t a i n e d
(with circle
is a theory of c e r t a i n
is a case of analogy, WB
quite
analogy
you must
%
(9.8)
just come
something
you at the beginning:
Boolean
of all i n t e n s i o n a l
Boolean
TR(M)
operation). algebras
some d i f f e r e n c e s variables.
in
and this
must
Indeed,
the
appear formula
200
~ ~
~
is in TAUT.
Hence,
Q~
is h e r e
perfectly
in the t h e o r y Now, junction
of o r d i n a r y
~B
(9.9)
to the and
~ (P
Obviously,
every
We see
logical granted
that
i.e.,
A
algebras.
theory
WT.
q)
---
theorem
of
we m a y
tions. (xXxVIII)
inference
hence
like
that
makes
any
sense
(XXXVII)
The
(p
formula
=_ q) ,
[~p
WT
~P
A aq],
~ T
is a t h e o r e m ~
the
is the
con-
P,
~p.
of
represents
WB
in
algebra
topological
is a s a m p l e
~
Qfl p ~
on the B o o l e a n
t a l k of
Here
!
and
equations:
~--
operator
Therefore,
~ ~
But n o t h i n g
the c o n n e c t i v e
interior WT,
q)
WB.
Boolean
four
(P<~
(9.10)
ly.
,
legitimate.
we p a s s
of
it is in
but
WT
converse-
a kind
of t o p o -
of s i t u a t i o n s .
Boolean
of t h e o r e m s
not
of
algebra
WT
which
Hence,
of
situa-
are not
in
WB:
(p-- q)=- D(p
(9.11)
(0 i) ~
(9.12)
(p-- q) =_ (q --_--p),
(9.13)
(P ~ q) -
Notice
that
tional
axioms
tity
(9.9)
formula
{ and
For
theories (another
are
WB
and
version
(p ~ q) ~
are,
details WT
(q m r ) ]
[~p_=,r1~p]_--~
interdefinable
(9.10)
other
--q),
[(p _= q) ^
[(p m q) -- p],
and
connective. Both
the
~
=i,
in
in fact, see
WT.
(p ~
[(p -
r),
i) ~ (p _---0)]
Therefore,
concerned
[9] and
with
addi-
the
iden-
[37]. (xxxIx)
are n o n - F r e g e a n .
of the F r e g e a n
~
(P<=~q),
axiom)
If we add
to t h e m
201
(9.14)
(p =
we obtain
WF.
[(p m
formula
its single Another the
about a p p a r e n t l y
q) -~ i]
V
is c h a r a c t e r i s t i c
axiom
~H"
equivalent
[ (p m
weaker
q) =_ 0]
for the B o o l e a n
is the c o n j u n c t i o n
version
of
~ H
formula
of
theory ~B
WH.
and
is the c o n j u n c t i o n
In fact,
(9.15). of
~T
and
formula
[(Up)
(9.16)
Thus,
WH
logical
is much
Boolean
any model(l)
It follows WB, creasing
stronger
algebra
W
[(~p)
WT
=- 0]. (XL)
and its c o n t e n t
of situations
is a m o d e l
that
WH
strength. TH(X)
are well-known. WB.
i]
than
of
So,
WH
is also
WT, WH and WF are
form
abou t
~
is a Henle
is that the topo-
algebra.
Indeed,
for
~ = < ~,F>
M
(9.17)
the
(p --- 0)
But, w h a t
(9.15)
This
V
i)
iff
A
a non-Fregean invarient
is a Henle
theory.
theories
All of them are e q u a t i o n a l
where
X
listed
I only discuss
hand,
properties
that
of inis, of
Properties
I do not have much
certain
in order
theories,
is a set of equations.
On the other
algebra.
of
WF
to say here
of theories
WT and
WH. F i r s t of all, this
respect,
both
the t h e o r y
theories WT
WT
and
WH
is the s m a l l e s t
are G - t h e o r i e s G-theory
over
and,
WB.
sequently,
(9.18)
the f o l l o w i n g
three
conditions
are e q u i v a l e n t
for any
T:
in
Con-
202 (a)
T
is an equational
(b)
T
is a G-theory over
(c)
T
is a theory such that for all formulas
(< ~ In general, ample,
~ )
is in
T
every G-theory
no equational
theory over
WT,
WB,
iff
( ~ ( ~ ~)
is in
~, T.
is equational but not conversely.
theory over
WB,
properly
contained
in
For exWT
is a
G-theory. Furthermore, quate for
WT
or
both entailments
one may easily construct uncountable models,
ade-
WH.
and
Hence,
~'WT
both theories
and
~WH
there seems to be a big difference
(9.19)
~WH
On the other hand, models
are concerned.
is uncountable. WT
~ WT
and
WH
I conjecture
and
WH.
infinite adequate models.
seems to be like
which may, however, ~ WT
~WF"
~
as far as adequate
that every model adequate
and
~WH
the Fregean
~WF
logic has.
Quite to the contrary.
,
In general, including
every proper ~WF'
to
~
universe of situations. immediately. Sb(~z )
Suppose,
rather)
where
There is nothing wrong
We simply learn that
(structural)
principles
Clearly,
elementary
concerning
WF
is
B,T,H or F.
is
Fregean
extension of the pure
the structure of the
one may raise a p h i l o s o p h i c a l
we choose an ontological Z
between
then we lose many
is the result of c o n t a m i n a t i n g
logic with some ontological
~WT
belong to the interval between
not the pure logic and that we owe much to the ontological axiom.
for
be omitted here.
If we jump from
valuable properties with that.
WT
However,
This is suggested by several minor differences
Entailments and
have adequate models.
between
has a countable
are q u a s i - c o m p l e t e
principle
~ Z
Are we d i s c o v e r i n g
problem (or, there-
203
by
the real structure of the universe of situations or only making a
convention to please ourselves some way? this question.
But do not hurry to answer
Any a n s w e ~ including that one firmly preferred by my-
sel~ does not matter here.
Another question seems to be primary.
Fregean logic is based on extremely strong ontological principle Fregean axiom) and has well known nice properties. first which properties of a logic principles accepted in
T.
~
T
The (the
So, we shall ask
are induced by ontological
For example, consider the replacement pro-
perty of logically equivalent formulas:
"logically equivalent for-
mulas are interchangable in all contexts salva veritate" as Quine put it in
[12], p. 159.
Well, interchanability is an intrinsic property
of identicals and holds for
~
and any extension of it.
be shared by logical equivalents?
Why must it
Because of the Fregean axiom, the
replacement property of logically equivalent formulas holds in Fregean logic.
It clearly fails for
But consider an entailment Two formulas T
iff
see by WT
~ ~
and
~ T
by the natural postulate where
T
T
is an invariant theory.
is any
~T'
i.e., in
T
T.
Thus, we
(invariant) equational theory over
~ T-logically equivalent formulas are interchanable in all
contexts salva identitate and salva veritate. in
(3.12).
are logically equivalent with respect to
is a tautology of
(9.18) that if
then
~
~
That is, if
~4=~ ~
is
then all formulas of the form
~[Vl~
~)[~I ~ ]
] _
are theorems of every theory over nice properties of
~T
or
T.
f[v/~ ] <=> ~[v/~ ]
This is an example of how some
depend on ontological assumptions in
i0. SCI and modal systems. It came as a surprise to me when I discovered in 1969 that
T. (xLI)
204
(i0.i)
the S C I - t h e o r y consequently,
WT
every modal
is an SCI-theory;
Now,
you are p r e t t y
res p o n s e
is still
our o p i n i o n s
sure that SCI
an e m p h a t i c
logic
is.
of my p o i n t
logical
two-valuedeness
ment
make
interested about.
Formerly,
it.
I thought extension
many
ghosts
be sure w h e t h e r modal
logic
modal
into
~
formulas
weaker modal (10.2)
of modal
modal
I suppose
modal With
and ade-
I can do in and
to show that it
logic
as an entail-
is that m o d a l i s t s
that
but a l m o s t
they are talking
entailment
c o u l d be de-
I see how w r o n g
entailment regret,
never
they are not a c t u a l l y
do not know w h a t
Now,
in
logic.
the trouble
~G.
at all.
systems
I was.
Too
so that one cannot
I no longer
them w i t h
defined
theories
defined
as
is a kind of s y n t a c t i c
and it also systems
e x i s t as well
equiv~lencies
This
All modal SO
The best
my
exactly
for e x t e n s i o n a l i t y
[9] that m o d a l
of
to compare
the strict
~5
in
You know
any p r e c i s e
in particular,
hence,
surround
it exists
and we may a t t e m p t
tions
to argue
But,
consider
as an entailment. (xLIII)
However,
to equate
So,
and,
fined as c e r t a i n intensional
Do you?
However,
logic.
the d i v e r g e n c e
talk m u c h about e n t a i l m e n t
to define
in e n t a i l m e n t
S4 (XLII)
reasoning.
one may try to c o n s i d e r
intensionalists
any effort
is.
of SCI and,
imposed on a language.
and other
S 5 = WH.
you do not have
whims
and,
system containing
by formal
logic
does not fit the i n t e n s i o n a l
S4
Unfortunately,
is to c o n t i n u e
First of all,
system
is a kind of m o d a l
NO.
But,
idea about what modal
defense
Lewis
in particular,
cannot be r e s o l v e d
what non-Fregean quate
equals
leads
to the result
are c o n c e r n e d
systems
we have the
contained
are not SCI-theories.
sets of formulas
of SCI.
idea
is
~(~4=~)
with
translation
of formulas
(i0.i)
above.
following
in the F e y s - v o n Also,
The
e v e r y modal
equa-
As
far as
result.
Wright
system
s y s t e m con-
205
taining
So,
p r o p e r l y c o n t a i n e d in
n e c i s s i t a t i o n rule ticular,
In view of that
S4
(10.2), you may ponder and
and closed under
is not SCI-theory.
In par-
the B r o u w e r s c h e s y s t e m is not SCI-theory.
S5
Thus,
(i0.i), again.
You c e r t a i n l y know
have been i n t r o d u c e d in 1930 by O. Becker and are
not Lewis systems, systems.
(G-rule)
S4
s t r i c t l y speaking.
