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,l + 1> yield t h e o c c w e n c e . P'
.
-
89
INFINITE RULES I N FINITE SYSTEMS
(1) - ( 4 ) a r e more a r l e s s obvious f r o m t h e d e f i n i t i o n o f I n f a c t , t h e y were mentioned because t h e y a r e h e l p f u l i n v i s u a l i z i n g
GE.
PROOF.
proof o f
( 5 ) . One way t o p r o v e ( 5 ) i s t o f i r s t a s s i g n
ordinals
the
to
d e r i v a t i o n s i n Fm and t h e n use t r a n s f i n i t e i n d u c t i o n o n t h e o r d i n a l
the p
to
has
a
show t h a t P ( U ) where P ( p ) I WF
through If p = 0
M
[if F occurs a t ( n , j ) i n GE and F
E WWnWj
F"
derivation i n
'
o f ordinal
2
t h e n e v e r y branch o f
p
eE
passing
c o n t a i n s an axiom].
t h e n P ( U ) h o l d s because ( a ) d e r i v a t i o n s o f o r d i n a l 0 a r e j u s t
t n e axioms o f
F and (b) because o f
p > 0
(1). I f
t h e n u s i n g t h e induc
t i o n h y p o t h e s i s and e i t h e r ( 3 ) o r (4) one can conclude
6 , CANONICAL
P(u).
FOR THE THEOREMS OF F"
DERIVATIONS
-
,
We s h a l l now proceed t o show how g i v e n an e f f e c t i v e d i s j u n c t o r @ on t h e formal Carnap t h e o r y F t h e spread tion o f E i n
m
FGR
6,
can be transformed i n t o a d e r i v a
( p r o v i d e d E had a d e r i v a t i o n i n
F").
-
Loosely
speak-
i n g t h e d e r i v a t i o n i s o b t a i n e d by a p p l y i n g t h e d i s j u n c t o r @ t o t h e
w.f.e. thus
o c c u r r i n g a t t h e nodes o f let Z
GE.
We need a f u n c t i o n f o r t h a t purpose:
be t h e ( p a r t i a l ) r e c u r s i v e f u n c t i o n such t h a t : C ( 0 ) = 0,
C ( < i > )= i, c(n*<j;) = Z{M) Thus
I(<Eo
,...,EP > )
= ((..
. (Eo
j
El) 63
@
. ...)
63 E p ) .
We a l s o need a f u n c t i o n which t e s t s whether we have reached o r n o t . Thus
an
axiom
l e t A b e t h e p a r t i a l r e c u r s i v e f u n c t i o n such t h a t : A ( 0 ) = 1, A
. . ..
( < i > )=.
A(
,
{ a 1 (i)
is) = A
(M)
* {U}
(i).
.
.,E > ) = 0 i f f a t l e a s t <Eo,. .,'E p> = 6 , ( ~ ) , P ) = 0 i s a n axiom o f F . Consequently i f A (GE(n) one o f t h e E o , ,E P then f o r a l l successors m o f n , A (GE ( m ) ) = 0.
Hence f o r
A(<Eo,.
90 6.1
E.
DEFINITION.
B A S I S STEP.
Fo4
each E E W we d e i i n e .the 6unct.Lon DE
a6 QoLLowb:
DE(0) = E .
S u p p h e that D E ( n )
RECURSION S T E P .
Case 1 .
G. K . LOPEZ-ESCOBAR
DE(n) = 0.
has been d e 6 i n e d .
i
Then 604
let
E N,
DE(n*) = 0
Case 2 .
DE(n) # 0.
Then compuie
A ( GE( n ) ) = 0 .
Subcase 2 . 1 .
GE(n) Then d e 6 i n e 604 & i E N
D E ( n * < i > ) = 0.
Subcase 2 . 2 .
A( 6,(n))
= 1. Then
DE(n*) =
d e 6 i n e do& a l l i E N
c ( GE(n*)).
I t i s an immediate consequence o f the d e f i n i t i o n t h a t DE w i l l be w e l l -
founded ( i . e .
I n : DE(M) # 0 )
t a i n s an axiom. Furthermore, i f
i s a t r e e ) i f f every branch o f E € W
f u n c t i o n i r r e s p e c t i v e o f whether D~
then
DE
is
a
total
6,
con-
recursive
i s w e l l founded o r not. Thus i f D E i s
w e l l founded, then a p p l y i n g the r e c u r s i o n lemma ( f i x e d p o i n t theorem)
of
r e c u r s i v e f u n c t i o n theory (see Rogers 1965) i t can be shown t h a t every w.f.e. o c c u r r i n g i n a node o f
D~
i s derivable i n
F&.
Rather than g i v e t h e ac-
t u a l proof we f i n d i t much more worthwhile t o e x p l a i n t h e r o l e o f t h e cond i t i o n s ( i ' ) - ( v ' ) of Section 3.Thus suppose t h a t we a r e a t a node n o f D E , t h a t F 6
U
and t h a t t h e occurrence
w - r u l e . Then i n writing
D~
n e i t h e r the parentheses nor
a):
Suppose t h a t we have a r e c u r s i v e l i s t i n g
...F. ..F
all
. ..
i s a c t i v e a t n because o f t h e
we have t h e f o l l o w i n g k i n d o f s i t u a t i o n (where we a r e
[O], F...F k, D e r F - ( I { z l ( k ) } ;
of
.. ., t h a t ...F...F [ k ] ) .
[l],
derivations i n
FER
of
i s , we have a z such t h a t f o r
In order t o be a b l e t o apply
91
I N F I N I T E RULES I N F I N I T E SYSTEMS
the r e c u r s i o n lemma we must be a b l e t o e f f e c t i v e l y determine a natural numb e r ? such t h a t F i r s t obtain
(1) (2)
...F . . . ) .
DerF"({^zj;
? can be obtained as
follows :
G E U such t h a t
G -I
...F...F
...F . . . F , [k]
G [k],
+F
k.
for a l l
Then using z and the e f f e c t i v e n e s s o f (2) (as formulated i n (v')) we determine a (3)
({zl}; G ) .
DerF-
Combining i t w i t h (1) we can determine a
(4)
Der
m
( { z2 } ;
...F .. .F) .
Then by repeated a p p l i c a t i o n s o f (i')
z2
- (iii')
such t h a t
( t h e exact number o f a p p l i c a -
t i o n s can be read o f f from n ) we can then o b t a i n a DerF" (
?I ;
.. .F. ..) .
The g i s t o f t h e above remarks i s t h a t that i s
if
in
z^
such t h a t
every branch of
6,
F then t h e r e i s a r e c u r s i v e d e r i v a t i o n o f E i n
t h e r e occurs an axiom of
f";
can
such t h a t
z1
FER t-
E. Thus, i n o r d e r t o prove Theorem 1 ( o f Section 3 ) i t E then i n every branch o f GE there occurs an axiom of F. But the l a t t e r i s t r u e f o r any ( f o r m a l ) Carnap theory (see Lemma 5.1). Thus the proof o f Theorem 1 i s now complete. s u f f i c e s t o show t h a t i f F" -I
7, A N
UNFAITHFUL A X I O M A T I Z A T I O NOF I N T U I T I O N I S M,
The o n l y (non - c o n t r i v e d ) case t h a t we know o f an u n r e s t r i c t e d w - r u l e which cannot be replaced by t h e r e c u r s i v e l y r e s t r i c t e d one i s i n i n t u i t i o n i s t i c number theory. Take, f o r example, f i r s t - o r d e r i n t u i t i o n i s t i c theory HA f o r m a l i z e d as i n Kleene 1952 and l e t sions o f HA w-rule.
HAm, HAER
number
be t h e exten-
obtained by adding t h e o - r u l e and t h e r e c u r s i v e l y r e s t r i c t e d
L e t m be t h e s t r u c t u r e
(w,+,-,O,').
shows t h a t f o r a l l sentences A o f
Then
an
easy
induction
HA:
(.1) i6 3 I= A ( L e . A h &tue i n m ) &en (.2) i6 "l I== 7 A, &en HA" I- I A .
HA" -I
A,
Furthermore, the methods used i n t h e l a s t chapter o f Kleene 1952 show f o r a l l sentences A o f HA: (.3)
i6
H A I R -I
A
&then
A .i~ fie.a-P.izabte.
Since there a r e sentences which a r e t r u e b u t not r e a l i z a b l e (e.g.
that
92
E . G. K . LOPEZ-ESCOBAR
vx [ 3 y T ( x , x , y ) v v y 7 T ( x , x , y ) ] ) i t follows t h a t HAm i s n o t a conservative extension of HA;,. The derivations of (.1) - ( . 3 ) a r e carried o u t in classical mathematics and t h u s n o t t o o relevant t o intuitionism and since H A i s supposed t o be a formalization o f ( a p a r t o f ) intuitionism, the counterexample i s also somewhat contrived. O n the whole i t i s much more natural t o use a n intui t i o n i s t i c metatheory when discussing i n t u i t i o n i s t i c s systems. I n t u i t i o n i s t i c systems a r e very often closed under Church's Rule (CR) and t h u s t h e f o l lowing argument o f Kreisel gives a very simple solution to the problem of replacing the OJ - r u l e by the recursively r e s t r i c t e d O J - r u l e (when the metatheory i s closed under CR). For simplicity l e t us consider H A , HA" and H A E R and l e t the meta theory M be closed under Church's rule. Then Kreisel's formulation o f the
-
-
-
problem i s t o show t h a t : (*)
{ A : II i - ( ( 3 c ) P r f H A ~ ( a o r ~ ' ) 5 j {A : M + ( 3 a ) P r f H A m ( a , r ~ l ) j GR
where P r f H A - and P r f H A "
GR
are (canonical) representations i n M of the
proof predicates of HA" a n d H A E R respectively. The proof of ( * ) i s a s fol 1ows : Suppose t h a t M l-(3a) P r f H A m ( a r r A 1 ) . Then since M i s closed under C R , M I- (3a),,PrfHAm(a, r ~ l ) . B u t then, because of the form of the predicates (
PrfHAm
3 a ) P r f H A m (a;'A7 GR
and )]
PrfHAm
. Thus
GR
M I-
,
M -I (
[(ja)GRPrfHA-(~,rA1)
3 a ) P r f H A m (a,'A').
-
GR
I n s p i t e of the above solution there i s s t i l l some room for improvement. I t seems t o us t h a t i f one i s to be completely f a i t h f u l t o the principles embodied in H A then one should t r y t o prove, i n t h e intLLit.ionhi6.tic me&t h e m y M, the following sentence:
Although we do n o t d o u b t t h a t i t i s rewarding t o be f a i t h f u l ,
in
this
particular instance we find t h a t i f we follow the i n t u i t i o n i s t i c metatheory (and thus look f o r solutions of the kind ( * ) - (***)) we do not r e a l l y de t e n i n e which chmaracteristics of the usual formalizationmof H A produce the r e s u l t t h a t H A i s n o t a conservative extension of HA,,
-
.
93
I N F I N I T E RULES I N F I N I T E SYSTEMS
The p o i n t o f view we p r e f e r t o adopt ( f o r t h e purpose o f t h i s paper) i s that
HA
whay
HA,
i s j u s t a formal system amongst many and then t o t r y t o f i n d
s e r v a t i v e extension o f
out
h a s t h e property t h a t HAm i s n o t a con-
as a f o r m a l system,
HA^^.
The d e f i n i t i v e answer s t i l l eludes u s . However we have reason t o
be-
HA
l i e v e t h a t i t has something t o do w i t h t h e f a c t t h a t a formal proof i n proof o r c o n s t r u c t i o n . That i s
i s an attempt t o represent the i n t u i t i v e formal proof i n HA
a
o f a sentence A i s c l o s e l y r e l a t e d t o t h e ( e f f e c t i v e )
c o n s t r u c t i o n t h a t v a l i d a t e s A . HAiR
s t i l l maintains a resemblance o f e f -
.
fectiveness, HAm does n o t (unless the netatheory i s i t s e l f c o n s t r u c t i v e ) I n order t o support t h i s b e l i e f we now introduce another f o r m a l i z a t i o n
for
f i r s t - o r d e r i n t u i t i o n i s t i c number theory which has t h e same c l a s s o f theorems as HA, b u t whose proofs try t o conceal as best as p o s s i b l e t h e
con
-
s t r u c t i v e content o f t h e theorem. I t i s f o r t h e l a t t e r reason t h a t we s h a l l c a l l i t U H A ( U n f a i t h f u l H e y t i n g ' s a r i t h m e t i c ) . I t has t h e property UHA"
that
i s a conservative extension o f UHA&. The formulae o f U H A a r e j u s t t h e formulae o f H A , however p r o o f s
in
U H A w i l l n o t i n v o l v e t h e formulae themselves b u t r a t h e r f i n i t e , non-empty, sequences o f formulae which we s h a l l c a l l c -bequents quents)
.
(read: confused se
AXIOMS OF U H A are those c-sequents i n which t h e r e occurs axiom o f HA.
STRUCTURAL RULES OF UHA a r e t h e r u l e s o f the form:
Fmm: Pho v,ided:
. ..pAn Ano . Aol
To conclude:
n
*
0 a p m W n od
{O,.
. .,n}
DECIDING RULES OF U H A a r e given by t h e schema:
Fhom:
Aot
To conclude:
*
Aor
.-r A n - l l
--
lAn-l
B I
B B
ANCIENT RULES OF U H A are given by t h e f o l l o w i n g schemata:
Fhom:
Ao,.
..,AM - l t B o
To conclude:
Rovided:
and
Aot
-..lAn-lrB1
AOt-*.tAn-l I C
Bg' *1 c .
i s a rule o f HA,
and correspondingly f o r t h e one premiss r u l e s o f
HA.
at
-
l e a s t one
94
E . G . K. LOPEZ-ESCOBAR
D e r i v a t i o n s o f c -sequent5 i n U H A a r e t h e n d e f i n e d i n t h e usual
tree
form. We say t h a t a f o r m u l a A i s p r o v a b l e i n U H A and express i t i n
sym-
U H A I-
bols: in
A, j u s t i n case t h e one - t e r m e d c-sequent
( A ) i s derivable
UHA.
7.1
Fox any 60hmLLea
LEMMA.
t h e ~ 0 U v w i n g c o n d i t i o n b atre
A 06 H A
e@L&Jdent:
(1) H A I(2)
A.
T h a t (1)
PROOF.
-
UHA k A.
(2) i s t r i v i a l .
To prove t h a t ( 2 )
P ( n ) stand f o r the following:
(1)
Hq.
06
A s t r a i g h t f o r w a r d i n d u c t i o n o n n shows t h a t P ( n ) h o l d s f o r
Suppose n e x t t h a t Then
(A).
let
II 0 a detLiwation i n U H A 0 6 length 5 n t h e n a t l e a s t t h e ,5amLLeae i n .the end C-beqUent a 6 n 0 pmuable i n
W I l [id
one
-
HA+
n
i s a d e r i v a t i o n i n UHA
o f t h e one termed
all
n.
c -sequent
A.
Thus as f a r as t h e c l a s s o f theorems i s concerned l i A and UHA a r e e q u i v a l e n t . The o n l y d i f f e r e n c e between t h e p r o o f i n H A
and
UHA i s
that
in
U H A t h e a c t u a l p r o v a b l e f o r m u l a can be h i d d e n w i t h extraneous formulae. I n
t h e f i n i t e case t h i s c h a r a c t e r i s t i c o f
U H A causes no problem because
it
i s r e l a t i v e l y s i m p l e t o search t h r o u g h f i n i t e s e t s . However i t does make a n enormous d i f f e r e n c e when we add t h e w - r u l e .
Fmm:
Ao,
.. . , A n - l ,
To c a n d u d e : For l e t
UHA“,
Aot
BxL
- - - rAn-l,
k < w
doh
WxB
U H A i R be U H A t w - r u l e ,
UHA
+ (recursively restricted
w - r u l e ) r e s p e c t i v e l y . By a p p r o p i a t e changes i n t h e p r o o f o f Lemma 7.1
we
o b t a i n t h e f o l l o w i n g lemma: 7.2
LEMMA.
F v h any nentence A
06
H A t h e ~ o U v w i n gc o n d i t i o n 6
atre
&pLiUdent:
(1) HA” I(2)
A,
U H A ” ~A.
On t h e o t h e r hand t h e spread c o n s t r u c t e d i n s e c t i o n 5 (see Lemma
g i v e s us t h e f o l l o w i n g :
7.3
LEMMA.
Foh a &? bwtencen A
06
HA
5.1 ( 5 ) )
INFINITE RULES I14 FINITE SYSTEMS
UHA"
id and o d y id U H A i R
I- A
UHA"
Thus we o b t a i n t h a t HA",
and UHA&
95
I-A.
are a l l e q u i v a l e n t
as
far
as t h e classes o f theorems are concerned.
W e doubt t h a t
m
UHA"
( o r i t s e q u i v a l e n t UH?$.R)
i s o f much use as f a r
as p r o o f - t h e o r y i s concerned, although there might be a p p l i c a t i o n s case o f the theory o f species where i t i s sometimes c l a s s i c a l w - complete f i r s t - order subtheory ( c f . like
U H A and UHA"
to
i n the have
a
Smorynski 1973). Systems
are made f o r t h e theory o f proofs and probably should
be considered i n conjunction w i t h the &-we11
- Mints - Simpson
Kreisel
customary
founded proofs discussed
in
1974.
REFERENCES, Grzegorczyk, A., 1958.
The ~
A. Mostowski, and C. Ryll-Nardzewski
n
i andc 2ke~ w - comjdcte LvLithe-tic,
Logic, v o l . 23, pp. Kent, C. 1967.
Thc Journal of Symbolic
188-206.
F.
RenMcRed w - hiLee doh c#&hctic,Notices
o f the American Mathemati-
c a l Society, v o l . 14, pp. 665. Kleene, S . C. 1952.
I n t r o d u c t i o n t o M e t m a t h c m a t i c a , 0. van Nostrand Publ. CO. New York, x + 5 5 0 pp.
K r e i s e l , G., 1974.
G. E. Mints, and S . G. Simpson
The U b e
06
Abb,Ouct
Laflguage LIZ Eleme~%&Vcy ! ~ k 2 i Z l c 7 , t / t 5 7 1 ~ & 5 : S o mPede
agogic E x a m p l e s , A r t i c l e i n Logic Colloquium, Lecture Notes i n Math-
ematics, vol . 453, Springer Verlag Publ
L6pez 1967.
- Escobar,
.
Co.
E. G. K. Rematlks OIZ a:? in~i~u.tmglanguage c 3 L t h coiz6ZtucZi.ve 60hmlLeas, Journal o f Symbolic Logic, vol . 32, pp. 305 318.
-
The
Nelson, G. C. 1971.
A ~u&tlw,thc~d.thic.t~dw - hiLec, Colloquium Mathematicum, v o l . 23,
1
- 3.
pp.
Ohya, T. 1970.
On
heCUlAiV2 h e S . & U d O l Z
06
pao,@
i l ? cl bLjA.?hi
COn64%L&Vk?
d i ~ L t & y long expesniovm, C o n c n t . Math. Univ. S t . Paul, v o l . pp. 4 3 - 52.
in18,
96
E . G.
K. LOPEZ-ESCOBAR
Rogers, H. 1967.
Theory of Recursive Functions and Effective computability, + 482 pp.
McGraw - Hi1 1 Publ. Company, x i x Routledge, N. A. 1950.
-
Gheek M a t h e m d c n . A r t i c l e i n Eureka, The Archimedeans' Jour
nal, October, pp. 3 - 4 . Shoenfield, J. R. 1959.
On a hen&Loted
w-hLLee, B u l l e t i n de 1'Academie Polonaise des
ences, Serie des S c i . Math., Astr.,
-
Sci
e t Phys., v o l . 7, pp. 405- 437.
Smorynski, C. 1973.
Chapter V
Of
Metamathematical Investigation of Intuitionis-
tic Arithmetic and Analysis, L e c t u r e Notes i n M a t h e m a t i c s , v o l . 344, Springer-Verlag Publ. Co. Smullyan, R. M. 1961.
Theory of Formal Systems, Annals o f Plathematics studies, 47, 142 pp., Princeton U n i v e r s i t y .
vol.
Takahashi, 1.1. 1970.
A theahcm
Takeuti, G. and 1963.
OIZ
M e 6 e c d ahdm ahitluneLLc lhlith thtlze w - u L e , J o u r n a l
of
the Mathematical Society o f Japan, v o l . 22, pp. 15 - 24.
A. Kino
On phedicaten ULith co11,5,ttu~ueLyi n d i n i t d y l o n g expke66iom, Journal o f t h e Mathematical Society o f Japan, vol. 15, pp. 176 - 190.
Department o f Mathematics U n i v e r s i t y o f Maryland C o l l e g e Park, Maryland, U . S . A .
ADDED IN P R I N T (Jan. 1977), ( 1 ) The use o f "Formal" i n been
2,
i s misleading. A b e t t e r choice w o u l d
have
"Recursively definable".
( 2 ) I would l i k e t o thank Professor A. T r o e l s t r a f o r p o i n t i n g o u t t h a t
D.
97
I N F I N I T E RULES I N F I N I T E SYSTEMS
P r a w i t z i n t h e a r t i c l e " C o m m e n t . s on Gentzen-type pfioceduteh and t h e c & ~ n i c d flation 06 .t)LLLth" (Proof Theory Symposium, L e c t u r e Notes i n Mathematics, v o l . 500) a l s o c o n s i d e r s t h e e f f e c t s o f a d d i n g a
non-
iterative classical disjunction t o i n t u i t i o n i s t i c logic. ( 3 ) G. C . M i n t s has shown t h a t t h e r e i s no p a r t i a l r e c u r s i v e f u n c t i o n
mapping f o r m u l a e
A o f intuitionistic arithmetic to locally
r e c u r s i v e p r o o f f i g u r e s o f H A t w - r u l e such t h a t :
3p E Rec P f i o ~ ~ ~ + ~ ( p->, A T ) ( A ) i s d e f i n e d and well-founded.
'I
correct
Non-Classical Logics, Model Theory and Computability. A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
SOME REMARKS by
L.
H.
DISCUSSIVELOGIC
ON
LOPES DOS SANTOS
S . Jaskowski (Jaskowski 1948) introduced
s e t o f a l l formulas i c S5, where
h i s discussive l o g i c D~ as the
A such t h a t M A belongs t o the nodal p r o p o s i t i o n a l l o g -
M i s the usual p o s s i b i l i t y operator. Generalizing, t h e d i s -
cussive l o g i c J(K) associated w i t h any modal l o g i c K may be defined as the set
{ A : M A € K). An axiomatics f o r
J(S5) i s presented
i n
C o s t a 1975,
and i n T. Furmanowski 1975 i s proved t h a t J ( S 4 ) coincides w i t h J ( S 5 ) .
In
the f i r s t p a r t of t h i s paper, we present axiomatics f o r system),
S4, S5, J(B) and J ( S 5 )
B ( t h e Brouwerian s p e c i a l l y appropriate t o r e f l e c t t h e i r
mutual i n c l u s i o n and non- i n c l u s i o n r e l a t i o n s . The p r o p o s i t i o n a l l o g i c J(S5) i s n o t closed under a l l r u l e s o f i n f e r ence which a r e u n i v e r s a l l y v a l i d i n t h e c l a s s i c a l p r o p o s i t i o n a l l o g i c , even under a l l those which a r e v a l i d i n the p o s i t i v e p r o k o s i t i o n a l
not
logic;
p a r t i c u l a r l y , i t i s not closed under m a t e r i a l detachement ( c f . JaEkowski 1948). (A But i f we p u t A + D ( A LtidcwdiwtQg impfie6 U ) f o r M A 3 U and A A U and
B are dincusniweLy conjoined) f o r M A & 0, where
i m p l i c a t i o n and & means c l a s s i c a l conjunction, then
3
means
classical
a l l v a l i d l a w s and
r u l e s o f the p o s i t i v e p r o p o s i t i o n a l l o g i c come t o be v a l i d i n J ( S 5 ) ,when= and & are replaced r e s p e c t i v e l y by +and t a i n e d ) . I n f a c t , N.C.A.
da Costa and
A (andusual d i s j u n c t i o n i s main-
L. D u b i k a j t i s
(see
Costa
and
D u b i k a j t i s , these Proceedings) showed how t o t r a n s l a t e J ( S 5 ) i n a language whose p r i m i t i v e connectives,besides c l a s s i c a l negation and c l a s s i c a l j u n c t i o n , are j u s t discussive i m p l i c a t i o n and discussive conjunction;
dis
-
they
a l s o showed how t o axiomatize i t by means o f t h e discussive counterparts o f t h e w e l l - k n o w n axioms and r u l e s o f t h e p o s i t i v e p r o p o s i t i o n a l c a l c u l u s 99
100
L. H. LOPES DOS SANTOS
( i n c l u d i n g Peirce law), plus a f i n i t e s e t o f axiom schemas r e l a t i n g negation t o the o t h e r connectives. I n the second p a r t o f t h i s paper, we extend
this
axiomatics i n order t o o b t a i n a n a t u r a l one f o r t h e discussive l o g i c assoc i a t e d w i t h t h e p r e d i c a t e l o g i c S 5 ( w i t h the Barcan formulas). F i n a l l y we note t h a t t h i s discussive l o g i c i s i d e n t i c a l t o the discussive l o g i c
asso-
c i a t e d w i t h the p r e d i c a t e l o g i c s4.
PART I ,
1, LANGUAGE, The Logicsconsidered i n P a r t I are t o be thought as f o r m u l a t e d i n a l a n guage whose p r i m i t i v e symbols a r e denumerably many
p,, p 2 ,
...,
propositional
letters
the connectives i (negation) and V ( d i s j u n c t i o n ) , t h e necess i t y operator L , and parentheses. The f o l l o w i n g d e f i n i t i o n s a r e adopted: Df.1.
MA = df.
Df.2.
(A
3
I L I A .
C) = df. (1A
v
C).
Throughout t h i s paper, the l e t t e r s A , C and
D, w i t h o r w i t h o u t numerical
subscripts, w i l l be employed as s y n t a c t i c a l v a r i a b l e s f o r formulas; f o r the sake o f b r e v i t y , the most e x t e r n a l parentheses o f a formula w i l l be
often
omitted.
2, THE
L O G I C S B, J(B) AND J ( S 5 ) .
The l o g i c B i s included i n J ( B ) , which i s included i n J ( S 5 ) . The f o l lowing theorems r e f l e c t a x i o m a t i c a l l y the s i t u a t i o n . THEOREM 1 .
B 0 axiornatizabLe by meam o d t h e &Uoeal(Ling axiom nchana and
hulu:
4A
(1)
LA,
(2)
L ( L A 3 A),
(3)
L ( L ( A 3 C)
(4)
L(A3 Lm),
(5)
Ada
(6)
0 a &utacl*oLogy,
=)
( L A D LC)),
LC 6hom L ( A = C ) indeh LLA @om L A ,
and
LA,
101
SOME REMARKS ON D l S C U S S l V E L O G I C
i n d m A dhom LA.
(7)
PROOF:
I t i s w e l l known t h a t B i s axiomatizable by t a k i n g as axioms
all
(LA =I LC),A=>,lA, and as r u l e s m a t e r i a l detachment and Godel's r u l e . Now l e t A E B; then A i s
t a u t o l o g i e s , a l l instances o f the schemas LA
3
A,L(A
3
C)
3
a t h e s i s o f t h i s w e l l -known axiomatics; we show e a s i l y , by i n d u c t i o n , t h a t LA
i s a t h e s i s o f the axiom system (1)-( 7 ) ; hence so i s A i t s e l f , b y
(7).
Conversely, schemas (1)- ( 4 ) and r u l e s (5) - ( 7 ) can be c l e a r l y derived in B ; t h e r e f o r e , a l l theses o f the axiom sys the w e l l -known axiomatics f o r tem (1)- (7) belong t o B
PROOF:
.
J (B) 0 axiomatizabLe by meanb ad (1)- (7) and:
THEOREM 2.
i n d e h A &om
(8)
-
L e t A E J(B);
LMA. then
MA E
B, by d e f i n i t i o n o f J(B),
f o r B i s closed under Godel's r u l e ; t h e r e f o r e , LM4
and
LMA E B,
i s a t h e s i s o f the a x i -
om system (1)- (8!, by Theorem l, and so i s A , by (8). On the o t h e r hand, l e t A be a t h e s i s o f t h e axiom system (1)- (8); i t can be e a s i l y shown, by i n d u c t i o n , t h a t f44 E B; t h e r e f o r e I t i s worth n o t i n g t h a t r u l e
-
A E J(B), (7)
by d e f i n i t i o n o f J(B).
i s superfluous i n the axiom
system
(1) (8) : THEOREM 3 . J (B) 0 axiamaAzabLe by meum ad (1)- ( 6 ) and ( 8 ) .
PROOF: By means o f (1)- (6) and (81, any formula A can be derived LA
from
as f o l l o w s :
(i)
LA ;
(ii) L(A
3 LF.24),
by ( 4 ) ;
(iii) LLMA , by (i), (ii) and (5); (iv) L ( L M (v)
LMA
(vi) A
3
W ) , by (2) ;
, by (iii), (iv) and (5);
, by (v) and
(8).
-
THEOREM 4 . J ( 5 5 ) 0 axiamatizabLe by meam v d (1) (6), ( 8 ) a n d (9)
L ( L A 3 LLA).
It i s w e l l known t h a t S5 can be axiomatized by t a k i n g as axioms a l l t a u t o l o g i e s , a l l instances o f the schemas LA 3 A, L(A 3 C) 3 ( L A 'LC),
PROOF:
102
L. H. LOPES DOS SANTOS
A 3 LMA, LA 3 LLA , and as r u l e s m a t e r i a l detachment and Gijdel ' s r u l e ; by i n d u c t i o n on the l e n g t h o f a d e r i v a t i o n o f A i n t h i s axiomatics, we show e a s i l y t h a t LA i s a t h e s i s of t h e axiomatics proposed f o r J(S5) i f A E S5.
Now l e t
C E J ( S 5 ) ; then MCES5, b y d e f i n i t i o n o f J ( S 5 ) , a n d L M C i s a t h e s i s o f
t h e axiom system proposed f o r J(S5);hence so i s C, by ( 8 ) . Conversely,if
C is
a t h e s i s o f t h e axiom system proposed f o r J ( S 5 ) , then i t i s easy t o show, b y induction, t h a t M C € S5; t h e r e f o r e C € J ( S 5 ) , ;b d e f i n i t i o n o f J ( S 5 ) . The l o g i c B i s indeed a pope,& p a r t o f J ( B ) , which i s a l s o a p h o p e h The non - i n c l u s i o n r e l a t i o n s can be thus established,taking
p a r t o f J(S5).
i n t o account Theorems 1, 2 and 4: THEOREM 5 .
B
0 n o t &Vbed undm the R d e ( 8 ) .
PROOF: With t h e h e l p o f simple Kripke diagrams, we may v e r i f y t h a t LM(Mpl 3 p,) € B and Mpl 3 p1 f? B
.
THEOREM 6 .
N o t & ivlsZances a6 ( 9 ) b d o n g t o
J(B).
PROOF: By means o f a simple Kripke diagram, we may v e r i f y t h a t J ( B ) , by d e f i n i t i o n o f J(B). ML(Lpl= LLp,) B B ; hence L(Lpl 3 LLpl)
3, THE
LOGICS
S4,
S5 AND J ( S 5 1 ,
The l o g i c s4 i s a subset o f S5, which i s a subset o f J(S5) .This chain o f i n c l u s i o n s may be a x i o m a t i c a l l y founded on Theorems 7 - 9 below. THEOREM 7 .
S 4 0 axiarnatizabLe by
mem o6 (2) and
(10) LLLA, id A 0 a M o L o g y ; (11) L ( L(A (12)
3
C)
3
L( LA
3
indeh C dhom L(A
LC));
3
C)
and
A.
PROOF: A l l theses o f t h i s axiomatics belong t o S4
, for
(2) and (10)-(12)
are c l e a r l y v a l i d i n s4. Proving t h e converse r e q u i r e s some lemmas.
The ~ o U o W i n g/uLee 0 vaeid i n t h e axiom bybteJ?i phopobed S4 i n Theahem 7 : i n 6 a A 6ham LA. LEMMA 7 . 1 .
PROOF:
T r i v i a l , by (2) and (12)
.
doh
103
SOMEREMARKS ON D I S C U S S I V E LOGIC
LEMMA 7 . 2 . The ~ o U o w i n ghu&e 0 v&d i n t h e axiom b y d t e m pkopoded doh 54 in Theohem 7: i n 6 e h L C 6hom L(A1= ( . . . = ( A n = C ) ) ...), L A j ( l < j < n ) .