Lewis p r e f e r r e d w e a k e r modal
s t r o n g e r modal systems are e x a m p l e s of how an inten-
sional mess may, u n c o n s c i o u s l y and rather involuntarily, theories of e x t e n s i o n a l Scott in
logic.
give b i r t h to
And I cannot help w o n d e r i n g w h e t h e r
[i] defends Lewis' p r o g r a m or his own.
There is a single reason for ruling out in systems as non-theories:
(10.2)
so many modal
none of these systems contain
m u l a t e d w i t h strict e q u i v a l e n c e in place of identity. the strict e q u i v a l e n c e c o n n e c t i v e b e h a v e ~ in
S4
s y s t e m ~ as an e x t e n s i o n a l i d e n t i t y connective,
(5.4)
for-
Thus, a l t h o u g h
and stronger modal
in almost all other
m o d a l systems it is not e x t e n s i o n a l with r e s p e c t to itself.
It was
p r o b a b l y i n t e n t i o n a l l y i n t e n d e d to be intensional. Lewis s y s t e m if
S3
S3
may present,
possibly,
an exception.
However,
a c t u a l l y is an i n v a r i a n t B o o l e a n theory then you will find
that the formula little.
~ (p ~ p)
is not in it.
But w h y to b o t h e r about
S3
This may puzzle you a
at all?
Rather forget it since
you may d i r e c t l y study all and every i n v a r i e n t theory in the interval between
WB
and
WT
and, c o r r e s p o n d i n g models.
those t h e o r i e s contain the formula ~ ~
(p ~
p).
Notice that some of These theories can
be easily e x t e n d e d to P o s t - c o m p l e t e theories w h i c h are, however, n e c e s s a r i l y non-Fregean. Alas!
Recall that
~ (p ~
p)
is in
WF.
(XLIV)
one c a n n o t kill the m o d a l logic by s h o o t i n g it w i t h formal
a r g u m e n t s as above.
Probably, modal logic cannot be killed at all.
Fortunately, we do not need any killing.
What we really need is a
complete s e p a r a t i o n of the n o n - F r e g e a n logic from the modal logic.
206
So, we are going to show that certain intensional desires, modal logic,
are flatly denied by
NFL.
typical for
(XLV)
ll. Notes on i n t e n s i o n a l necessity. Belief-sentences may b e l i e v e that Clearly,
~
are p a r a m o u n t to intensional contexts. and not
belief-contexts
b e l i e v e that
do not fit to NFL.
~
although
Someone ~_~ ~ .
You may easily r e c o n s t r u c t
the M o r n i n g - E v e n i n g - S t a r p a r a d o x with b e l i e f - s e n t e n c e s and see that they violate r e p l a c e m e n t laws of equals by equals.
The c o n c l u s i o n is
that there is no functional d e p e n d e n c e of
(ii.i)
(the situation)
that
x
b e l i e v e s that
p
on
(11.2)
How
the p e r s o n
(ii.i) depends on
(11.2)
x
and
(the situation)
that
p.
I do not know.
Similar d i f f i c u l t y is p r o v i d e d by the intensional n o t i o n of necessity.
Furthermore,
you will clearly see that this notion is not
r e p r e s e n t e d in SCI by the square c o n n e c t i v e discuss some p r o p e r t i e s of equations. ges w i t h two kinds of e q u a t i o n s i d e n t i t y c o n n e c t i v e or Suppose that
T
~
But,
first let us
Now, we c o n s i d e r open W - l a n g u a -
q ~ *
where
~
is either the
identity predicate. (xLVI) is a c o n s i s t e n t i n v a r i e n t B o o l e a n theory con-
taining all formulas of the form:
Iii.31 Then,
? the theory
T
?I
contains all formulas of either form:
207
(11.4)
(92 92)
ili.5)
(11.6)
Thus,
granted
T,
we may roughly say that as far as equations are con-
cerned all and only all true equations d e s c r i b e the same situation, vis.,
the unit of the B o o l e a n algebra of situations.
You may see it
also in m o d e l s and e v e r y t h i n g is p e r f e c t l y all right if only you do not listen to the i n t e n s i o n a l ghosts. is s o m e t h i n g wrong w i t h
(11.4), that
T h e y w h i s p e r to you that there (11.4)
could be a c c e p t a b l e only
if r e s t r i c t e d to analytic e q u a t i o n s and you had b e t t e r get rid of NFL and resort to some i m p o s s i b l e world.
But,
I tell you:
do not make
the s l i g h t e s t c o n c e s s i o n to i n t e n s i o n a l shosts if you w a n t to keep an e x t e n s i o n a l position.
R e m e m b e r the d i s t i n c t i o n b e t w e e n reference and
sense and do not confuse what you are speaking about w i t h how you are s p e a k i n g about it.
This kind of c o n f u s i o n is very close to the use-
and m e n t i o n confusion.
An a n a l y t i c sentence and a n o n - a n a l y t i c true
sentence may refer to the same.
There is n o t h i n g w r o n g with this.
You know w h a t formulas of the form
~
mean a n y t h i n g like the formula
is analytic~
<
show that the square c o n n e c t i v e
O
mean in SCI.
They do not
Moreover,
I will
cannot imitate the m e t a t h e o r e t i c
p r e d i c a t e of a n a l y t i c i t y a c c o r d i n g to the following postulate:
~
(11.7)
is true iff
~
is analytic.
W h a t e v e r a n a l y t i c formulas are, they c e r t a i n l y c o n s t i t u t e a consistent i n v a r i a n t theory to
T,
(11.8)
~
~
~
,
T.
Then,
the s y n o n y m i t y r e l a t i o n relative
is defined as
the e q u a t i o n
~
~
is in
T.
208
Since
"~ T
distinct
depends
theories
termines
N
m a y define
the e q u a t i o n a l
of
if and only
additionally
that
is an e q u a t i o n a l
T
theory.
(11.9)
Thus,
we
suppose
postulate
a set
T
T-admissible
(7.2),
Thus,
in this case
to
WT
WH
then c l e a r l y
holds.
However,
not both. such that Again,
WH,
(11.9)
t
) = 1
we
T,
assumptions
on
kernel
suppose
set of a n a l y t i c a l
Hence,
reduces
T
formulas,
are con-
A truth-valuation
t
TAUT,
the c o n d i t i o n
not in
exist
T.
and only
the p r o b l e m w h e t h e r following
as above,
does
the
question.
there exist
a
T.
WT and WH.
(11.7)
t
Suppose, such that
are G-theories.
then (ii.i0)
T = TAUT. (ii.ii).
cannot be realized.
truth-valuations
T = WH
(11.7)
~
~. is in
each of them satisfies if
and,
such that
that both
there e x i s t
T
to the
is no t r u t h - v a l u a t i o n
recall
similar
Hence,
truth-valuations
iff
the p o s t u l a t e
the p o s t u l a t e Quite
~
there
Moreover,
in
T = equational
~. is in
formulas,
three cases:
by
the
for some e q u a t i o n
truth-valuation
t(~
and
whenever
of a n a l y t i c a l
Then,
theory c o n t a i n e d
to truth.
in our discussion.
We c o n s i d e r
What de-
if
can be r e a l i s e d
(ii.ii)
that two
relation.
itself.
i.e.,
that T - a d m i s s i b l e
(11.7)
it follows
Obviously,
is related
t(~i, ~ ) = 1
these will m a t t e r
Giv e n
T,
t(~ ) = 1
(11.10)
T.
The r e m a i n i n g
cerned w i t h how a n a l y c i t y
only
synonymity
is e q u a t i o n a l
the theory
is said T - a d m i s s i b l e
T
equational
kerne__~l of
if
in
the same
is the largest
T
called T
on e q u a t i o n s
either
for every
t
If
Passing
T
is
such that
(11.10)
or
but
or
(11.9)
(ii.ii)
truth-valuation
is s a t i s f i e d
WT
(ii.ii)
but t
is not.
is denied.
reasoning
will
show that the o r d e r i n g
connective
209
cannot
imitate
entailment.
ible t r u t h - v a l u a t i o n
(11.12)
and suppose But,
then
that
t
such
Ask w h e t h e r
) = 1
is
or
WT
the c o n n e c t i v e
a T-admiss-
that
t(~ ~ ~
T
there exists
~
iff
~(
WH.
~
#
You may also
is h a r d l y
take TAUT
to be called
for
an o r d e r i n g
T. con-
nective. Finally,
you m a y v e r i f y
is no i m i t a t i o n
of e i t h e r
Let me c o n c l u d e synonymity valence prov e
here.
Actually,
"~ T"
useful
Rather,
we will
they
are in a o n e - o n e
want
to p o i n t out a p a r t i c u l a r
some they
remarks lead
t h e o r y of NFL.
related
modulo
a role
to the to equi-
T
and
But this p o i n t m a y be
since
theories.
theories
in the d i s c u s s i o n
theories.
relations
with equational of some
relation.
immediately
far from s y n o n y m i t y
property
connective
or s y n o n i m i t y
of e x p r e s s i o n s
correspondence
w h i c h may play
the identity
jump to strong e q u a t i o n a l
we do not d e p a r t very
G-theories
with
senses
in m o d e l
that
equivalence
Obviously,
representing
to be very
omitted
logical
this c h a p t e r
relations
classes
yourself
We
over c e r t a i n
of how NFL and FL
are related. Consider formulas
a theory
of the
mula is in
(W.._---- i)
T = WF
valued
over
and,
relative
(shortly,
WT. (xLVII)
to
conversely. WF
V
(~ ~
T
contains
all
and d e f i n e
ZOV-formula)
0=)
We say that all
relative
T. First,
If
form
(11.13)
then
T
it is easy to see that:
~to
formulas
are
zero-one
to be a z e r o - o n e - v a l u e d T
iff the d i s j u n c t i o n
for(11.13)
210
(11.14)
T
is e q u a t i o n a l
if e v e r y
(or, G-theory)
t h e o r e m of
is a Z O V - f o r m u l a
Subsequently, which
consider
are t h e o r e m s
a G-theory
(ii.15)
Thus,
Fregean
the F r e g e a n
relation
T
largest
kernel
kernel
to
formula
of
of
contained
T = equational
T
is w h a t
T
T.
it the F r e g e a n
G-theory
in
relative kernel
in
determines
of
T.
kernel
to
T
T.
It is
Therefore,
of
T.
the s y n o n y m i t y
,,2 T"
One may also
sider
relative
and call
12. P o s s i b l e
between
i.e.,
the set of all Z O V - f o r m u l a s
of
and is the
T,
if and only
semantics
shift to semantics
extensionality
for example,
world
and SCI.
and try to point
of SCI and i n t e n s i o n a l i t y
possible-world
semantics
out d i f f e r e n c e s
of m o d a l
logic.
for s e n t e n t i a l
Con-
modal
sys-
tems o n l y . ( x L v I I I ) The simple pages
of
[42], w o r k s
some
stronger
this
sense
lar to that sentential Boolean ment. axio m
possible-world
modal
algebra
(10.2).
only.
Boolean
G-theories.
The u n i v e r s e
run over
to SCI
set
21
forces
the r e l a t i o n
for
it only w o r k s
of simple K r i p k e m o d e l s
is the p o w e r
two
theorem
The r e a s o n here
the
considered
in 1971,
on the set
I
in
is simi-
as
as the only d i s t i n g u i s h e d
as shown by the late D. P e n k o v
of SCI
on the first
completeness
If a p p l i e d
and its the unit serves
Moreover, (5.4)
systems
variables
sketched
in the sense of a w e a k
for i n v a r i a n t for
semantics,
ele-
the logical
in Kripke m o d e l s
to be transitive. Certainly,
the i n t e n s i o n a l
is very c o m p l i c a t e d
and d i f f i c u l t
background
of p o s s i b l e - w o r l d
to d e s c r i b e
at least one idea of p o s s i b l e - w o r l d
semantics
and analyse. for m o d a l
semantics
However,
systems
can
211
be d i s t i n c t l y
identified.
It is the r e q u i r e m e n t
tinguished
element
in the model,
Of course,
I am unable
to tell w h a t
for this.
Yet,
it the s e m i - F r e g e a n
I call
seen as the r e s u l t to p l e a s t
only
Validity complicated and also
fail
some
intensional
for the s i g n i f i c a n t
To see this c o n s i d e r These
Boolean
Boolean
designed
theories
modal
algebras
the b i n a r y
circle o p e r a t i o n
as follows:
Ia = a o 1
modal
algebra
A
(12.2)
Ia
Obviously, then
ness
< A,F >
theorem
Boolean
(12.3)
f-set
I(a+
of
is not in
it fits a p a r t i a l
models
=DF
not every m o d a l
A
all
A
systems,
Moreover,
of modal
it is
semantics
are not G-theories.
in the sense of
algebras
[44].
with
an
to SCI define
ultrafilter
F
of a
iff
whenever
contains
of SCI.
a ~ i.
f-sets.
In fact,
algebra where
WB*
formulas
of the
form:
~(~@)
But,
if it does
the class of all
provides
~WB*
(~=_ ~ ) -
seen in the
b)
(Boolean)
is a m o d a l
for the e n t a i l m e n t
theory c o n t a i n i n g
F
algebra
semantics
where
which
To adapt modal
and we see that a is an
accomplished
(0 ~ i) s u p p l e m e n t e d
I.
Then,
[43].
algebras
unary o p e r a t i o n
a o b
axiom,
it may be
for w e a k m o d a l
semantics
additional
(12.1)
since
is c l e a r l y
fact that m e t h o d s
non-trivial
are n o n - t r i v i a l
postulate
algebraic
for those ( i n v a r i a n t )
is r e s p o n s i b l e
postulate,
of the F r e g e a n
semantics,
modal
ghost
algebra.
ghosts.
of the s e m i - F r e g e a n
in L e m m o n ' s
just one dis-
the unit of the B o o l e a n
intensional
of s e m i - a b o l i t i o n
possible-world
responsible
viz.,
to have
the c o m p l e t e -
is the s m a l l e s t
212
Again,
a brief
comparison
for the e x t e n s i o n a l i t y illustration axiom which aboliton
of
(12.1)
is an i n t e n s i o n a l
between affair
that u l t r a f i l t e r s
account
in m o d a l
Hallden
completeness
semantics,
of m o d a l
model).
Although
of m o d a l
logic
systems
algebra
suppose,
we know w e l l
in m o d a l
algebras
a class
Thus,
one may
semantics
completeness
models,
carried
Cresswell's
This
Kripke
algebra
perhaps,
logicians.
possibly
Cresswell
out in c o l l a b o r a t i o n result
set-theoretical
a detailed with
is c o m p l e t e l y constructions
F
a pro-
in
the p o s s i b l e -
algebras
with
procedure
SCI
since
in
also
[46]
it w e a k e n s
logic,
the
the
essentially of C r e s s w e l l
L. B l o o m t hows
misleading
a
in the sense of
analysis
Stephen
But,
correspond,
modified
about NFL as an e x t e n s i o n a l
class.
algebras?
the c o n v e r s e
is very e m b a r r a s i n g
f-
Is it p o s s i b l e
enrich
it fits the w h o l e
results
we may r a i s e
in B o o l e a n
actually
postulate
filters(i)
[44].
on m o d a l
gener-
Certainly,
a complicated
Then,
It
(Fregean
algebraic
models which
based
and,
However,
a genuine
the s e m i - F r e g e a n
(12.2).
with
But the con-
of the class of all
enough,
so that
logic.
remarkable
of how formal,
M.J.
semantics
from modal
annuls
use of f-sets
Surprisingly
thesis
Boolean
into
semanticists.
and,
[43] and M a k i n s o n
to S C I - m o d e l s
by e x p l i c i t
theorem.
methodological distinct
the p r o p e r t y
try to m o d i f y
to be possible.
possible-world
full
are taken
[45]).
and i d e n t i t y
if they exist,
(relational)
sense,
circle operation. seems
form,
the s t r u c t u r e
with
of
[39],
as yet by modal
to that of L e m m o n
an a p p r o p r i a t e
world
(see
lead to interesting,
in the field o c c u p i e d
to define
but
in c o n n e c t i o n
a l i e n to i n t e n s i o n a l
the use of f-sets
it may
algebras
for example,
of the unit of the t w o - e l e m e n t
b l e m similar
of the F r e g e a n
and the e x t e n s i o n a l
of m o d a l
is a p r o p e r b l e n d of t w o - v a l u e d n e s s
sets of a m o d a l
argument
It is also a good
semi-abolition
sometimes;
cept of an f-set is c o m p l e t e l y
even
is a strong
axiom. (xLIx)
It is true
alization
(12.3)
of SCI and its theories.
of the d i f f e r e n c e
of this
and
that
and an e x a m p l e
can s u p p o r t
superficially
213
a d e c e i v i n g philosophy.
13. Cresswell models To simplify the discussion
for SCI.
of Cresswell models,
tion and conjunction will be explicitly m e n t i o n e d connectives
from
the value of texts, tions
Let operations tains
I f
in
to at
pi(f). f
set
21
j
2 I.
(13.1)
j
< 2 I,
I
H (j)
and
j
in
then
I
I
and
and
g
in
2 I,
G2(f,g).
Subsequently,
GO,
G2 >
=
f
f
a unary and two b i n a r y
GI,
21
also con-
the algebra
and the structure
=
g
in
<
iff the following conditions
are
2I:
pj (G0(f,g))
= 1
iff
f = g
pj (G l(f))
= 1
iff
fj = 0
pj(G 2(f,g))
= 1
iff
fj = gj = 1
Cresswell models of
21
f. or, in complex con3 we form the set H (j) of all func-
for all
is said to be a c-model on
(13.3)
in
and
is called a c-algebra on
for all
is in
f. = i. 3 be, correspondingly,
Thus,
G =
satisfied
f
is the set of all
will be denoted as
Gl(f)
°
If
21
such that
GI, G2, G O on
I, the powerset
{0,i~.
For each
G0(f,g),
(13.2)
as truth-functional
of the SCI-language.
Given a non-empty functions
only the nega-
are represented
here as c-models. (LI)
play the role of d i s t i n g u i s h e d
elements
in c-models.
Elements If
h
214
is an h o m o m o r p h i s m say t h a t
h
if
is in
h(~)
fied by
satisfies
h
in
(13.4)
to f i n d
a formula
H (j)
Let
Then,
X ~ ~
iff
in
SCI-language
we h a v e
and e v e r y
for
some
homorphism
in
Sth(C)
then
relationship
be a m o d e l
and
Q
A
of
M.
Define
X
in
Q
(13.5)
Here,
D (a)
Q
two p r o p e r t i e s :
in
is the v a l u e
= 1
of
D
D (a) ~ D (b)
If
Q
is a d m i s s i b l e
Consequently, Gi,
G2,
more,
G0
is in
is a k i n d then
D
the o p e r a t i o n s on the
operations
f-image
a
at
(13.7)
(13.7)
iff
satis-
any if
is c o n t a i n e d
To a n a l y s e SCI-models.
2Q
as
it, w e w a n t First,
Let
of s u b s e t s
to
formulas
I,
SCI-models.
A
(13.1)
Sth(C).
for SCI.
from
as in
~I,j>,
X
to o u r
then w e
of the
we
M = ~A,F> universe
follows.
For
every
A,
F
condition
on
is in
G
theorem:
set
set
family
from
(13.6)
The
<
theorem
P x ( D (a))
has
also
D
C
the
C
set of all
non-empty
h
c-models
the m a p a
be the
c-model
be a n o n - e m p t y
and every
in the c - m o d e l
of c - m o d e l s
certain
to the c - a l g e b r a
the C r e s s w e l l
for e v e r y
I
to o b t a i n
~
Sth(C)
is a c o m p l e t e n e s s
some
show how
the
C.
j
This
from
of
G I, G 2, G O
a.
Q
is in
Then
is o n e - o n e
A
in
Q
admissible
if
a ~ b.
of s e p a r a t i o n
~,
call
and,
whenever
~,
X.
o 2 Q,
property
and
embeds
of
A
A
define
denoted
m a y be e x t e n d e d
imposed
as
into
on
Q.