PROOF: By i n d u c t i o n on n .
n = l ; from L ( A 1
CASE 1.
t h i s and L A , , CASE 2.
3
C ) , we d e r i v e
L(LA1
3
L C ) , by (11)-(12); f r o m
we d e r i v e L C , by (12).
n > 1; from LA1 and L ( A 1 =
(... = ( A n =
C ) ) ...), we d e r i v e L ( A 2
= ( ... 3 (An= C ) ) ...), by Case 1 above; from t h i s , i n d u c t i o n and LA (1 2 j 2 n ) , we d e r i v e L C .
hypothesis
j
COROLLARY. 16 C 0 a tau-totogicd cottbequence
d h v a b l e @om LA . (15 J Theohem 7 .
j
5 n)
06
A1,
..., A,,
i n t h e axiom hystem p o p o d e d
LEMMA 7 . 3 . L ( L A 3 L LA) 0 a v&d doh S4 i n Theahem 7 .
schema
06
then LC 0 60h
S4
in
t h e axiom dydtem p h o p o s e d
PROOF: L e t A be any formula, l e t C he t h e formula 1 A V A . By (10) a n d Lemma 7 . 1 , L (A = ( C 3 A ) ) i s a t h e s i s o f t h e axiom system r e f e r r e d above. From t h i s , by (11)-(12),we d e r i v e (i) L ( L A
3
L ( C =A)).
The formulas (ii) L ( L ( C = A )
(iii)
=
L(L( LC =LA)
L(LC = L A ) ) , 3
L(LLC =LLA))
a r e instances o f (ll), and
(iv) L ( L ( L L C
= LLA)
a(LLC
3
LLA))
i s an instance o f (2). From ( i ) - ( i v ) , by t h e C o r o l l a r y o f Lemma 7 . 2 , we der i v e L(LLC = ( L A
3LLA)).
3
L L A ) ) ; from t h i s , by (11)-(12),we g e t L ( L L L C S L ( L A
But L L L C i s an instance o f (10); therefore,L(LA
t h e s i s o f t h e axiom system
LLA) i s
a
r e f e r r e d above, by (12).
COROLLARY. The 60UoOuling hu&c 0 v a l i d i n t h e a x i o m a t i U p o p o s e d doh S 4
i n Theohem 7 : i n d e h L L A
dhom
LA.
Now we a r e a b l e t o complete t h e p r o o f o f Theorem 7 . I t i s w e l l
known
104
L . H. LOPES DOS SANTOS
t h a t S4 i s a x i o m a t i z a b l e by means o f a l l t a u t o l o g i e s , a l l i n s t a n c e s o f t h e schemas LA
=A
and L ( A
C)
3
L(LA
3
L C ) , m a t e r i a l detachement and Gtidel's
A € S4; i t i s n o t d i f f i c u l t t o show, by i n d u c t i o n , t h a t
r u l e . Now l e t
LA
i s a t h e s i s o f t h e axiom system proposed f o r S 4 i n t h e body o f T h e o r e m 7 , hence so i s A, by Lemma 7.1.
w i t h t h e h e l p o f Lemma 7.3;
THEOREM 8. S5 0 axiomdtizable by meam
Ln6m L A
(13)
06
(2).
(10)-(12)
and
MLA.
6hom
PROOF: S i m i l a r t o t h e p r o o f o f Theorem 7, c o n s i d e r i n g t h a t
a x i o m a t i z e d by means o f a l l t a u t o l o g i e s ,
=
L(A 3 C )
LEMMA 8 . 1
= L C)
L(LA
Lemmas 7.1-7.3
.
and
A
3
S5
can
be
a l l i n s t a n c e s o f t h e schemas LA=A,
LMA, m a t e r i a l detachement and Gtidel's r u l e .
a r e t o be c a l l e d f o r , as w e l l as:
L (A
3
LMA) 0 a v a l i d nchema a6 t h e axiam nyntem phvpaned 6vh
S5 Ln Thevhm 8.
PROOF:
I t i s e c s i l y v e r i f i a b l e t h a t a l l i n s t a n c e s o f t h e schemaML(A3LMA)
belong t o S 4; consequently, by Theorem 7, t h e y a r e a l l d e r i v a b l e by o f (2)
and
(10)-(12);
t h e r e f o r e , a l l i n s t a n c e s o f t h e schema
means
L(A
LMA)
a r e theses o f t h e axiom system proposed f o r S 5 i n Theorem 8, by (13). THEOREM 9 . J ( S 5 ) 0 axiamdtizable by meam ad (21, (14) PROOF:
Ln6m A
7.1-7.3
and
MA.
I t i s known t h a t J ( S 5 ) i s a x i o m a t i z a b l e by means o f (1)-(2)
(11)-(12)
(9),
6ham
(10)-(12)
and 8.1,
and
( 1 4 ) ( c f . Theorem 4 o f ' C o s t a
,
(4),
1975). Given Lemmas
i t i s easy t o show t h e e q u i v a l e n c e o f b o t h axiom systems.
The axiom system p r e s e n t e d f o r J ( S 5 ) i n T h e o r e m 9 i s r e a l l y an extens i o n o f t h e one p r e s e n t e d f o r S 5 i n Theorem 8, f o r any a p p l i c a t i o n o f r u l e (13) i s an a p p l i c a t i o n o f r u l e (14). The l o g i c S 5 i s indeed a p r o p e r p a r t
o f J ( S 5 ) , s i n c e we h a v e : 'THEOREM 10.
S 5 i~ not d o b e d u n d a h d e (14).
PROOF: Simple K r i p k e diagrams show us t h a t PI
3
e
M P ~ S5.
M(pl
Mpl)
€ S5 a n d
105
SOME REMARKS ON DISCUSSIVE LOGIC Although 5 4 i s a proper p a r t o f S5,J(S4) J(S5).
AS
i s n o t a proper
part
a m a t t e r o f f a c t , J(S4) i s identical t o J(S5). T h i s
t i t y , a l g e b r a i c a l l y e s t a b l i s h e d i n T. Furmanowski-1975, i s a n
of
iden-
immediate
consequence o f Theorem 9 and THEOREM 1 1 . nnd
J ( S 4 ) i b axiamatizabLe b y menflb 0 6
(2),
(10)-(12)
(14).
PROOF: L e t A € J ( S 4 ) ; t h e n MA € S 4 , by d e f i n i t i o n o f J ( S 4 ) , a n d M € S5, f o r s 5 i n c l u d e s 54; so A € ~ ( s 5 ) by , d e f i n i t i o n o f J ( S 5 ) ; therefore, A i s a t h e s i s o f t h e a x i o m a t i c s proposed f o r J ( S 4 ) , by Theorem 9.Conversely, l e t A be a t h e s i s o f t h i s a x i o m a t i c s ; by i n d u c t i o n , we show t h a t M A € 5 4 ; therefore
A € J ( S 4 ) , by d e f i n i t i o n o f J ( S 4 ) .
I f m i s a m o d a l it y
and K i s a modal l o g i c , t h e s e t {A: mA € K} i s
c a l l e d t h e m-fragment o f K. By d e f i n i t i o n , J ( K ) i s t h e M-fragment o f f o r any
K . I t i s n o t d i f f i c u l t t o v e r i f y t h a t B and
K
,
S5 a r e t h e L-frag-
ments r e s p e c t i v e l y o f J ( B ) and J ( S 5 ) . Nevertheless, t h e analogous assumpt i o n c o n c e r n i n g S 4 i s n o t t r u e : t o g e t h e r w i t h Theorem 11, i t would l e a d us t o t h e f a l s e c o n c l u s i o n t h a t S4 c o i n c i d e s
4, THE LOGICS
s4,
s5 AND
w i t h S5.
J(B)
The l o g i c s s4 and s 5 n e i t h e r a r e subsets o f J ( B ) n o r have i t as a subset.
p1 belongs t o J ( B ) and does n o t belong n o r o f S 4 . On t h e o t h e r hand, L(Lpl=- LLp,) belong t o S4 and i t does n o t belong t o J ( B ) ; so J ( B ) i n c l u d e s n e i t h e r S4 n o r S5.
On t h e one hand, t h e f o r m u l a Mpl= t o s 5 ; so
J ( B ) i s a subset n e i t h e r o f S 5
PART I 1
1, THE
AXIOM
SYSTEM J*,
We s h a l l c o n c e i v e t h e axiom system J* as expressed i n
a
language
whose p r i m i t i v e symbols are: a denumerable i n f i n i t e s e t o f i n d i v i d u a l v a r i ables; a nonvoid s e t o f n - a r y p r e d i c a t e l e t t e r s , f o r each n a t u r a l number n ,
106
L . H. LOPES DOS SANTOS
n 2 1; the c l a s s i c a l connectives i and V; t h e necessity operator L; t h e ex-
i s t e n t i a l q u a n t i f i e r 3 ; and parentheses. The s e t o f formulas i s r e c u r s i v e l y d e l i m i t e d as usual. The l e t t e r s x and q w i l l be employedas s y n t a c t i c a l vari a b l e s f o r i n d i v i d u a l v a r i a b l e s o f t h e object-language. standard r e s t r i c t i o n ,
to
the
the notation A ( x l y ) w i l l r e f e r t o the f o r m u l a
Subject
ob-
t a i n e d from A by r e p l a c i n g each f r e e occurrence o f x i n A by an o f q. F i n a l l y we i n t r o d u c e t h e f o l l o w i n g d e f i n i t i o n s :
D f . 1.
(A
D f . 2.
( A & C)
=df. l ( 1 A V l C ) ,
D f . 3.
(A
=df. ( ( A
D f . 6.
(1A V C),
=df.
C)
( A A C)
C) & (C
,
=df.
(MA
3
=
( M A & C).
df.
( A ->C)
3
1LlA
MA =
D f . 4. D f . 5.
C)
3
occurrence
df.
3
A)),
C),
J* h a s t h e f o l l o w i n g a x i o m schemas and r u l e s :
J*l.
L A , id A . i + a .tautology.
J*2.
L(LA
J*3.
L(L(A
3
J*4.
L(A
LMA).
J*5.
L n d e h C &tom L ( A
J*6.
1.ndehA dhom MA.
J*7.
L(A(x/y)
J*8.
1n6eh L ( 3 x A
3
3
A). C)
=
The n o t a t i o n J* I-
3
L(LA
3
LC)).
C ) and A.
3xA). C)
Ahom
L(A
3
C), id
x h
nv-t dhee i n C.
A means t h a t A i s a t h e s i s o f J*.
d i 6 C U A b i V t Logic amoiated w i t h t h e mod& phedicate logic 5 5 ( w i t h t h e B a a c a n d o m u l a ) .
THEOREM 1 2 . J* i d an adqyuate axiamalics doh t h e
PROOF: I n Theorem 6 o f Costa 1975 an axiomatics f o r t h e d i s c u s s i v e l o g i c associated w i t h t h e modal p r e d i c a t e l o g i c S5 i s p r e s e n t e d ; t h e e q u i v a -
107
SOME REMARKS ON DISCUSSIVE LOGIC
l e n c e between t h i s a x i o m a t i c s and J*, w i t h r e s p e c t t o t h e u s u a l d i f i n i t i o n s o f t h e q u a n t i f i e r s , can be e a s i l y shown.
2, THE AXiOM SYSTEM 3* The language o f t h e axiom system
J*
must comprise as p r i m i t i v e s y m b o l s
t h o s e o f J * w i t h t h e e x c e p t i o n o f t h e n e c e s s i t y o p e r a t o r , which i s t o b e r e p l a c e d by t h e b i n a r y d i s c u s s i v e c o n n e c t i v e s +and
A. In
;i* we
retain
D e f i n i t i o n s 1-3 and add: D f . 7.
LA =
( 1 A ->
D f . 8.
MA =
ILIA,
D f . 9.
OA =
l(A VIA)
df. df. df.
3*
The s e t o f theses o f
l(A V lA)),
i s c h a r a c t e r i z e d by t h e f o l l o w i n g axiom sche-
mas and r u l e s : Ax.Dl.
A 4( C + A ) .
Ax.02.
(A
Ax.03.
((A +C)
Ax.04.
(A A C) + A .
Ax.05.
( A A C) -> C .
Ax.06.
A ->(C
->(A
Ax.07.
A ->(A
V C).
Ax.08.
C ->(A
V C).
Ax.09.
(A
Ax.1.
A -.llA
4
(C
AD))
4( ( A 4 C )
->A)
+A.
A C)).
->
((C
Ax.2.
1 1 A ->A.
Ax.3.
l ( A V lA)->
C.
Ax.4.
l ( A V C) ->
l ( C V A).
Ax.5.
l ( A V C) ->
(1A A 1C).
Ax.6.
l ( l 1 A V C) ->
+U)
+ ( A +D)).
->U)
->((A
l ( A V C).
V C)
->D)).
L. H . LOPES W S SANTOS
108
->D) ->
Ax.7.
( l ( A V C)
Ax.8.
l((A V C) V
Ax.9.
l((A->
Ax.10.
l ( ( A A C) V
Ax.11.
1(1(A V C) V
Ax.12.
- I ( I ( A ->C)
Ax.13.
l ( l ( A A C) V
Ax.14.
l(A(x/y)
Ax.15.
i(3 xA
U ) -> l ( A U ) ->
C) V
=I
3
U)->
D).
((lA4C) V
D)).
V (C V
(A A 1(C V (A->l(C
U ) ->
V
a)). D)).
U) V l(1C
(l(1A V
U ) -> ( A - > l ( l C
V
U ) ->
D)).
a)).
(A A l ( 1 C V 0 ) ) .
3 x A ) ->
C ) ->
V
V
3xi(A
R1.
ln6eh C 6~om A ->C
R2.
Indm 3 x A ->C
l(A 3
3
A).
i d x 0 n o t 6hee i n C .
C)
and A. i d x A nut dhee i n C.
6hom A ->C,
The notation j*I- A means that A is a thesis if
3,
EQUIVALENCE BETWEEN J* AND
J*
J*,
Two axiom systems A and A ' are said to be equivalent with respect to a given set of definitions when: (i) this set is enough to assure general and univocal translability of the language of one system to the language of the other; (ii) any formula of A is a thesis of A if and only if it abbreviates in A ' , according to the definitions, a thesis of A ' ; (iii) any formula o f A' is a thesis of A' if and only if it abbreviates in A, according to t h e definitions, a thesis of A. We will prove that J* and j* are equivalent with respect to Definitions 5-7. Condition (i) above is obviously fulfilled by Definitions 5-7. If A is a formula of J * , let TI-A be the formula of 5* abbreviated by A according to Definition 7; if C i s a formula of J*,let T r C be the formula of J* abbreviated by C according to Definitions 5-6. Conditions (ii) and ( i i i ) may now be so rewritten: (ii') for any formula A of J*, J* I- A if and only if f* I- TFA; (iii) for any formula C of i*, *: I- C if and only if J*I- T r C . Proving their f u l f i l l m e n t requires s o m e lemmas. LEMMA 1 . The @%xdng
h d e A v&d
in
5.:
in6eh A ->U
d h a m A ->C
109
SOME REMARKS ON D I S C U S S I V E L O G I C
and C
3
D. i s w e l l k n o w n t h a t t h e r u l e i s d e r i v a b l e by means o f Axioms 01-
PRO0F:It
02
and R1.
LEMMA 2. j* IPROOF:
L(A(x/y) 3 3 x A ) .
B y A x i o m 4 and D e f i n i t i o n 1,
-
J* I- 1 ( A = A ) ->
(i)
by (i), Axiom 3
l ( A V 1A);
and Lemma 1 ,
( i i ) j* I- l ( A 3 A ) + O ( A ( x / y ) a 3 x A ) ; by Axiom 14, ( i i ) and Lemma 1 ,
z*
(iii)
+ O(A(x/y)=3xA).
I- i ( A ( x / y ) = 3 x A )
A p p l i c a t i o n s o f D e f i n i t i o n s 7 and 9 on ( i i i ) complete t h e p r o o f
LEMMA 3. The @ % ~ 4 ~ L hlLee n g b detLivabte i n :* : i n d e t L( 3 x A L ( A 3 C ) i d x h not &ee i n C. PROOF:
By D e f i n i t i o n s 7 and 9, L ( A
(i) by Axiom 3
C) i s
i ( A 3 C ) -> O(A 3 C ) ; and D e f i n i t i o n 9,
-
J* I-
(ii)
3
= C ) 6aom
O(AzC)->
O(3xA3C);
so we d e r i v e f r o m ( i ) . w i t h t h e h e l p o f (ii) and Lemma 1, l ( A 3 C ) -> O( 3 x A 3 C ) . (iii) L e t u s suppose t h a t x i s n o t f r e e i n C ; t h e n i t i s n o t f r e e i n
O(3xA
3
C);
t h e r e f o r e , we can d e r i v e 3 x l ( A 3 C ) -. O( 3 x A 3 C ) (iv) f r o m ( i i i ) , by R2; f r o m ( i v ) we get, by Axiom 15
->O( 3 x A
3
C ) , which i s
L( 3 x A
LEMMA 4 . Undemtanding L me vaeid i n j*
M a
.
PROOF:
3
C),
and Lemma 1, l ( 3 x A = C )
by D e f i n i t i o n s 7 and 9.
dedined symbot
06
:*,
t h e ochernab J * l - J * 6
C f . Costa and D u b i k a j t i s , t h e s e Proceedings, P a r t 11,
7, 9-11, 13, 20, 23, 26-29. LEMMA 5 .
16 J* I-
A,
then
PROOF: S t r a i g h t f o r w a r d by
j*
I-
Lemmas 2-4.
TFA.
Theorems
110
L.
, then
16 j*I- C
LEMMA 6 .
Understanding ->
PROOF:
H.
LOPES DOS SANTOS
J* I- TrC.
and A as d e f i n e d symbols o f J * , Axioms 01-A15
and R1-R2 a r e e a s i l y shown t o be v a l i d i n J*. 16 0 0 a .taLc*oLogicd comcquence
LEMMA 7 .
hiwabLe
dhom LA and
*;
LC i n
PROOF: L e t us suppose t h a t
06
A and C, t h e n LD 0 d e -
D i s a t a u t o l o g i c a l consequence o f A and
C ; by
Lemma 4 ,
(i)
j* I- L(A = ( C =D)).
(ii)
J* I- L(L(A
(iii)
j*
By ( i ) ,
-
+ L ( L ( C ID)=
=D))
L ( LA = L ( C ID),)),
I
L( LC DLU)):
Lemma 4,
( i i ) and
-
(iv)
I (C
I- L ( L A = L(C ID)).
J*
From LA, ( i v ) and Lemma 4, we d e r i v e (v)
L(C
= D);
f r o m (iii), ( v ) and Lemma 4 we g e t (vi)
L( LC
L D f r o m LC and ( v i ) , by Lemma
f i n a l l y we d e r i v e LEMMA 8.
J*
LD);
I
I- L((A->
j* I-
C) G ( M A
I C))
4.
and
L ( ( A A C ) E. ( M A & C ) ) .
PROOF: T r i v i a l , by Lemma 7 and Theorems 30-33 i n P a r t
I1 o f Costa a n d
D u b i k a j t i s , t h e s e Proceedings. LEMMA 9 .
*;
I- L ( ( T r T r A ->
J * I-
TrTrC)
L((T?TrA A T r T r C )
z r r T r ( A + C))
and
T r T r ( A A C)).
PROOF: C o n s i d e r i n g t h a t T r T r (A 4 C ) and T r T r ( A A C ) a r e r e s p e c t i v e l y
(MTrTrA
3
T z T r C ) and ( M T r T r A & T r T r C ) ,
t h e lemma i s an i m m e d i a t e con-
sequence o f Lemma 8. LEMMA 1 0 . The &UouLing h d e Lh waeid i n LfA
3
C).
5.:
L n 6 c h L ( 3 x A =_ 3 x C )
@om
LOGIC
SOME REMARKS ON D l S C U S S l V E
PROOF: From L(A
(i)
111
C ) we d e r i v e , by Lemma 7,
L(A = C )
and
L(C 3 A ) .
B u t Lemma 2 assures t h a t
s* I- L(C
(ii)
and j* I- L(A
3xC)
3
3
3xA);
so we d e r i v e (iii)L(A
3xC)
3
and
L(C
3
3xA)
and ( i i ) , byLemma 7. From ( i i i ) , by Lemma3, we g e t from (i) (iv)
L( 3 x A 3 3 x C )
and
L( 3xC
f r o m t h i s , by Lemma 7, we d e r i v e
LEMMA 1 1
.
3
L( 3xA
3xA); f
3xC).
The 6oLtotuLng hLLee 0 dehivable i n
3.:
i n d e h L ( MA
MC)
L(A E C)
dhom PROOF:
From
(i)
L(A
?
C ) we d e r i v e , by Lemma 7,
L(lA= iC)
and
L(iC = l A ) ;
by Lemma4, (ii)
J* I-
L(L(1A 2 1 C ) = L ( L l A Z L l C ) ) ,
(iii)j * I- L(L(1C 2 1 A ) 3 L ( L1C 3 L l A ) ) . By t h e same lemma, we d e r i v e
(iv)
L(L1A
from ( i ) - ( i i i ) ;
3
LlC)
J*
L(L1C X L l A )
f r o m ( i v ) , by a n o t h e r a p p l i c a t i o n o f the1,emma
L ( l L 1 A E l L l C ) , which i s LEMMA 1 2 .
and
I- A
L( MA-MC)
,
get
id and o n l y id j* I- T Z T r A.
PROOF: W i t h t h e h e l p o f Lemmas 7-11,
we show e a s i l y t h a t J * I - L ( A z T r T r A ) ,
by i n d u c t i o n on t h e l e n g t h o f A.(Remember t h a t T r T r l C , T r T r (C V 0 ) Tr"Tr3xC a r e r e s p e c t i v e l y
lTrTrC,
lemma f o l l o w s f r o m t h i s r e s u l t by
.
and
( T r T r C V T Z T r D ) and 3 X T r T r C ) . T h e Lemma 7.
LEMMA 1 3 . 16 D 0 a t a u t o L o g i c a t comequence hivable dhom LA and LC in J* PROOF:
we
by D e f i n i t i o n 8.
06
A and C, t h e n L D
de-
I t i s enough t o make o b v i o u s m o d i f i c a t i o n s i n t h e proofofLemma 7.
112
L . H. LOPES DOS SANTOS
LEMMA 14. J* PROOF:
I- L( L A
(1A ->l(A
V 1A))).
I t i s easy t o v e r i f y t h a t t h e c w s i d e r e d schema i s v a l i d i n
the
p r e d i c a t e l o g i c S5, which i s included !n t h e s e t o f theses o f J*.
LEMMA 1 5 . The ~ o ~ u ~ u 4i nd ge 0 u a e i d i n J* : indm L(3xA :3xC) 64om L(A
= C).
PROOF:
I t i s e n o u g h t o make o b v i o u s m o d i f i c a t i o n s i n t h e
proof o f
Lemma 10.
LEMMA 1 6 .
L(LTrTFA ETrTrA).
J* I-
PROOF: Considering t h a t T r TFLA i s (1Tr T? A ->lTr
T r A V 1 T r T r A)), t h e 1 emma
i s an immediate consequence o f Lemma 1 4 ,
LEMMA 1 7 . The 6oaRowing 4 d e L(A
= C).
s L C) 64om
0 detLivabLe i n J*: i n 6 e h L( L A
PROOF: From L(A E C), by Lemma 13, we g e t
(i)
L ( A = C)
and
L ( C = A);
from ( i ) and conveniently chosen instances o f J * 3 , we d e r i v e
=I
( i i ) L(LA
LC)
and
L(LC
3
LA),
by J * 5 . From ( i i ) a n d Lemma 13, we d e r i v e L ( L A
LEMMA 1 8 . J *
I- A
i6 and o n l y i6 J*
I-
=
LC).
T r TFA.
PROOF: By t h e same method employed i n t h e p r o o f o f
Lemma 12,
with
the
help o f Lemmas 13-17.
THEOREM 1 2 . J* I- A i6 and ondy id
-
3* I-
J* I- C i6 and o d y i6 J* IPROOF:
I f J* I- A, then
i* I- T
?*
I-
Tr C
.
~ A ,by Lemma 5; conversely,
then J * I- T r T r ” A , by Lemma 6, and i f and o n l y i f
TEA;
J* I - A ,
i f 3 * I-TFA,
by Lemma 18; hence J * I- A
T ~ A .We show s i m i l a r l y t h a t ;*I- C i f and
only if
J* I- T r C , w i t h t h e h e l p o f Lemmas 5, 6 and 12. We have thus seen t h a t j* i s e q u i v a l e n t t o t h e
discussive l o g i c J*
SOME REMARKS ON DlSCUSSlVE
associated w i t h t h e p r e d i c a t e l o g i c S5
.
113
LOGIC
I t i s worth noting t h a t
3*
i s
also equivalent t o the discussive l o g i c associated w i t h the predicate l o g i c S4.
T h i s d i s c u s s i v e l o g i c i s indeed i d e n t i c a l t o J*, a s we may
by t h e same methods employed i n P a r t I i n o r d e r t o show t h a t J ( S 4 ) cides
verify coin-
with J(S5).
REFERENCES C o s t a , N . C. A. da 1975,
Remmhn on Jao-kowski d i 6 c u s o i v e l o g i c , Reports o n Mathematical Logic, 4, 7-16.
Costa, N . C . A. da 197+,
and L. D u b i k a j t i s
On J G k o w s k i d i s c w n i u e Logic,
t h e s e Proceedings
.
Furmanowski, T. 1975,
Remmhn on di6cubniue poponLLLonul CaecLLeub, Studia Logica, 34, 39-43.
Jackows k i , S. 1948,
Rachunek zdaM &a ~ y ~ . t e m &dedukcyjnych u n p m x z n y c h , S t u d i a Soci e t a t i s S c i e n t i a r u m Torunensis, S e c t i o A, I , nP 5, 57-77. ( A n E n g l i s h t r a n s l a t i o n o f t h i s paper appeared i n S t u d i a Logica, 24 (1969), 143157.)
Centro de Logica. Epistemologia e Hist o r i a da C i s n c i a Uqiversidade Estadual de Campinas Campinas, G o Paulo, B r a z i l .
PART I 1 MODEL THEORY
Non-Classical Logics, Model Theory and Computability, A . I . Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
SOME DIRECTIONSI N MODEL THEORY ( * I by
MIROSLAV BENUA
T h i s paper a t t e m p t s t o expose and a l e r t a s t u d e n t o f model t h e o r y t o new areas of r e s e a r c h . We, o f course, have t o make t h e u s u a l d i s c l a i m e r , t h a t because o f t h e tremendous g r o w t h o f model t h e o r y n o t h i n g c l o s e t o comp l e t e n e s s can be done i n an h o u r ' s l e c t u r e . To g i v e an example, we do discuss t h e i n t e r e s t i n g concept o f r e c u r s i v e model t h e o r y
proposed
not i n
Barwise and S c h l i p f 1976. (We m e n t i o n i t because o f a s t i l l unsolved problem which i s s i m p l e t o s t a t e and w a r r a n t s g r e a t e r d i s s e m i n a t i o n :
every
if
c o u n t a b l e model o f a t h e o r y i s r e c u r s i v e l y s a t u r a t e d , i s t h e t h e o r y
w-cate-
gorical ?) The c h o i c e h a s been made f i r s t l y by t h e f r e s h n e s s o f t h e m a t e r i a l and secondly by t h e d e s i r e t o b r i n g o u t as many c o n n e c t i o n s t o o t h e r f i e l d s ( s e t theory, p r o b a b i l i t y , computer s c i e n c e ) as p o s s i b l e . We avoided well-kown d i r e c t i o n s l i k e t h e outcomes o f M o r l e y - B a l d w i n l L a c h l a n c i t y work, f o r c i n g ,
therefore categori-
e t c . The vague problem i n t h e l a s t s e c t i o n i s aimed a t
t h e p h i l o s o p h i c a l l y minded members o f t h e audience; however, we
wish that
a p o s s i b l e outcome were p r e c i s e and t e c h n i c a l . To g i v e an i d e a o f t h e work i n model t h e o r y done i n t h e p a s t we
dis-
cuss b r i e f l y t h e m a j o r problems (and a g a i n we j u s t choose) model t h e o r y h a s been f a c i n g f o r some t i m e . More complete l i s t s may be f o u n d i n
Friedman
1975, and f o r model t h e o r y i n p a r t i c u l a r i n Robinson 1973.
(*) ro
P r e p a r a t i o n o f t h i s p a p e r was p a r t i a l l y s u p p o r t e d b y Fundagao d e Ampa-
5
P e s q u i s a do Estado d e Sao P a u l o (FAPESP)
e P r o j e t o s (FINEP), B r a z i 1 .
117
and
F i n a n c i a d o r a d e Estudos
118
MIROSLAV BENDA
THREE OUTSTANDING PROBLEMS Vaught's conjecture: The numbeh 0 6 c o u n t a b l e models 0 6 u t h e o h y in 5 w OR 2w. Foh a mohe g e n e h a t q u e s t i o n h e p l a c e " f h e o h y " by
a sentence
06
1
w1 w
.
T h i s c o n j e c t u r e has s t i m u l a t e d and i s s t i m u l a t i n g a l o t Our work on modeloids,on
of
research.
which we r e p o r t e d b e f o r e t h e c o n g r e s s , w a s p a r t i a l l y
i n f l u e n c e d by t h e d e s i r e t o p r o v e t h e c o n j e c t u r e . The f i r s t g e n e r a l
result
i s due t o M o r l e y (see M o r l e y 1970). The number o f c o u n t a b l e models o f a s e n tence
of
L
w1 w
i s 5 w 1 o r 2w
.
M o r l e y ' s r e s u l t and i t s e x t e n s i o n s
now w e l l understood thanks m a i n l y t o t h e work
are
o f Vaught (Vaught 1974),
B u r g e s s - M i l l e r (Burgess and M i l l e r 1975),and o t h e r s . C l o s e s t t o p r o v i n g t h e c o n j e c t u r e came G. Sacks whose work i s n o t y e t p u b l i s h e d b u t an account i t may be found i n H a r n i k and Makkai 1976. H a r n i k and M a k k a i , i n f a c t ,
of
proved
i n d e p e n d e n t l y a r e s u l t c l o s e t o Sacks' which i s easy t o s t a t e : a P C w l c l a s s whose c o u n t a b l e members have o n l y c o u n t a b l y many automorphisms has e i -
5 w or
ther
2 w c o u n t a b l e models. Because o f i n d i c e r n i b l e s ,
no f i r s t - o r -
d e r t h e o r y s a t i s f i e s t h e assumptions o f t h i s theorem. Sacks' r e s u l t hasmuch m i l d e r c o n d i t i o n on t h e automorphism s a t i s f i a b l e i n some E C A
classes.
t h i s r e s p e c t a r e s u l t o f M. Rubin and S.Shelah (see Rubin and Shelah
In
1975)
( v e r y ) weakly complements t h e above: I f a t h e o r y T has Skolem f u n c t i o n s and a l i n e a r c j r d e r i n g o f t h e u n i v e r s e t h e n i t has 2w c o u n t a b l e models. ( T h i s e x tends o u r e a r l i e r r e s u l t ' t h a t such a t h e o r y has 2 w models.) Most models o f t h e s e t h e o r i e s have 2 u automorphisms.
0 4 c a h d i n a l i t i e s 06 o p e c t h a c l o s e d undeh comple-
The spectrum problem: A ApeCthUm is t h e n e t
d i n i t e modePo mentation ?
06
a sentence
@.Ahe
T h i s problem was posed by H. Scholz i n 1957. M o s t o f t h e
research
w e n t i n t o establishing t h a t the spectra a r e very e f f e c t i v e sets o f n a t u r a l numbers. T h i s d i r e c t i o n was c u l m i n a t e d by Jones and Selman who
showed
t h a t t h e s p e c t r a a r e e x a c t l y t h e s e t s o f n a t u r a l numbers which a r e r e c o g n i z a b l e by n o n - d e t e r m i n i s t i c , T u r i n g machines t i m e bounded by 2", s t a n t , x l e n g t h o f i n p u t (see Jones and Selman 1972). T h i s l i n k s famous
P = NP problem t o t h e spectrum problem: I f P = NP t h e n
c a con-
the
now
spectr.
a r e c l o s e d under complementation.0n t h e m o d e l - t h e o r e t i c s i d e t h e problem was
SOME D I R E C T I O N S ON MODEL THEORY
recently investigated by Fagin (Fagin 1976 contains basic references)
119 who
among other things proved t h a t f o r any sentence I$ the spectrum of I$ or l+ i s c o f i n i t e . I n f a c t the s e t of sentences which a r e eventually true in a l most a l l models of cardinality n i s a complete and decidabletheory(studied e a r l i e r by Gaifman). Morley's conjecture: L e t T b e a t h e v h y . The Lahgeh t h e u n c v u n t a b l e c a h d i n a l , t h e mVhe m o d e l s 0 6 T v b t h a t c a h d i n a l i t y .
All the r e s u l t s we know of in t h i s direction are due t o Shelah and i t i s d i f f i c u l t t o give a simple account of them. R a t h e r we r e f e r the reader to his Tarski's Symposium lecture and t o Shelah 197+ ( s p e c i f i c a l l y t h e s e c tion on p. 283).
FINITE MODEL
THEORY,
Classical model theory has the ultimate r e s u l t about f i n i t e m o d e l s .
Two f i n i t e models are elementarily equivalent i f f they a r e isomorphic. Of course, t h i s i s an overstatement, as we have seen in the discussion o f the Scholz' problem. B u t the f a c t remains t h a t most methods and r e s u l t s of model theory apply almost exclusively t o i n f i n i t e models. The f i r s t concentrated e f f o r t on f i n i t e models was i n i t i a t e d oy Keisler: we s h a l l report
on some of his r e s u l t s which open unexpected connections. The basic idea i s t o study i n f i n i t e models which appear f i n i t e in some model of mathematics. The idea of using non-standard models t o get standard r e s u l t s was used by A.
Robinson and even e a r l i e r by C. Ryll-Nardzewski. B r i e f l y , l e t M be mathematics, by which we mean the collection of objects and relations in the public domain of mathematicians. This can be considered as a n ill-defined structure; model theory showed how we can get en extension of i t , M* , which c o u l d be characterized by saying t h a t the following holds about M*: whateveh c o u l d b e , i n .
S p e a k i n g p r e c i s e l y , M * i s saturated or a t l e a s t tol-saturated a n d t h e phrase " v b j e o t c v d d be" means t h a t the existence of the o b j e c t i s n o t forbidden by a f i r s t order formula. (These explanations are the main reason f o r the slow acceptance of non-standard methods). So M* h a s i n f i n i t e n a t u ral numbers because there i s no f i r s t - o r d e r formula which says t h a t e v e r y n a t u r a l number i s e i t h e r 0 or 1 or ... . M * i s richer than M in some
120
MIROSLAV BENDA
aspects b u t i n o t h e r s i t i s p o o r e r .
M* does n o t r e c o g n i z e
the
set
s t a n d a r d n a t u r a l numbers, f o r example; t h e o b j e c t s i t does r e c o g n i z e
of are
c a l l e d intmnd. A theorem a b o u t an i n t e r n a l o b j e c t can be t r a n s f o r m e d i n t o an i n f o r m a t i o n a b o u t p r o p e r t i e s o f s t a n d a r d o b j e c t s ( t h i n k o f M* as an u l -
M-
t r a p o w e r ) . K e i s l e r p u t s i t by s a y i n g " M * - ~ i n i t e mode& e x k i b d t h e
i n g behavioh od d i n i t e mode&". B u t t h e i n t e r n a l model t h e o r y o f M* i s t h e same as i n M so we
have
n o t g a i n e d much. And i t i s h e r e where we s h o u l d ask o u r s e l v e s t h e q u e s t i o n : "Why do we want t o s t u d y f i n i t e models". The answer t o t h i s i s c l e a r ,
fi-
n i t e models a r e everywhere. L e t us t a k e an example:
EXAMPLE, We have s u b j e c t s and want t o t e a c h them t o
distinguish
black
f r o m w h i t e . We can g e t o n l y s t a t i s t i c a l r e s u l t s so we s h o u l d s t u d y l a r g e r groups and we i n f a c t make t h e l e a p t o s t u d y an i n f i n i t e group A o f s u b j e c t s b u t one which i s f i n i t e i n M*. L e t
F ( a ) be t h e p r o b a b i l i t y t h a t a sub-
j e c t ~l e a r n s t h i s on t h e f i r s t t r i a l , and assume t h a t
F
we can t r a n s f e r t h e r e s u l t s back t o M . Q u e s t i o n s which about t h i s s i t u a t i o n a r e :
i s internal sothat m i g h t be
asked
( 1 ) what i s t h e p r o b a b i l i t y t h a t a s u b j e c t l e a r n s
t h e t a s k a f t e r two sessions (assume he does n o t f o r g e t i t ) . ( 2 )
what
i s
t h e average p r o b a b i l i t y o f l e a r n i n g t h e t a s k a f t e r two sessions. The answer t o ( 1 ) i s G(u) = F ( u )
+
(1 - F(a)).F(a),
and t o t h e second q u e s t i o n , ZG(a). 1 aEA IAl
o r i f we d e n o t e M * ) we h a v e
JG du
p(a)
=
&, t h e u n i f o r m measure on P(A)
(taken i n
.
O f course, t h e t a s k o f model t h e o r y i s n o t i n f i n d i n g
F(a) n o r
i n making t h e computations. The t a s k i s t o a b s t r a c t f r o m such examples
even and
seek r e s u l t s about t h e a b s t r a c t i o n s . K e i s l e r proposes t o s t u d y models where A i s i n t e r n a l t o measure on
P(A)
(see K e i s l e r 1976)
(A, p ,
Fi)i
I
M*, M * - f i n i t e b u t i n f i n i t e ; p i s a p r o b a b i l i t y Fi, i E I , i s an i n t e r n a l M * - f i n i t e
( t a k e n i n M*) and
sequence o f f u n c t i o n s o f v a r i o u s numbers o f arguments f r o m A i n t o
Io,1]*
121
SOME D I R E C T I O N S ON MODEL THEORY
( t h e r e a l s o f M*). We t a k e a h i n t f r o m t h e example and b u i l d o u t o f t h e f u n c t i o n s t e r m s u s i n g c o n t i n u o u s f u n c t i o n s f r o m R" i n t o
i n t e g r a t i o n w i t h r e s p e c t t o t h e measure (see K e i s l e r f o r d e t a i l s ) . a powerful language s t r o n g e r t h e n
Fi
and
R , s u p , inf
L
L w w b u t weaker t h a n
:if
w1 w
This i s
Fi admits
o n l y 0 and 1 as v a l u e s and we i n t e r p r e t 1 as "tfLue" t h e n we h a v e a n o r d i nary r e l a t i o n a l s t r u c t u r e
and any f i r s t - o r d e r s t a t e m e n t about i t
expressed by s a y i n g t h a t a c e r t a i n t e r m has v a l u e 1:
use
can
1 - x i n s t e a d o f n e g a t i o n , and sup i n s t e a d
instead o f conjunctions,
t h e e x i s t e n c i a l q u a n t i f i e r . Thus,the continuous f u n c t i o n s
be
multiplication of
a c t as connec-
t i v e s , s u p and inf as q u a n t i f i e r s . The i n t e g r a l i s a k i n d o f q u a l i t a t i v e q u a n t i f i e r not expressible i n L w w "about h a l f o f t h e elements s a t i s f y
.
I t a l l o w s us t o make statements
like
by w r i t i n g
$'I
+ / $ ~ I J
2
( 4 = Q ( x ) i s a term here = 1 i f Q i s t r u e and pression
0 otherwise); i f t h i s
i s - 1 we, as b e f o r e , t h i n k o f i t as t r u e and i t does mean
exthat
about h a l f , a c c o r d i n g t o 1-1, elements s a t i s f y Q . The s i m p l e s t most u s e f u l r e s u l t i n m o d e l t h e o r y Lllwenheim
r
theorem. To f o r m u l a t e t h e r e s u l t i n t h e p r e s e n t s e t - u p K e i s l e r
d e f i n e s as s u b s t r u c t u r e o f
85 A
i s t h e Skolem-
i s i n t e r n a l and
u,(a) =
( A , IJ
, Fi)
a structure
p8
i s t h e measure:
if
aEB
-
a 18 =
( A , u B , FJ
where
0 i f not.
So o n l y t h e exprcssons u s i n g i n t e g r a t i o n ( t h e q u a l i t a t i v e q u a n t i f i e r ) may have changed meaning. When t h e y do n o t we c a l l t h e s u b s t r u c t u r e elementary:
01 18 and
OZ
i s a n e l e m e n t a r y substructure o f any
al...an E A
(repeat A
a
i f f o r any term
t ( q .. . w n )
s e e t h e reasons above) t e v a l u a t e d i n
i s i n f i n i t e s i m a l l y close t o t evaluated i n
02 18 ( a t a l . . .a,).
SKOLEM-LOWENHEIM THEOREM ( K e i s l e r ) :
Let ti = \ A \ log
n 181 = n a n d
+
111 l o h a t
= 0 021 B
and
01 = ( A , I J , F ~ ) ~I 0 ab bedone. L e i n be n u c h n2 0 Thgn t h e h e in a 8 C_ k Auch t h a t
- z
IAl
.
0 a n eLementa/ry oubnthuctwre
06
m.
Tn duct aemobt
122
MIROSLAV BENDA
~ ( € 3 ) = 0. The i m p o r t o f t h e theorem t o s t a n d a r d models i n a
Note t h a t
s p e c i a l s i t u a t i o n i s t h i s : assume we have a c o n s t a n t term
oln = (n,
u n , d,...
no such t h a t i s
6,)
tarn. Then g i v e n tan=tmn f oIr B some B c n
such t h a t t Oln=
">no
then
t
and
E>
models
0 , there
is
and 1Blcn.E.
The p r o o f o f t h e theorem uses t h e weak l a w o f l a r g e numbers Stone-Weierstrass theorem 2
and t h e The theorem may be proved w i t h o u t t h e a s s u m p t i o n
.
n / I A l = 0 b u t t h e p r o o f i s e a s i e r w i t h i t . Under t h i s assumption a l m o s t a l l 6 : n + A a r e one-to-one. The computation o f t h i s i s s i m p l e and p r e sents a t i n y b i t o f t h e use o f non-standard p r o b a b i l i t y used t h r o u g h o u t K e i s l e r ' s work so we show i t i n f u l l : we t h i n k o f t h e f u n c t i o n d : n -->A as choosing n - t i m e s an element f r o m A r e p l a c i n g i t a f t e r each c h o i c e . The p r o b a b i l i t y o f choosing a one-to-one sequence i s 1 ). m
1.(1-
as
(
I
1
-
m
~
(1
- 2) ...(1 - 2 ) L (1 - L ) " . IAl
IAl
IAl
=) e-1 f o r i n f i n i t e m
B u t t h e l a w o f l a r g e numbers i s used i n t r y i n g t o show t h a t
( /t(x) d p p f o r almost a l l k-th choice
i s close t o B E A,
181 = n . 8 i s o b t a i n e d by n c h o i c e s f r o m A . W i t h x, = t ( k - t h c h o i c e ) . The ex-
we a s s o c i a t e a random v a r i a b l e
p e c t a t i o n o f each
E
(/.t(x)dp)'
xk
is
= (f.t(x)dp)"
a n d s i n c e we a r e r e p l a c i n g , x k ' s a r e independent. The l a w
of
large
numbers says t h a t 1 n
I k 2")
the
because t h e m a c h i n e
cannot even r e a d t h e i n p u t . We, t h e r e f o r e , c o n s i d e r t h e number o f s t e p s r e q u i r e d t o f i n i s h t h e computation b u t i n dependence o n t h e i n p u t .
The m o s t
125
SOME D I R E C T I O N S ON MODEL THEORY
n a t u r a l a b s t r a c t s e t t i n g would be a l g e b r a s on A* quences o f members o f
the set o f f i n i t e
se-
A.
I n o r d e r t o g i v e an example o f t h e use o f c o s t subalgebrasweshow t h a t t h e y were i m p l i c i t l y used i n an o l d p r o o f o f Ryll-Nardzewski 1952.
THEOREM ( Ryll -Nardzewski ) . Peanv a h i t h e t i c .LA n v t binLt&y axivmcLtizabte.
PROOF: Assume t h e c o n t r a r y . Then we have f i n i t e l y many axioms and
because
the
what we
t h e o r y has d e f i n a b l e Skolem f u n c t i o n s ( v i a t h e w e l l - o r d e r i n g )
have, e s s e n t i a l l y , i s f i n i t e l y many f u n c t i o n s and axioms w i t h o u t q u a n t i f i e r s which t e l l us t h e p r o p e r t i e s o f t h e f u n c t i o n s . The c o n s e q u e n c e
of
t h i s i s t h a t any s e t i n any model o f a r i t h m e t i c which i s c l o s e d u n d e r t h e f u n c t i o n s i s a model o f t h o s e f i n i t e l y many axioms. The n a t u r a l numbers t o g e t h e r w i t h t h e f u n c t i o n s g i v e r i s e t o a c o s t a l g e b r a somewhat
like
the
a l g e b r a i n Example 2. I f we t a k e f i n i t e l y many numbers, among them 0, e v e r y number can be expressed as a term i n t h e f u n c t i o n s and t h e f i n i t e l y "generators".
We d e f i n e t h e
cost
o f computing
6
many
a t n as t h e com-
p l e x i t y o f n (as a term) p l u s 1. A simple, b u t c r u c i a l , o b s e r v a t i o n i s t h a t there are f i n i t e n-subalgebras f o r every
n € N. Now e v e r y t h i n g s a i d up t o
t h i s p o i n t i s expressible i n arithmetic, i n p a r t i c u l a r t h e f o r m u l a
cp(n)
s a y i n g t h a t t h e r e i s a bounded n - a l g e b r a . T h i s f o r m u l a has t h e p r o p e r t y that O(0) A ( w n ) ( $ ( n ) - >
$(n+l))
h o l d s i n a l l models o f t h e f i n i t e l y many axioms. Consider t h e m o d e l
of
these axioms which i s generated i n a non-standard model o f t r u e number t h e o r y by an i n f i n i t e element w, u s i n g t h o s e f i n i t e l y many f u n c t i o n s . e v i d e n t t h a t i n t h i s model any therefore
l$(w)
It
i s
w-subalgebra i s i n f a c t t h e whole model,
holds there.
T h i s theorem c o u l d be f o r m u l a t e d f o r a b s t r a c t c o s t a l g e b r a s .
I t would
claim t h a t a cost algebra s a t i s f y i n g c e r t a i n conditions ( t h e cost d e f i n e d by c o m p l e x i t y o f terms and e x p r e s s i b l e i n f i r s t o r d e r e t c . ) h i l s an elementari l y e q u i v a l e n t c o s t a l g e b r a where subalgebras and n - s u b a l g e b r a s
are the
same f o r some n i n t h e c o s t s c a l e . More u s e f u l d i r e c t i o n though w o u l d b e t o keep t h e c o s t s c a l e f i x e d and e x p l o r e t h e c o s t f u n c t i o n s which a r e t i e d t o t h e c o m p l e x i t y o f terms. I n t h i s c o n n e c t i o n an axiom t o c o n s i d e r i s C(6t g("))
2 C(dog,")
126
M I R O S L A V BENDA
assuming the operations a r e closed under substitutions. A somewhat more concrete problem i s t o characterize the cost a l g e b r a constructed by Silver in more general terms; perhaps the f a c t t h a t algebras associated with Skolem functions on L , a r e J6nsson i s r e l e v a n t here (see the proofs of 7.4.9 and 7.4.10 in Chang and Keisler 1973).
SAMPLINGS, There a r e two kinds of tools: those of general use and those f o r s p e c i f i c operations. The method o f w h a t we c a l l samplings has been used so f a r in the l a t t e r capacity b u t we think t h a t i t s r i g h t place i s in t h e f i r s t category among ultraproducts, indicernibles e t c . The method consists ofextracting information about a structure from small samples of the s t r u c t u r e in the sense described below. The f i r s t r e s u l t of t h i s kind was obtained by Kueker; i t i s related below. The general problem in t h i s area i s b e s t expressed in Shelah 197+ : "Kueker i n defined some f i l t e r s .We c a n e a s i l y suggest more". We would l i k e t o add t h a t rather than f i l t e r s more samplings should be constructed; they more o r l e s s canonically d e f i n e the filters.
...
...
a he* S E P ( A ) such t h a t 60% euehy 6iand any d g e b m on A 06 countabLy many o p e a d h n b w e c a n dind which includes a and 0 closed undeh .the ope hat ion^.
DEFINITION. A OampLLng 05 A 0 nite
o e S
a EA
The definition may be stated equivalently f o r structures andelementary substructures instead of algebras and subalgebras. The LBwenheim-Skolem theorem ( f o r c o u n t a b l e l a n g u a g e s ) i s equivalent t o saying t h a t P ( A ) , the countable subsets of A , f o r m s W 1 a sampling o f A . More generally, i f w 5 K < I A l then P K ( A ) i s a sampling o f A .
EXAMPLE 1 .
EXAMPLE 2. Chang's conjecture f o r saying t h a t the s e t s of order 'type in Chang and Keisler 1973).
(K+,K) is forms a sampling of
(K++,K+) K+
equivalent t o (see 7.3.4
K++.
I t i s c l e a r t h a t there a r e no interesting samplings of countable
sets
127
SOME D I R E C T I O N S I N MODEL THEORY
because a c o u n t a b l e s e t w i t h a "successor" f u n c t i o n has no p r o p e r elementaof
r y s u b s t r u c t u r e . I n t u i t i v e l y , sampling i s a c o l l e c t i o n o f s m a l l subsets
A and t h e r e a r e so many o f them t h a t one g e t s an i d e a o f t h e s e t f r o m t h e p r o p e r t i e s o f t h e samples. To make t h i s c l e a r e r we a s s o c i a t e w i t h e v e r y sampling a f i l t e r o f l a r g e s e t s o f samples.
DEFINITION. L e X the
6iXtetl
(6
whae
I
(A,
S be a ( n o n - i x i v i d ) sampling on S genmrated by 6 d o o e d undm 6 , , ..., 6, ,... I
6, ,..., 6, ,. .. )
A . By
FS
we
denote
-id an d g e b m .
S = P,,,,(A),
I n t h e case when
06
FS
i s j u s t the f i l t e r generated
by
P w , ( A ) . I n general, i t i s countably c o m p l e t e
c l o s e d unbounded subsets o f
and normal. To express n o r m a l i t y and o t h e r t h i n g s i t i s c o n v e n i e n t t o i n t r o duce a q u a n t i f i e r : €
for
S
we w r i t e
I ... 6 ...1
(WS6)(
FS
€
(3p)
1(vS6)i write
...A . . . )
if
;
Ws and
IS mean i n t h e case o f
t h a t t h e s e t i n q u e s t i o n i s closed-unbounded and s t a t i o n a r y
S = Pu,(A)
respectively.
N o r m a l i t y t h e n means t h a t i t i s p o s s i b l e t o s w i t c h ( p a r t i a l l y ) W and WS:
( W x € A ) ( VS6) R ( f i )
R SAxS
where
. This
3
( WS6) ( x € 6) R
(x6 )
i s t h e p r o p e r t y on which most theorems ( b u t n o t a l l ,
see below) hinge. I t enables us t o p r o v e a r e s u l t on o m i t t i n g o f t y p e s w h i c h l o o s e l y speaking says t h a t i f a t y p e can be sampled by subtypes a l l o f which can be o m i t t e d t h e n t h e t y p e i t s e l f can be o m i t t e d (see Benda 1 9 7 t b f o r det a i l s : t h e t y p e i s c o u n t a b l e so i t has no sampling b u t we can r e l a t i v i z e the notion). Another r e s u l t proved u s i n g n o r m a l i t y i s e s s e n t i a l l y t h e f i r s t
result
i n t h i s f i e l d i f we do n o t c o u n t t h e p o w e r f u l samplings p r o v i d e d by
super-
compact c a r d i n a l s (see Kueker 1972):
THEOREM ( K u e k e r ) .
01 a bR/ructuhe on A
m
I= @
L e X @ be a oentence
.
Then:
06
Lwlw
, S
a oampfing od A and
( W S 6 ) m16 I=@
I n Barwise 1974 t h e s e r e s u l t s a r e extended. A use o f sampling i s i m plicit
a l s o i n t h e work o f Shelah on Whitehead c o n j e c t u r e
(Shelah 197+)
.
128
MIROSLAV BENDA
He d e f i n e s t h e r e what amounts t o a sampling o f a s t r o n g l i m i t
cardinal A
( s a t i s f y i n g some a d d i t i o n a l p r o p e r t i e s ) b u t he does n o t use t h e f i l t e r defunctions f i n e d above; i n s t e a d o f c o u n t a b l y many f u n c t i o n s he uses K ( < A ) w i t h t h e e f f e c t t h a t t h e f i l t e r becomes
K-complete.
The q u a n t i f i e r
asso-
ciated with i t satisfies:
if
( 3 S6)(A16
i s free)
then
A i s free
f o r an a l g e b r a A ( s a y a group) o f c a r d i n a l i t y
A.
I t w o u l d b e u s e f u l t o k n o w how t h e p r e s e r v a t i o n o f d i f f e r e n t
p r o p e r t i e s v a r i e s from sampling t o sampling. S p e c i f i c a l l y ,
i s the theorem
above f a l s e f o r some o f t h e c a n o n i c a l samplings we d i s c u s s e d b e f o r e ? These q u e s t i o n s may l a t e r appear as n a i v e as t h e q u e s t i o n whether u l t r a f i l t e r s o n w
have d i f f e r e n t p r o p e r t i e s . Samplings p l a y a r o l e i n s e t t h e o r y as w e l l . Assume t h a t we
have
a
model o f ZF w i t h a s e t A i n i t and we want t o e x t e n d i t e l e m e n t a r i l y so t h a t A s t a y s i n t h e e x t e n s i o n b u t t h e image o f A i s p r o p e r l y l a r g e r t h a n A ( s i t u a t i o n s l i k e t h i s occur f r e q u e n t l y i n l a r g e cardinal questions). are the properties if
6will
P ( . ) i s a p r o p e r t y such t h a t f o r e v e r y sampling S ( n o n - t r i v i a l )
we have
(Vs6)
P(6)
What
have i n t h e e x t e n s i o n ? A p a r t i a l answer i s t h i s : of A
t h e n P ( A ) i s t r u e i n t h e e x t e n s i o n . L e t us denote b y
QA t h e p r o p e r t i e s d e f i n e d i n t h e l a s t sentence. QUESTION:
Is
a t y p e ? When i s i t complete ?
Note t h a t i f t h e axiom o f determinateness h o l d s
QWl
i s a
complete
t y p e . (There m i g h t be some problems w i t h t h e axiom o f c h o i c e h e r e b u t t h e a l t e r n a t i v e d e f i n i t i o n o f a sampling i s e f f e c t i v e ) . ’ A s i m p l e f a c t w e n o t i c e d i s that
Qwl #
ordinal while
CAN
because
Owl c o n t a i n s t h e p r o p e r t y b e i n g a c o u n t a b l e
does n o t c o n t a i n t h e p r o p e r t y . Is ,Q
MODEL THEORY LIVE
=
Qwg?
UP TO ITS NAME?
J. S i l v e r once r e l a t e d . t o me a s t o r y a b o u t a man f r o m an i n d u s t r y who c a l l e d him w i s h i n g t o c o n s u l t him on model t h e o r y w i t h p r o s p e c t s o f i t i n m a r k e t i n g . Blunders l i k e t h i s a r e f a i r l y common,
using
people t h i n k ( r i g h t -
l y ? ) t h a t model t h e o r y r e f e r s t o t h e i m p o r t a n t and fundamental a c t i v i t y o f
SOME D I R E C T I O N S ON MODEL THEORY
129
getting a t f a c t s and predictions bymeans ofmodels.The model theory we study i s , of course, n o t suited, by i t s own d e f i n i t i o n , f o r producing quantitative r e s u l t s sought in applied a r e a s . B u t i t i s notalways the numbers t h a t weseek, sometimes we need a q u a l i t a t i v e judgment. This thought i s most e f f e c t i v e l y expressed in the following quotation from R. Thom 1969 ( p . 3 3 3 ) : "But a s s o o n a s we r u n i n t o d i f f i c u l t i e s , c o n t r a d i c t i o n s , feel ( l i k e i n E l e m e n t a r y P a r t i c l e t h e o r y now) o r when w e overwhelmed by t h e mass o f e m p i r i c a l d a t a t h e n t h e need a r i s e s f o r some c o n c e p t u a l g u i d a n c e i n o r d e r t o c l a s s i f y t h e d a t a and t o f i n d o u t t h e most s i g n i f i c a n t phenomena. I f s c i e n t i f i c p r o g r e s s i s t o be a c h i e v e d by o t h e r means t h a n p u r e c h a n c e and l u c k y g u e s s i t r e l i e s n e c e s s a r i l y o n a q u a l i t a t i v e understanding of t h e process studied". Many problems which need a q u a l i t a t i v e understanding could be readily suggested; we shall describe one which seems intimately connected withmodel theory and what we discussed above. The problem i s to imitate o n g e n e r a l structures the process by which natural s c i e n t i s t s obtain information about the.world we l i v e in. The value of better understanding of t h i s process i s cl ear. Let us be more s p e c i f i c . I n the f i r s t approximation we s e e t h a t the information i s obtained by taking f i n i t e samples of the s t r u c t u r e (i.e. the Universe). This i s a process we understand t o some degree i f we take countable samples and we also understand t h a t , in general, a f i n i t e sample has almost no information about the structure from which i t i s taken. If we take, f o r an example, an i n f i n i t e l i n e a r order , a f i n i t e sample of i t g i v e s absolutely no information a b o u t the order because f i n i t e l i n e a r orders a r e t h e same everywhere. This example i s unfair t o l i n e a r orders because in f a c t all structures exhibit t o a large degree such behavoir. (Ramsey's theorem being partly responsible f o r t h i s . ) B u t i f we pull a real out of a model of s e t theory we know t h a t i t may contain a l o t of information about the model.It may happen t h a t the real i s O d which codes u p information about a f a i r l y large part of the whole s t r u c t u r e . The key t o finding out a b o u t the model from the real i s t o i n t e r p r e t i t the r i g h t way. And t h i s i s e x a c t l y t h e methodology used in science. The f i n i t e sample i s interpreted, generalized, idealized into a pattern which i s subsequently checked on the universe by an experiment. Cat1 t h i s pkocess be usefully modelled on abstract s t r u c tures ? This would involve defining the possible interpretationsandspeci-
130
MI ROSLAV BENDA
fying t h e permissible experiments f o r v e r i f y i n g t h e conclusions o f t h e s e i n t e r p r e t a t i o n s . T h i s e s s e n t i a l l y asks f o r some measure of t h e amount of i n formation about a s t r u c t u r e which can be obtained from small p a r t s of i t . We a r e q u i t e f a m i l i a r with t h e r e v e r s e procedure, i n j e c t i n g i n f o r m a t i o n i n t o a s t r u c t u r e . A t i p i c a l example i s the proof of Morley's Upward C a t e g o r i c i t y r e s u l t using Keisler's two cardinal theorem. We have a s t r u c ture which i s not s a t u r a t e d and we i n j e c t t h i s information i n t o i t so t h a t , t o p u t i t antropomorphically, the s t r u c t r u r e "bnom" t h a t i t i s not s a t u r a t e d and "hemembehb" i t a f t e r passages t o equivalent s t r u c t u r e s . Can we turn t h i s technique a r o u n d ? Perhaps we should not b e t o o frugal i n t h e begining and allow countable samples because i t i s not f i n i t e n e s s which i s important; i t i s the i n t e r p r e t a t i o n s of the sample. And h e r e we come t o t h e beginings of model theory which s t a r t e d with the s t u d y o f i n t e r p r e t a t i o n s ; however, t h e s e a r e i n t e r p r e t a t i o n s of s y n t a c t i c a l o b j e c t s by semantical ones. The i n t e r p r e t a t i o n s under discussion a r e just t h e o p p o s i t e , they seek t o i l l u m i n a t e a semantical information by s y n t a c t i c a l means.This i s e x a c t l y t h e p r a c t i c e used i n science. The processes of t h e r e a l w o r l d l i k e the behavior of t h e e l e c t r o n o r the development of an e m b r i o a r e i n t e r preted by formal means l i k e H i l b e r t spaces and s t a b i l i t y theory r e s p e c t i v e l y ( t h e popularity of t h e c a t a s t r o p h e theory i s mainly due t o t h i s kind of i n t e r p r e t a t i o n ) . What model theory has s t u d i e d and s t u d i e s i s the r e l a t i o n of the t h e o r i e s encountered a b o v e , o r t h e o r i e s simply invented, t o a b s t r a c t s t r u c t u r e s which pass a s a s u b s t i t u t e f o r t h e r e a l world. Perhaps by turning some of i t s a t t e n t i o n t o t h e f i r s t s t e p ( r e a l world - theory and checking) w i l l model theory f u l l y l i v e u p t o i t s name.
REFERENCES
I
Barwise, K. 1974,
J.
M o b t o m k i ' n coUupbing 6unCtion, Fund. Math. 8 2 , 95-103.
Barwise, K. J . and J . S. Schlipf 1976,
An inthoduction .to h e c ~ i v d ybatwrated and hebplendent mod&, The Journal of Symbolic Logic, 41, nQ 2 , 531-536.
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SOME D I R E C T I O N S ON MODEL THEORY
Benda, M. 197+a,
Modeloih,
197+b,
CompacAnan doh o m W n 5
i n preparation. 06
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Burgess, J. and D. M i l l e r 1975,
Rematrkn on invahiavLt
duchiptiwe n e t theohg, Fund. Math. 90, n0 1,
53-75. Chang, C. C. and H. J. K e i s l e r 1973,
Model Theory
Amsterdam.
H.
Friedman, 1975,
, North-Holland,
One Iiundhed and .tm phubeemn i n mathemmatical l o g i c , T h e J o u r n a l o f Symbolic L o g i c , 40, n0 1, 113-129.
Fagin, R. 1976,
P h o b a b U e n on ljinite mod&,
The Jounal o f Symbolic Logic, 41,
n0 1, 50-58. Hanick, V. and M. Makkai 1976,
Vaugkt nelztcncen and Xhe c o u d n g theohem,
The J o u r n a l o f
Sym-
b o l i c Logic, 41, n0 1, 171-187. Jensen, B. R. 1972,
The dine n&um3.u~e06 com&ucLibLe kiehmchy,
Annals o f
Math.
Logic, 4, n0 3, 229-308. Jones, N . 0. and 1972,
A.
L. Selman
TwLing mackineh and t h e npectha ad @ & - o h d m 6ohmuean, P r o ceedings o f t h e 4 t h Symposium on Theory o f Computing, Denver, 185-1 96.
K e i s l e r , H. J. 1976,
Model Theory of Finite Structures, ( l e c t u r e n o t e s ) , S p r i n g o f 1976.
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Kueker, D. W . 1972,
Liiwenhh-Skobem and intehpalatian theahem6, B u l 1 . Amer. M a t h . SOC., 78, 211-215.
Loeb, P. A . 197+,
C o n v m i a n 6hom non,5tandahd t v
ntandahd
mCabUhe
~ p a c e n ,t o
appear. Morley, 1970,
M.
D.
The nwnbeh 06 countabbe mad&, nP 1, 14-18.
The J o u r n a l o f Symbolic Logic,35,
Robinson, A. 1973,
Meiamathematicd phobbemn,
The J o u r n a l
of
Symbolic L o g i c , 38,
nP 3, 500-516. Rubin, M. and S. Shelah 1915,
On fineahey ahdehed made&, N o t i c e s AMS, o c t . , A646.
Ryl 1 -Nardzewski
1952,
, C.
The h a k k 0 6 t h e axiam 06 i n d u d a n i n elementahy a h i t h n i e t i c Fund. Math.,
39,
,
239-263.
Shelah, S. 197+,
A compactnu6 theahem d V h n i n g d a h cahdin&,
i h e e algebhah,
W h i t e h e a d phabbem, and t h a n A u e h A u h , ( s e m i n a r n o t e s ) Louvai n. Silver, 197+,
J.
How t o eLikncLte t h e dine nthuotuhe dhom t h e wahk a6 J e a e n , t o appear,
Thom, R. 1969,
Topobogicd made& i r l bialogy,
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133
SOME DIRECTIONS I N MODEL THEORY
Department o f M a t h e m a t i c s U n i v e r s i t y o f Washington S e a t t l e , Washington, U.S.A. and Departamento d e M a t e m i t i c a U n i v e r s i d a d e d e Sao P a u l o Sao Paulo, S P . ,
Brazil.