2 Q.
operations D(A).
arbitrarily(l)
Furtherover
the
215
whole f,g
powerset in
2Q
except
t h a t we
PF(Gl(f)) PF(G2(f,g)) PF(G0(f,g))
Finally,
let
H (F)
b e the
It is i m m e d i a t e l y
(13.9)
Q.
on
universe
The
model
but
semantical
A)
D(A) C(M).
ing b a s i c
then and, We
over put hence,
say t h a t
fF = gF = 1
= 1
iff
f = g
the
the Q
of
M
for
all
the
"C(M)"
same
2Q
to the
and,
is i n d u c e d
gF = 1.
C(M) of
of
original
A.
A.
But,
M.
C(M).
a c-
In fact,
sends
is a h o m o r p h i s m
how-
Hence,
SCI-language ~
In
a submodel
It is n o t
model
by
The m a p
r
as
contains
C(M).
of the
admissible
a great
cardinality
inherited
by
the
presents
subsets
c-model
are
that
neglects
sets
the k e r n e l
M
such
G I, G 2 H (F) >
= D (r ( ~ ))
into
2Q
structure
(homomorphism
~(~)
in
set of all
isomorphic
~
g
of a d m i s s i b l e
called
St?(C(M))
Str(M)
is the
A c-model,
set of
formulas
to the a l g e b r a
and we h a v e
the
follow-
obtained
as
= Str(M)
formulas C(M)
satisfied
f r o m some
by
r
SCI-model
if
to the
equality:
(13.10)
where
iff
m a y be of the
properties
is a v a l u a t i o n
algebra
Q
D(A),
an S C I - m o d e l ,
= 1
notation
Q = P(A),
is
fF = 0
= ~ 2 Q, GO,
an a d m i s s i b l e
whose
iff
that
the c l a s s
case,
ever we chose
into
.
For example,
the e x t r e m e
of
Q,F
Obviously,
variety.
r
following:
= 1
set of all
seen
C(M)
is a c - m o d e l
some
the
2Q
(13.8)
set
require
in
M. M
will
be
216
called
special. (LII)
hence,
one may hope
These
c-models
that they suffice
SCI.
Indeed,
shows
that the class of special
tinuum if non plete
a close
at most)
then,
theory
Z
isfies build
all
models
because
ized w a s h e r
c-models
sees
definite,
By
is a subset of
TR(M)
Compare,
in particular, poss i b l e
(13.10),
worlds
many q u i t e
that y o u c a n n o t
expect
SCI-theories.
fruitful. a golden
Thus,
of c o m p l e t e n e s s
we go in a s e m a n a t i c s a logical
calculus
beyond(l)
and prove
the
Now we may
holds
same property. based
on our
for special
superstructrLres
in
C,
c-
i.e., M
relation M
the set
does if
Other let by all
TR(C(M))
not hold,
M
in
is finite
clearly
see that The
between
in
lesson
c-models
or,
is and
does not seem to be very theorem
just the b e g i n n i n g
theorems
C,
satisfied
things happen.
relationship
ad hoc
a ten foot c o m p u t e r -
a c-model
SCI-model
with
built
I do not know.
Given
semantics
but
~
(13.4)
we see that a c o m p l e t e n e s s
crown on a semantics
the r e l a t i v i t y
true
impossible
The c-model
Then,
it has the
and you will
any smooth
in it.
handkerchief?
C(M)
model
a com-
ask w h y b u i l d
but the c o n v e r s e
the F r e g e a n
there exists
job.
for every
for example,
of conIn fact,
is i m p l i c i t l y
however.
formulas
theorem
respect
(13.10),
theorem
seriously
of
for SCI.
satisfy
(13.4)
do the
and,
to Z has the one the c a n o n i c a l over M z sat-
are s o p h i s t i c a t e d
be the set of all
general.
The
just a single
homomorphism, s
with
viz.,
that
in s e m a n t i c s
p r o o f of his
theorem
is not
Mz
In view of
One may
are more
~
as kernels
(of the c a r d i n a l i t y
of SCI,
but does not
their kernels
to w a s h
observations
that
model
SCI-models.
over SCI-modelso
their
X
C(Mz).
using
Special
TR(C)
in
So, one clearly
semantics
c-models
some valuation,
formulas
a c-model
(LIII)
such
(quotient)
property:
of the C r e s s w e l l
the c o m p l e t e n e s s
X
SCI-models
to some e x t e n t
by the r e g u l a r i t y
over
Lindenbaum-Tarski desired
inspection
provides
X ~ ~
involve
forces
for a logic of it.
Already
us to ask how
the c o m p l e t e n e s s
theorem.
a kind of c o m p l e t e n e s s
theorem
is not
far can
To c o n s t r u c t for it
217
(not to speak about w e a k wasting
theorem)
appears
very o f t e n
semantics
it uses only p o w e r s e t s
21
no p o s s i b l e - w o r l d s argument
worlds
semantics
cannot
models
as
(6.4)
or
an even more
as u n i v e r s e s
provide
acute
the s e n t e n t i a l
can be c o u n t a b l y
against
(9.19)
not to be a deep m i n d e d
presents
model
as the s t r o n g e s t
infinite.
any t h e o r e m
because
about
philosopher
to k n o w
at
finiteness
and infinity.
the p o s s i b l e - w o r l d
appears
to be a real p r o b l e m k i l l e r questions
concerned
with
since
The
fight w i t h
may never end.
But,
some u n e l a b o r a t e d (i) one b a d l y
Modal
intensional I must
logicians
done at that.
a x i o m and,
well.
But,
Fregean
logic
(2) which
logic.
so?
But,
suggests
and NFL.
Secondly,
and
models.
is boring,
unprofitable
So,
I conclude
that NFL is a kind of m o d a l
Thus,
This
and, with
FL
intensional
ghosts
or the
from the devil.
logic
story.
and you will
this
one.
certainly
quite
you must
in the non-
meet
a fact
connective
NFL into FL as shown
is s o m e t h i n g
this
find there many
get rid of the i d e n t i t y
to t r a n s l a t e
I know
Indeed,
and d e e p e r
to the F r e g e a n
then you w i l l
and
consider
you m u s t dig d e e p e r
that we may e a s i l y
Now,
either
logic
logicians
is a longer
of m o d e r n
if you do that
again.
minded
else comes
its r e l a t i o n s h i p
It is s u f f i c i e n t
by Cresswell,
think
everything
w h y is this
facts.
semantics
not a l l o w one to ask
infinite
this paper.
Extensionally
look at the h i s t o r y
relevant
as that
remarks.
ghosts
finish
important
be-
remarks.
as the only e x t e n s i o n a l Fregean
it does
countable
14. Final
One need
that the d i f f e r e n c e
between
So,
infinite
all.
is as n e a r l y
run
Possible-
countable
they do not exist
Since
I see this
this kind of semantics.
and u n c o u n t a b i l i t y
important
case.
variables
tween c o u n t a b i l i t y
first
just
time.
Possible-world
over,
completeness
different
in
[49]
from
218
diffuse seem.
i n t e n s i o n a l ghosts.
But things n e v e r are so s i m p l e as they
You k n o w that t r a n s l a t i o n s may be b e t t e r or w o r s e and,
t r a n s l a t i o n s m a y not be t r a n s l a t i o n s translations
is w h a t are their
of C r e s s w e l l ' s
at all.
invariants.
translation you will
connective
at any price.
But,
W h a t always m a t t e r s
the real p a r a d o x is that it is imposat all.
is to e q u a t e
Also,
of s i t u a t i o n s rather,
equivalence.
u n l e s s one a g r e e s
[50].
It r e s e m b l e s ,
universals
a theory
in a sense,
and some ad hoc
w h a t Quine c a l l e d r e i f i c a t i o n of
R e i f i c a t i o n of s i t u a t i o n s
in some W - l a n g u a g e ,
being objects
is p e r f o r m a b l e w i t h i n
b a s e d on n o n - F r e g e a n ~WB
w h a t we are used to t a l k i n g about
this
"translation"
I b u i l t a t h e o r y of r e i f i c a t i o n of s i t u a t i o n s
If b a s e d on a logic as s t r o n g as
logic,
t h e o r y is the b a s i s
of course.
it a u t o m a t i c a l l y
in p r o b a b i l i t y theory,
c o n s t i t u t i n g a B o o l e a n a l g e b r a in o r d i n a r y
leads to viz.,
(4)
Non-Fregean
situations described w h i c h obtain.
logic c o n t a i n s
to be an a b s o l u t e m o n i s t necessary
fact.
But,
type. equa-
naturally.
the exact t h e o r y of facts,
s e n t e n c e s or,
If one a c c e p t s
Fin-
for a t r a n s l a t i o n of C r e s s w e l l ' s
e x c e p t i o n a l l y well,
in true
events
sense.
t r a n s l a t i o n does not p r e s e r v e e n t a i l m e n t p e r f e c t l y .
tions are t r a n s l a t a b l e
Also,
or,
and also the r e l a t i o n s h i p b e t w e e n F r e g e a n F U N K T I O N E N and
their W E R T V E R L A U F E .
This
one c a n n o t get rid
that t h o u g h t is a b o u t nothing,
S t i m u l a t e d by C r e s s w e l l ' s
c o n s t r u c t i o n by S l u p e c k i
ally,
W h a t can be done
stops t a l k i n g w i t h sentences.
(3)
in
It m a y be con-
to get rid of i d e n t i t y
sible to get rid of the i d e n t i t y c o n n e c t i v e it w i t h m a t e r i a l
about
If you ask for i n v a r i a n t s
find only a few.
s i d e r e d as a t r a n s l a t i o n o n l y if one w a n t s
the b a d
in o t h e r words,
i.e.,
situations
the F r e g e a n a x i o m then one is c o m p e l l e d
in the sense that there e x i s t s o n l y one and
So, y o u see that F r e g e a n a x i o m s h o u l d be a b o l i s h e d .
o n l y the n o n - F r e g e a n
logic allows us to r e p e a t w i t h
standing Wittgenstein's
thesis
facts
Indeed,
and not objects.
that the real w o r l d what
full u n d e r -
is a t o t a l i t y of
I t r i e d to do in
[51] was a formal
219
reconstruction
of the o n t o l o g y
of TRACTATUS,
p r e t e d by Dr.
B. Wolniewicz. (LIV)
idea of w h i c h
logic OT has
suits
At that time,
to be b a s e d on.