Non-Classical Logics, Model Theory and Computability, A . I . Arruda, N.C.A. da Costa and R. Chuaqui (eds .) 0 North-Holland Publishing Company, 1977
A SEMANTICALDEFINITIONOF by R O L A N D 0 8.
PROBABILITY
CHUAQUI
there are
two
subjects t h a t come t o mind. On t h e one hand, the axiomatic foundations
When one speaks about the Foundations o f P r o b a b i l i t y ,
of
the Calculus o f P r o b a b i l i t y , which i s a w e l l
- developped,independent
mathe-
matical d i s c i p l i n e ; on t h e o t h e r hand, t h e study o f p o s s i b l e i n t e r D r e t a t i o n s f o r p r o b a b i l i t y statements. As i s w e l l -known, there are several c o n f l i c t i n g i n t e r p r e t a t i o n s o f these statements h e l d by t h e d i f f e r e n t schools i n
the
subject. I n t h i s paper, based on the i n t e r p r e t a t i o n o f P r o b a b i l i t y r e l a t i n g i t t o t r u t h , I present a new d e f i n i t i o n o f a p r o b a b i l i t y measure i n seman-
t i c a l terms. Thus, most o f the content o f t h e paper deals w i t h
the
second
o f the foundational subjects mentioned above. Expressions o f the form ' I t i s probable t h a t . . . I same way as, although more guardedly than,
guide b e l i e f
'It i s true that...'.
in
the
Thus, both
expressions have gerundive f o r c e . (For an i l l u m i n a t i n g discussion see Lucas 1970, Ch. I,11). This k i n s h i p n a t u r a l l y leads t o attempt a d e f i n i t i o n P r o b a b i l i t y s i m i l a r t o T a r s k i ' s semantical d e f i n i t i o n o f t r u t h
of
(see Tarski
1935).
I b e l i e v e t h a t f o r many common language uses o f t h e word ' p r o b a b l e ' these I s h a l l
there i s no adequate measure o f p r o b a b i l i t y . However, w i t h
not be concerned. My aim i s t o a t t a i n a d e f i n i t i o n o f a probabilitymeasure, s a t i s f y i n g the axioms o f the Calculus, t h a t can account f o r al1,or a t l e a s t most, o f i t s a p p l i c a t i o n s i n c u r r e n t s c i e n t i f i c and s t a t i s t i c a l p r a c t i c e . My conception i s connected w i t h Carnap's d e f i n i t i o n (see Carnap 1950 o r
Carnap and Jeffreys
1971)
-
a c t u a l l y , t h e p i c t u r e on page 297 o f h i s
book i n s p i r e d some of my ideas
-
because I use model - t h e o r e t i c a l 135
1950
methods.
136
ROLAND0 6 . CHUAQUI
However, s i n c e I see P r o b a b i l i t y i n t h e same c a t e g o r y as t r u t h and n o t l o g i c a l t r u t h , t h e r e a r e i m p o r t a n t d i f f e r e n c e s i n t h e two conceptions. I n p a r t i c u l a r , Carnap
i n t e n d s h i s measure t o be d e f i n e d on t h e space o f a l l mod-
e l s , whereas I l i m i t t h e space t o t h o s e models t h a t p i c t u r e a p o s s i b l e o u t come. Which outcomes a r e p o s s i b l e i s determined by t h e laws o f t h e phenomenon i n v o l v e d . Thus, ' p o s s i b l e ' i s meant h e r e i n t h e sense o f p h y s i c a l pos
-
s i b i l i t y and n o t l o g i c a l p o s s i b i l i t y . out-
I n o r d e r t o o b t a i n a p r o b a b i l i t y measure i n t h e space o f p o s s i b l e comes,
I use an equal l i k e l i h o o d o r e q u i p r o b a b i l i t y r e l a t i o n . Being
based
on e q u i p r o b a b i l i t y , my d e f i n i t i o n i s a l s o r e l a t e d t o t h e c l a s s i c a l concept i o n . However, my equal l i k e l i h o o d r e l a t i o n w i l l be w e l l - d e f i n e d
and w i l l
n o t s u f f e r f r o m t h e p i t f a l l s o f t h e c l a s s i c a l one. The paper has an i n t r o d u c t o r y f i r s t s e c t i o n t h a t g i v e s a g e n e r a l
des-
c r i p t i o n o f t h e procedures i n v o l v e d , and a second s e c t i o n w i t h some m a t h e m a t i c a l p r e l i m i n a r i e s . I t c o n t i n u e s i n a t h i r d s e c t i o n w i t h t h e formal d e f i n i t i o n o f t h e s i m p l e p r o b a b i l i t y s t r u c t u r e s and t h e c o r r e s p o n d i n g p r o b a b i l i t y r e l a t i o n , n o t i o n s which a r i s e when t h e r e a r e no
equi-
sequences o f
outcomes. The n e x t s e c t i o n g i v e s necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f a p r o b a b i l i t y measure c o m p a t i b l e w i t h an e q u i v a l e n c e
relation
h a v i n g t h e same p r o p e r t i e s as t h e equal l i k e l i h o o d r e l a t i o n d e f i n e d p r e v i
-
o u s l y . T h i s s e c t i o n o f t h e paper c o n t a i n s t h e most i n t e r e s t i n g newmathenati c a l r e s u l t s o f t h e paper: necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e
ex-
i s t e n c e o f a measure on a f i e l d o f s e t s , i n v a r i a n t under a group o f t r a n s formations. The f i f t h s e c t i o n d i s c u s s e s compound p r o b a b i l i t y which a r i s e when t h e r e a r e sequences o f outcomes, and ways
structures, of
defining
a
p r o b a b i l i t y measure f o r them. The paper concludes, i n s e c t i o n s i x , w i t h ex-
A reader n o t interested i n the technical material could read sections one a n d s i x t o
amples and methods f o r a p p l y i n g t h e p r o b a b i l i t y models g i v e n . understand t h e main i d e a s
.
A b r i e f o u t l i n e o f an e a r l i e r v e r s i o n o f t h e s e i d e a s appeared i n Chuaqui 1975, a l t h o u g h some o f them were a l r e a d y i n Chuaqui 1965.
1, GENERAL DESCRIPTION, We assume a language L w i t h some s e t o f n o n l o g i c a l c o n s t a n t s . F o r
the
moment we l e a v e t h e e x a c t d e s c r i p t i o n o f L undetermined. I t m i g h t b e a f i nitary f i r s t
- order
language, an i n f i n i t a r y language, o r a language o f some
137
S E M A N T I C A L D E F l N l T t O N OF P R O B A B I L I T Y
other type.
F o r t h e d e f i n i t i o n o f t r u t h , we u s u a l l y
i n t e r p r e t L i n hela-
-titiand Aybtemb c€ = , w h e r e A i s a n o n e m p t y s e t and R1,...,Rn-l i s a sequence o f r e l a t i o n s o v e r A o f t h e a p p r o p i a t e t y p e so as t o match t h e n o n l o g i c a l symbols i n L . F o r any sentence T a r s k i ' s method,
' $ 0 &ue
i n d'
.
I$
we can define,by
I n a system o f t h i s t y p e
everything
r e l a t i n g t o t h e symbols i n L i s determined. Thus, f o r e v e r y sentence o f L , @ i s t r u e o r f a l s e i n LJZ.
Systems r e p r e s e n t , i t may be s a i d , p o s s i b l e s t a t e s
o f t h e w o r l d . When we i n t e r p r e t L i n Ce, we determine c o m p l e t e l y which sentences a r e t r u e and which a r e f a l s e . When p r o b a b i l i t y statements apply, t h e p r e c i s e s t a t e o f t h e w o r l d which o b t a i n s i s n o t c o m p l e t e l y determined; t h e r e f o r e we must change t h e n o t i o n o f an i n t e r p r e t a t i o n o f t h e language L . Our i n t e r p r e t a t i o n s w i l l no l o n g e r be s i n g l e systems b u t c e r t a i n c l a s s e s o f systems w i t h some a d d i t i o n a l ties.
proper-
The C a l c u l u s o f P r o b a b i l i t y i s u s u a l l y a p p l i e d t o happenings o r o c c u r rences, which may be experiments, o b s e r v a t i o n s , o r n a t u r a l phenomena. i n t e r e s t s us i n a l l t h e s e cases i s t h e henu&
What
o r alLtcame o f t h e phenomenon.
The " t h e o r y " ( i n a wide sense o f t h e term) i s what determines which a r e t h e p o s s i b l e outcomes. I n some cases t h e r e i s o n l y one p o s s i b l e
outcome;
o t h e r s , many. I t i s i n t h e s e l a t t e r cases t h a t t h e C a l c u l u s o f
in
Probability
i s important. L e t us t a k e as o u r f i r s t example t h e choosing o f a sample S o f s i z e from a f i n i t e p o p u l a t i o n P . When we say 'S
m
has n r e d t h i n g s ' we mean t h a t
one o f t h e p r o p e r t i e s o f t h e outcome was t h a t t h e sample had n r e d objects. The same outcome has many d i f f e r e n t p r o p e r t i e s , which can be
described
in
many d i f f e r e n t wayswe can t h i n k o f a n i d e a l a p p r o x i m a t i o n - o f an outcome,nanel y a r e l a t i o n a l system t h a t r e p r e s e n t s a p o s s i b l e model o f t h e s i t u a t i o n in-
volved. I n t h e case we a r e l o o k i n g a t we can schematize t h e comes as systems
Rl,...,RH-l
a s =, where
possible
out-
P i s a f i x e d f i n i t e set,
a r e f i x e d subsets o f P t h a t r e p r e s e n t t h e p r o p e r t i e s w.e
are
i n t e r e s t e d i n ( f o r i n s t a n c e , ' r e d ' ) , and S i s any subset o f P o f m members ( t h e sample). For each subset S o f m members t h e r e i s a correspending system
as;
hence t h e s e t o f p o s s i b l e outcomes c o n s i s t s o f a l l models
o f t h e f o r m d e s c r i b e d above.
as
To speak about t h e s e outcomes we need a language L with n o n l o g i c a l s y m -
-
-
-
R1,...,Rn-l,S. We now use as i n t e r p r e t a t i o n s f o r L n o t systems, b u t classes, K o f r e l a t i o n a l systems a p p r o p i a t e f o r L . K, then, w i l l be t h e bols
s e t o f p o s s i b l e outcomes.
138
ROLAND0 6. CHUAQUI
t h e dimple
I n t h e case t h a t we s h a l l f i r s t c o n s i d e r , i . e . d-iXLLot(lheJ,
phobabLlLty
K t o be a s e t o f r e l a t i o n a l systems w i t h a common
we can t a k e
u n i v e r s e . These s i m p l e casesmay be c h a r a c t e r i z e d as t h o s e i n which t h e r e a r e no sequences o f outcomes. F o r i n s t a n c e , i n t h e example j u s t m e n t i o n e d , M i s t h e s e t o f systems
as. We
s h a l l l a t e r analyze cases i n which sequences o f
o u r p r o b a b i l i t y s t r u c t u r e s w i l l be more complicated.
outcomes appear; t h e n
However, i n o r d e r t o b u i l d t h e s e compound p h o b u b i l i t y n,iXuctwre~ we
shall
need t h e s i m p l e ones. P r o p e r t i e s o f outcomes a r e u s u a l l y c a l l e d ewe&.
These events
can
r e p r e s e n t e d by sentences. I f two sentences a r e l o g i c a l l y e q u i v a l e n t , r e p r e s e n t t h e same event.
be they
classes
Thus we may t a k e events as e q u i v a l e n c e
o f sentences determined by t h e r e l a t i o n o f l o g i c a l equivalence. Thus, i f we t a k e @ t o be t h e sentence ' t h e r e a r e n r e d t h i n g s i n S ' , t h e euent $J
o b t a i n s i s t h e c l a s s o f sentences l o g i c a l l y e q u i v a l e n t t o @ .
We
that shall
d e f i n e a p r o b a b i l i t y measure on t h e s e e q u i v a l e n c e c l a s s e s . However,
when
t h e r e i s no danger o f c o n f u s i o n we s h a l l speak o f t h e p r o b a b i l i t y o f Q meani n g t h e p r o b a b i l i t y o f t h e corresponding equivalence class. We know t h a t f o r r e l a t i o n a l system i s true i n
a.
( o r 6&e)
in M , if
For
a
and sentences @ we can d e f i n e @
K, we say t h a t Q ORhue M. i n K. We understand t h i s
simple p r o b a b i l i t y structures @
i s t r u e ( o r f a l s e ) i n e v e r y r e l a t i o n a l system i n
S i m i l a r l y , we s h a l l d e f i n e t h e phobabLlLty
06
@
p r o b a b i l i t y as s o r t of a measure o f t h e degree o f " p a r t i a l t r u t h " o f a sentence. Thus, i f @ i s t r u e i n
K,
i t s h o u l d be t h e case t h a t t h e p r o b a b i l i -
K, t h e p r o b a b i l i t y o f
ty o f @ i n M i s one; i f @ i s f a l s e i n
s h o u l d be z e r o ; and i n a l l o t h e r cases t h i s p r o b a b i l i t y s h o u l d
in
@
be
K
between
z e r o and one. T h i s degree o f " p a r t i a l t r u t h " i s vaguely p r o p o r t i o n a l t o t h e s i z e o f t h e s e t o f models o f set i s
K i t s e l f ; and i f
@
@ t h a t are i n
i s false i n
K. I f
@
i s t r u e i n M,
this
K, t h i s s e t i s empty. The n a t u r a l
t h i n g t o do, then, i s t o o b t a i n a p r o b a b i l i t y measure d e f i n e d on a c o l l e c t i o n o f subsets o f
K t h a t i n c l u d e s t h e s e t s o f models i n K o f sentences
of t h e a p p r o p i a t e language. Having done t h i s , we may d e f i n e t h e p r o b a b i l i t y of @ in
K as t h e measure o f t h e s e t o f models o f
o r eqLLiphobubXLty r e l a t i o n between events, i . e .
K.
@ that are i n
I n o r d e r t o d e f i n e t h i s measure I s h a l l i n t r o d u c e an ~ Q U &
likelihood
between e q u i v a l e n c e c l a s s e s
of l o g i c a l l y e q u i v a l e n t sentences. T h i s e q u i p r o b a b i l i t y r e l a t i o n i s d e r i v e d from a symmettry r e l a t i o n between s e t s o f models, i . e . t i v e l y , two s e t s o f models a r e nymmc?Ahd,
subsets o f
M. I n t u i -
i f t h e y a r e symmetrical w i t h r e -
s p e c t t o t h e laws o f t h e c o r r e s p e n d i n g phenomenon. F o r d e t e r n i n i n g
these
139
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
06 L ~ ~ a ~ d ~ o h m a t(ifoum n c t i o n s ) that atre "invahiant undw t h u e Lawb". Two subsets o f K w i l l be symmetrical i f one
symmetries, we consider the ghoup
can be transformed i n t o t h e o t h e r by one o f these functions. Then, two sentences ( o r , more accurately, two equivalence classes o f l o g i c a l l y equivalent sentences) models i n
9 and $ are e q u a l l y l i k e l y i n K , i f t h e corresponding sets o f M o f @ and are symmetrical.
The laws o f the phenomenon determine t h e simple p r o b a b i l i t y
structure
M ; M, i n i t s t u r n , gives t h e group o f transformations. L e t us s e e h o w t h i s group, c a l l i t G K , i s obtained i n our example.
GK i s a subgroup
group o f permutations o f t h e universe, P. For a permutation
6
of
the
t o be i n GK ascan
i t must s a t i s f y some a d d i t i o n a l requirements. Each r e l a t i o n a l system
K,
be decomposed i n two p a r t s . One t h a t i s the same f o r a l l s y s t e m s i n namely the constant p a r t part
= cP,S>.
cK
= ;
and another, the
6
Now, t h e f i r s t requirement on
variable
i s t h a t i t should n o t
M o u t s i d e o f K. That i s , i f we apply 6 t o the v a r i a b l e , transforming i t i n t o another system asl,v ( S l i s p a r t o f a system, 02 s, v then the s e t o f elements d ( x ) f o r x i n S, i . e . the image o f S by d ) , take a system i n
s h o u l d a l s o be t h e v a r i a b l e p a r t o f
%,V and call i t a A@tem. The members of wA are w h a t are called nhimpte i n d i n i t e Aequencu w i t h are referred t o as 5i& t m n i n A . In case n E w , the members of A' vLite n - t m i e d nequencen again w a h t m in ~ A. If 6 E ' A , we write 6 = whose <do,. ..,6n-1> ; in particular < a , b> i s the two - termed sequence terms are a , b . The symbol S A denotes the A& 06 & d i n i t e hequencu w L t h t m i~ n A ( i . e . $A = u {'A: n E w 1 ) . An n - a h y h w o n R on A i s any subset of ' A . For n-ary relations R we write Rxo x n - ] instead of < x o,..., xn-l> € R. If R i s binary, aRb stands f o r < a , b> € R. A heRatioMae A Y A ~i s ~ a system a of the form d = < A , R , a > , where: A i s a nonempty s e t called the domain or u n i w m e of the system ul, R = i s a system of relations on A,and a = J i s a system of elements of A called the did.tingLLinhed demem2 0 6 02. I and J should be d i s j o i n t ; 1 u J i s called the index A & a j a . We also I, J. IJe use german l e t t e r s for relational write & = < A , R i r a - > J systems and the corresponding capital i t a l i c l e t t e r f o r t h e i r universes. (For d e t a i l s see Tarski 1954). I t i s useful t o define the A ~ W tqpe ofY a relational system. If A = , where the m i ' s are natural numbers f o r every i E I , then a relational system c E = < A , R i , aj> i s of sim ila rity type I, A i f , f o r each i E I , Ri i s an mi - a r y relation. We say t h a t two r e l a tional systems are n h i f they are of the same type. and K c_ I u J , we c a ll the system obI f & = F A , R ~ , U , >a ~I , tained by omitting a l l relations and distinguished elements whose index i s n o t i n K the r e s t r i ct i o n of cZ t o K , in symbols: 06
If
a
,...,
aI
K = j
.
L E I ~ K j, € J n K '
,
We now consider the concept of iAomohphiAm of two s i m i a r a J a n d & = i Let oZ= < A , R i ,
systems. be t w o
similar relational systems. Then & and.% are isomorphic i f t h e r e i s
a
143
SEMANTICAL D E F I N I T I O N OF PROBABILITY
6
one-one f u n c t i o n
(i)
f o r every
A
from
i
I
E
onto B such t h a t :
x o ,..., x m , - , € A we have, RixO
and
,... , 6 ( x m
i f and only i f S i 6 ( x 0 )
(ii) f o r every j € J ,
6
Under these conditions
q)= h j .
,...,x
1
~ , - ~ h
h
i s c a l l e d an Domoxpkiom and & the homohpkic &I-
age 06 Ce undetl {, i n symbols ,$ = 6 * a . I t i s c l e a r t h a t i f d i s a system and 6 a one - one f u n c t i o n w i t h domain A , then there i s one and o n l y
one r e l a t i o n a l system
such t h a t
C$
8 = 6 *Ce. 6
systems w i t h a comnon universe A , and
A , then Bd denotes the b e 2 i n symbols: p16 = { d * Q
06
I f B i s a set o f
relational
i s a one-one f u n c t i o n w i t h domain
nyotems obfained by .tlze a o t i o n
06 6
in
B,
:Ce EB}.
We a l s o use some terminology from General Algebra.
F o r more
details
about t h i s and r e l a t i o n a l systems see Henkin, Monk, and Tarski 1971,Prelimi n a r i e s and Ch. 0. For any
or
a E w
a = w , an a - atly p c & t i d
o p w o n Q on a s e t A
i s a f u n c t i o n w i t h domain included i n “A and range i n c l u d e d i n
.
A If
002 = “A , then we say t h a t Q i s a t o t i d a - u h y opehation, o r s i m p l y an a - m y op&an, on A. A t o t a l 0 - a r y o p e r a t i o n on A represents a d i s t i n guished element o f A . A pahtiae @ e b m i s a system d o f the form CZ=
Q = i s a system o f p a r t i a l op Ce i s an d g e b h a i f every p a r t i a l operation i n t h e system
where A i s a nonempty s e t and e r a t i o n s on A.
i s t o t a l . An equivalence r e l a t i o n R on A i s a congmencc h M o n on
a
if
such t h a t Q. i s a - ary, aiRbi f o r every ’iE a , and J Qju and Q.b e x i s t , we have Q.a R Q.b, f o r every j € I . It i s well-known J J J t h a t i t i s p o s s i b l e t o obtain, f o r each congruence r e l a t i o n R , the y u o f i e n t a,b E “A
f o r every
(pw5.a.t)
Cel R
@ebm
.
Ile a l s o consider c e r t a i n kinds o f algebras and p a r t i a l B o o k z n @ebha
(BA) i s an algebra
B
=
the b i n a r y operations o f j o i n and meet, pleme*on,
A
-
i s the unary operation o f
corn-
and 0 and 1 are d i s t i n g u i s h e d elements o f B . B A ‘ s a r e c h a r -
a c t e r i z e d by w e l l
A { x : x a A}
algebras.
, where V , A are
- known axiom systems.
t h e nupmnmum o r
I f A _c B , we designate by V I x : x E A } ,
A i f they e x i s t i n B . When we x € %, we w r i t e V I x , : i E w } , A { x i : i E d . I f a l l suprema and i n f i m a e x i s t f o r countable subsets A , we say t h a t B is coumkbty complete. The notions o f atom, ,id&, and countably c o m d e t e
have an i n f i n i t e sequence
in@num
of
144
ROLAND0 B. CHUAQUI
i d e a l i n a BA w i l l a l s o be needed. I f 1 i s an i d e a l i n a BA B , we w r i t e B / I f o r the q u o t i e n t algebra determined by I . For more d e t a i l s about BA's see S i k o r s k i 1969.
A p a r t i c u l a r k i n d o f BA i s a 6 i e L d 0 6 n u b n e t n 0 6 a n e t X, lF = - , 0 , X > , where F contains subsets o f the s e t X , and t h e op-
,
and a p a r t i a l countable operation C(see
Tarski 1949). A mdid algebha (CA) i s a GCA i n which t h e p a r t i a l operat i o n s are t o t a l . A p a r t i c u l a r k i n d o f GCA i s t h e d i s j u n c t i v e BA ed w i t h t h e BA B
.
E
=
a
~ =b 0,
"xi
CiE
= c
h associat-
a+ b = c
i s obtained as f o l l o w s :
and
if
: i E d = c and xi A x . = 0 f o r a l l i < j E w . J (See Tarski 1949, def. 15.14). By a 6 i n i t e L y a d d i t i v e ( p h o b a b L L t y ) meabuhe o r a BA IB we
if
only i f
v{x,
understand,
as usual, a f u n c t i o n p s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s :
8
and DO p-1 5 l o , 1 ]
1.
DO
2.
i f a,b E B and a A b = 0 , then
3.
p ( 1 ) = 1.
p =
;
p(a
+ b)
= p(a)
+ p(b);
We say t h a t p i s a countable uddditive ( p m b a b U y ) meabue or,simply, a mecuue, if i n a d d i t i o n p s a t i s f i e s : 4.
I f x E "B
and
Ciao
xi e x i s t s , then
A measure 1.1 i s s t r i c t l y p o s i t i v e i f :
p(CiEwxi)
= ZiEw
p(xi)
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
5.
~ ( a =) 0 implies
a = 0,
f o r every
145
a 8 B.
Throughout t h i s paper we s h a l l consider two d i f f e r e n t f i r s t - order languages of s i m i l a r i t y type A = , J > , a 6irZitatg language LA and . The language L* has a d e n u m e r a b l e s e t of an in,$ivtitahg language L* W1W
vuhiublw, and
1"
WlW
has a s e t of v a r i a b l e s of c a r d i n a l i t y
wl.
The con-
n.tatant6 a r e divided i n t o l o g i c a l
and nonlogical ones. Both languages have logical constants A , V , I , v , 3 and = , standing f o r (finite)conjuncLLon, dibjunOtiOn, negation, u n i v e u d and e x b t e n t i d quuntibication, and idena y . In a d d i t i o n the i n f i n i t a r y language LA has l o g i c a l constants A and WlW
v standing
f o r denumerable conjunctions and d i s j u n c t i o n s , r e s p e c t i v e l y . The nonlogical constants f o r both languages a r e the phedicatw ( o r heXation q t n b o b ) and the i n d i v i d u a l comRavLtn. W i t h every p r e d i c a t e a n a t u r a l number i s c o r r e l a t e d , which i s c a l l e d t h e hank of t h e symbol. For every i E 1 there i s a p r e d i c a t e of rank mi For each j E 3, an individual constant. The correspondence i s such t h a t d i f f e r e n t indices correspond t o d i f f e r e n t symbols. The i d e n t i t y symbol, though regarded a s a l o g i c a l c o n s t a n t , is i n cluded i n the set of binary p r e d i c a t e s ( p r e d i c a t e s of rank two). The v a r i ables and t h e individual constants a r e t h e i n d i v i d u a l h p b o b . The exphunion.4 of L* a r e f i n i t e sequences of symbols; t h e expressions f o r L;, a r e t r a n s f i n i t e sequences of symbols of length l e s s than wl.Among
.
expressions we d i s t i n g u i s h t h e domula. The s i m p l e s t , s o - c a l l e d atomic bomLLean, a r e obtained by combining M a r b i t r a r y individual symbols by means of a p r e d i c a t e of r a n k n ; compound formulas a r e b u i l t from simpler ones by means of s e n t e n t i a l connectives and y u a n t i @ x exphwniokih ( i . e . q u a n t i f i e r s followed by v a r i a b l e s such as y u o r 3 v ) . An occurrence of a v a r i a b l e i n a formula may be e i t h e r &tee o r bvund; a formula in which no v a r i a b l e occurs f r e e i s c a l l e d a nentence. The s e t of sentences of L* i s designated by n Sb , and t h e s e t of sentences of L* b j Solo . 01
We assume known T a r s k i ' s d e f i n i t i o n of n a t i s 6 a c A o n and .thLL.th (for s e e K e i s l e r 1971). When t h e r e i s nodanger L* s e e Tarski 1954, and f o r L* 01W
of confusion we s h a l l use the same metalogical symbols f o r both languages. An a n i g n m e n t i n o2 i s a function from the v a r i a b l e s i n t o A . For x an assignment i n a r e l a t i o n a l system CL of s i m i l a r i t y type b and $I a formula
of
LA o r LA
W1W
we.assume defined:
cx c
@
1x1
( i . e . x nuh>di,w @ i n
a);
146
ROLAND0 E.
CHUAQUI
and, also, i f @ i s a sentence we have:
le TWO
(oh L
L*
1W
)
have,
0 h e in
(i.e. @
a,$
r e l a t i o n a l systems
a).
o f s i m i l a r i t y type
etemev~tahiey
are
e q u b d e n t , ifthey s a t i s f y e x a c t l y the same sentences o f L* ( o r
); i.e.
W 1W
b= @
P
02:
if and o n l y i f f o r every sentence @ o f L'
i f and o n l y i f
OZ C @
&
!=
( o r Lilw)we
@.
For any class lK o f r e l a t i o n a l systems o f s i m i l a r i t y type common universe A , any formula @ of i n A , we define:
blodK,x($) =
{ a :a
L*
M, d
E
( o r L*
W1W
!=$
I f @ i s a sentence, we w r i t e ModK(@)f o r Mod
mod&
06
me .in
@ &that
M.
6 and w i t h ) , and any assignment x
[XI} . K, x
06
( @ ) ; ModK(@)i-4 t h e
I f C i s a s e t o f sentences and @ i s a sentence, then @ i~ a c o n h e -
quence 06 C i f
@ holds i s a l l systems i n which a l l sentences o f
and we w r i t e
C C $
and we w r i t e
C @
.
.
01
system o f deduction, and we w r i t e C t- $ i f $ h dehivabte @om C
Aheohem i f i t i s d e r i v a b l e from the empty s e t , and we w r i t e c o n s u l t K e i s l e r 1971). By the w e l l first
- order.
- known
. This
.@0a L*
I- $ ( f o r
W1W
Completeness Theorem o f
l o g i c we have f o r every C 5 S* and every @ 8 S,'
and o n l y i f C t- @
hold,
C
0 v a l i d i f i t i s a consequence o f t h e empty s e t , For both languages L* and' 1 we choose a standard @
i s n o t t r u e f o r L*
W1W
"weak" completeness i n t h e sense t h a t f o r every @ E Stlw,
C @
if
still
have
C
; however we
finitary
I=@ i f and
only
if I-@.
We c a l l two sentences eqLLivden.t i f I-
@
, J > , t h a t s a t i s f i e s the f o l lowing conditions:
.
(i) For each d,g€ lK, CR 1 J = $ I J (ii) L e t K be the l a r g e s t subset o f I U J
dl K
=$I
such t h a t
such t h a t f o r any
a,&€ K,
K ; then i f there i s a formula w i t h one f r e e v a r i a b l e
6
there i s a j E
J such t h a t b
$I
161 , then
i s the only element o f A w i t h C n l K I= @ = a
j * I t i s c l e a r t h a t c o n d i t i o n (i) i m p l i e s t h a t a l l systems i n
JK have the to
same universe and t h e same d i s t i n g u i s h e d elements. I t i s a l s o easy
see
t h a t there i s a l a r g e s t s e t K w i t h t h e p r o p e r t i e s r e q u i r e d f o r (ii). Cnl K w i l l be c a l l e d the common paht 06 lK; f o r f u r t h e r reference we s h a l l desig-
B E lK , By
=
06
-
dl ( I U J) K ;- a n d , f o r any {CRY : G! E B). Condition (ii) expresses the f a c t t h a t any de-
nate by CRY ( t h e vahiable paht
Cn)
f i n a b l e element i n the common p a r t o f lK i s designated by an i n d i v i d u a l constant. I t i s t o be remarked t h a t the only c o n d i t i o n e s s e n t i a l f o r the r e s t
our work i s t h a t a l l systems i n lK have a common universe.
The
other
of re-
quirements seem t o be n a t u r a l , b u t they are n o t indispensable. I n the f i r s t place, i t would n o t be n a t u r a l t o have as one o f our p o s s i b l e choices designation o f an element o f t h e universe, Hence, the i n d i v i d u a l
the
constants
should have an i n v a r i a n t i n t e r p r e t a t i o n i n a l l t h e systems. The naturalness o f t h e second c o n d i t i o n w i l l be c l e a r from the d e f i n i t i o n o f the transformations t h a t preserve the "laws o f t h e phenomenon". This ghoup
06 &mmtiom GK,
contains a l l permutatiors
6
group
o f the
of
common
148
ROLANDD B. CHUAQUI
universe A t h a t satisf.y simultaneously the f o l l o w i n g c o n d i t i o n s :
(1)
For any 1 7 7 ,= < A,Ri,
a ,> j
jsj
(a) < A , R i $ a j > i E I ,
< A , R ~ t ~ - ' ( a j ) >i (b)
(2)
< A , R i p d(aj)>LE 1,
J
5
j E j= < A , R i , b j > L E I , j € J
'
f o r some permutation g o f J
d ( a j ) > i E 1 , j a j
5
for
any o f these f i e l d s .
I now reformulate these ideas i n order t o have a more perspicuous mathematical problem.We consider t h e f i e l d o f subsets o f E$,JFy=
-
given by,
Fy
=
{By: 8 E F}.
Fy i s obviously isomorphic t o IF. Thus i f we d e f i n e a measure ? on IF by,
we can o b t a i n p on
~ ( 3 =) !(By).
Fv
150
ROLAND0 6 . CHUAQUI
L e t G; =
{6*: 6
€ GK}
the isomorphic image o f formations o f
.
IF,,
The group G;
B
-GK*
, where A* by 6,
a,,
a p p l i e d on an element
determines an equivalence r e l a t i o n C
i f and o n l y i f
t i c e t h a t if 6 € G; Also we have f o r €3,
C €
F,
B-
C
Thus, what we need i s a measure if
-
8
-Gt
C, then
=
-Gt
of
M y gives
Fv by:
On
6
f o r some
d*(B) = C
, then 5
a,,
i s , then,a group o f t r a n s -
G;
6*((R,,).
(no-
€ G;
g* f o r some g € GK and j * ( B ) = B g ) .
i f and o n l y i f on
L(B) = v(C),
IF,,
such
B,,
-Q
.
C,,
that:
for all B ,
C €
F,,
.
i s what i s c a l l e d a measure on IF,, imatLiant undm t h e g m u p 06 ,ttam 60hmatiom G; (see Tarski 1949, p. 229). We s h a l l discuss i n the n e x t s e c -
p
t i o n necessary and s u f f i c i e n t c o n d i t i o n s f o r the existence o f such measures.
4, EXISTENCE OF
MEASURES,
Our task now i s t o f i n d a m e a s u r e on t h e f i e l d
i n v a r i a n t under the group o f transformations
Necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e existence o f f i n i t e l y
measures i n v a r i a n t under a group o f f u n c t i o n s are given i n Tarski 1949. 231. Hence I s h a l l concentrate on (countably a d d i t i v e ) measures. To discuss t h i s k i n d o f measures i t i s convenient t o have complete f i e l d o f sets. I f our language i s
L*
w1
(i.e.
a
=
.
G; additive
p.
countably
i t contains countable
IF,, i s countably complete. But i f our l a n L* ( f i n i t a r y ) we work w i t h t h e s m a l l e s t countably complete f i e l d o f subsets o f M y generated by IF,, . conjunctions and d i s j u n c t i o n s )
guage i s
Thus, the mathematical problem, i n a l l i t s g e n e r a l i t y , i s t o measure p on a countably complete f i e l d o f sets
IF
=
,in-
v a r i a n t under a group o f permutations G o f the u n i t s e t X ;namely p should satisfy:
(*)
if A, B €
F
and
A = 6*B
f o r some
6
€
G, then
p(A) = ~ ( 8 ) .
A measure p t h a t s a t i s f i e s ( * ) i s c a l l e d G - i n w a h i a n t . A l l r e s u l t s about these measures presented i n t h i s s e c t i o n are a l s o v a l i d f o r quasi -groups i n s t e a d o f groups (see Tarski 1949, p. 142). I n order t o solve t h i s problem o f existence o f measures, I s h a l l
prove
151
SEMANTJCAL DEFINITION OF PROBABILITY
B i s a BA and R an B t h a t i s R - invahiant,
some theorems t h a t apply t o a r b i t r a r y B A ' s . Thus, i f equivalence r e l a t i o n on B we need a measure
on
i.e., a, b E B
2
aRb
and
u(a) = u(b).
imply
To f i t our problem i n t h i s framework, we define the equivalence r e l a t i o n on
IF by: A
i f and o n l y i f t h e r e i s an
B
By ( * ) our measure 1-1 must be
6
8 G such t h a t 6*A = B .
%-invariant. I f
u
is
-G-invariant,
then i t a l s o has t o be i n v a r i a n t under the f o l l o w i n g equivalence r e l a t i o n : A
dG
i f and o n l y i f t h e r e are Y,Z E F'
B
6 = C.
L < U
A measure
u
Z.
L'
is
Yi +G Zi
and
>-invariant
such t h a t A = C i < o
i
f o r every
Yi,
< w.
i f and only i f i t i s
zG- i n v a r i a n t .
This equivalence r e l a t i o n cG has t h e f o l l o w i n g p r o p e r t i e s :
(i)
a cangkuence & W o n 06 IF
is
=G
i f A , 8 , C,
(a)
i f Y , Z E wF
(b)
all
(ii)
D
A t B
then
i
E j E
€
F with
ei;C
A
< F , t , Z>, namely,
n B=0=C n D
and A
+ D,
n
w i t h Yi
Y . = 9, = Zi
then
U,
=
j
Yi
ZG
n Z J. and Yi C i < U Zi .
i s ( ~ i n i t c L g ke&inLng, ) namely, i f A , Y o , Y,, A = Yo + Y, , then there are Z o , Z1 E F
LG
and
B = Zo
+
,
Z,
and Y,
Yo eG Z o ,
=G Z1
6C,
.
CG
B cGD, for
Zi
B E F, A cG B such t h a t
(For a discussion and p r o o f o f these p r o p e r t i e s see Tarski 1949, pp.
144,
145, 222). There are some elements o f t h a t an element elements o f
F, Y
A € F €
F t h a t must have measure zero.
L e t us say
i s G - n e g l i g i b l e i f t h e r e i s a sequence o f d i s j o i n t
wF, such t h a t A
zG Yi
f o r every
be i n v a r i a n t a l s o f o r the equivalence r e l a t i o n
i
p ( b ) o r p ( 6 ) > p ( a ) .
may be v e r y d i f f e r e n t f r o m
To a v o i d t h i s problem we need t o impose a s t r o n g e r r e q u i r e m e n t t h a n i n v a r i ance on t h e measure p :
We say t h a t a measure p i s n f i o n g l y R-inuatLiant i f f o r any
a, 6 E B
we have, c € 8 such t h a t aRc and c
p ( a ) 5 p ( b ) if and o n l y if t h e r e i s a
5 6.
Here, t h e second 5 i s t h e usual p a r t i a l o r d e r i n g i n B .Me w r i t e ' a 5 b ' f o r 'there i s a
c
e 8 such t h a t aRc and c 5 b ' . We have t h e f o l l o w i n g :
R
THEOREM 4 . 4 . Let B be a BA and R a 4.thiotey 13un&ue, 4 e 6 i n i n g , c o n g u e n c e helation on 6 . Then t h e m 0 a n f i o n g L y R - invahiant memute on B id and ovtey is 6 0 4 m y atom4 a , 6 E B, we have aRb. hloheoue4, id
t k i b kkmt condition 0 6LLe6.ieeed t h e 4 f i o n g L y R - inuahiant meanuhe
0
unique.
PROOF:
By Lemma 4.1 i t i s c l e a r t h a t t h e measure d e f i n e d i n t h e p r o o f
of
4.2 i s s t r o n g l y i n v a r i a n t i f a l l atoms a r e e q u i v a l e n t ( i t i s enough,in t h i s case, t o d e f i n e
p(a) = 1 / m
, where tn
i s t h e number o f atoms,for any atom
a )* Suppose, now, t h a t p i s s t r o n g l y R - i n v a r i a n t , and with
p ( a ) -< ~ ( b ) Then. . aKc 5 b
a,b
are
atoms
f o r some c . B u t c has t o be an atom and,
so, n e c e s s i t y f o l l o w s . I t i s easy t o see t h a t t h i s measure i s R
- invariant
even
if
not
all
atoms a r e e q u i v a l e n t , though i n t h i s case i t i s n e i t h e r s t r o n g l y R - i n v a r i a n t n o r s a t i s f i e s (+) o f 4.3.
Notice, also, t h a t the condition
l e n c e f o r a l l atoms c o u l d be r e p l a c e d by
'2
of
equiva-
i s a simple ordering',
i.e.
' ~ 5 o ~r b6 5 R a f o r a l l a , b E B ' . L e t us now c o n s i d e r t h e g e n e r a l case. IF i s a c o u n t a b l y complete of subsets o f a s e t X , G
a group o f t r a n s f o r m a t i o n s g f
IF, and I
field
a subset
F . ble want a measure p t h a t i s G - i n v a r i a n t and t h a t vanishes e x a c t l y . ble c a l l measures t h a t s a t i s f y t h i s l a s t r e q u i r e m e n t I - p o n h i u e ( i . e . p i s I - p o s i t i v e i f , p ( A ) = 0 i f and o n l y i f A E I ) . I n
of
on t h e elements o f I
o r d e r t o f o r m u l a t e necessary and s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f such measures we s h a l l use t h e requirements o b t a i n e d i n K e l l e y 1959,for t h e
155
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
existence o f a s t r i c t l y p o s i t i v e measure on a Boolean Algebra. We need some d e f i n i t i o n s , which apply t o any countably complete
x
Let
=
<xi
:
i < n> be a f i n i t e sequence i n
where tn i s t h e l a r g e s t i n t e g e r k 5 n
io < i, < a6
... < ik-l
:
rem, i t i s p o s s i b l e t o p r o v e t h e f o l l o w i n g f o r (1)
i f a , b € A, t h e n
(2)
i f a € A,
then
a A b exists, a 5 m h, where
h = (X/I)/R
i s finite.
, By a theorem o f F i l l m o r e 1965, i t i s p o s s i b l e t o p r o v e t h a t d i s i s o morphic t o a subalgebra o f < C ( S ( I D ) , R ) , + , C > where D i s t h e BA o f i d e m m u l t i p l e elements o f
d . From c o n d i t i o n (iv) on B , we deduce t h a t
ID
a l s o has t h e p r o p e r t i e s i n c l u d e d i n t h a t c o n d i t i o n . Hence, by K e l q e y ' s 1959 measure on
ID. We t r a n s f e r t h e mea-
s u r e t o t h e o p e n - c l o s e d s e t s o f t h e Stone space
S(ID), e x t e n d t h e measure
there i s a
strictly
positive
t o t h e c o u n t a b l y complete f i e l d o f s e t s generated by them, and
define
the
11 on t h e measurable f u n c t i o n s . II i s d e f i n e d on a l l o f C(S(D),Z). T r a n s f e r back, now, II t h r o u g h t h e isomorphism t o 6i and then, u s i n g 16.11 o f T a r s k i 1949, t o IF. T h i s measure on IF has t h e d e s i r e d p r o p e r t i e s . integral
The n o t i o n o f a s t r i c t l y p o s i t i v e measure s h o u l d be m o d i f i e d when want t h e measure G - i n v a r i a n t . A measure t h a t i s G - i n v a r i a n t must on a l l G - n e g l i g i b l e s e t s . B y a r e s u l t o f B r a d f o r d 1971, p. 194, we
we
vanish can
see
t h a t most group o f t r a n s f o r m a t i o n s o f i n t e r e s t produce non - empty G-negli
-
g i b l e s e t s . T h i s i s t h e c a s e , f o r i n s t a n c e , when f o r e v e r y n E w t h e r e a r e d i s Yo,... such t h a t X = CiCnYi and Yi z' Y . f o r e v e r y i , j < n. ,Yn-l G j To t a k e i n t o account t h i s f a c t we i n t r o d u c e t h e f o l l o w i n g n o t i o n :
joint
We say t h a t a measure
A €
F, Let
p(A) = 0
u
on
IF
positive i f for
i s G- strictly
any
implies t h a t A i s G - negligible.
NG be t h e s e t o f G - n e g l i g i b l e elements. I n Chuaqui 1977,
proved t h a t NG i s a c o u n t a b l y complete i d e a l i n
IF.
it i s
Thus, f r o m 4.5.we
get:
THEOREM 4.6. Le* IF be a c o u n t a b l e coi~iple*e d i e l d 06 oubnetn od a b e i X, and G a ghoup 0 6 Ltanhdornunationh 06 IF Then t h e d o U o w h g conditionh
.
and j o i n t l y nuddicient 604 t h e exbintence 06 a countably addiLive, G - &vatLiant, and G - 6.thicLLy pob&iUe rneame on IF: (i) X f NG (ii) IF / NG h a .the K&ey phopehty and b weahey c o u n t a b l y d i n M b u Live. I t s h o u l d be p o i n t e d o u t t h a t (i) i s n o t enough f o r t h e e x i s t e n c e o f a G-in-
ant
MtCedAdhY
v a r i a n t measure. F o r a couterexample see Chuaqui 1973. S i m i l a r l y as f o r f i n i t e B A ' s , t h e measures o b t a i n e d by
these
theorems
a r e n o t , i n g e n e r a l , unique. We need t h e c o n d i t i o n o f s t r o n g i n v a r i a n c e . measure
p on
IF
i s strongly G
- invariant
i f f o r every
A
,
B
€
F
A
with
157
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
A,B
1 NG , we have: u(A) ~ p ( 8 i) f and only i f there i s a C 8 F such t h a t A gGCcB.
The f o l l o w i n g theorem
can be obtained from 2.10 o f Chuaqui 1969:
THEOREM 4 . 7 . Lct B be a countably complcte dield 06 nubseZ5 0 6 a n c t X, and G a ghoup 06 .tham~omnatiom06 B . Then, doh t h e e x h t e n c e o d a counta b 4 a d d i t i v e , n.thongly G - i n v a h i a n t , and G - n&k,t4?y p o n U v e meawe p on F , t h e 6oUowing c o n d i t i o m me n e c e s o m y and nu6@,&2nt: (i) X B NG (ii) doh any
, A, B E
F,
$hue 0 a C E
F,
nuch t h a t
2
A
B S G C ~ A . Mokeoveh, undeh t h e s e conditioMd t h e
4.7.
oh
C 5B
meauhe p 0 unique. Condition ( i i ) i n
can be replaced by:
n~ztih~ies t h e countable chain c o n d i t i o n ( i . e . e v a y n e t 0 6 dinjoint elements 0 6 F 1 NG 0 at mast c o u n t a b l e ) , and 608 any a E F / NG , a # 0 , t h u e 0 a sequence 06 d i n j o i n t e l e m e n t s x 8 wF 1 NG and a 6equence y E wF 1 NG buch t h a t X / NG = Ci < wxi and xi sG yi 5 a 6ok e v m y i < w .
(ii F'/) NG
There a r e several o t h e r a l t e r n a t i v e s t o (ii'), t h a t are
obtained, j u s t
as i s (ii'), from p r o p e r t i e s of simple c a r d i n a l algebras (see Tarski p.
1949,
1 1 7 - 120 and Chuaqui 1 9 7 + ) . The main problem t h a t needs f u r t h e r study i s t h e r e l a t i o n
l o g i c a l theory and t h e measure. I b e l i e v e t h a t the methods
between
the
developed
in
S c o t t and Krauss 1966, might be useful i n t h i s connection, because once
we
have a measure p on
F we can p u t some o f our simple p r o b a b i l i t y s t r u c t u r e s
i n the framework o f t h e i r p r o b a b i l i t y systems w i t h s t r i c t i d e n t i t y . Unfortunately, t h e c o n d i t i o n on t h e i r p r o b a b i l i t y systems t h a t requires the measure on the BA t o be s t r i c t l y p o s i t i v e , excludes some o f the mostnat u r a l simple p r o b a b i l i t y s t r u c t u r e s from t h e i r framework. For instance, l e t
us take the simple p r o b a b i l i t y s t r u c t u r e K formed by r e l a t i o n a l systems le = < C , + , a , U > a E C,where i C , + > i s a c i r c l e i n t h e planeandOchooses 0 i s t h e constant p a r t o f M . G K contains one element o f C. PK i s obthe t r a n s l a t i o n s o f the c i r c l e . A n a t u r a l p r o b a b i l i t y measure t a i n e d from t h e usual Lebesgue measure. It i s easy t o see t h a t t h i s measure does n o t s a t i s f y Gaifman Condition, namely:
ROLAND0 B. CHUAQUI
158
PK ( 3 v
a(,)) = 1
-
and nupFEC( w ) PK(Vue
O(ta))= 0,
C ( w ) i s t h e s e t o f a l l f i n i t e subscts o f
where
s e n t i n g 0, and ta
5, COMPOUND
C
, 3 i s the symbol repre-
the i n d i v i d u a l constant t h a t denotes
a
.
PROBAB I LITY STRUCTURES.
I n t h i s s e c t i o n I s h a l l f o r m a l i z e the s t r u c t u r e s t h a t appear when t h e r e are sequences o f outcomes, as i n the u r n example o f s e c t i o n one. The o f succesion
o f outcomes plays a fundamental r o l e i n these cases.
order In
urn model, there were o n l y two succesive choices, so t h a t the b a s i c was the system < {O
, 11, 5
P
=
IT
:
>. I n the general case, we may have any a r b i t r a r y
. For t h i s system, l e t z = i n : n < t3 t E T I , where n < t means n 5 t and n # t. The p o s s i b l e ‘IT= < T , 5 >
p a r t i a l l y ordered system and
the
order
succesive outcomes are determined by a f u n c t i o n
M w i t h t h e f o l l o w i n g prop-
erties :
Do K = { b : i s a function, Do6 E P, and f o r each R: e Dad, d ( 4 E “ 5 13) (ii) For each E Do M , M i s a simple p r o b a b i l i t y s t r u c t u r e
(i)
6
n =
, and
distinguished e l -
for
6,
g E Do
M, then
mk =
for
6,
g E Do
M, then
a4 = u? J
with
m? j
. .
Do M , M6 i s the s e t o f p o s s i b l e outcomes provided t h a t the
E
IK d i2
should be i n
6
SUC-
6
E DoK , t h e s e t o f p o s s i b l e cutcomes determined by the preced-
Lesive choices given by
a r e r e a l i z e d . The choice a t t o f an
i n g choices. Requirements (iii) and (iv) are c o n d i t i o n s o f coherence:
the
same index should represent the same symbol, and thus the corresponding r e should
l a t i o n should have the same a r i t y ; a p a r t i c u l a r i n d i v i d u a l constant always represent t h e same i n d i v i d u a l .
A compound p r o b a b i l i t y s t r u c t u r e i s , then, a p a i r
with the
p r o p e r t i e s s p e c i f i e d above. The s e t o f p o s s i b l e (compound) outcomes set
H,,.,
,
K>
‘
=
IF : f
Intuitively,
i s the
defined by: E
U
QoM-l
and f o r each t € T, F ( t ) E KFl
.
U represents the order o f succesive outcomes ( i n general,
159
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
i t w i l l be time) and each f u n c t i o n
H,
F €
,a
T, K >
sequence o f
such t h a t each outcome depends on a l l preceding ones. For each F E H
,F ( t )
i s a r e l a t i o n a l system i n K F J z
outcomes,
. The
uni-
F ( t ) i s denoted 1 F ( t ) ( . The universe o f F i n symbols 1 F ( , i s the union o f I F(t)l f o r t E T . The universe o f < 71, K>, i n symbols IIKl,
verse o f
%
i s the union o f
IF 1
F E PI
for
.
The languages we have t o consider f o r these compound p r o b a b i l i t y s t r u c tures must be more complicated than f i r s t - o r d e r languages. We must be able to talk
about f i n i t e sequences o f outcomes. In order t o do t h i s ,
f i n i t e sequences o f elements o f T
. Thus,
our language has f o u r
we
need
types
of
VL4.VLl={uo,v1, . . . I contains the v a r i a b l e s used t o r e f e r t o o b j e c t s i n I K ;1 V12 = {to,tl, ...I
vatLiabLes given by t h e denumerable sets VLl,VL,,V13,and
r e f e r t o e l e m e n t s o f T ; VL3 = {no,fi,,,...l r e f e r t o n a t u r a l
VL,, = C x o , x ,
%)
of
,... 1
numbers ;
and
r e f e r t o f i n i t e sequences o f elements o f T (i.e.elements
. The L o g i d
corn&&
, V,
Y ) the t e r beand T , which stand f o r
i n c l u d e (besides 1 , A
=
nary r e l a t i o n s S and P I which stand f o r a d d i t i o n and m u l t i p l i c a t i o n tween n a t u r a l numbers; the i n d i v i d u a l constants
a
L , which stands f o r 5 between elements o f T ; and the t e r n a r y r e l a t i o n V a l , such t h a t V a l ( x , n ,t) i s i n the numbers 0 and 1 ; the b i n a r y r e l a t i o n
t e r p r e t e d as: t h e n a t u r a l number n i s i n t h e domain o f t h e sequence x and x evaluated a t n i s t. The 6 h d k t L t g .type f o r a language c o r r e s p o n d i n g t o
i E I > , J>, where I = U {J : 6 € D O M I . 6 For each i E I , we have an
6 =
is
i E I and
6
Pi ,and f o r each j EJ,
an i n d i v i d u a l constant c . . The s e t o f i n d i u i d d con,5mh i s denoted CN, J Ul, U CN i s denoted IS1 , and VL3 U IU,'T) i s denoted IS3. The n o t i o n o f dtomic
born&
i s defined by:
(a)
I f a , B € IS1 u VL2 U I S 3 U V 1 4
(b)
I f t E V L , and ao,.
. . ,a mi-l
, then a
E IS1
= 8 i s an atomic formula.
then P . t a o ...am,-l i s an I
atomic formula. (c) (d)
,p E
If n , m
IS3
, then Smp
and Pmp are atomic formulas.
I f x € V L 4 , n € I S 3 , and .tEVL2, then
Val(xnt) i s an atomic
formula. (e)
If R1,
R 2 E VIZ,then L i l t 2 i s an atomic formula.
160
ROLAND0 B. CHUAQUI
For the connectives we define, as usual, -I@ , (@A $), and (4 v $)
as
formulas i f @ and J, are formulas. For s i m p l i c i t y we consider o n l y f i n i t a r y languages. We a l l o w q u a ~ 6 i c ~ over a n any type o f v a r i a b l e w i t h t h e f o l lowing r e s t r i c t i o n : Vu@ i s a formula i f and o n l y i f @ o n l y
For u E V l p and I$ a formula,
contains i n d i v i d u a l symbols i n IS1 and a t most one v a r i a b l e i n VL, o n l y appearing f r e e i n @
. For
the o t h e r types o f v a r i a b l e s t h e r e are no r e s t r i c -
tions. We d e f i n e n a d h d a o t i o n
,
F € H,T,K> Vll
u
u V13 u V14
Vl,
t E Vl,,g(f)
6orniueabin m e m b m 06 M,T,K, . For each F i s a f u n c t i o n g w i t h domain if such t h a t : i f v E VLl g ( v ) E IFI;
06
an a n n i g n i i l e n t Liz
I
if n € V13,
E T;
g(n) €
and i f
W ;
, g(x)
x E V14
EYT
.
I f g i s an assignment i n F and a a v a r i a b l e , g ( y ) stands f o r t h e assignment t h a t coincides w i t h g i n every v a r i a b l e except, p o s s i b l y , i n a where
.
i t assigns T I f a i s an i n d i v i d u a l constant o r a v a r i a b l e , we have: a F = g ( a ) , if a i s a v a r i a b l e ; aF = a . , i f a i s c.; aF = 0 , i f a i s 9 4 J J 9 U; aF = I , i f a i s i 9 I f t e T, and Pi. i s a p r e d i c a t e , we d e f i n e :
.
PI(')
= Ri,
P!()'
=
I
where
Ri i s t h e i& r e l a t i o n o f F ( t ) , i f i
E IF,T ;
0 , otherwise.
Now we can define by r e c u r s i o n g n a d h & h @ i n F, for any formula @, assignment g , and I$ [g] F E H,T,K> ; i n symbols t h i s w i l l be F
F t=
F @[g] i f and o n l y i f a = B g
1) I f @ i s
a = 6 , then
2)
If @ i s
Piaho...an7i-1
3)
I f $ i s Smnp ( o r Pmnp), then F != @
F
, then F I=@
22
5)
nF = pgb). 4 I f @ i s V d ( x n t ) , then F C @ [g] i f F F main o f x and x evaluated a t nF i s 9 9 4 I f @ i s Ltlt2,then F !=@;9] i f and
6)
If @ i s l$
4)
,
then F I=
*
[g] i f and o n l y if
i f and o n l y i f m
( o r mF
4
.
F 9
and o n l y i f nF i s i n F 9
+
F F n g=pg the
t
9'
F
F
o n l y i f tl 5 t2
4
@ rq1 i f and o n l y i f n o t
9
F I=
. $b].
do-
161
SEMANTICAL DEFII.IITlON OF PROBABILITY
If @ i s ($ A
7)
F b e Cgl
e),
F b @ r g ] i f and o n l y i f F!=$b]
then
*
F b @ [g]
I f @ i s ($ V O ) , then
8)
F
0
Cgl
*
, then
If @ i s Ww$J with
9.1)
There i s a v a r i a b l e t € V t 2 f r e e i n $ ; i n t h i s F b@ [g] i f and o n l y i f f o r every T € IF(g(.t))l
w 8 Vtl
Otherwise we define,
IF1
T €
F C @ [g]
with
and o n l y i f f o r every
t h e r e are two cases: case
we
, F I=$ Lg(,")]
i f and o n l y i f
a € Vt2 (Vt3 or T
E T (U o r
For an assignment g and a formula @ f o r every
or
for
have,
.
every
, FC$J[g(:)].
If @ i s W a$
10)
F b $ b ]
i f and o n l y i f
9)
9.2)
and
F € M,T,K>
, F !=@ [g] .
I n order t o d e f i n e
P ( @ ) ,
Vt,),
then F!=@[g]
if
3) F b $[g(:)].
,
I=@ [g]
i f and o n l y i f
the p r o b a b i l i t y o f a sentence @ ( o r ,
more, accurately, o f the equivalence c l a s s o f sentences l o g i c a l l y equivalent to
@) i n
of
K>
< H , K > , we have t o d e f i n e a p r o b a b i l i t y measure on t h e subsets o f the form: IAad ( 4 ) =
IF:
H
F !=@I.
and
With t h i s measure, we d e f i n e P ( @ ) j u s t as i n s e c t i o n three. The problem, again, i s t o d e f i n e a reasonable p r o b a b i l i t y measure on these Of
lH< T , K > *
For each
6
€
DoM,K
6
i s a simple p r o b a b i l i t y s t r u c t u r e and
subsets then,
if
possible, we can proceed as i n s e c t i o n t h r e e t o g e t a measure on an approp r i a t e f i e l d o f i t s subsets. This f i e l d o f subsets o f M i s obtained a s f o l -
6
lows: We take a l l formulas t h a t have j u s t v a r i a b l e s i n VL1, one v a r i a b l e a i n V t 2 , and symbols i n the s i m i l a r i t y type o f
IK
6'
O m i t the v a r i a b l e a a n d
we get a sentence @ o f a f i r s t - o r d e r language. Then t h e f i e l d o f o f the form ModK ( @ ) f o r t h i s type o f sentence i s t h e one needed.
6
subsets L e t US
c a l l t h i s f i e l d B16 and t h e corresponding measure !.I 6' I do n o t have a general s o l u t i o n f o r passing from these measures !.I
to
a measure on the subsets o f ,& ,,
in
>,,
.
It i s easy t o do i t , however,
6
two important cases:
CASE I. The order type o f finite.
r
i s f i n i t e o r w , and f o r each
6
€
Do M,K6 i s
162
ROLAND0 B. CHUAQUI
Consider
B the f a m i l y o f subsets o f M,T,K>
o f t h e form A=IF:F E
H
we F I Poj = j} f o r some j E Do IK. The f i e l d o f subsets o f M are i n t e r e s t e d i n i s generated by B , and i f we d e f i n e a measure on B i t c a n and
be extended t o t h i s f i e l d . We d e f i n e
u ( A ) as the product o f t h e
numbers
u j l (Ij(t)l) ~ for
a l l R: E P a j . Proceeding as i n t h i s case, we can a p r o b a b i l i t y measure f o r the u r n model discussed i n s e c t i o n one. CASE 11.
6,
of
g E Pa
B
is arbitrary,
K, M6
=
IKg
P ,T times, i . e .
.
but simply ordered,
L e t us c a l l
for
every
IK6, P. Then H,T,K, i s t h e d i r e c t power
= TP = I F : DaF = T
P.l,T,K>
and
define
s i r e d measure i s the product measure o f t h e
u
6
and D 0 F - l c P}.
The de-
' s . By the usual considera-
t i o n s i n P r o b a b i l i t y Theory, we can o b t a i n B e r n o u i l l i ' s law
of
l a r g e nun-
bers.
6,
EXAMPLES A N D
APPLICATIONS,
We s h a l l f i r s t study a few p r o b a b i l i t y models and
a n a l y z e l a t e r how
they may be applied. If2 have already seen the model f o r choosing
a sample
from a f i n i t e population. For f u r t h e r reference, I s h a l l c a l l t h i s example, Example 1. EXAMPLE 2.
Distribution o f
b a l l s i n t o n c e l l s . There are several
h
ways
o f s e t t i n g up t h e simple p r o b a b i l i t y s t r u c t u r e s depending on the s t a t i s t i c s t o be obtained. Which o f these a p p l i e s i s determined by the evidence. 2a) Maxwell-Boltzmann s t a t i s t i c s . I n t h i s case each p a r t i t i o n ( i d e n t i f i a b l e ) b a l l s i n t o the
/r
of
the
c e l l s i s e q u a l l y l i k e l y . The s e t lK can be
considered as c o n s i s t i n g o f a l l s t r u c t u r e s Cno = < A,Po,. contains a l l p a i r s o f numbers < i , m > w i t h
i
where A
and m < n ( i . e . A = ) L X n ,
i f we adopt t h e usual s e t - t h e o r e t i c a l conventions)'; Pm c o n s i s t s o f a l l parts, w i t h m as second coordinate, and 0 i s any subset o f A t h a t i s a f u n c t i o n w i t h domain t h e s e t o f a l l numbers l e s s than h . Each elenents o f
...
A, < i , m >
represents t h e f a c t t h a t t h e b a l l
i is in
c e l l m. Po, Pn-l are t h e n c e l l s and 0 i s the p a r t i c u l a r p a r t i t i o n chosen. I t has t o be a f u n c t i o n , because each b a l l can be i n o n l y one c e l l . The group GK c o n s i s t s o f a l l permutations f u n c t i o n f o r every f u n c t i o n 0 w i t h domain n o t c o n t a i n a l l permutations o f
A.
4 .
6
o f A such t h a t
Thus, i n t h i s case
6*0 GK
is a does
2b) B'dse - E i n s t e i n s t a t i s t i c s . I n t h i s case t h e b a l l s are n o t i d e n t i f i a b l e
163
SEMANTICAL D E F I N I T I O N OF PROBABILITY
The systems
a0 i n
K , f o r t h i s s t a t i s t i c s , a r e o f the same s i m i l a r i t y type
as before, i. e . d o = < A,Po,.
.. Plz- 1 , 0 > . The
A, such t h a t i f < i , m > € 0 ,
i s a l s o the same, b u t 0 i s a subset o f every < j , m > f o r some
j < i i s also i n 0 (i.e.
for
.. Pn-1'
common p a r t < A , P o , . f o r each m
a € c
where C i s the s e t o f p o i n t s i n the c i r c l e , F resents t r a n s l a t i o n s i n the c i r c l e ,
tF
,
i s the s e t o f forces,
a d d i t i o n o f forces,
6
tc rep-
i s t h e con-
tinuous f u n c t i o n t h a t associates each i n i t i a l f o r c e w i t h a f i n a l
position,
and I i s t h e s e t c o n t a i n i n g the i n i t i a l f o r c e ( I contains one elements o f
F 1. In t h i s example, i f g E GK and the d i s t i n g u i s h e d elements a r e t r a n s formed by
CJ
to
g ( a ) ( a € C), then these new elements should
s a t i s f y the
GK s a t i s f i e s c o n d i t i o n s
l ( a ) and l ( b )
same sentences as the o l d ones j i . e .
i n s e c t i o n 3). These transformations i n GK are isometries
o f the
Their e f f e c t on t h e semigroup o f forces i s more complicated. the functions g i n GK
6(x)
do n o t d i s t i n g u i s h elements x , y
= 6(y). Thus, g t r e a t s
a l i k e . As
6
i s continuous,g
of
circle.
In p a r t i c u l a r , F such t h a t
F as a c i r c l e by d e a l i n g w i t h these two p o i n t s i s a l s o an isometry o f t h i s new
"circle".
Hence, i f one takes two i n t e r v a l s A , B o f t h e same length o f t h i s " c i r c l e " o f forces, the class o f s t r u c t u r e s t h a t has 1 i n A i s e q u a l l y l i k e l y t o the class t h a t has I i n
B. Which f i n a l p o s i t i o n s a r e equally l i k e l y depends
164
ROLAND0 9. CHUAQUI
on t h e f u n c t i o n
6.
I f t h e r o u l e t t e i s balanced, we should have an
sends equal i n t e r v a l s o f forces i n t o equal i n t e r v a l s o f p o i n t s ,
d
and
that then,
equal i n t e r v a l s i n t h e c i r c l e o f p o i n t s are equiprobable. I f t h i s i s n o t the case, we might have p o i n t - i n t e r v a l s o f d i f f e r e n t l e n g t h e q u a l l y l i k e l y . When we add a v a r i a b l e i n i t i a l p o s i t i o n o r
a variable
direction
of
force, t h e a n a l y s i s i s s i m i l a r , though more complicated.
A s i m i l a r method can be a p p l i e d t o the case o f t h e throw o f a c o i n o r a How-
die, because t h e outcome depends on the f i n a l p o s i t i o n o f t h e o b j e c t .
ever, the s i t u a t i o n i n t h i s case i s even more complicated so we w i l l n o t a t tempt a d e s c r i p t i o n . I t i s u s e f u l t o n o t i c e , t h a t i n t h e case o f a t r u e c o i n o r d i e , t h e models can be s i m p l i f i e d thus:
c A , I > where A i s t h e
set
of
faces and 1 contains the one t h a t obtains. We pass now t o examples o f compound s t r u c t u r e s . We have
a l r e a d y dis-
cussed an instance o f th,e u r n model. I s h a l l c a l l i t Example 4. A l l o t h e r cases o f u r n models can be represented by s i m i l a r p r o b a b i l i t y s t r u c t u r e s . For these examples, we d e f i n e a p r o b a b i l i t y measure by Case I o f s e c t i o n 5.