OT and is the g e n e r a l
OT for short,
and w e a k e s t
as inter-
I only had a general
Now,
I may say that NFL
extensional
and t w o - v a l u e d
logic. (5)
For many years
poin t of view structed, the sole
and,
is r e s p o n s i b l e
presented inventor
connective
except ~os,
and investigated.
seen
from
have been
suggested
according
to w h i c h
the
as t a u t o l o g i e s logics
borators,
know NFL as I know
Stephen
formulas of SCI.
(6)
be c o n s i d e r e d
tential well
tendency axiom.
is not all.
as models
variables
of the F r e g e a n
and i d e n t i t y
Thus,
~,p,
logicians
This p ~
is
(p
~
p)
a methodology systems
to do.
Yet,
my friendly
young
preI colla-
It is e s s e n t i a l l y
as a general
due
logic w i t h
not d i s c r e d i t which
Be it right or wrong, than m o d e r n
logic.
This may not be easy.
as c o n s i s t i n g
again,
of a single
makes
do.
This
as in the Cress-
it hints
in a sense, is c e r t a i n l y
So it s e r i o u s l y
calls
fact?
But,
Of
to some
by the F r e g e a n very
strong
for an ex-
Why do we tend to d e s c r i b e
necessary
may
sentential
is the point.
NFL at all but
t e n de n c y
which
we may get rid of sen-
similarly
is codified, this
are s t r u c t u r e s
logic w h i c h
connective
We m a y and we r e a l l y
in our t h i n k i n g
planation.
(1957)
all those
ghosts.
NFL as I prefer
SCI-models
superflous.
fact does
and m u c h o l d e r
world
this
trivially
this
This
of formal
L. B l o o m and D o n a l d J. Brown.
translation.
course,
in particular.
good properties.
Yet,
variables
to be
Moreover,
it now w i t h o u t
is con-
pretend
But,
p ~
to them that today N F L m a y be seen d i s t i n c t l y exceptionally
I cannot
intensional
This
logic
H. G r e n i e w s k i
are sets of t h e o r e m s
them to see and i n v e s t i g a t e never
(1948),
(1960s).
more or less,
[53] w h e r e
the n o n - F r e g e a n
connective,
J. ~os
and M. J. C r e s s w e l l
worshipped,
logic as an entailment.
However,
of NFL and i d e n t i t y
evidently
would
viewed
for the w a y
has a long history:
A. N. Prior
vented
I have
anyway,
the I feel
220
that when
following
suggested
b y the
Moreover,
one may
SCI-models but
this
fact that say even
variables
jects.
One
scribed
by a sentence.
Hence,
should
we
Stevens August
use
however,
Institute 1972.
Elements
with
should
I am not
sure
of Technology
is n o t
is a l s o so good.
semantics
run over
b y a name.
any
it a n d
if w e a r e
totality
of o b -
of non-Fregean
entailment
logic
I hope we will able
uses
are objects;
a new kind of language this new
which
It c a n o n l y b e d e -
semantics
non-Fregean
to you how good
try to use
This
of these models
an a d e q u a t e
a W-language
something.
translation
actually
a situation
Thus,
shown
lose
the c o n v e n t i o n a l
task of using(!)
I have
I think we
Frankly,
that
do not
denote
face the
entailment. fore,
cannot
we
Cresswell's
is n o t a d e q u a t e .
sentential
theories
tendency
with
is. be
to d o that.
in it. a new
Theresuccessful.
221
S u p ~ l e m e n I.
The F r e g e a n
scheme
b u t also to f o r m a t o r s tors,
predicates,
Accordingly,
(*)
of d i v e r s e
quantifiers
To be sure,
is e i t h e r
neither
be d i s c u s s e d
Fregean
here.
models there
(section is some
II. logic, then,
one has
sentences
and sional forms
quite
~
are of equal
that this distinct
formator).
or a function.
will
the s y n t a x of W - l a n g u a g e s
(sec-
[13])
(set-theoretical) clearly
the area of m a n y - v a l u e d
to have more
than
two values.
is the r e f e r e n t i a l
from logical
scheme
suggests
many-valuedeness.
shows
previously.
by the idea of the e t e r n a l
tainly here
to all referents.
find y o u r s e l f
in some
that one can prove
in a language
with
in p a r t i c u l a r ) ,
(1.3),
identity
granted
formator
independent
world.
conis, not
Other-
r e a l m of
of any l a n g u a g e y o u w o u l d cer-
It m u s t be e m p h a s i z e d
a two-valued ~ .
~
is constructed.
Subsequently,
impossible
scheme
You are a d v i s e d
wise,
prior
the logic
of exten-
The a s s i g n m e n t
before
or logic and,
The
that the a s s i g n m e n t s
that the F r e g e a n
t h e o r y of truth.
one as m e n t i o n e d
(and p r o p o s i t i o n s ,
But
(ontological)
to jump to c o n c l u s i o n s
senses
that
(*).
as far as the c o n s t r u c t i o n
This
y o u m i g h t be s t a g g e r e d
suggest
my c o m p r e h e n s i o n .
status
is concerned.
a secondary
or an o b j e c t
[ii],
of the F r e g e a n
to the c o r r e s p o n d e n c e
however,
in
are allowed
beyong
Elaboration
logic
truth
func-
nor their c o u r s e s - o f - v a l u e s
both
and
connectives,
d i c t u m s h o u l d read:
that we have e n t e r e d
to r e a l i z e
is, probably,
III.
functions
and sentences
(description
of their c o n v e n t i o n a l
significant
many-valuedeness, latter
a situation
to names e.g.,
ontological
4 of this paper
One may think
since
categories,
Nevertheless,
2) and the s t r u c t u r e
not only
and u n i f i e r
the famous F r e g e a n
everything
tion
applies
t s
extensional
One m a y d e f i n e
logic
a kind of
222
synonymity relation T:
if both
equation guage,
~,
~, ~
~
r e l a t i v i z e d to a given
are terms or formulas then
~
is in
T.
Then,
~
(consistent)
~ ~
~
Moreover,
g r e a t e s t c o n g r u e n c e in the class of all c o n g r u e n c e s
ulas and,
~
is in
T
a b s t r a c t i o n class of
then ~
~
if
~
~
also is in
modulo
m e a n s that the
is a congruence of the lan-
c o n s i d e r e d as free a l g e b r a i c structure.
guage w h i c h satisfy the condition:
theory
r-~T. Then,
~
~'J T
of the lan-
where ~, ~ T.
Set
is the
are form-
S T ( ~ ) = the
T-senses of terms and
formulas c o n s t i t u t e an a l g e b r a i c structure, similar to the language the and known as L i n d e n b a u m - T a r s k i c a n o n i c a l structure (the q u o t i e n t of the language)
with respect to
T.
defines a reference a s s i g n m e n t a c o m p l e t e theory rM(~ ) = rM(~)
TM
over
On the other hand,
rM
T;
and, each model
any model M
T M = the theory of
of
M.
T
M induces
Then,
iff
~ r-~ ~ , i.e., (~ ~) is in T M. Moreover, TM we can find a m o r p h i s m f from the structure of T-senses onto the structure of M-referents, general: IV.
that is, into the model
f ( s T ( ~ )) = r M ( ~ ) . In view of this,
Compare
M
such that in
[6], p. 21.
the basic laws of e n t a i l m e n t f o r m u l a t e d by
Scott must be slightly m o d i f i e d and the reader can easily do that. Scott only c o n s i d e r s a n t e c e d e n t s of
~
finite e n t a i l m e n t r e l a t i o n s and,
are s u p p o s e d to be finite sets of formulas.
the other hand, he allows the succedents of finite sets of formulas. of g r e a t e s t interest.
Of course,
~
follows G. G e n t z e n closely. ful o n l y to prove, Moreover,
Also,
The c o n d i t i o n a l a s s e r d o n
of Scott involves m u l t i p l e sequents
On
to be a r b i t r a r y
We drop the finiteness c o n d i t i o n since,
is not very deep.
his
finite e n t a i l m e n t relations are
case of NFL, we w i s h to o b t a i n it as a corollary. difference
therefore,
in the
the second relation
(with many succedents)
This a p p r o a c h seems, however,
since he to be use-
if true, the so called cut e l i m i n a t i o n theorem.
if Scott's e n t a i l m e n t r e l a t i o n deals with a genuine dis-
223
junction the m u l t i p l e sequents reduce to simple sequents one succedent);
compare the rule
(V)
in
[i], p. 798.
(with exactly For the theory
of e n t a i l m e n t relations adopted here and its h i s t o r i c a l aspects the reader may c o n s u l t
[7] and
[8].
Entailment relations
be r e p l a c e d by the c o r r e s p o n d i n g c o n s e q u e n c e o p e r a t i o n formulas.
It may happen,
such that
VI.
~ ~ ~
on sets of
Cn(X)
iff
X ~
c u r i o u s l y enough,
for every formula
~
~
that
there exist formulas .
Notice also that dis-
e n t a i l m e n t s may have a common set of tautologies(!).
L a n g u a g e s of kind
v a r i a b l e s and terms
VII.
Cn
for some p a t h o l o g i c a l e n t a i l m e n t
has no t a u t o l o g i e s but,
tinct(!)
can always
They are c o n n e c t e d as follows:
is in
V.
~
Equations
W
may involve even m a n y sorts of nominal
(names).
~
~
b e l o n g to W - l a n g u a g e s under discussion,
which is carried on in the meta-language.
The m e t a - l a n g u a g e also in-
volves the notion of identity and equations w h i c h we render rather differently.
There, we use the c o n v e n t i o n a l
d i s t i n c t i o n between
~
and
=
symbol of equality.
only e m p h a s i z e s
The
the d i s t i n c t i o n be-
tween language and m e t a - l a n g u a g e .
VIII.
The c o l l e c t i o n of all t r u t h - v a l u a t i o n s
A p p l y a c o m p a c t n e s s a r g u m e n t as in IX. unifier)
X. [18] and
is a Boolean space.
[8].
If the language involves b o u n d v a r i a b l e s
(quantifiers and
then the natural p o s t u l a t e should be s l i g h t l y modified.
Historical motivation [19], p. 492.
for this t e r m i n o l o g y may be found in
Henle algebras may be seen as Boolean algebras
with an a d d i t i o n a l unary o p e r a t i o n
I
such that
224
(*)
Ia = 1
Conditions
if
(*) a n d
(**)
and,
la = 0
if
are
equivalent
if w e
agree
(4.9)
a o b = I ( a -~
Henle
algebras
class
TBA of topological
supplemented
are
with
and
algebraically
simple
Boolean
(compare circle
[20]).
When
operation
[9], pp.
algebras,
operation
I
definability
(**)
quasi-topological arises
Xl. nected
in t h e
~
defining
sense
of
If w e p a s s
to Q T B A or,
a o b
(4.7.1)
(a o b) u
operation
sets becomes
[22],
more
o)
F
Boolean
= Ia A
TBA one may
algebras
Ib
are mutually shown
also
(Lemmonls
the
in
[21]
i.e.,
exist
in
~
in
iff
a binary by
(**) ;
that mutual
class
extension
last equation
use
definable
for the m u c h b r o a d e r
f-sets
moreover,
QTBA
algebras
of [23]).
(***).
~
is w e l l - c o n -
equivalently,
= 1
then
arbitrary
the condition
(*)
a o b = 1
Boolean
o r c o d = i.
algebras
for t h e e x i s t e n c e
(with of
f-
intricate.
The condition
is in
is,
in t h e
that:
b)
It h a s b e e n
(c o d)
then
class
They
algebras
in TBA;
if
XII.
I.
[21].
Boolean
(*)
circle
of
the
is a l s o v a l i d
be
an e x t r e m e
IIa = Ia
f r o m T B A if w e d r o p
Let
that
such
a
and,
that:
and present
I(a ~
instead
23-24
a ~ i.
Ia = a o i.
I1 = 1 Ia ~
QTBA
b)
interior
(***)
see
a = 1
and,
(4.7)
replace
~
c a n be r e l a x e d . (4.7)
Write
by:
is a c o n g r u e n c e
of
=A ,
a = b F
whenever
225
(4.7.2)
Then,
if
a = b F
(4.7)
contracted
deserve
to n o r m a l
of a g e n e r a l i z e d
~*
respect
one:
model
terms,
known
models,
as Boolean
XIV.
This
con t a i n s
strange.
simply
~
and
tion
[
over ~
algebra
can be
The c o n t r a c t i o n
the e x a c t
that
one may set
~,
everything
generalized
that if put
interpreta-
the great
logic,
for
models
They
General-
in structures
FL) ; see
(normal)
in w h i c h
models.
is Boolean. occur
[20] and
model
[24].
the f-set
< ~,F >,
finite
Then obviously, A - F
also has
of this kind are i n t u i t i v e l y
diversity
of models
reveals
(and NFL).
even more. r(~ ) = * (relevant)
Sometimes,
algebra
exist models
models
the o p e r a t i o n
Fregean
the c o m p l e m e n t
of the SCI
r
model
one element.
unless
*
[.
de-
the same p r o p e r t i e s
It also means,
algebras,
There
To be sure,
But note
under
models
is c h a r a c t e r i s t i c
in e ve r y
has e x a c t l y
One may e x t e n d
ing
Models
This means
them g e n e r a l i z e d
(for the F r e g e a n
the real w e a k n e s s
XV.
ent in
call
b a s e d on B o o l e a n
F
one element.
M.
in the F r e g e a n
is not to say that
cannot be a B o o l e a n exactly
to
F.
is the c o n t r a c t i o n
have
provide
if the u n d e r l y i n g
models
where
~*
structures.
This p r o p e r t y
at least two elements.
or infinite,
very
seen that
and,
large even
ised F r e g e a n
~*
One may also c o n s i d e r
are equal
can be very
models.
truth and entailment.
algebraic
is in
connective.
It is e a s i l y
-~
and
b
Generalized
If
is i s o m o r p h i c
---[ are the same.
and
~
~.
that normal m o d e l s
tion of the i d e n t i t y
o
then
~
in semantical
the F r e g e a n model.
then also
class of g e n e r a l i z e d
to satisfaction,
are simple
and
F
factor by
~
of any normal m o d e l
XIII.
is in
to be called normal.
normal models
o
a
we get a m u c h b r o a d e r
fined by
with
and
Using
the c o n v e n t i o n
and see that given in the l a n g u a g e
it appears
convenient
concerna valua-
has a refer-
to c o n s i d e r
226
pairs
< _~, r >
where
_~
is a model and
They are also called models,
sometimes.
r
a v a l u a t i o n over
_~.
In fact, we were c o n c e r n e d
in III with this kind of model.
XVI.
C o n s i d e r the formulas
(Pl ~
where
Pi' p=3
P2 ) A ...
~n
A (Pi ~
Pj)A--.
are v a r i a b l e s and
s a t i s f i a b l e in
M
for
n = 1,2,...
A (Pn ~
1 z _ i ~ j <_ n+l.
iff the u n i v e r s e of
M
contains
:
Pn+l )
Clearly,
~n
at least
n+l
is
elements.
XVII.
XVIII.
(21
In principle,
every e n t a i l m e n t can be defined that way,
For example,
TFA = the set of all formulas of the form:
(". =-> ( 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 <
WT
or
you have
semantical
of
Yl
y
A
...
a case
is a
(9.1),
(9.2)
as
and
(9.3).
( y ~ z) ~ (x ~ z ) ,
-x~ y
if
and
for
x ~
(x I o yl ) ~ ... or
Xm~
( x ~ y)
y m
and,
this
infinite
(well-connected)
add
...) :
v(x m o y m )
axiom
set w i t h
again.
recall
~A,F 2
1
Ym"
of d i f f e r e n c e ,
M =
x~ ya
(n,m = 1,2,
of a n a l o g y ,
theorem:
iff
~
stand
compare
a case
with
m a y be a x i o m a t i z e d
to
of i m p l i c a t i o n s
If y o u
To see h e r e
z
x~ y ~
let
(ultrafilters
predicate:
(z I o Zl) n ... ~ (z n o z n) ~
then
Finally,
x ~ (yu - y ) ~
(ordinary)Boolean
f-sets
algebras
corresponding
( x ~ z),
x,
those
involve
of t h e s e
identity
(yn -y) m
the
which
axioms
with
with
XI a n d n o t e
is a m o d e l topological
the
then
M
is
Boolean
algebra.
XXXIX. following law
formal
[(p^ q) ~
[(p ~q) the
To p o i n t
~
second
p]
~
facts
p]<~
the d i f f e r e n c e
may prove
[(p~ q) ~
[(pv q) ~
formula
is a c e r t a i n
out
in
ambiguity
q]
(9.7) in
to be of
q]
is in
is in and WB
between
the with
WT
some WB.
but
first
and
interest. However,
not
one
respect
WB
in
in
WT, The
the Boolean
the e q u a t i o n
WB.
(9.13).
Compare Thus,
to the o r d e r i n g
also there
of the
231
Boolean a l g e b r a of situations. be the set of all e q u a t i o n s equivalence
g~=~ ~
is in
On the other hand,
~ ~ ~ TFT
let
A
(or,
B)
of the S C I - l a n g u a g e such that the
(or, TAUT).
Then
WB = TH(A)
and
WT = TH (B) .
XL.
Instead of
(9.16), one may also use
one of the following formulas: [(p=_.q)~-O)],
XLI.
(p--Up)~
(pro O p ) < ~
[(p~l)
As y o u might expect,
regular syntactic p r o p e r t i e s
~
(together with
(~ p ~
[I - p ) ,
(P~q)
T ) any --
(p--O)].
e q u a t i o n a l d e f i n i t i o n s have all the
(translatability,
n o n - c r e a t i v i t y and eli-
minability)
with r e s p e c t to the a b s o l u t e n o n - F r e g e a n e n t a i l m e n t in any
W-language.
But given a W - l a n g u a g e w i t h q u a n t i f i e r s and e q u a t i o n s of
both kind,
one may ask for p r o p e r t i e s of so called standard d e f i n i t i o n s
of the form
u ~
where
~
~
(v I ..... Vn) <~-~ ~ [v I ..... Vn,U]
is a new formator and we know that for all
there exists exactly one
u
such that
prove that standard d e f i n i t i o n s tions in models.
~[Vl,...,Vn,U].
Vl,...,v n It is easy to
a c t u a l l y define s e m a n t i c a l l y new func-
However, we cannot prove that s t a n d a r d d e f i n i t i o n s
have all the regular s y n t a c t i c p r o p e r t i e s unless we s u p p l e m e n t the e n t a i l m e n t w i t h suitable o n t o l o g i c a l principles. p r o b l e m of d e f i n i t i o n s
in n o n - F r e g e a n
in c o l l a b o r a t i o n w i t h Mr. M. Omyla, University.
As yet, we know,
standard definitions
fiers
~
and
~
logic are being now w o r k e d out
a d o c t o r a l candidate at W a r s a w
for example,
that a nice theory of
requires Boolean a s s u m p t i o n s
also a Q - p r i n c i p l e w h i c h states,
The details of the
r o u g h l y speaking,
(like in WB) and that the quanti-
are signs of g e n e r a l i z e d meets and joins in the
Boolean algebra of situations.
232
XLII.
For the proof,
model of
WT
filter of
whenever
A.
A
o b s e r v e first that is any a l g e b r a in
S4
tion with this, that M c K i n s e y found in h y p e r c o n t i n u u m power,
a d e q u a t e for
(countable or uncountable)
XLIII. compare
But suppose, ~m
TBA
is a g e n e r a l i z e d
and
F
is any ultra-
Use then the c o n t r a c t i o n o p e r a t i o n and the w e l l - k n o w n
M c K i n s e y - T a r s k i c h a r a c t e r i s a t i o n of
WH
with
~
~m
in
[38].
Notice,
in connec-
[39] a g e n e r a l i s e d model of
WT.
Infinite models adequate for
are due to Scroggs
[40].
is the m o d a l entailment.
If you w a n t to
or some e x t e n s i o n of it you have to use a n o t h e r
t e c h n i q u e of R. Wojcicki,
again.