EXAMPLE 5.
An example o f much i n t e r e s t i s when we have the same experiment
repeated several times. To be s p e c i f i c , l e t us take the t o s s i n g o f coin
M
times. We take
Cel = < A , C1>
I
a
true
KO w i l l be t h e s e t o f j u s t two models where A = { h , t}, C1 = {lz} and C2 = It 1
= < n , 5 >.
and U l 2 = < A , C >;
5 E Ua lK, M = K O . 5 To d e f i n e a p r o b a b i l i t y measure f o r t h i s example, we use the method d i s -
( h = heads, .t = t a i l s ) . For every cussed f o r Case I 1 i n s e c t i o n 5.
I s h a l l now discuss b r i e f l y how these p r o b a b i l i t y models can beapplied. My views here are very much i n f l u e n c e d by Lucas' views(see Lucas 1970)
es-
p e c i a l l y i n Chapter 5. When applying P r o b a b i l i t y 'co p a r t i c u l a r cases,
we
have t o d i s t i n g u i s h two f a c t o r s : t h e framework and the evidence. I n my way o f s e t t i n g up things t h e framework i s given by the p a r t i c u l a r
probability
s t r u c t u r e s chosen, and t h e evidence i s what leads us t o these s t r u c t u r e s . We s h a l l see f i r s t the case analyzed by J e f f r e y s 1961, p. 15.,discussed a l s o by Lucas 1970, p. 50.: "Suppose that
I know
that S m i t h i s a n Englishman, b u t otherwise
n o t h i n g p a r t i c u l a r a b o u t him. He i s v e r y l i k e l y , on t h a t e v i d e n c e , have a b l u e r i g h t eye. But s u p p o s e t h a t I am i n f o r m e d t h a t h i s
know to left
eye i s brown - t h e p r o b a b i l i t y i s changed c o m p l e t e l y " .
I agree w i t h Lucas t h a t
t h i s i s n o t a p r o b a b i l i t y o f asingular
165
SEMANTICAL D E F I N I T I O N OF P R O B A B I L I T Y
p r o p o s i t i o n whose subjects i s Smith. Instead, i t i s t h e p r o b a b i l i t y of t a k i n g a sample o f one element o f a population. In t h e f i r s t case, t h e s e t lKE o f possible outcomes c o n s i s t s o f s t r u c t u r e s <E,B,S > where E i s t h e
set o f
Englishmen, 6 The b l u e - e y e d Englishmen,and S a s e t o f one element. I n the
ME
second case, i . e . when we know t h a t Smith has a brown l e f t eye, t h e s e t o f possible outcomes c o n s i s t s o f s t r u c t u r e s where
€3h
i s the class
o f Englishmen w i t h a brown l e f t eye. L e t I$ be t h e sentence " t h e r e i s in S
and x i s i n
a b l u e eye i s
x
an
E " . I n the f i r s t case t h e p r o b a b i l i t y o f Smith having
PK (I$), w h i l e i n t h e second E
it i s
PK
probabi l i t i e s a r e q u i t e d i f f e r e n t .
E
(I$),a n d
t h e s e two
The case o f the throw o f a c o i n can be analyzed as f o l l o w s . From p r e v i -
ous experiences, physical laws, e t c . ... we suppose a t f i r s t t h a t t h e c o i n used i s a t r u e c o i n w i t h equal p r o b a b i l i t i e s , as explained i n E x a m p l e 3. Further evidence might l e a d us t o the conclusion t h a t t h i s i s n o t the r i g h t model and we change i t . Among the important items o f f u r t h e r
evidence i s
t h a t provided by s u c c e s s i v e t r i a l s a t throwing the coin. Here,using compound p r o b a b i l i t y models, as those i n Example 5,
we may use B e r n o u l l i ' s
theorem
I f a sequence
obtains
and estimate t h e P r o b a b i l i t y o f a c e r t a i n sequence.
t h a t would be very improbable according t o our i n i t i a l assumptions,we r e j e c t t h i s i n i t i a l model and look f o r another. My procedure terms,
w i t h t h a t appearing i n Lucas 1970 Chapter V
agrees,
i n general
which t h e reader may
consult f o r f u r t h e r d e t a i l s . The o n l y s t a t i s t i c a l methods t h a t I have n o t y e t thought are some types o f Bayesian s t a t i s t i c s .
Prima
facie,
out c a r e f u l l y
they seem n o t
t o be
j u s t i f i e d , because they i n v o l v e p r o b a b i l i t i e s o f p r o b a b i l i t y hypotheses t o gether w i t h p r o b a b i l i t i e s o f events and t h i s , i n my system, would i n v o l v e a mixture o f o b j e c t - language and metalanguage.However there might be methods o f dealing w i t h t h i s mixture. A p o s s i b i l i t y i s given by procedures
similar
t o those appearing i n S c o t t and Krauss 1966, s e c t i o n 6. I b e l i e v e t h a t the methods I have presented i n t h i s
adequate d e f i n i t i o n o f a p r o b a b i l i t y measure t a k i n g i n t o
paper provide account t h e
an main
c h a r a c t e r i s t i c s o f p r o b a b i l i t y statements.
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ROLAND0 6 . CHUAQUI
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D e h W a h h h e i t n b e g h i b b i n d e n 6ohmadibiehten Spmchcn, Studia Philos. (Warsaw).Vol. 1, 261 - 405 (English t r a n s l . i n Logic,Semantics and Metamathematics, Oxford U. Press, 152-278).
1949,
Cardinal Algebras , Oxford U n i v e r s i t y Press, New York.
1954,
ConttLibwtionb t o t h c t h e m y VOI.
06
mod&
I,II, Indagationes Mathematicae,
16, 572-588.
l n s t i t u t o de Matemitica Universidad C a t o l i c a de C h i l e Santiago, C h i l e .
and Departamento de M a t e m i t i c a
Universidade Estadual de Campinas Campinas. Sao Paulo, B r a z i l .
Non-Classical Logics, Model Theory and Computabi l i t:!, A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.) 8 North-Holland Publishing Company, 1977
THE EQUIVALENCE OF SOME
AXIOMS
OF
STRONGIWIFIITY
b y L U l Z PAUL0
DE ALCNJTARA
I , I NTRODUCTI O N , As pointed o u t by Mostowski 1967 t h e e x i s t e n t i a l assumptions
known as
'axioms o f i n f i n i t y ' can be j u s t i f i e d by two general p r i n c i p l e s : (I)
P r i n c i p l e o f t r a n s i t i o n from p o t e n t i a l t o a c t u a l i n f i n i t y .
(11)
P r i n c i p l e o f existence o f s i n g u l a r sets.
An e a r l y a p p l i c a t i o n o f t h e f i r s t p r i n c i p l e i s Dedekind's argument
for
the existence o f i n f i n i t e sets. Also, t h e f i r s t p r i n c i p l e allows us t o formulate the axiom o f inaccess i b l e numbers and L g v y ' s r e f l e c t i o n schema f o r Zermelo
- Fraenkel
s e t theo-
ry. The second p r i n c i p l e allows us t o formulate s t i l l stronger axioms. L e t us suppose t h a t i n c o n s t r u c t i n g sets by means o f t h e usual set-theo r e t i c operations we o b t a i n o n l y sets w i t h a p r o p e r t y P. I f good reasons why a l l s e t s should have the p r o p e r t y P we
t h e r e a r e no
are f r e e t o a d d t o
the axioms a statement i m p l y i n g the existence o f sets w i t h o u t the
property
P.
For example we apply t h i s second p r i n c i p l e i n t h e f o r m u l a t i o n
of
the
axiom s t a t i n g the existence o f medsurable c a r d i n a l s . I n t h i s paper we study some statements r e l a t e d w i t h the p r i n c i p l e o f e x istence o f s i n g u l a r sets, which are e q u i v a l e n t t o r e f l e c t i o n p r i n c i p l e s . We work i n t h e syStem BG o f Bernays t i o n and choice. 169
-
Gddel w i t h t h e axioms o f founda-
170
L U l Z PAUL0 DE ALCANTARA
We employ the terminology and t h e n o t a t i o n o f Drake 1974.
I n t h i s s e c t i o n we given an account o f some r e s u l t s a n n o u n c e d
i n de
Alcantara 1974.
A. L6vy 1960 proposes an axiom schema (M) t o be added t o t h e Z e r m e l o
-
Fraenkel axioms which p o s t u l a t e s t h e existence o f a t l e a s t one i n a c c e s s i b l e number i n the range o f every normal f u n c t i o n d e f i n e d id.f.a.0.)
THEOREM 1 .
(M)
h equivalent t o .the 6oUow&g Achema :
Euehy nvtonal dunction d.6.a.o. wkich atre inaccennible.
THEOREM 2. (M")
a l l ordinals
(Lgvy)
(MI)
(M)
for
and proves t h e
has cYrb&Wy
h g e @xed pointn
(Lgvy)
h e q u v d e n t t o t h e conjunction 06: Evetly notonal d u n d o n d.6.a.o. h a at t e a t one heguRatl nwnbm i n .i& hange.
@ Thehe I) exht
a t r b M y
h g e inaccennible n w n b m .
LGvy's p r o o f o f theorem 2 i n v o l v e s a n o t i o n o f inaccessible number whose equivalence w i t h the usual one seems demonstrable o n l y on the assumption o f the axiom o f choice (see Shepherdson 1952). Assuming the axiom o f choice i t i s p o s s i b l e t o prove t h a t (M)is equiva-
.
l e n t t o (M")
We need a lemma:
LEMMA 1 .
(M") hpk%A: (3) Evehy namal d u c t i o n d.6.a.a.
has a2 l e a t one hegULah
6ixed
point. PROOF:
6,
5
Let
6
be a normal f u n c t i o n d.f.a.0.;
let
6'
be t h e d e r i v a t i v e o f
the normal f u n c t i o n which counts t h e f i x e d p o i n t s o f 6 . Since 6 i s d.f.a.0. then 6 ' i s a l s o d.f.a.0.. By (M") t h e r e e x i s t s
i.e.,
such
that
6'(5)
= f3
i s regular.
a
171
AXIOMS O F STRONG I N F I N I T Y
THEOREM 3 .
(M)
.iA
eqLLiude& -to (MI').
PROOF: Obviously (M) i m p l i e s (M"). I n order t o prove t h e converse we g i v e some p r e l i m i n a r y d e f i n i t i o n s : L e t 5 be a c a r d i n a l , and -t5
t (0) = x 5
a sequence defined by
tp.
t ( n + 1 ) = 2
5
We p u t
L e t g be a f u n c t i o n defined on the c l a s s
g(5) =
of
c a r d i n a l numbers by
*
5
Finally l e t
O n + O n b e a f u n c t i o n defined by
6:
for all
5
.
We have g ( 5 ) >
6(0) = g(0)
6(5
+
1) = g(d(5))
d ( X ) = sup
P L"
f o r an i n a c c e s s i b l e
AXIOMS O F S T R O N G I N F I N I T Y
177
REFERENCES, Boffa, M. 1970,
de pLLinsance .indEtieune cmdinul inhini donnT, B u l l e t i n de l a S o c i g t e Mathgmatique d e
Sun L’enhenible den en5embLen h&Edit&aiien.t UM
B e l g i q u e , 22, 3 8 9 - 392. de A l c a n t a r a , L. P. 1974,
On kiew
ctxioiii
06
oche~iic&t
nthomj L n 6 i ~ L i t yin axiotiiatic
nef
theaay
( a b s t r a c t ) , The J o u r n a l o f Symbolic L o g i c , 39, 410. Drake, F. 1974,
S e t Theory,
North
- Holland,
Amsterdam.
Kruse, A . H. 1965,
G?utliendiccl: i i i i i v w e s cciid .the sups Compositio Mathematicae, 17, 96 - 101.
COlllJJ~e*~ riiodeb
06
Shephadoun,
Lgvy, A. 1960,
Axiatii
nclwnic~tct 0 5 n t r i u i i g i i i i i i i t y
iii
uxioittcLtic net theutry,
Pacific
J o u r n a l o f Mathematics, 10, 2 2 3 - 238. Mostowski, A. 1967,
Recerd t r e ~ d ! Ain ne,t theatry, i n Problems i n t h e Philosophy of
Mathematics, N o r t h
- Holland,
Amsterdam.
Shepherdson, J . 1952,
l n n m I?lad& 225
- 237.
doh Se.t
Tlieotry, The J o u r n a l o f S y m b o l i c
Logic,
Departamento de Matematica Universidade Estadual de Campinas Campinas, Sao P a u l o , Brazil.
14,
Non-Classical Logics, Model Theory and Computability. Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
A.I.
PRESENT PROBLEMS ABOUT INTERVALS IN RELATION-THEORY AND LOGIC, b y ROLANU F R A Y S S E .
0 1NTRODUCTI ON I
Everybody knows
AND SUMMARY ;
t h e two
t o t a l ordering r e l a t i o n .
d e f i n i t i o n s o f an
Firstly, the abhotde
i n t e r v a l i n a chain,
intehvd
c l o s e d under i n t e r m e d i a r i t y : i f x and g > x belong every z such t h a t
to
, which
the
or
i s a Set
interval,
then
x < z < y belongs t o i t . Secondly t h e h e h t i v e i n t e h v d :
g i v e n two elements a and b > a , whose s e t Ia,bl relative interval i s the set o f t i o n o f the r e l a t i v e interval o f
bound,
i s called the
the
x ' s such t h a t a < x < b . Analogous d e f i n i x ' s such t h a t x < a, o r such t h a t
the
More g e n e r a l l y , g i v e n an a r b i t r a r y subset F o f t h e base E o f
x > a. ordering
F w i l l be a subset o f E - F, D such t h a t , i f t belongs t o F, t h e n a l l x ' s o f U a r e t. Any a b s o l u t e i n t e r v a l D i s a r e l a t i v e i n t e r v a l , by t a k i n g t h e bound F = E - D ; and c o n v e r s e l y any r e l a t i v e i n t e r relation, a
~ ~ d a t i wintehwae e w d h bound
maximal by i n c l u s i o n among s e t s
v a l i s o b v i o u s l y c l o s e d by i n t e r m e d i a r i t y , But t h i s e q u i v a l e n c e
t h e n i s an a b s o l u t e
between d e f i n i t i o n s w i l l
not subsist i n
interval
.
generaliza-
tions t o arbitrary relations. I n t h e y e a r s 1950, t h e a u t h o r proposed t h e (see, f o r i n s t a n c e , F r a f s s e 1973, p. 107). tions
A and
following
generalizations
Recall that, given
B o f common a r i t y n , a b i j e c t i o n
6
two
base I A l , t o a subset G o f I B I , i s c a l l e d a t o c d hoamohpkism o f A 8, when
6
i s an isomorphism o f t h e r e s t r i c t i o n A / F o n t o B/G. F o r
l o c a l isomorphism, i t i s s u f f i c i e n t , and o b v i o u s l y necessary, s t r i c t e d t o every s e t o f < n
elements o f i t s domain F , be a 179
rela-
F of
f r o m a subset
being
that local
the
towards
6
a re-
isomor-
180
ROLAND
phism. R e c a l l t h a t
6
FRA'I~SSE
i s c a l l e d a L o c d aLLtomo/Lpkinm o f
6
A, when
lo-
is a
c a l isomorphism o f A towards A. These n o t i o n s , as t h e f o l l o w i n g n o t i o n s o f i n t e r v a l s , e x t e n s i b l e t o any muLt.iheL&on,
which i s a f i n i t e
are
sequence
immediately of
relations
t h a t a r e c a l l e d i t s c o m p o n e ~ ,w i t h a common base; t h e n t h e a h i t y
of
the
m u l t i r e l a t i o n w i l l be t h e maximun o f t h e a r i t i e s o f i t s components.
Gener-
a l l y we w i l l c a l l them r e l a t i o n s , by a b b r e v i a t i o n ,
compo-
except when new
n e n t s a r e e x p l i c i t l y added. Given a r e l a t i o n A o f base E = I A l , a subset D o f E i s c a l l e d an A-ubboLLLte i n t e n u d , o r s i m p l y an A - i n t e r v a l , when any l o c a l automorphism o f t h e r e s t r i c t i o n A/D, extended by t h e i d e n t i t y on E - U , g i v e s a l o c a l automorphism o f A: we s h a l l say s i m p l y t h a t i t i s e x t e m i b e e by t h e i d e n t i t y on E - 0. Note t h a t i t i s s u f f i c i e n t t o c o n s i d e r l o c a l automorphism o f A/U on domains o f p 5 n-J elements, and t o e x t e n d them by i d e n t i t y on n - p e l e ments o f E - D .
If A
I f A i s a chain, we r e f i n d t h e usual i n t e r v a l .
is
a
U i s an A - i n t e r v a l i f f , f o r any element t o f E - D , e i t h e r a l l elements o f D a r e < t , o r a l l a r e >.t, o r a l l p a r t i a l o r d e r i n g r e l a t i o n of base E , t h e n
a r e incomparable t o 2 . Given a r e l a t i o n A o f base
E, t a k e a subset F o f E. Then a s u b s e t D o f
E - F i s c a l l e d an ( A , F ) - i n t e , t ~ ~ L , o r an A - h e h L i w e intwvul bound, when
with
F as
a
D i s a maximal s e t , by i n c l u s i o n , among s e t s 0' such t h a t any
l o c a l automorphism o f A/D',
extended by i d e n t i t y on
F, g i v e s a
local
au-
D' have, f o r any e l ement t o f F, a l l t h e i r elements < t o r a l l >t;and by t a k i n g f o r D a maxi m a l D' , we g e t a s e t c l o s e d f o r i n t e r m e d i a r i t y , and so a u s u a l interval. Any a b s o l u t e i n t e r v a l D i s o b v i o u l y a r e l a t i v e i n t e r v a l , by t a k i n g t h e tomorphism of A. I f A i s a chai'n, t h e p o s s i b l e s e t s
bound E - D .
However t h e converse i s f a l s e , a l r e a d y f o r
a
partial
i n g r e l a t i o n A. F o r i n s t a n c e , g i v e n u a n d b > a (modu1o.A) t h e s e t X'S
such t h a t a < x < b i s a r e l a t i v e i n t e r v a l P , w i t h
Now suppose t h e r e e x i s t x,y i n incomparable t o y and
{u,b}
as
D, and t o u t o f D, w i t h u < x
v ,
ble
t h e i d e n t i t y on I A I
-
Eh-l;
and
by
b u t n o t e x t e n s i b l e by t h e i d e n t i t y on IAI-€h-l
i n c r e a s e d by t h e element w . So t h a t t h e s e t o f elements h a v i n g an even rank i n E h , f o r i n s t a n c e , i s n o t an A - f i n i t e - V a l :
c o n t r a d i c t i o n p r o v i n g o u r con-
c l u s i o n ( 1 ) . The c o n c l u s i o n ( 2 ) f o l l o w s f r o m 1.1. 0
5 , THE
SUBVAL,
AN INTERMEDIARY
BETWEEN INTERVAL
NOTION
AND F I N I T E - V A L , Given a r e l a t i o n A, a subset 0 o f t h e base I A l w i l l be c a l l e d a n
d
subvnL when, f o r any l o c a l automorphism
IAl
- D , either Q
i s n o t e x t e n s i b l e by t h e i d e n t i t y on G, o r
by t h e i d e n t i t y on any subset o f D*, phism o f 3.1.
6
A-
D*
o f A/D, and any subset G o f
=
i s extensible
o b t a i n e d f r o m G by any l o c a l
automor-
A/D*.
Any i n t e h v d a h e x t e h v d -in a n u b v d .
D be an e x t e r v a l and E t h e base, Q be a l o c a l automorphism o f AID, w i t h do-
D T h i s i s obvious f o r an i n t e r v a l . L e t so t h a t E - 0 i s an i n t e r v a l . L e t main F and codomain
F ' , and l e t g be a l o c a l automorphism o f A/(€-D),
domain G and codomain G ' . Suppose t h a t on G: we have t o p r o v e t h a t The u n i o n g U IF'
6
6
6
i s a g a i n e x t e n s i b l e b y t h e i d e n t i t y l G , on
U IG b e i n g a l o c a l automorphism, and
i s a l o c a l automorphism and so
u
6
U g, we g e t
lG G I .
E - D an i n t e r v a l ,the u n i o n
g by composition ( s e e a l s o 1.5).
Noreover g U I F and t h u s g - l U I F a r e l o c a l automorphisms: with
with
i s e x t e n s i b l e by t h e i d e n t i t y
6 u lG,. 0
by
composition
T h e r e e x i s t s a subval which i s n e i t h e r a n i n t e r v a l , n o r a n e x t e r v a l . T a k e t h e c y c l i c o r d e r i n g r e l a t i o n a l r e d y d e f i n e d i n 2,from t h e t o t a l o r d e r i n g o f n a t u r a l i n t e g e r s : f o r any i n t e g e r u, t h e s e t o f i n t e g e r s ? u i s a subval. cause i f u = 0, t h i s i s t h e whole base; i f u ? 1, among i n t e g e r s
any
local
Be-
automorphism
u , i s e x t e n s i b l e by t h e i d e n t i t y on t h e s i n g l e t o n
{Ol, i f f
i t preserves t h e usual t o t a l o r d e r i n g between i n t e g e r s ; and i n such a case, i t i s e x t e n s i b l e by t h e i d e n t i t y on a l l t h e i n t e g e r s < u.
Now
it i s
suffi-
192
ROLAND FRA'I'SSE
2
c i e n t t o take u
2
2, i n o r d e r t h a t t h e s e t o f i n t e g e r s
u be n e i t h e r
an
i n t e r v a l n o r an e x t e r v a l : we a l r e a d y n o t i c e d i n 2 t h a t t h e o n l y i n t e r v a l s o f t h e c y c l i c o r d e r i n g a r e t h e empty s e t , t h e base and t h e s i n g l e t o n s . Anq n u b v d b
3.2.
R
binite-ud.
0 L e t n be t h e a r i t y o f o u r r e l a t i o n A; we know t h a t , i n o r d e r f o r D t o i t i s s u f f i c i e n t t h a t sequences o f l e n g t h p 5 n f a l l i n t o
be a f i n i t e - V a l ,
a f i n i t e number o f e q u i v a l e n c e classes,each of t h e s e c l a s s e s b e i n g sodefined, t h a t e q u i v a l e n t sequences g i v e a l o c a l automorphism e x t e n s i b l e by on any s e t o f M - p elements o u t o f Q; and a l l t h a t f o r e a c h p pose
2
identity
M. Now sup-
D be a subval. F o r each q 5 n, c o n s i d e r as e q u i v a l e n t any two subsets (where E i s t h e base) w i t h c a r d i n a l q , when t h e r e s t r i c -
G and G' o f E - D
a r e i s o m o r p h i c . Take a r e p r e s e n t a n t i n e a c h
o f these
e q u i v a l e n t classes, and l e t H be t h e f i n i t e s e t , u n i o n o f t h e s e
represent-
t i o n s A/G and A/G'
ants. Now c o n s i d e r any two sequences u,u i n D, w i t h a same len,gth 5 M , as e q u i v a l e n t when t h e t r a n s f o r m a t i o n o f u i n t o u i s a l o c a l automorphism o f A , e x t e n s i b l e by t h e i d e n t i t y on H . As H i s a f i n i t e s e t , t h e r e a r e f i n i t e l y
D
many c l a s s e s o f e q u i v a l e n t sequences. I t i s s u f f i c i e n t now t o n o t e t h a t ,
b e i n g a s u b v a l , t h e t r a n s f o r m a t i o n o f u i n t o an e q u i v a l e n t s e q u e n c e u o f length
p 5 M,
if' i t i s
e x t e n s i b l e by t h e i d e n t i t y on H , i s a l s o e x t e n s i b l e
by t h e i d e n t i t y on any subset o f E - Q
with cardinal
a subset G t h e r e e x i s t a subset G ' o f H w i t h A/G'
n - p , since f o r i s o m o r p h i c t o A/G
There e x i s t s a f i n i t e - V a l which i s n e i t h e r a subval
nor
the
such
.0
comple-
mentary s e t o f a subval. To see i t , n o t e t h a t , f o r a t o t a l o r d e r i n g r e l a t i o n A
, the
A-subvals a r e e x a c t l y t h e A - i n t e r v a l s and t h e A - e x t e r v a l s .
Indeed
f o r any s e t D which i s n e i t h e r an i n t e r v a l n o r an e x t c r v a l , t h e r e e x i s t s x ,
q, i n D and z , R o u t o f D w i t h x < z < q < t o r R < x < z < q. I n any case t h e t r a n s f o r m a t i o n o f x i n t o q i s e x t e n s i b l e by t h e i d e n t i t y o n t h e s i n g l e t o n {t} b u t n o t b y t h e i d e n t i t y on { z } .
Now t h e u n i o n o f t w o
intervals
a d m i t t i n g a n o t h e r i n t e r v a l between them and a n o t h e r a f t e r , f o r a t o t a l d e r i n g , i s a f i n i t e - V a l which i s n e i t h e r a subval n o r t h e complementary
orof
a subval. There e x i s t s a subval whose complementary s e t i s n o t a s u b v a l . Indeed, t a k e t h e p a r t i a l o r d e r i n g r e l a t i o n o b t a i n e d f r o m two t o t a l o r d e r i n g s A and
8, w i t h any element o f I A l uncomparable t o any element o f 181. Take a median i n t e r v a l 8' o f 8, and c a l l D t h e u n i o n o f I A l and 18'1. Now Q i s a subVal, s i n c e any l o c a l automorphism i n D, e i t h e r t r a n s f o r m s an e l e m e n t o f I A l i n t o an element o f I B ' I , o r conversely, and t h e n i s n o t e x t e n s i b l e t o
193
PRESENT PROBLEMS ABOUT INTERVALS
any i d e n t i t y out of D; o r i t i s t h e union of a local automorphism of A and of one of B ' , and then i s e x t e n s i b l e t o the i d e n t i t y on E - D (where E i s the base). However E - D i s n o t a subva1,sincethe transformation of an element of I B I , a n t e r i o r t o B ' , i n t o an element p o s t e r i o r t o R ' , i s e x t e n s i b l e by t h e i d e n t i t y on t h e s i n g l e t o n of any element of I A l , b u t not on t h e singleton of any element of 18'1. F i n a l l y , note t h a t the notion of subval i s not preservedby f r e e i n t e r p r e t a b i l i t y , as i s t h e notion of f i n i t e - v a l . Let C be a chain, o r t o t a l ordering, A a unary r e l a t i o n on the base I C I , w i t h value + f o r elements of a C - i n t e r v a l , and B another s i m i l a r unary r e l a t i o n , corresponding t o a C-interval d i s j o i n t from the former, and even l e t t i n g a t l e a s t one element between them, and a t l e a s t one element a f t e r them. Mow t h e union of these Ci n t e r v a l s A and R i s n e i t h e r an i n t e r v a l nor an exterval f o r C , thus i t i s not a C-subval. However i t i s an ( A €3 C)-subval, where A 8 C i s t h e multirel a t i o n of components A , 8 , C. Indeed i f 6 i s a local automorphism of A €3 C i n s i d e t h e union of C-intervals A and 8 , then 6 cannot transform an e l e ment of t h e C-interval A i n t o an element of t h e C-interval 8 , b e c a u s e A takes value + in t h e former a n d - i n t h e l a t t e r . So 6 i s the union o f a local automorphism of C i n s i d e A and a l o c a l automorphism i n s i d e B , and then 6 i s e x t e n s i b l e by any i d e n t i t y o u t of t h e union of i n t e r v a l s A and B
.
of an
PROBLEM: I f A i s f r e e - i n t e r p r e t a b l e i n a chain, the complementary set A-subval i s i t an A-subval?
4 , THE
RELATIVE
INTERVAL,
Note t h a t t h e empty s e t i s a r e l a t i v e i n t e r v a l , with t h e whole base as i t s bound; t h e whole base i s a r e l a t i v e i n t e r v a l , the unique one w i t h t h e empty s e t a s i t s bound; t h e s i n g l e t o n of any element u , with t h e base minus u as a bound ( o t h e r bounds can e x i s t ) . Note t h a t f o r a given bound F, t h e condition about s e t s D ' in our introduction: "any local automorphism of A/D' i s e x t e n s i b l e by the i d e n t i t y o n F 'I, i s an inductive condition. P r e c i s e l y , i f f o r a same F, we have an ascending chain of s e t s D' ( f o r i n c l u s i o n ) , t h e i r union i s s t i l l a 0 ' . Indeed i f n i s t h e a r i t y of A , f o r being a l o c a l automorphism of A / D ' and being e x t e n s i b l e by t h e i d e n t i t y on F, i t i s s u f f i c i e n t t h a t a b i j e c t i o n 6 h a v e a l l i t s r e s t r i c t e d b i j e c t i o n s t o any s e t with cardinal p 5 M , s a t i s f y i n g the same property, reduced i t s e l f t o s u b s e t s of F with cardinal n - p , and
194
ROLAND
FRAYSSE
...,
that f o r p = 1,2, n. As an immediate Consequence, given a subset F o f the base, a n y s i n g l e ton o u t o f F being obviously a s e t 0’ w i t h t h e above mentionned p r o p e r t y , there e x i s t s an (A,F)-interval
which includes t h i s s i n g l e t o n . I n o t h e r words,
$he union 06 a&? (A,FJ-intenw&
0 IAl
- F.
Other consequence: id G_c F, any ( A , F ) - i n t e h w d 0 i n c l u d e d
i n an
(A,G) -int&t~d.
Foh a b i n m j h e l u t i o n A, and a divLite nubnet F
4.1.
06
L2.b bane, Mehe
arre divLiteRy many (A,F)-intehv&. 0 Consider as e q u i v a l e n t any two elements u,v o f I A l
-
F,when t h e t r a n s -
formation o f u i n t o w i s a l o c a l automorphism o f A, e x t e n s i b l e by t h e ident i t y on F. Since F i s f i n i t e , t h e r e a r e f i n i t e l y many classes i n t h i s equiv-
alence. Then any i n t e r v a l w i t h
F as a bound , as soon as i t contains an e l -
ement u , has t o c o n t a i n a l l the c l a s s o f u; as i t i s seen
6
bijection
because
a
i s a l o c a l automorphism o f A i f f i t s r e s t r i c t e d b i j e c t i o n s
to
any p a i r o f elements o f t h e domain, a r e themselves l o c a l automorphisms. 0 Note t h a t the statement does n o t extend t o t e r n a r y r e l a t i o n s . For i n stance, take the c y c l i c o r d e r i n g defined i n 2, from
the usual
chain
n a t u r a l i n t e g e r s , and take f o r F the s i n g l e t o n o f 0. Each s i n g l e t o n integer
uf- 0
s a t i s f i e s the condition o f e x t e n s i b i l i t y t o
o f
of an
F o f any lo-
cal automorphism ( n e c e s s a r i l y reduced t o t h e i d e n t i t y on u ) ; and t h e addit i o n o f another i n t e g e r
w
# Oand # u i s impossible,the
transposition
(u,w) being a l o c a l automorphism o f the c y c l i c ordering, i n e x t e n s i b l e
the i d e n t i t y on 0. So any s i n g l e t o n o f an i n t e g e r bound
+0
i s an
by
interval with
F.
4.2. Foh a binahy h e L d o n A, and a bu64et F 04 .LO b a e , i6 U, V ahe each a n (A, F) - i n t e t t v d , and id doh any dement x 0 6 $he union U U V , ,thehe e d t a g 06 $he i n t e M e c t i o n U il V , t h e t t a m ~ o h m a t . i o n06 x i n t o g being a Local automohpkinm 06 A, t h e n U = V.
o According t o the maximality o f r e l a t i v e i n t e r v a l s , f o r i n c l u s i o n , i t i s s u f f i c i e n t t o prove t h a t the union U u V i s an (A,F)-interval, o r simp l y s a t i s f i e s t h e e x t e n s i b i l i t y c o n d i t i o n . I t i s even s u f f i c i e n t t o p r o v e i t from a l o c a l automorphism o f A/(U U V ) , w i t h a domain o f o n l y one e l e -
ment
. Let
x and z be two elements o f the union U U V , t h e
transformation
o f x i n t o z being a l o c a l automorphism. By hypothesis t h e r e e x i s t s an e l ement g o f the i n t e r s e c t i o n , the transformation
o f x i n t o y being
a
lo-
PRESENT PROBLEMS ABOUT INTERVALS
195
cal automorphism, and so the transformation of y i n t o z. S i n c e x and y belong both t o U , o r both t o V , t h e f i r s t l o c a l automorphism i s e x t e n s i b l e by the i d e n t i t y on F; same r e s u l t f o r the second, and f i n a l l y f o r t h e transformation of x i n t o z. o PROBLEM: For a t e r n a r y r e l a t i o n , e x i s t e n c e of two d i s t i n c t i n t e r v a l s , with a same bound, each r e s t r i c t i o n in the union, with cardinal 2, h a v i n g an isomorphic r e s t r i c t i o n in t h e i n t e r s e c t i o n . More s t r o n g l y , each f i n i t e r e s t r i c t i o n i n the union, having an isomorphic r e s t r i c t i o n i n t h e i n t e r section.
Note t h a t 1 . 7 does not s u b s i s t f o r r e l a t i v e i n t e r v a l s . For i n s t a n c e , take a base E of four elements u , x , y, z; t h e s e t F = {u}; t h e unary r e l a tion A with value + f o r x, z, u and - f o r y; t h e binary symmetrical r e l a t i o n R with value + f o r ( u , x ) , ( u , z ) and t h e i r converses, value - i n o t h e r cases. Now the p a i r {x,yl. i s an (AB,F)-interval: t h e element z c a n n o t be added s i n c e the transformation of x i n t o z i s a local automorphism inextens i b l e t o the element U . Take U = {y,z,u} and consider the r e s t r i c t i o n t o 27 of A 8 , the bound F being unchanged. Now t h e i n t e r s e c t i o n of U and the i n t e r v a l { x , y l reduces t o t h e s i n g l e t o n { q l : i t i s n o t an (AB/U,F)-interVal, being not maximal f o r i n c l u s i o n , s i n c e iii t h e p a i r { y , z l , t h e t r a n s formation of y i n t o z i s not a l o c a l automorphism.
5, CLOSURES O F
A RELATION; COMPACT R E L A T I O N ,
These c l o s u r e s , already defined i n F r a f s s e 1974, p . 135, extend t o rel a t i o n s t h e c l o s u r e procedure whereby t h e chain of real numbers i s derived from the chain of r a t i o n a l s . Given a r e l a t i o n A of base E , we d e f i n e an A-@3e4 as a s e t 3 of nonempty A-intervals ( a b s o l u t e i n t e r v a l s ) s a t i s f y i n g the following c o n d i t i o n s : ( 1 ) any A-interval including an element of 3 i s a l s o an element o f 3 ; ( 2 ) t h e i n t e r s e c t i o n of two elements of 3 i s an e l c n c n t of 3 ( i t i s an A-interval , by 1 . 2 ) . A maximal A - f i l t e r w i l l be c a l l e d an A-uLtm6i-eteh. Any A - f i l t e r m a y be extended t o an A - u l t r a f i l t e r . An A - u l t r a f i l t e r i s s a i d t o be L k i u i d i f i t c o n s i s t s of a l l A-intervals including a s i n g l e t o n . I f t h e A - u l t r a f i l t e r i s not t r i v i a l , t h e i n t e r s e c t i o n of a l l i t s elements i s empty, and each e l ement i s i n f i n i t e . For a given A-interval 27 and A - u l t r a f i l t e r 3 , e i t h e r U i n t e r s e c t s each element of 3 , and i s t h e r e f o r e an element o f 3 , o r t h e r e
196
ROLAND
FRA'I'SSE
e x i s t s an element of 5 d i s j o i n t from D. Consequently, i f 3 and 3' are distinct A-ultrafilters,thereexist elementsD of 3 a n d D ' of 3' with D and 17' disjoint. Let us complete the base € by embedding.itin the s e t E* of a l l A - u l t r a f i l t e r s : by identifying each t r i v i a l A - u l t r a f i l t e r with the element of € generating i t . With each non-trivial A - u l t r a f i l t e r 3 l e t us associate a relation A ( 8 ) with base € ( a ) and the same a r i t y as A . The bases E ( 3) are assumed t o be d i s j o i n t from € and from one another. The relations A ( S) are subjected t o the following condition, which may always be s a t i s f i e d : f o r any f i n i t e subset F of € ( b ) and any element 0 of 3, there e x i s t s a t l e a s t one isomorphism of the r e s t r i c t i o n A ( 3 ) / F o n t o a r e s t r i c t i o n of A / D . Once the relations A ( 8 ) have been chosen (some of them may have empty bases), the c h u t e At of A i s unavbiguously defined as follows on the union €+ of E and the s e t s E ( 3) f o r a l l 3. Let It be the a r i t y of A a n d xl, ..., x , elements of €+. I f some x . belongs t o E , we replace i t by X I 4. = x4. and say t h a t i t i s fixed. Now g r o u p a l l xi lying in a same s e t € ( 3 ) t o gether, and consider t h e i r images under a local isomorphism of A (3) towards A/D, where 0 i s an element of 3. Denote these images by xIi; we s t i p u l a t e t h a t d i f f e r e n t members D of d i f f e r e n t u l t r a f i l t e r s 5 be pairwise d i s j o i n t and contain no fixed elements xi. We then s e t A+(xl, x,) = A ( x ' ~ , x ' ) ; t h i s value i s independent of the s p e c i f i c intervals D and isomorphisms chosen (by 1 . 6 ) . Let us recall two problems of the above reference:
...,
...,
PROBLEM 1 . Given a relation A and a nontrivial A - u l t r a f i l t e r 3 , assoc i a t e an empty relation with every other nontrivial A - u l t r a f i l t e r . Does there e x i s t a nonempty relation A ( 3 ) giving as closure a logical (element a r y ) extension of A ? PROBLEM 2. Let 5 and be two nontrivial A - u l t r a f i l t e r s , A ( 3 ) and A ( 6 ) relations each of which yields a logical extension of A via closure (the relations associated with a l l other A - u l t r a f i l t e r s being empty).Is the extension obtained by considering A ( 3 ) and A ( q ) a logical extension of A ?
5.1. ement of pact i f , each 3,
Starting from a s e t €, consider a s e t of f i l t e r s 2 on E , each e l such a f i l t e r 3 being a subset of €. Call t h i s s e t of f i l t e r s camf o r any choice function 6 such t h a t 6 ( 2 ) i s an element of 3 f o r there e x i s t f i n i t e l y many 3 ' s such t h a t t h e u n i o n of t h e s e t s
PRESENT PROBLEMS ABOUT I N T E R V A L S
d ( 3) i s
197
E . Fa4 any n e t E , t h e b e t oh aPl LLeRhndilReh) o n E i n a c o m p a c t
6 e R . 1 n d e e d s u p p o s e t h e c o n t r a r y : t h e r e e x i s t s a c h o i c e f u n c t i o n d such t h a t , f o r
a n y f i n i t e s e t U o f u l t r a f i l t e r s , E u = E minus t h e union of 6 ( 3) f o r a l l of U, i s nonempty. So t h e s u p e r s e t s of the € [ , I S c o n s t i t u t e a f i l t e r on E ; take a f i n e r u l t r a f i l t e r : i t would be an u l t r a f i l t e r on E , d i s t i n c t f r o m a1 1 ul t r a f i l t e r s (communi cated by FI. J e a n ) . Coming back t o r e l a t i o n s , we say t h a t a r e l a t i o n A i s c o m p a c t when the s e t of a l l A - u l t r a f i l t e r s i s compact. By t h e preceding r c m a r k , any unary r e l a t i o n o r m u l t i r e l a t i o n i s compact. By t h e same a r g u n e n t , i t i s seen t h a t any chain i s a compact r e l a t i o n . A hellLtion A L8 compact i56, camplemeiztahlj h e t vah.
604
any h i n i f e u n i o n 0 6 A - i n t e h u a h , t h e 06 A - i n t e h -
(hedehently t o t h e babel .in a & X t e uiuon
Let us prove f i r s t l y t h e following lemma. CnCe E die b a e
0 5 A;
nuppo4e t h a t , do4 any A-intehual 0, t h e
cample-
dinitc union o6 A-intehua&. T h e n a n y Ua,tmd.iet e a a n E, once heduced t o A-in-tehu&, g i v e n an A-uetha6iLteh. nieil*cutj 4 e L
E-
D
i 4
a
Note t h a t , f o r t h e c o n s e c u t i v i t y C on natural i n t e g e r s , t h e only C-int e r v a l s being t h e empty s e t , t h e whole base and s i n g l e t o n s , an:' non t r i v i a l u l t r a f i l t e r , once reduced t o C - i n t e r v a l s , gives only t h e base: t h i s i s not a C-ul t r a f i l t e r . 0 Let 3 be an u l t r a f i l t e r on E , and A t h e s e t of a l l A - i n t e r v a l s b e longing t o 3. Suppose A i s not an A - u l t r a f i l t e r : t h e r e e x i s t s ar A-interVal D with D B A and D i n t e r s e c t s any element of A. The c o m p l e m e n t a r y s e t E - U E 3. By hypothesis E - D i s a f i n i t e union of A-intervals: so one of them U* E 3 and consequently U* 6 A. F i n a l l y D and D* a r e d i s j o i n t s e t s ; on another s i d e U i n t e r s e c t s any element of A, and so D i n t e r s e c t s D*: c o n t r a d i c t i o n . o
Proof of the proposition :
0 Suppose t h a t , f o r any f i n i t e union of A - i n t e r v a l s , t h e c o m p l e m e n t a r y s e t i s s t i l l a f i n i t e union of A-intervals. I f A i s not compact, t h e r e i s a choice function 6 such t h a t , f o r any f i n i t e s e t U of A - u l t r a f i l t e r s , EU = E minus union of 6 ( 3 ) ' s f o r a l l 3 o f LI, i s nonempty. Then the s u p e r s e t s o f EUis c o n s t i t u t e a f i l t e r on E . Take a f i n e r u l t r a f i l t e r , and reduce i t t o A-intervals: t h i s gives an A - u l t r a f i l t e r , by t h e above lemma. Call i t V
ROLAND FRA'I'SSE
198 and n o t e t h a t
6(v) 6 v.
F o r each f i n i t e s e t U o f A - u l t r a f i l t e r s ,
i s by
EU
h y p o t h e s i s a f i n i t e u n i o n o f A - i n t e r v a l s : one o f them belongs t o V. T a k i n g U = {Vl, we g e t
EU = E -
€ - 6(V)
cluded i n
6 (v); t h e r e
v which i s 6(v)-: c o n t r a d i c t i o n ; so A i s
e x i s t s an element o f
, thus d i s j o i n t from
incom-
pact. o f A-
Conversely suppose A compact; suFpose t h e r e e x i s t s a f i n i t e s e t intervals
D, with
€ minus u n i o n o f D ' S b e i n g n o t a f i n i t e u n i o n o f A - i n t e r -
v a l s . To each A - u l t r a f i l t e r 5 , a s s o c i a t e
6( 3) b e i n g
a 0 belonging
to
5;
o r 6(5) element o f ? a n d d i s j o i n t f r o m each D. W i t h t h i s c h o i c e f u n c t i o n , no f i n i t e u n i o n o f 6(3) can g i v e €: c o n t r a d i c t i o n . 0
6
I
PROBLEMS ABOUT I N T E R V A L S t SUMS t PRODUCTS 8 I N C O N N E C T I O N
WITH
EHRENFEUCHT'SLOGICAL
STUDY O F ORDINALS
I
I n F e f e m a n 1357, s e v e r a l i m p o r t a n t r e s u l t s due t o E h r e n f e u c h t a r e r e lated.
1)
id
Vd I-
(a) ( d o h n
whehe
$d
(w2,.
..,wn)
= $( 6
:
06 in&?kence: k) --f
I .t(w,),
P
+
$
$
w2,.
then
then
I-
+
PI(@) + $ ;
pn($)+
n-
;
.., w n ) .
We conclude this s e c t i o n with the remark t h a t t h e downward Skolem-Tarski theorem holds f o r both L and L P Pn *
111,
$J
Lowenheim-
PROJECTIVEBOOLEAN ALGEBRAS,
1 I NTRODUCTI O N , I
In t h i s s e c t i o n we d e f i n e a new kind o f Boolean a l g e b r a s , c a l l e d n-proj e c t i v e Boolean a l g e b r a s , which a r e g e n e r a l i z a t i o n s of t h e S u s l i n algebras introduced by L . Rieger in 1955 ( c f . Rieger 1955). Our work, however, i s patterned a f t e r a recent paper by E . E l l e n t u c k (Ellentuck 197+) i n which he s t u d i e s the S - a l g e b r a s o f Rieger based on his previous paper on S u s l i n l o g i c (Ellentuck 1975). R i e g e r ' s idea with t h e S u s l i n algebras was t o provide a s t r u c t u r e i n which one could model nn1 a n a l y s i s .
1. BASIC
nt
a n a l y s i s . Our algebras a r e intended t o help model
RESULTS,
Let B be a Boolean algebra.
209
PROJECTIVE LOGIC
The joim and nieeA of B w i l l be denoted r e s p e c t i v e l y by Sup and 7ng. The iizditzite j o i n of t h e family {bi 1 i E I } i s denoted by Sup bi or simply by Sup bi
i
is7
i f i t c l e a r which s e t 7 i s .
I f Q i s a map from [neq!" i n t o 6 we s h a l l use the n o t a t i o n Pit($) f o r the following element of 73, provided i t e x i s t s : Pi*($)
=
and, s i m i l a r l y ,
SUP
F
Ini( $ ( w ) , wEF
where, a s u s u a l , F runs over The symbols P,(@) and and t h e previously defined confusion.
DEFINITION 1 2 . (W
- PBA
ijoh n h h t )
F u l l n and G over
FU.eek
.
P;($) w i l l be used both f o r t h e above suprema formulas o f Lpn, b u t t h i s should l e a d t o no
A u - B C J C J ~d~ gU e~ b~ t ~ ~ U M w - I 3 4 O j ! L d W t 600tea~d g c b h n id Lt 0 it - PBA dot C V U i ~ i E W.
Formula ( 4 ) i s a very powerful d i s t r i b u t i v e law and not a l l algebras closed under Pit and P i s a t i s f y i t . In f a c t , t h e r e a r e complete B o o l e a n algebras i n which ( 4 ) f a i l s . In our work, however, we need t h i s d i s t r i b u t i v i t y t o t i e t h i n g s up ( s e e f o r example condition ( 6 ) below), and we a r e thus forced t o introduce i t a s p a r t of t h e d e f i n i t i o n . To g e t an example i n which ( 4 ) f s i l s s e e Morais 1976. Another way t o see the importance of ( 4 ) i s t h e next proposition which presents t h r e e e q u i v a l e n t formulations of ( 4 ) .
PROPOSITION 13. 7 6 B 0 c( Bootenti d g e b h a i n t o B , tt,t -I$ be t h e tNnp deijined by (-
whehe - 0 t h e nytnbat eqUiWdent:
604
Q)( w )
=
-$ (w)
c a t i i p L ~ i e n t d L oi ~n ~ B
aid
Q 0 a riiap
. Theit tlze
6hotti
[Aeq]'
60ttCJdt7g
ah&
210
RICARDO
MORAIS
Now, using these equivalences, we can get several properties of projective Boolean Algebras, namely:
PROPOSITION 1 4 . A u - B o o L ~ ~dMg e b h a B 0 n-PBA i6 m d Only i6 d a s e d u n d e h the P i a p e h a t a h and (4) holds. PROPOSITION 1 5 .
16
M > 1
PROPOSITION 17. Evehg
whehe
w
PROPOSITION 1 8 . PROOF:
Now
M-
PBA, then B 0 ( n - 1) - PBA.
an example.
The cornple*e B a o l e a ~d g c b m 2
= {O
,I} 0 w - PBA.
Since 2 i s complete we have just t o show ( 4 ) holds i n 2 .
P,($)
=
0
i f f sup In6 - $ ( w ) . = G uEG
.
B 0
- PBA ~ a t i n 6 i e A :
iotoak-n add
iff
i f f (by Proposition 3 )
holds
M
aMd
8 0
Sup In6 $(u) = 0
F uEF ( 3 G E FuRe;)(Wu
1 iff Pi( - $ ) = 1
i f f (WF E FuRen)(3w E F)(@(u)=O) E
iff
G)($(u) = 0 ) i f f In6 Sup $(u) = O G uEG
- P i ( - $ ) = 0 and therefore (6)
211
PROJECTIVE L O G I C
The most i m p o r t a n t example o f an w - PBA however i s g i v e n by t h e
fol
-
lowing: The Lindenbawl dgebaa L
THEOREM 1 9 . w
- PBA.
PROOF:
Let
1
@
I
06
P
,the w - pfihujedue Logic
LP
denote t h e e q u i v a l e n c e c l a s s o f t h e f o r m u l a @ i n L
P
.
i n t o L and d e f i n e a map $J from [bey]" P by choosing f o r each W E [hey]' a r e p r e s e n t a t i v e f o r -
L e t @ be a map f r o m [hey]" i n t o f o r m u l a s of L
P
mula @(u) o f t h e e q u i v a l e n c e c l a s s
(8)
P,(@) = !pn(@)l and hence
T,
We t h e n show
(li(w).
,
F i r s t we have t o p r o v e t h a t t h e f o r m u l a choice o f t h e map
Pn o p e r a t o r .
i s c l o s e d under t h e
P
Pn(@) does n o t depend on t h e
@.
I t s enough t o show t h a t f o r any o t h e r map
'Ju E [ ~ e y ] " I- @ ( w )
+
$(u) t h e n
I-
JJ
:
if
Pyz(@)+ P , ( ~ J ) .
By (R4) t h i s f o l l o w s f r o m (9)
VF E F a n I-
AF@(u)
+
pn($)9
which i n t u r n f o l l o w s from, (WG E F u R e V : ) ( V F E
Fan)
(by R5), I-
AF@(w)
+
VG$(u).
B u t t h i s i s t r i v i a l s i n c e by P r o p o s i t i o n 5, g i v e n any F and G , F n G # @ . T h e r e f o r e (9) h o l d s . Now t o f i n i s h t h e proof of Theorem 19 we have t o show t h a t t h e d i s t r i b u t i v e law (4) holds i n L
P'
We s h a l l need,
(10) P i ( @ ) = I P p 4 which i s e q u i v a l e n t t o ,
S U P I AG @ ( u ) I = I P i ( $ J ) I G and so we have t o prove: (i) ( W G E F f i i )
I
I- h G @ ( V )+
Pi(@)
and
(ii) I f (WG E Fufl;)
I- h G $ J ( u + )
11 t h e n
I-
P i ( @ )+ $
.
212
RICARDO MORAIS
PROOF of (i):
From p r o p o s i t i o n 5 g e t
(WG E FU.eei) (WF E F d n )
+
AF l$(v)
+
VG l $ ( v )
now a p p l y (R4) and use P r o p o s i t i o n 2. S t a r t w i t h the hypothesis
PROOF of ( i i ): (WG € FULL;)
I-
1
$J
+ VG 1 $(v),
t h e n a p p l y (R5), and use P r o p o s i t i o n 2. F i n a l l y ( 8
,
(10) and P r o p o s i t i o n 2
give
Pn(@)
=
-
P,*(
- 0)
and t h e r e f o r e ( 6 ) h o l d s , which i s e q u i v a l e n t t o ( 4 NOTE: L
PM
E v i d e n t l y e x a c t l y t h e same p r o o f shows t h a t t h e Lindenbaum a l g e b r a of
(denoted L
PM
) i s n - PBA.
3, FREE n - PROJECTIVE BOOLEAN ALGEBRAS, DEFINITION 2 0 .
An nP
-
BooLean d g e b t a 0 a a - ho-
Izornornotpkm b-een
momohpkinm t h a t pk,hedmve~t h e Pn opehatoh. An W P - homomotpkinm 0 a u - honiomohpkintn
&at p u e h v e n Pn d o t evehy n
DEFINITION 2 1 .
L e t B be m
- genehaten B i d
(a)
G nP
(b)
G dheely
E w.
n - PBA and G
c
B. Tken:
B 0 t h e nm&ent
n - PBA containing G
.
nP- genmaten B i d G nP-genehaten B UJ~C! in a d d i t i o n given m y o t h m n - PBA B' and m y map h : G + B' t h e h e i b an nP - homomohpkidm H : E + B' w h i c h extend6 h
.
- net
06
gen -
An n - PBA 0 a dhee nP- d g e b h a i d contaia n P - n e t ad g e n ma to a . S . i m . 2 d y , dedine a 6hee ~ P - u Q e b h a .
a
dhee
(c)
S .in i. 2 dy dedine W P - neA 06 genehatom and 6hee
WP
ehato4,5.
DEFINITION 2 2 .
I f i s a common p r a c t i c e . i n any t e x t about " f r e e " s t r u c t u r e s t o
first
t a l k about i t s uniqueness and a f t e r w a r d s t o prove i t s e x i s t e n c e .
The
l o w i n g two p r o p o s i t i o n s a r e proven i n t h e same way i t i s u s u a l l y
done f o r
general Boolean a l g e b r a s . See f o r example Halmos 1963, p. 42.
fol-
213
PROJECTIVE L O G I C
PROPOSITION 23. 76 B 0 u dhee n P - d g e b h u , G t h e he,t 06 6hee nP-genefu7Xoh.S and h .the given map 6honi G into .the n - PBA B',then t h e nP - hamomahpkinm H : B + B' t h a t extendh h 0 unique. PROPOSITION 24. Any &oo 6hee n P - d g e b h a whohe .the hame catr&&y atre nP-0oma5pkic.
h d
0 6 genmatom
have
Now t o p r e s e n t an example o f a f r e e U P - a l g e b r a ( t h e e x i s t e n c e o f a f r e e n P - a l g e b r a i s proved s i m i l a r l y ) we proceed as f o l l o w s . F i r s t d e f i n e a phOpOh.iJ%onCdl o g i c LK f o r each c a r d i n a l
K
and t h e n
show t h a t t h e Lindenbaum a l g e b r a L~ o f LK i s a f r e e U P - a l g e b r a
we
with
K
generators. LK i s g o i n g t o have a s e t o f
{Pa j
c1
< Kl
K
variables
,
and t h e p r o p o s i t i o n a l c o n n e c t i v e s 1 and A o p e r a t o r s P,? and P;
. As
in L
PM '
we i n t r o d u c e
and l e t t h e s e t o f f o r m u l a s be t h e l e a s t s e t such t h a t
.
(a)
pa
(b)
if @
(c)
i f 0 i s a c o u n t a b l e s e t o f f o r m u l a s t h e n A @ i s a formula.
(a)
i s a f o r m u l a f o r each o r d i n a l
c1
0 ,
1
.i~ an n - phojeotiwe 6 i e l d
06 b&,
whehe
A,
6Zand6 doh "boLd6ace A". PROOF:
( f o r a d e t a i l e d p r o o f p l e a s e see M o r a i s 1976).
We w i l l show t h a t
i s c l o s e d under t h e Pn,
b u t t h i s i s not enough,
however, t o p r o v e t h a t . i t i s n - p r o j e c t i v e because t h e d e f i n i t i o n n - p r o j e c t i v e algebra s t a r t s w i t h a o - a l g e b r a .
B u t i t i s easy t o see
t h e same argument used below can be r e p e a t e d t o show t h a t L I ~ + ~i s
o f an that closed
under t h e P1 o p e r a t o r , and t h i s i n t u r n i s a g e n e r a l i z a t i o n o f c o u n t a b l e
219
PROJECTIVE L O G I C
unions and i n t e r s e c t i o n s ( c f . Kuratowski and Mostowski 1968, p. 341). L e t now @ be any map f r o m
x
E
1
i n t o Qn+l,
[AQQ;'
1
P,($) can be g i v e n b o t h by a Jn+l and a By P r o p o s i t i o n 3
x
P,($)
E
JA+l
a n d we
show
that
predicate.
has two e q u i v a l e n t f o r m u l a t i o n s , namely:
(a)
( 3 F E F U . e e n ) ( W w E F)(x E $ ( w ) )
(b)
(WG E FU.eei)(3v E G ) ( x E @ ( w ) ) .
and
We a r e g o i n g t o use (a) ( r e s p e c t i v e l y (b)) t o show t h a t g i v e n by a
+
F i r s t , since w onto
i s countable, there i s a
[bey]"
F and
s t i t u t e the sets
1- 1 r e c u r s i v e map
by means o f X
{O
G by t h e i r r e s p e c t i v e
tw
E [bey]"
, 1 1 , and as we d i d f o r
I g(w)
= 1) E
The e x p r e s s i o n range
(Wm
.
I n addition,
= 0
{ w 8 [neq]'
ik
.
)'i+lp r e d i c a t e .
@(k) E
[(tlange g
=
{O, 11
.
g(n1) = 1).
otl
I g(w)
= 1) E
1 4n+1 c $ + ~f o r
T h e r e f o r e i f we w r i t e W1 and
sub-
11 i s w r i t t e n Fulln
( ~ 6 1 ) ( ~ 5 * ) " ' ( Q 6 , ) ~ Q ' ~ ) ( 9 ( 6 1 1 m* ,* .
F i n a l l y since
we
we t h i n k t h e domain o f g as w .
@
F a n and g ( k ) = 1) + x E @ ( k ) ]
g = {O,
E w)(g(m)
The e x p r e s s i o n
which i s
X from
c h a r a c t e r i s t i c functions
T h e r e f o r e (a) i s e q u i v a l e n t t o ( 3 9 E "u) (Wk E w )
and
is
P,*($)
E
and so we can t h i n k t h a t t h e domain o f $ i s w . L e t _v be t h e
[bey:"
i n t e g e r a s s o c i a t e d w i t h w C [beq]" g : [hey;"+
x
1 ( r e s p e c t i v e l y iln+l)p r e d i c a t e .
1
i s equivalent t o
A,i)4
= 1)
a l l k we can make
x
E $(k)
a
31 f o r q u a n t i f i c a t i o n o v e r r e a l s and WO
and 30 f o r q u a n t i f i c a t i o n o v e r numbers, t h e statement ( a ) now reads:
31 WO [ ( W O A,31
... "
Q1 Q'O) j L 3 1
- Kuratowski
we s i m p l i f y t h e above t o 31 3 1 VO [ ( A
...
v n+1
) +
v
Q'l,QO]
.
n+1
n Then u s i n g t h e T a r s k i
...
a l g o r i t h m s ( c f . Rogers 1967,
1
p.
307)
R I CARD0 NORA I S
220
which i s a
zi+l predicate.
Now using (b), since
G €
FuRel
is
we, s t a r t w i t h
[
W130
... 2'1 20) +
(WOA W 1
u n
and end up w i t h a $+1
predicate.
JLi and s i n c e @ ( k )
V1
E
1 4n+l
1
... 21 2 ' 0 1
n+ 1
came
I t i s c l e a r by now t h a t one o f t h e most i n t e r e s t i n g notions t h a t
up along t h i s work was t h a t o f
Full;
and i t s counterpart FLLeen
symmetry between these two classes o f
. The
,
c ;n+l
generalizing Ellentuck's F u l l s e t s , sets
n o t o n l y helped c u t t i n g a l l our proofs i n h a l f b u t also, and more s i g n i f i c a n t l y , w i t h o u t t h i s symmetry most o f our p r o o f s
-
c o u l d n o t have
come
through, s p e c i a l l y our l a s t theorem i n which the simultaneous use o f
FuUn
FuRe;
and
was fundamental.
For these reasons we foresee an i n c r e a s i n g use o f these n o t i o n s i n
the
f u t u r e s t u d i e s o f p r o j e c t i v e sets.
To conclude t h i s work, among several i n t e r e s t i n g q u e s t i o n s f o r which a l l t h i s machinery i s applicable, we s e l e c t e d two t h a t we are p a r t i c u l a r l y i n t e r e s t e d i n i n v e s t i g a t i n g , namely: (1) (2)
L or L i f any? Pn P ' If. M. i s 3 universe o f s e t s and B i s an n - p r o j e c t i v e Boolean a l gebra, what can be accomplished i n s i d e t h e Boolean valued m o d e l
What k i n d o f i n t e r p o l a t i o n theorem holds i n
MB ?
REFERENCES Addison, J. A. and Y . Moskovakis 1968,
Some comequenca Nat. Acad. Sci.,
06
t h t axiom o d de6&abLe d e L m i n a t e n a s ,
B e l l , J. L. and A. B. Slomson 1969,
Proc.
Vol. 59, 708- 712.
Models and Ultraprqducts, North
- Holland,
Amsterdam.
Ellentuck, E. 1975,
The ~owzdatiom06 S w f i n Logic, The Journal o f symbolic Logic, v o l . 40, 567-575
PROJECTIVE
197+,
22 1
LOGIC
Fhee SwL& d g e b m , S u b m i t t e d t o Czech. Wath. J o u r n a l .
Fenstad, F. 1971,
The a x i a m
0 6 deLetuninateflcbb,
Proceedings o f t h e Second Scandinavian
L o g i c Symposium, Ed., J . E. Fenstad, N o r t h - H o l l a n d , Amsterdam,41-61. Halmos, P. 1963,
Lectures on Boolean Algebras, Van Nostrand K e i n h o l d Company, London.
Kechris, A. 1973,
i'leubwre and categohy i n eddeotiwe denchipLLwe
bet
theahy, Annals
of
Math. L o g i c , V o l . 5, 3 3 7 - 384. K e i s l e r , H. J. 1971,
Model Theory for Infinitary Logic, N o r t h - H o l l a n d , Amsterdam.
Kuratowski, K. and A. Mostowski 1968,
Set Theory, N o r t h - H o l l a n d , Amsterdam.
L u s i n , N. 1925,
S w l Lcb emembLcn p o j e o t i w e h de M. H e i d Lebcngue, C. R. Acad.
Sci.
de P a r i s . Morais, R. 1976,
Projective Logic, Ph. D. T h e s i s , Eutgers U n i v e r s i t y , U.S.A.
Moschovakis, 1970,
Y.
DeLmtirzaflcy und
pmLJ&Otld&Mgb
ad t h e co~dizuum,in Mathematical
Logic and Foundations of Set Theory, E d . Y . B a r - H i l l e l , H o l l a n d , Amsterdm, 24 - 62.
North-
Rieger, L . 1955,
Cancmning Suofin d g e b m (S - k e g e b m ) and t l z e i n (Russian), Czech. I l a t h . J o u r n a l , Vol. 5. 99
htphtbtntathion
- 142.
Rogers, H. 1967,
Theory of Recursive Functions and Effective Computability, MacGraw H i 11. Sierpinski, W. 1925, S w l une &abbe d'emembLcb, Fund. Math. Vol. 7, 2 3 7 - 2 4 3 .
-
l n s t i t u t o de Matematica U n i v e r s i d a d e F e d e r a l do R i o de J a n e i r o R i o de J a n e i r o , RJ.,
Brazil
Non-Classical Logics, Model Theory and Computability, A.I. Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
TYPES, WITH MODEL COMPLETENESS, AND RELATED PROPERTI ES ,
SOME THEOREMS ON OMITTING
APPLICATIONS AMALGAMATI
TO
ON,
by C H A R L E S C .
PINTER
1, INTRODUCTION, Many t o p i c s of c u r r e n t i n t e r e s t i n model theory involve m o d e l s which omit designated s e t s of types. For example, t h e e x i s t e n t i a l l y closed models of a theory T a r e p r e c i s e l y those which omit a c e r t a i n s e t of types. The Same i s true f o r t h e g e n e r i c models of T f o r f i n i t e f o r c i n g , t h e completing models of T ; t h e models which a r e analganation bases f o r T , and so on. I n f a c t , i n p o r t a n t p r o p e r t i e s of a theory T hold i f f a l l the models of T omit c e r t a i n given types: t h i s i s t h e case of a theory being model complete, f i ni t e l y f o r c i n g complete, having the amalgamation property, the congruence extension property: and many o t h e r s .