His note[41]
will also show you how
little i n f o r m a t i o n on r e l a t i o n of e n t a i l m e n t s is p r o v i d e d by translation maps b e t w e e n c o r r e s p o n d i n g sets of tautologies.
XLIV.
There is an i n v a r i a n t e q u a t i o n a l B o o l e a n theory, p r o p e r l y
c o n t a i n e d in
WT
and very close to F e y s - v o n W r i g h t s y s t e m
WT* = TH(Sb( ~ ~))
where
place of
~p.
~
~ p m
Opc---~Qp
and
obviously.
~ WH*
is like
(p ~ q)~--~ ~ ( p ~ q).
S
o
WT*
and
S
o
is c o n s i d e r e d as a part of
sult
[23] the system
SO
QTB
or, more precisely,
( ~ i) --~- 1
SO
and
It contains
q u a s i - t o p o l o g i c a l Boolean algebra.
contain exactly the WT*.
WT*.
Moreover,
By L e m m o n ' s re-
On the other hand, ~A,F>
where
Clearly,
A
is
is a
Why not?
you will m e e t first the d i f f e r e n c e
b e t w e e n the unit of a B o o l e a n algebra and u l t r a f i l t e r s
But,
WT*
We c o n c l u d e that Lemmon's alge-
braic s e m a n t i c s does not fit e x t e n s i o n a l t w o - v a l u e d logic.
Thereby,
So ,
q u a s i - t o p o l o g i c a l B o o l e a n algebras w i t h the
(of SCI) of e x a c t l y those models
(4.7) holds.
in
may be thought as a "theory" of a l g e b r a s in
unit as only one d i s t i n g u i s h e d element.
You may think about that.
Put
These formulas are not in
there is another r e l a t i o n s h i p b e t w e e n
the theory
with
is not a G-theory.
But, b e l i e v e it or not,
same e q u a t i o n s if
~T
S o.
in it such that
you will taste some w h i m of intensionality.
I am not i n t e r e s t e d in it.
Even more,
if you are really inter-
233
ested in n o n - F r e g e a n logic then I advise you to forget all modal systems w h i c h are not SCI-theories. $4,S 5
and the like.
XLV.
Also, you b e t t e r forget the labels
Why start c o u n t i n g w i t h
4?
M o d a l logic is an e s t a b l i s h e d trend in formal logic of today.
It starts w i t h i n t e n s i o n a l notions of necessity,
p o s s i b i l i t y and the
like and, c o n s t r u c t s formal systems w h i c h a l l e d g e d l y formalize these nitions but involve m a n y perplexities, Eventually,
m o s t l y c o n c e r n e d with identity.
it lives in the mad p h i l o s o p h y of intensional entities,
n o n - e x i s t e n t objects and essentialism.
Clearly,
any e n d e a v o u r to make
NFL s u b o r d i n a t e to modal logic is simply an a r r o g a n t offence. You may be sure that the fountain of modal logic will never dry up.
However,
it is only a kind of naivete to attempt to b u i l d an in-
t e n s i o n a l formal logic. p r o p o s e then,
W h a t e v e r i n t e n s i o n a l formal logic you m i g h t
sooner or later,
I bet someone will invent an intension-
al c o n s t r u c t i o n in n a t u r a l l a n g u a g e w h i c h b r e a k s all your logical rules. hand,
And no h a r m w i l l be done to our discourse. cannot h a p p e n w i t h NFL.
If one starts w i t h true p r e m i s s e s and
breaks the laws of the n o n - F r e g e a n or later,
logic then one will arrive,
sooner
at false conclusions.
XLVI.
In open W-languages,
the set
LA
of logical axioms is to be
a u g m e n t e d w i t h axioms for identity predicate: axioms.
This, on the other
~ ~ ri-
and invariance
Note that we m u s t also have the "mixed" invariance a x i o m
scheme:
where more,
L"
~ ' 2 ' ~i'
the theory
WB
~2
are any terms
(nominal formulas).
is d e f i n e d as in SCI.
However, we define
as in XXXIX.
To get
where
are any formulas, both sentential or both nominal.
~
XLVII.
, ~
WH
Further-
we add the principle:
((~, ~
WT
)~,~)~0)
We may suppose the u n d e r l y i n g language to be open.
If, how-
234
ever,
the language involves q u a n t i f i e r s then
er a s s u m p t i o n on
XLVIII.
(11.15)
r e q u i r e s a strong-
T.
P o s s i b l e - w o r l d semantics
for modal systems with q u a n t i f i e r s
forces us to c o n s i d e r fiction stories in which,
for example,
Sherlock
Holmes was a s c h o l a s t i c logician at O x f o r d and Dr. W a t s o n i n v e n t e d p o s s i b l e - w o r l d semantics. ing.
Some tales of this kind may even be amus-
Yet, this is not a science and logicians are not n e c e s s a r i l y the
b e s t fiction writers.
XLIX.
The s e m i - F r e g e a n p o s t u l a t e seems to be a m i x t u r e of a mathe-
matical desire for s i m p l i c i t y and some intensional whim.
The Boolean
unit is d i s t i n g i s h e d by p r o p e r t i e s of Boolean o p e r a t i o n s only.
Thus,
a logical m a t r i x based on a B o o l e a n algebra and c o n t a i n i n g the singleton El} as the set of d i s t i n g u i s h e d e l e m e n t s may be r e p l a c e d n a t u r a l l y by the u n d e r l y i n g Boolean algebra.
In this case we get rid of logical
m a t r i c e s in favour of mere algebras and, a superficial s i m p l i f i c a t i o n actually is attained.
However,
the s e m i - F r e g e a n postulate.
this is not a c o m p l e t e d e s c r i p t i o n of
For example,
Kripke m o d e l s c o n s t i t u t e a
rather complex kind of r e l a t i o n a l structure with the power
21
as the
u n d e r l y i n g universe;
21
is
also,
the unit of the Boolean a l g e b r a
used in modal logic as the only d i s t i n g u i s h e d element. L.
A l t h o u g h I am s t r o n g l y o p p o s e d to the i n t e n s i o n a l formal logic,
n o n - e x i s t e n t e n t i t i e s and p r o p o s i t i o n s at the same time,
i n d e p e n d e n t of any language and,
to the Fregean axiom in e x t e n s i o n a l
thereby attack the general index method in semantics.
logic I do not It may be con-
sidered as an e n d e a v o u r to "dilute" the semantics of u n a m b i g o u s extensional
logic to d e s c r i b e a m b i g u i t i e s and v a g u e n e s s of our t h i n k i n g
in natural language
(compare
[47],
[48]).
m e t h o d is closely tied to intensional
Unfortunately,
logic.
the index
Possible-worlds
seman-
tics in m o d a l logic is the first a p p l i c a t i o n of index m e t h o d w h e n the
235
Fregean logic was a p a r a d i g m of e x t e n s i o n a l and t w o - v a l u e d logic. What I can say right now is o n l y m e t h o d o l o g i c a l advice:
d e v e l o p first
the n o n - F r e g e a n logic as a more general p a r a d i g m of e x t e n s i o n a l and t w o - v a l u e d logic and, only then try to use the index method.
LI.
The o r i g i n a l d e f i n i t i o n of C r e s s w e l l models and the c o n c e p t
of s a t i s f a c t i o n is f o r m u l a t e d w i t h i n the c u s t o m a r y f o r m a l i s m of possible-worlds
semantics.
One may easily t r a n s f o r m C r e s s w e l l models into
c-models using one-one c o r r e s p o n d e n c e s b e t w e e n r e l a t i o n s in I ~
P(I) ~.
~here 2 I.
. . ~P(I)
P(I)
and maps from
P(I) ~ . . . ~ P(I)
is the set of all subsets of
I~ and, b e t w e e n
P(I) P(I)
and
The final form of c-models allows us to define s a t i s f a c t i o n by
means of h o m o r p h i s m s as in case of our models. the S C I - l a n g u a g e to a c - a l g e b r a 21
to
such that for all ~ , ~
G 2 ( h ( K ), h ( ~ ) ) LII.
and
:
G
is a map
h(~)
h(~)
A h o m o m o r p h i s m from h
of all formulas into
= Gl(h(~)),
h(~)
= G 0 ( h ( ~ ), h ( ~ ) ) .
One may b e g i n w i t h a g e n e r a l i z e d SCI-model
in a quite analogous w a y c-models Moreover,
=
C(M)
one may c o n t r a c t each such
isomorphic to some c-model
C(M*)
and obtain,
w h i c h also satisfy
C(M)
where
M
(13.10).
to a structure w h i c h is M*
is the c o n t r a c t i o n of
M.
LIII. Nz
Actually,
the C r e s s w e l l proof uses the g e n e r a l i z e d S C I - m o d e l
w h i c h consists of the S C I - l a n g u a g e and the theory
sequently, LIV.
However,
Standarte, at all.
the special c-model
Herr G r u p p e n f u h r e r der SL Actually,
(i.e., Symbolic Logic) it does not m a t t e r here,
A n o t h e r story is that the subtle logician, OT
and, sub-
C(Nz).
Georgie Top, disagrees.
s u g g e s t e d that
z
John Myhill,
should be rather f o r m u l a t e d w i t h i n the set theory
(based on FL, of course).
Evidently,
is at w o r k in M y h i l l ' s mind.
the r e i f i c a t i o n of situations
In fact, he sketched a c o n v e n t i o n a l
236
(set-theoretical) OT
and not the
proved in
m o d e l of OT
itself,
OT.
But, one o b t a i n s the c o n s i s t e n c y of
that way.
[52] in a g e o m e t r i c a l way
C o n s i s t e n c y of
OT
(using the Cantor set).
has been
237
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2)
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3)
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4)
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5)
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7)
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8)
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9)
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I, Princeton,
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ii)
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14)
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16)
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: SCI and Modal Systems, an abstract to appear in The Journal of Symbolic Logic 37(1972).
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Lewis, C.I., Langford, G.H.: Publications, 1959.
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Rasiowa, H., Sikorski, PWN Warsaw, 1963.
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Kagan, J.D.: An Axiomatization of Topological Boolean Algebras to appear in Zeitschrift fur Mathematische Logik und Grandlagen der Mathematik.
22)
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Lemmon, E.J.: An extension algebra and the modal system Notre Dame Journal Of Formal Logic 1(1960), 3-12.
24)
Scott, D. : A Proof of Independence of the Continuum Hypothesis, Mathematical Systems Theory 1(1967), 89-111.
25)
Brown, D.J., Suszko, R.: Abstract Logics, Dissertationes Mathematicae, Vol. 102(1973).
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Stupecki, J.: Uber die Regeln des Aussagenkalkuls, 1(1953), 19-40.
27)
Hiz, H.: Complete sentential calculus admitting extensions, Summer Institute of Symbolic Logic in 1957 at Cornell University, Vol. 2, 260-262.
28)
Suszko, R.: Concerning the method of logical schemes, the notion of logical calculus and the role of consequence relations, Studia Logica II(1961), 185-214.
29)
Wojcicki, R.: Logical matrices adequate for structural sentential calculi, Bull., Acad. Polon. Sci. (Set. Sci. Math. Astronom. Phys.) 17(1969), 333-335.
30)
Bloom, S.L., Brown, D.J., Suszko, R.: Some theorems on abstract logics, Algebra i Logika Sem. g(1970), 274-280.
31)
Suszko, R.: Adequate models for the non-Fregean sentential calculus (SCI), in Logic, Language and Probability, edited by R.J. Bogdan and I. Niiniluoto, D. Reidel, Pub., (1973),pp. 49-54.
32)
Bloom, S.L., Brown, D.J.: Classical abstract logics, tationes Mathematicae, Vol. 102(1973).
33)
Los, J.: The algebraic treatment of the methodology of elementary deductive systems, Studia Logica 2(1955), 151-211.
34)
Los, J., Suszko, R.: On the extending of models II, Fundamenta Mathematicae 42(1955), 343-347.
35)
Hallden, S.: On the semantic non-completeness of certain Lewis calculi, the Journal of Symbolic Logic 16(1951), 127-129.
36)
Diamond, A.H.: Simplification of the Whitehead-Huntington set of postulates for the algebra of logic, Bulletin of the American Mathematical Society, Vol. 40(1934), 599-601.
R.:
Symbolic Logic,
2nd ed., Dover
The Mathematics of the Metamathematics,
T,
Studia Logica
in Disser-
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Suszko, R., Zandarowska, W.: Lewis' systems S 4 and S and the identity connective (in Polish), Studia Logica 295(1971), 169-177.
38)
McKinsey, J.C., Tarski, A.: Some theorems about the Lewis and Heyting calculi, The Journal of Symbolic Logic 13(1948), 1-15.
39)
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40)
Scroggs, S.J.: Extensions of the Lewis system of Symbolic Logic 16(1951), 112-120.
41)
Wojcicki, R.: On reconstructability of classical propositional logic in intuitionistic logic, Bull. Acad. Polon. Sci. (Ser. Sci. Math. Astronom. Phys.) 18(1970), 421-422.
42)
Kripke, S.A.: Semantical considerations on modal logic, Acta Philosophica Phennica 16(1963), 83-94.
43)
Lemmon, E.J.: Algebraic semantics for modal logics, The Journal of Symbolic Logic 31(1966), 46-65 and 191-218.
44)
Makinson, D.C.: A generalisation of the concept of a relational model for modal logic, Theoria 36(1970), 331-335.
45)
Lemmon, E.J.: A note on Hallden-incompleteness, Journal of Formal Logic 7(1966), 296-300.
46)
Cresswell, M.J.: 347-372.
47)
Montagne, R.: Pragmatics in R. Klibansky ed., contemporary philosophy-la philosophie contemporaine, Florence 1968, 102-122. :
48)
Classical intensional
Universal grammer,
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The Journal
Notre Dame
logics, Theoria 36(1970),
Theoria 36, 1970,
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49)
Cresswell, M.J.: Functions of propositions, bolic Logic 31(1966), 545-560.
5O)
Suszko, R. : Reifikacja sytuacji (Reification of situations), Studia Filozoficzne 69(1971), 65-82.
51)
: Ontology in the Tractatus of L. Wittgenstein, Dame Journal of Formal Logic 9(1968), 7-33.
52) A.M.S. 53)
The Journal of Sym-
Notre
: Consistency of some non-Fregean theory, Notices of 16(1969), abstract 664-23.
Prior, A.N.: necessary?
IS the concept of referential opacity really Acta Philosophica Fennica 16(1963), 189-198.
A REALIZABILITY INTERPRETATION OF THE THEORY OF SPECIES William W. Talt
I•
The main purpose of this paper is to define a realizability interpretation for
second order intuitlonistic logic and to prove that every deducible sentence of this system is realizable.
As a consequence of thisj we shall obtain Girard's 1970 Nor-
mallzability Theorem for second order intuitlonistic logic•
The ideas of this pa-
per extend in a straightforward way to the intuitlonistlc simple type theory (logic of finite orders); but I will not discuss this extension here.
2. 2.1.
(3 The Theory.4~of Species• For each of the symbols cs=i 9 0 j
i, ...~ ~ c o n t a i ~
infinitely many ~-varlables~
denoted by ....
The i-variables are intended to range over individuals and are also denoted by
The O-variables are intended to range over p r o p o s i t i o ~
and~ for n)O, the n-vari-
ables over n-ary species of individuals (i.e. properties of n-tuples of i~ividuals). The non logical constants o f ~
are individual a ~
function constants.
An n-ary f u ~ -
tion constant (n)0) is intended to denote an n-ary operation on individuals.
The
individual or i-terms are built up in the usual way from i-varlables and individual and function constants~ and are denoted by Ti , S~
...~ s~ t, s0~ . . . .
The formulae of ~ are built up in the usual way from atomic formulae Xntl...t n n~,
by means of the operations
A= B for a=i, 0p I, • . . •
~A
We shall see in 2• that the remaining intuitionistic logical
241
operations can be defined in terms of these• are distinctt
If A is a formula~ n >
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~
= Z u ~ w )) =
Zxyz]
,% G+vmt ~ Zuv't) m Zxyz]
Then~ ~ la Dedekind 1888~ we can derive Peano's postulates for number theory from the premises ~/xy(ExyV-~Exy)
%~x-;Eox'
~/xy(Ex'y' D gxy)
243
Let A
-
n
be obtained from A by replacing each part X tl...t n of A b y ~ x n t l . , . t n .
As noted by Kreisel~ the result of Godel 1932-3 extends to show that if A is deducible in classical second order loglcp then A" is deducible i n ~ , here that A is a formula o f ~ o ) that each of ~ Ij ~ E a n d ~ l ~E
(We are assuming
One need only note that ~ I A - = ~ A - is derivable I n 2 2 is preserved under the translation A-~A" t and that
translates as
~/XC~A"
T~-
AET~/xO~3When ~ = i~ this is again an instance of ~{E. But~ when ~ = np it is
VXnA -
T n-
A-[Tn-/ Xn] However~ we do have the application
~
XnA -
~ T n"
A'["~Tn-/Xn] of ~
E, from which we obtain ACTn/ xn]" using ~-iTnt'l,OOtn ~ TnJl,,Otn, Thus~ in this sensej classical second order logic is embedded i n ~ ,
This
result applies also to simple type theory.
2.3.
It will be convenient and suggestive to introduce another notation for natural
deductions i n ~ .
Accordingly~ we introduce the notion of an A-term for each formula
A~ by induction. a~A means that a is an A-term,
Let V0~ VI~ ... be an infinite llst of symbols.
X0~ etc. will denote arbitrary V n.
XA~A
X~ Y~ Z~
244
X A is called an A-variable.
b ~ B =~ ) ~ A b ~ A D B
b~ B ~ C~
~x% IMX%
A m B~ a ~
A ~
(ca)~
c~Vx% => (cZ%t= The free occurrences of variables in ~ variables Y(~ in A,
reap,
c and those in a and Tcc~ reap,
B[T~/X~]
are XA itself and the free occurrences of
The free occurrences in ~ A b
occurrences of X A and X ~
B
and XX~5 are those in b other than
The free occurrences in (ca) and (cT (I) are those in We must impose a
Restriction on kX Ct, kX~b is a ~X~B term only when X (z is not free in A whenever is free in b, With each natural deduction D of A we associate an A-term ~D~ as follows:
n
l! vx%/ I: o A~B
A
=
(
~B
)
B
•
T~
= (Vx~A T~) •
I AET~/ Note that discharged and undischarged premises transform into bound and free vari-
245
(20)
Uh >
J
lira
h(y,z) d
x)
y_<x<_z
O
~t ,
la paetie absolument continue du membre de droite de (20)
est une mesure inf6rieure h la pattie absolument continue au sens large de la mesure
Posons, quel que soft
xCR,
~(x)
(21 )
~h"
lira
=
y_<x
0<[z-yl~ et admettons pour ~vitev toute ambiguit6 que dans
R.
y< z
Uh(K)>- IK ~ dye' 6tant un compact pour lequel
bees O
quels que soient
I1 suffit pour ~tablir (20) de montrer que
(22) K
e(y,z) > 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 <
inf
h(y,z) < c+E
y_< x_< z
O<]z-y]_<~ pour tout nombre
0< a < _ Cro,
en tout point xCK.
Donnons-nous maintenant un hombre 0 < cr < cr
O
est un recouvrement de
K
form6d'intervalles v~rkfiant
Si
]Yk ' Z k [ ( l _ < k < n )
0 < zk-y k < (~
il vient n
= ~,
h(Yk, Zk)
h(Yk,Zk)
(Yk, Zk )
k=1
soit encore, si tousles intervalles
[Yk ' Zk]
rencontrent
K,
(1 _< k _<_n),
246
again not much more than a reformulation of Girard's idea).
The isomorphism between natural deduction and the lambda calculus with type structure~ utilized in 2.3 - 4~ was first noted in Curry-Feys 1958 for the case of positive implicational logic.
3.
The Lambda Calculus and Realizability.
3.1.
The terms of the lambda calculus/are built up from the variables V n and the
constant K by means of the operations )~XD
(ab)
of lambda abstraction and evaluationp reap.
The notions of a free occurrence of a
variable~ and of substitution for free variables are defined as usual.
The rule of
lambda conversion is
(~Xb)a
The relations a ~ > 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,
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