O n e o f t h e o b j e c t s of this paper i s t o show t h a t many s u p e r f i c i a l l y u n r e l a t e d r e s u l t s of model theory a r e , i n f a c t , consequences of the same simple theorems on omitting types. By s y s t e m a t i c a l l y developing c e r t a i n r a t h e r elementary observations on omitting types, we can r e c a p t u r e - and unify-many known r e s u l t s , f i n d a number of new ones, and almost t r i v i a l i z e some c l a s s i c a l r e s u l t s . As an example of t h e l a t t e r , we show t h a t Lindstroai's t h e o rem on model completeness is a consequence of a very simple observation on omitting types i n t h e o r i e s which a r e c a t e g o r i c a l i n some i n f i n i t e power. In Section 2 of this paper we develop s e v e r a l p r o p e r t i e s o f omitting types, and i n Section 3 we i l l u s t r a t e t h e i r uses with a v a r i e t y of examples. Our standard reference t o model theory w i l l be Chang and Keisler 1973. 223
224
C. C. PINTER
We assume throughout t h a t L i s a countable, f i n i t a r y , f i r s t - o r d e r guage; a .themy T i s a c k d u c t i v e l y closed s e t o f sentences o f L . p l i f y n o t a t i o n , we w i l l l e t 7 designate any f i n i t e sequence (u1, v a r i a b l e s , and, i f quence (al, symbols
a
...,a n ) o f
@(V)
lan-
To sim-
...,w n )
of
i s a s t r u c t u r e , we w i l l l e t Cr d e s i y a t e any f i n i t e semembers o f U l . For any formula @
.1 and @ j
, we
will
the
use
w i t h t h e i r obvious meanings, provided t h a t t h e f r e e
v
v a r i a b l e s o f $ occur among
and t h a t t h e sequence
a
matches t h e sequence
lJ.
2, SOME L e t u(; theory o f L
or4
THEOREMS
{on(;)
=
, we
:
n
OMITTING I?
be a sequence o f formulas o f L . I f T i s a
say t h a t T Lac&y
which i s c o n s i s t e n t w i t h T ,
n
U}
TYPES,
o m i h u i f f f o r every formula
I$(;)A T u n ( ; )
i s consistent w i t h T
6 w. L e t C bc a countablc s c t o f sequences
u o f formulas; i f
T
(V)
@
for
some
locally
u 8 C, wc w i l l say t h a t T i s Z - compLeAe. A model aI= T w i l l be s a i d t o V M LC ~ i f f a omits cach u I? C . The h i t t i n g - types Theorem
ornits every asserts :
.
C 76 (2.1) 16 T h C - compL&te, t h e n T h a a m d e L which ofii& compLeAe Rheafiy: T h C - campLett id5 T h a a tiiodel wkich o n i i h
Ile say t h a t T i s 1 - c o r z n h t e n t i f f T C_ T',bfhere
DEFINITION.
c o n s i s t e n t and C
- complete.
C l e a r l y , T has a nodel vl;tittiw C more, i f T i s
, that
.id6 T
0 a
T'
is
0 Z - carznhtent. FurtherT 0 C - cmpLeAe.
complete t h e o r y T 0 C - c a ~ b h t e n t.id:
-- -
PROOF: (iii) (i) T
B
T
.
(ii) (i) i s immediate. (iii): Every model o f T i s a model o f a complete extension
of
i s , o f a'complete, C - complete theory. Thus, f o r eachformula $(;)I
SOME THEOREMS O N OMITTING TYPES
every model Ol t==
225
T s a t i s f i e s the sentence
(ji)+(i)
* V i e W ( 3 G ) [ + (;)
A l~i(;)J
By compactness, we have (iii). 0 EXAMPLE, Let T be a consistenttheory in the language L" of w-logic.Then, every extension of T has an w - model i f f every extension of T i s w-complete i f f f o r each formula @(;) there are integers il, i such t h a t
..., Q
The next two theorems have a great many applications, which will be developed in Section 3 of t h i s paper.
PROOF : (i) i s immediate, using the contrapositive of the definition of"T 10cally omits u " . (ii) : He will show t h a t i f T i s C - complete and @, i s any sentence which i s consistent with T , then T U { @ I i s Z - complete; (ii) will follow immediately from t h i s . Hell, suppose T i s C - complete and q(;) is consistent with T U { @ } . Then I)(;) A @ i s consistent with T I so f o r some n 8 w ,$(;) A @ A l u n ( ; ) i s consistent with T . Thus $(;) A l u n ( ; ) i s
consistent with T U {@I. 0 ( 2 . 4 ) THEOREM
The doaRoLuing me equivalent:
(i) T h C - cornpdete (ii) Fa& & nentencen tokich or;l.LtA c .
@
,Tt
@
idd
@
0 h u e i n e v # y model
06
T
We will show t h a t (ii) holds i f f T i s the intersection of a l l i t s complete, c - complete extensions; our r e s u l t n i l 1 then follow by (2.3). Suppose (ii) holds, and l e t { T :~ i € I} be the s e t of a l l the c o m p l e t e ,
PROOF:
C
- cornplete
theories containing T
. Plow,
T
5
2 T ; furthermore,if
@€
?Ti,
226
C. C. PINTER
, hence @ E T . Conversely, T = T. ; c e r t a i n l y @ € T i m p l i e s t h a t @ i s t r u e i n e v e r y A. model o f T o m i t t i n g C . On t h e o t h e r hand, i f @ i s t r u e i n every model o f T o m i t t i n g Z , then € Ti f o r each i € I, so @ Ti 0 t h e n @ i s t r u e i n e v e r y model o f T o m i t t i n g C
suppose
€a
.
D E F I N I T I O N , L e t T be C - c o n s i s t e n t . The i n t e r s e c t i o n o f a l l t h e complete, C
- complete
n o t e d by
T
e x t e n s i o n s o f T i s c a l l e d t h e C - comp.&,t,Lano f T , and i s
C
.
TC i s t h e s m a l l e s t C - c o m p l e t e t h e o r y c o n t a i n i n g T . We w i l l
see
dein
t h e n e x t S e c t i o n t h a t f o r c i n g - companions, m o d e l - companions, and o t h e r f a m i l i a r c o n s t r u c t i o n s o f model t h e o r y a r e s p e c i f i c examples o f 1-completions. I t i s obvious t h a t any m u d e l
06
T wkich v m h C hub .to be a modeL
v6
Tz. From t h i s f a c t , we i m m e d i a t e l y deduce: (2.5)
THEOREM, T C I- @
c.
4 5
@
A h e i n ewehy made[ a6 T which o m h
I n many cases which we a r e l e d t o c o n s i d e r , t h e c l a s s J o f a l l t h e mode l s o f T which o m i t C t u r n s o u t t o be an elementary c l a s s . F o r such cases, the following i s useful: (2.6)
THEOREM, SUppVbe Rhe d k b b 3 of, ClRe Rhe mod&
06
T which o m d C
0 an elementaht; d k b b . Then TC = Th (31, and Tc A a d o m d z e d b y
whete PROOF:
each
vi
T *U { ( . W ; ) v i i b
IDi(;)
:
0
E C)
a 6ini.te d i b j u n c L i o n .
T C = Th(7) b y ( 2 . 5 ) . Thus, each model o f T C o m i t s C , and o u r
r e s u l t f o l l o w s f r o m a s i m p l e a p p l i c a t i o n o f compactness. 0 I n t h e sequel, i t w i l l be u s e f u l t o n o t e t h a t , i f
T i s C-consistent,
t h e r e i s an obvious i n d u c t i v e procedure f o r c o n s t r u c t i n g t h e C - c o m p l e t i o n TC O f T. o f consistent To c o n s t r u c t T C , we d e f i n e a sequence U o E UIc t h e o r i e s as f o l 1ows : (2.7) (i) U o = T. (ii) Given UK, d e f i n e UK+l by: f o r each u E C and each f o r m u l a a ( ; ) , i f UK +I)(;) + o n ( ; ) f o r each n E w , t h e n 1(3;)$(;) E UK+l
...
.
(iii) I f
c1
i s a l i m i t o r d i n a l , U, =
F i n a l l y , l e t TC be t h e d e d u c t i v e c l o s u r e o f
u
U Y. UK<W U ~ Our assumption t h a t Y b < d
.
Then f o r a l l
such b = l2( dl + d 2 ) .
d E D,
that
250
J. R. SHOENFIELD
and
d < a ->
(2)
d < b.
a E D, a
Since
i s transcendental over D .
b I D;
From (1) and (2),
so b i s t r a n s c e n d e n t a l o v e r D . Hence we have a unique f i e l d isomorphism of
D(a)
on D and takes a i n t o b.
D(b) which i s the i d e n t i t y
and
We m u s t s h o w t h a t t h i s i s o m o r p h i s m preserves o r d e r . An element o f D ( a ) i s o f t h e f o r m polynomials into
I t w i l l t h e r e f o r e be s u f f i c i e n t
6(b)/g(b).
(3)
0
c
6(a)->
0
6
f o r every polynomial If
d ( a ) / g ( a ) w, h e r e
s
6,
i t holds f o r a l l
i t holds
for
6,6,.
6
If
6
i s constant, then
pose
6
f o r some
has degree
t o prove
d.6
d € D.
H e n c e we
(1) h o l d s f o r
6
H e n c e we may s u p p o s e t h a t i s clear. If
(3)
d E D; so
2 2 . Since
with
i s 1. I f
reducible.
6(x) = x-d
are this
with coefficients i n D .
(1) h o l d s f o r
6,.
g
and
6(b)
may suppose t h a t t h e l e a d i n g c o e f f i c i e n t o f and
6
w i t h c o e f f i c i e n t s i n D ; and t h e isomorphism c a r r i e s
6
(3)
6
i s o f degree
f o l l o w s from
(1) and
1
i s ir-
,
then
(Z).Now
i s i r r e d u c i b l e , i t has no r o o t s
sup-
in
D
Then i t has no r o o t s i n A o r B ; f o r t h e s e r o o t s would be a l g e b r a i c D and hence i n D . Since t h e l e a d i n g c o e f f i c i e n t o f
l a r g e values o f
x in D
. Combining
these f a c t s
6
6,
i s 1, 6 ( x ) > 0
for
with the f a c t t h a t A and
B a r e r e a l closed, we see t h a t 6 assumes o n l y p o s i t i v e v a l u e s f o r o r B . Thus (3) h o l d s . Q.E.D.
argu-
ments i n A
The f o l l o w i n g c o r o l l a r i e s o f Theorem 2 a r e proved l i k e t h e
corre-
sponding c o r o l l a r i e s o f Theorem 1.
COROLLARY 1 .
RCF i b model? c o m p l e t e .
COROLLARY 2 .
RCF 0 campLett.
COROLLARY 3 .
RCF 0 an ,axiomdzaLLon
R
.
1 6 a sentence $ (06 t h e language 06 o h d u e d t h e n it i b t h u e i n e v e h y cloned 6 i d d .
COROLLARY 4 .
thue i n R ,
06
.
over
i b
251
QUANTIFIER ELIMINATION I N FIELDS COROLLARY 5 .
RCF 0 decidable.
We g i v e an a p p l i c a t i o n of Corollary 1 . In o r d e r t o s o l v e H i l b e r t ' s 17th problem, Artin proved the following r e s u l t . Let d(xl , . .,x,) be a p o l y n o m i a l with r a t i o n a l c o e f f i c i e n t s which assume only non-negat i v e values when i t s arguments a r e r e a l . Then 6 i s a sum of squares of rational functions w i t h rational coefficients.
.
To prove t h i s , A r t i n considered t h e f i e l d G of r a t i o n a l functions o f with rational c o e f f i c i e n t s . If the conclusion i s f a l s e , 6 i s not a sum of squares in G . A t h e o r e m o f Artin and Schreier then t e l l s us t h a t t h e f i e l d G may be ordered so t h a t 6 < 0. A r t i n t h e n used t h i s t o show t h a t 6 must assume a n e g a t i v e v a l u e f o r some real arguments. This p a r t of t h e proof follows r e a d i l y from C o r o l l a r y l . F i r s t , the theory
x1
, . . . ,x,
of real closed f i e l d s shows t h a t G can be extended t o a real closed f i e l d and R i s a s u b f i e l d of H. There i s a sentence I$ w h i c h s a y s t h a t 6 a s s u m e s a negative value; and t h i s sentence i s t r u e i n H, s i n c e 6 < 0 . I t follows t h a t 4 i s t r u e i n R. F i n a l l y , we can o b t a i n an axiomatization f o r R i n t h e l a n g u a g e o f fields. W e take t h e axiomatization in the language of o r d e r e d f i e l d s , and r e p l a c e each formula o 5 Z by 3 x ( o + x 2 = Z ) .
H;
4, OTHER
CASES,
We s h a l l not t r e a t t h e o t h e r two cases i n d e t a i l , s i n c e t h e additional material i s mostly a l g e b r a i c . This material i s discussed i n Robinson 1 9 5 6 f o r d i f f e r e n t i a l l y c l o s e d f i e l d s and i n Kochen 1 9 7 4 f o r v a l u e d e only make a few remarks which w i l l enable t h e reader t o connect fields. W t h e material i n Kochen 1974 w i t h our methods. The language of valued f i e l d s can be formulated i n several ways. However i t i s done, i t i s possible t o make a s s e r t i o n s in t h i s language a b o u t the r e s i d u e c l a s s f i e l d and t h e value group. For example, l e t $(xl, x,) b e a f o r m u l a o f t h e f i e l d language. Then there i s a formula
...,
..
I$*(xl,. ,xn) of the valued f i e l d language with t h e following p r o p e r t y . Let F be a valued f i e l d ; u l ,..., a i n t e g e r s o f F ; F t h e r e s i d u e ,..., t h e c o s e t s o f a l , . . . , a i n F. Then class field;
an
252
R. SHOENFIELD
J.
+(a,,. . .,a
i s t r u e i n F i f f +*(al
I n t h e v a l u e d f i e l d language
,...,an) i s
t r u e i n F.
we can f o r m t h e t h e o r y
of
T
valued
f i e l d s which s a t i s f y H e n s e l ' s Lemma a n d h a v e r e s i d u e c l a s s f i e l d s c h a r a c t e r i s t i c 0. To make t h i s i n t o a t h e o r y a d m i t t i n g QE,
we m u s t
of add
new s y m b o l s f o r e a c h f o r m u l a a b o u t t h e r e s i d u e c l a s s f i e l d o r t h e v a l u e group.
Thus i f
I$(
xl,.
.., x n )
i s as above, we add a new
relation
symbol
P and a new axiom P(X1'
..., xn)-
+*( X1,
...' X n ) .
We c a n now p r o v e t h a t t h e r e s u l t i n g t h e o r y T* admits
QE, and
u s e ' t h i s t o o b t a i n , f o r example, t h e Ax-Kochen r e s u l t s on A r t i n ' s c o n j e c t u r e . Making more use o f t h e r e s u l t s t i z a t i o n o f the f i e l d o f
i n Kochen 1974, we can g e t an axioma-
p - a d i c numbers. However, t h e problem o f
finding
an a x i o m a t i z a t i o n o f t h e f i e l d o f meromorphic s e r i e s w i t h c o e f f i c i e n t s i n a f i n i t e f i e l d i s s t i l l open.
REFERENCES, Kochen, 1974,
S. The
modeL t h e o h g
06
l o c u e d i e l d b , L o g i c Conference K i e l 1974,
L e c t u r e Notes i n Mathematics, S p r i n g e r - V e r l a g , 384-425. Robinson, A. 1956,
Complete Theories, N o r t h - H o l l a n d P u b l i s h i n g Co., Amsterdam.
Shoenfield, 3 . R. 1970,
A themem on quunti6ieh e l h i u z a t i o n , Symposia Mathematica, v o l . V,
173-1 76.
Department of Mathematics Duke U n i v e r s i t y Durham, N o r t h C a r o l i n a , U.S.A.
PART I11 COMPUTABILITY
Non-Classical Logics, Model Theory and Computability, A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
THE DECISION PROBLEM OF THE CONGRUENCE LATTICES OF PSEUDOCOMPLEMENTEDSEMILATTICES(*)
ON
by
H.
P . SANKAPPANAVAR
1, INTRODUCTION, The s t u d y o f t h e d e c i s i o n problems f o r v a r i o u s c l a s s e s o f ( b o t h a l g e b r a s and r e l a t i o n a l s t r u c t u r e s )
-
which began i n t h e 1930's
-
Church gave t h e f i r s t u n d e c i d a b i l i t y r e s u l t (see Church 1936) c r e d i t a vast
c l a s s e s o f s t r u c t u r e s which a r e , classes o f structures
-
with o t h e r
congruence l a t t i c e s , subalgebra
t i c e s , l a t t i c e s o f subspaces o f geometries, except f o r
lat-
l a t t i c e s o f v a r i e t i e s o f alge-
b r a s , automorphism groups, endomorphism groups, e t c . i n t h e e a r l y 1960's,
until
t h e d e c i s i o n problems f o r
i n a n a t u r a l way, a s s o c i a t e d
f o r example,
when
has f o r i t s
l i t e r a t u r e , and an e x c e l l e n t survey o f t h e work done
1965 has appeared i n Ershov e t a1 1965. However,
much l a t e r ,
structures
-
were c o n s i d e r e d o n l y
r e s u l t s o f Tarski a n d
Grzegorczyk which were p u b l i s h e d i n 1949 and i n 1951
respectively
of (see
T a r s k i 1949, and Grzegorczyk 1951). Tarski proved t h a t the ( f i r s t - o r d e r ) theory o f the spaces o f two
- dimensional
p r o j e c t i v e geometries
lattices
of
sub-
( w i t h p o i n t s h a v i n g homo-
geneous r a t i o n a l c o o r d i n a t e s ) i s undecidable ( T a r s k i 1949).
The problem o f
whether t h i s t h e o r y i s r e c u r s i v e l y i n s e p a r a b l e f r o m t h e s e t o f i t s f i n i t e l y r e f u t a b l e sentences appears t o be s t i l l open. ~~
~~~
~~
Grzegorczyk
(in
the
above
~~~
(*) This work was essentially done in 1973 at the University o f Waterloo, Waterloo, Ontario, Canada a n d was supported by a Province o f Ontario Graduate Fellowship. 255
256
H. P. SANKAPPANAVAR
mentioned paper) ccnsidered the d e c i s i o n t a i n classes o f algebras which are
problems f o r the t h e o r i e s o f cer-
associated w i t h t o p o l o g i c a l spaces:
he
showed the e s s e n t i a l u n d e c i d a b i l i t y o f t h e t h e o r i e s o f closure algebras, o f Brouwerian algebras, o f t h e algebras o f bodies, o f t h e algebras o f convexity and o f t h e semi - p r o j e c t i v e algebra. He a l s o
deduced Tarski's
mentioned above from one o f h i s r e s u l t s . I n 1962 Kargapolov study o f d e c i s i o n problems f o r l a t t i c e s
of
subgroups by
result
initiated
theory o f subgroup l a t t i c e s , and hence a l s o o f t h e congruence l a t t i c e s , Abelian t o r s i o n
- free
reduced groups i s undecidable
the
showing t h a t t h e of
( c f . Kargapolov 1962).
As c o r o l l a r i e s he mentions t h e u n d e c i d a b i l i t y o f t h e theory o f subgroup l a t t i c e s o f groups and o f Abelian groups. In 1970 Kozlov proved t h a t t h e theory o f l a t t i c e s o f subgroups o f f i n i t e Abelian p - g r o u p s
separable ( c f .
Kozlov 1970). As consequences Kozlov
i s recursively i n -
l i s t s a l s o t h e un-
d e c i d a b i l i t y o f t h e t h e o r i e s o f l a t t i c e s o f subgroups o f Abelian p - groups, o f f i n i t e Abelian groups, o f t o r s i o n l i a n groups.
- free
Abelian groups and o f f r e e Abe-
He asks whether t h e u n i v e r s a l theory o f l a t t i c e s o f subgroups
o f Abelian groups i s decidable.
Also i n T a i t s l i n 1970 i s e s t a b l i s h e d
h e r e d i t a r y u n d e c i d a b i l i t y o f t h e theory o f the l a t t i c e o f subgroups d i r e c t sum o f two i n f i n i t e c y c l i c groups and hence t h a t o f
the
the
o f the
theory of
subgroup l a t t i c e s o f any c l a s s o f groups which contains such a d i r e c t sum; he a l s o gives some p o s i t i v e r e s u l t s ,
such as the d e c i d a b i l i t y o f t h e theo-
r i e s c f subgroup l a t t i c e s o f f i n i t e Abe:ian
groups w i t h
f i n i t e Abelian p - g r o u p s w i t h
)L
d i r e c t sums o f
f i n i t e c y c l i c groups.
h
isomorphic
T a i t s l i n proved i n 1968 ( c f .
h
generators,
of
generaLors and o f Abelian groups which are Moving on t o
rings,
T a i t s l i n 1968a) t h a t the theory o f t h e
lat-
t i c e o f i d e a l s of a polynomial r i n g over a f i e l d w i t h a t l e a s t two unknowns i s h e r e d i t a r i l y undecidable, w h i l e t h a t o f a polynomial r i n g i n one unknown i s decidable; i n f a c t , he observed the d e c i d a b i l i t y o f the i d e a l l a t t i c e o f a Dedekind domain.
I n the same year T a i t s l i n published a n o t h e r paper
( T a i t s l i n 1968b) i n which be proved t h a t the theory o f the p a r t i a l l y dered s e t o f simple i d e a l s o f a polynomial r i n g i n a t l e a s t t h r e e
or-
unknowns
over a f i e l d i s h e r e d i t a r i l y undecidable and he mentions the c a s e o f t w o
un-
knowns as an open problem.It i s a consequence o f the r e s u l t s p r o v e d i n R a b i n 1964 t h a t the theory o f congruence l a t t i c e s o f countable Boolean
algebras
i s decidable. More r e c e n t l y , B u r r i s and Sankappanavar have examined t h e dec i s i o n problems o f the t h e o r i e s o f l a t t i c e s o f subrings o f r i n g s w i t h u n i t y ( t h e case o f Boolean algebras i s a p a r t i c u l a r case), o f congruence l a t t i c e s o f s e m i l a t t i c e s , semigroups and unary algebra, and o f l a t t i c e s o f v a r i e t i e s
257
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
( cf.
B u r r i s and Sankappanavar 1975). Boolean algebras are the o n l y non- t r i v i a l proper s u b v a r i e t y (see
e.g.
Sankappanavar 1974) o f t h e v a r i e t y o f pseudocomplemented s e m i l a t t i c e s ,
and
the class o f Boolean algebras i s a subclass o f t h e c o n g r u e n c e - d i s t r i b u t i v e pseudocomplemented s e m i l a t t i c e s . Since t h e theory o f congruence l a t t i c e s o f countable Boolean algebras i s decidable as mentioned above, a question n a t u r a l l y arose whether the theory o f ( a s l i g h t l y l a r g e r class o f )congruence l a t t i c e s o f congruence- d i s t r i b u t i v e pseudocomplemented s e m i l a t t i c e s i s a l s o decidable. I n t h i s note we show t h a t t h i s theory i s r e c u r s i v e l y inseparable (and hence undecidable).
From t h i s we deduce an u n d e c i d a b i l i t y
result of
Ershov and T a i t s l i n 1963 which i n t u r n i s an improvement on an undecidabili t y r e s u l t due t o Grzegorczyk ( c . f .
Grzegorczyk 1951). I t i s a l s o
observed
t h a t the theory o f Heyting l a t t i c e s and t h a t o f f i l t e r l a t t i c e s o f pseudocomplemented s e m i l a t t i c e s are r e c u r s i v e l y inseparable.
2, PSEUDOCOMPLEMENTEDSEMI
LATTICES
AND THEIR
CONGRUENCE LAT-
TICES,
, 0 > i s a A - n PmieaU;ice Luith zeLo i f f A i s a b i L , and 0 i s a d i s t i n g u i s h e d element o f L , s a t i s f y i n g x A y = y A x, x A(y A 2) = ( x A y) A 2, x A x = x , and
An algebra f =
x A a = 0
61
=
{x
5. be
€ L: a
the
o f an element a i n L i f f x i s the and i s denoted by a*.
An
i s a peudacamptemented nem.Lld-tice i f f < S , A , O >
w i t h zero and
*
partial An
5 x 5 b}.
algebra is
a
i s a unary operatior! on S such t h a t a* i s
the pseudocomplement o f a i n S . The c l a s s o f a l l psecdocomplemented semil a t t i c e s i s a v a r i e t y and i s denoted by PCS. I f S € PCS, we w r i t e S i s a PCS, and i t s congruence l a t t i c e i s denoted by Can S. study o f t h e congruence l a t t i c e s o f
For
a
systematic
pcs's one should r e f e r t o Sankappanavar
1974. We mention here only those concepts and r e s u l t s needed f o r our
pur-
pose. I n t h e sequel S denotes an a r b i t r a r y PCS. An element a i n S i s cloned i f f a** = a
, and
t h e s e t of suchelements i s denoted by B(S),while i t s com-
plement i n S by N ( S ) . The elements o f N(S) are s a i d t o be non - d a h e d , o r e q u i v a l e n t l y , an element a i s non - closed i f f a < a** R ( S ) i s a sub-
.
algebra o f S
and a l s o forms a Boolean algebra i n t h e usual sense.
It
is
258
H. P . SAEIKAPPANAVAR
well-known t h a t t h e c l a s s o f Boolean algebras can be d e f i n e d by a s e t i d e n t i t i e s involving only A
,*
and 0
, hence
of
as a s u b v a r i e t y o f t h e v a r i e t y
PCS. For c € B(S) d e f i n e D C ( S ) = { x € S : x** = c } . I t i s known t h a t Dc(S) i s a subalgebra o f S. For 0 i n Con S, ( e ) , denotes t h e r e s t r i c t i o n o f 0 t o B ( S ) . I t should be noted t h a t (e)B can be regarded e i t h e r as a PCS- congruence o r as a B A - congruence on B(S) considering B(S) as a PCS o r as a Boolean algebra ( i n t h e usual sense) r e s p e c t i v e l y . The mapping ** : S S, b b b * * , i s Q ( S ) o r simply Q , i te. a PCS - homomorphism whose kernel i s denoted O ( S ) = { < x , p: x , q € S and x** = q**}. For a € S we d e f i n e t h e congruence li ( o r ( a ) - )on s as f o l l o w s : +
<x,q>€
LEMMA
1.
ii
iff
Foh a E S,
x , y e S.
x I\ a = q A a,
(a),
= ((a**)- ) B
2 . 16 a , B 8 Con S me huch ,then a = B . LEMMA
.t(2dt
CY
Q, B 2 Q a d ( a ) , = (B),
The proofs o f t h e above lemmas a r e n o t hard and hence omitted. We
also
need t h e f o l l o w i n g p r o p o s i t i o n which i s proved i n Sankappanavar 1974. PROPOSITION
Con S (0) W X W Y . ( x ' < q**
Let
.iA -+
d i n - t % i b m v e i6d S b c l t i b d i t b oh q 5 x ) .
the
condition
x 5 q
We say t h a t S i s congmence- dintzibu-tiwe i f f Con S i s d i s t r i b u t i v e . K be a c a r d i n a l . A PCS S i s a K - Boolean G e b m i f f S i s congru-
e n c e - d i s t r i b u t i v e and lUc(S)l 2 K f o r every c € B(S). I t f o l l o w s t h a t a PCS i s a Boolean algebra i f f i t i s a 1 - Boolean algebra. As and V denote t h e e q u a l i t y r e l a t i o n on S and S x S r e s p e c t i v e l y . e(a,b) denotes t h e congruence generated by < a , b>
3, LOGICAL TERMINOLOGY
AND THE BASIC
THEOREM,
By a lungage we mean a f i r s t - order language w i t h e q u a l i t y ( f o r
these
and o t h e r r e l a t e d n o t i o n s see Shoenfield 1967) which has only a f i n i t e nunber o f non- l o g i c a l symbols. I f L i s a language, a theohy T i n L i s a
set
o f sentences o f L which i s closed under l o g i c a l deduction. A sentence u i n L i s &bLt@ k e d h b l e i n a theory T i f f t h e r e e x i s t s a f i n i t e model o f T i n which l u
i s t r u e . We denote by
t h e s e t o f a l l sentences
which are t r u e o f a l l t h e f i n i t e models o f T , and by T
f
in
L
the set o f all
259
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
f i n i t e l y r e f u t a b l e sentences i n L . lie say t h a t T i s xecwmiveLy inbepaha-
bee i f f t h e r e e x i s t s n o r e c u r s i v e s e t o f sentences A i n L such t h a t T and A i s d i s j o i n t f r o m T r a b l e t h e n T and T f i ,
5
A
It i s clear t h a t i f T i s recursively insepa-
f'
a r e undecidable.
L e t L be a language w i t h one b i n a r y p r e d i c a t e symbol P language(not n e c e s s a r i l y d i f f e r e n t f r o m L ). L e t
,
another
LI
,y)
6 ( x ) and P(X
be f o r L J w i t h one and two f r e e v a r i a b l e s r e s p e c t i v e l y . F o r e v e r y s t r u c 6 t u r e M~ o f L I w i t h u n i v e r s e M I , we d e f i n e a s t r u c t u r e o f L induced by and p - which i s denoted by M1(6 ,p ) - as f o l l o w s :
mulas o f
M1(6
.p)
=
where
0
= {A E
Al : M I k 6 ( ( n ) l
R
= { E M I
and : 6
, t E 0 and MI
b p ( A ,Z)j
The f o l l o w i n g theorem- w h i c h i s t a k e n f r o m B u r r i s a n d 1975- i s t h e b a s i c t o o l i n t h e n e x t s e c t i o n . THEOREM 3 . 1 .
L e X T be a theoxy .&I a Language L ulith t h e popeL@ that T
0 kecwmively h e p a h a b L e . LeA T I be a theahy in 1, and p ( x , y) m e ma 6ohmLLean i n L, nuch that (1)
Sankappanavar
. k n w n e that
6(x)
doh evehy 6inite model N 06 T thetle e x h b a &hLte model MI induced nLtuc2uhe MI (6 p ) N , and
06
T~ nuch that t h e
(2)
60'1- wetly model M1 a6 T I t h e induced nLtuctwle M 1 ( 6 model 06 T .
Then T I 0 heCWL4iV@
4,
in
a
LmepahabLe.
ELEMENTARY THEORY
Let LI
,p )
OF CONGRUENCE L A T T I C E S I
denote t h e language o f l a t t i c e s , i . e .
L I has two
non- l o g i c a l which
symbols A and V . One can w r i t e i n L I formulas Caatom(x), I N LA ( x ) say r e s p e c t i v e l y t h a t "x 0 a c o a t o m : ' , " x 0 A-.ihtleducibLe". kle denote by
DCON t h e c l a s s o f a l l ( d i s t r i b u t i v e ) congruence l a t t i c e s
o f congruence- d i s t r i b u t i v e PCS'S and ~h (DCON) denotes DCON
the theory
of
i n L I , i . e . t h e s e t o f a l l sentences i n L I t h a t a r e t r u e o f !ICON.
260
H . P . SANKAPPANAVAR
THEOREM 4 . 1
Th(DC0N) -in trecwlniu&y imepatrable.
PROOF: L e t T f denote t h e t h e o r y o f an i r r e f l e x i v e , symmetric b i n a r y r e l a t i o n R . I t i s shown i n Ershov 1965 t h a t T f and T' a r e r e c u r s i v e l y i n f separable. L e t 5 be t h e sentence 3X3Y3Z(X # Y &
X
# z
Y #
Z)
and l e t T be t h e t h e o r y a x i o m a t i z e d by T' u e x t e n s i o n o f T', T and T
f
{c l
. Then
since T i s a f i n i t e
a r e r e c u r s i v e l y i n s e p a r a b l e and so
T qualifies
t o be t h e t h e o r y T o f Theorem 3.1. L e t M = < A , R> be a f i n i t e model o f T a, b E A
such t h a t € R (and hence
, so
/A1 2 3.
a # b
gebra E R , .tabA tab =
< a , b> E R
'ab ; and < a , c> E R t h e n
A tab = {a) ;
(iv) i f a , b , c , d E A s u c h t h a t { a , b }
r
and E R, t h e n and (v)
if
a, b E A
tab A
x
=
n
{ c ,d} = B
,
w i t h < a , b> E R
x A tab=
and x E 2A
then
{ a }
if
x={al,
{ b l
if
x = { b } ,
tab i f
x
x
= { a ,b l ,
x # {al,{bl,{a,bl.
A {a,b} if
We t a k e t h e 0 E 2 A as a d i s t i n q u i s h e d element i n S and d e f i n e f o l 1ows : (i) i f
b E
zA , b*
(ii) i f
b E
A,
, b*
= b'
*:
S
+
S
as
;
= { a , 6)" where
Then i t i s easy t o v e r i f y t h a t
< a , b> E R
t u b A t c d = 0;
$ = <S, A ,
b = Lab
* , O>
.
i s i n d e e d a PCS
with
PSEUDOCOMPLEMENTED S E M I L A T T I C E S
B ( S ) = 2A and N ( S ) = t L a 6 : < u , b > € I?). Me a l s o n o t e t h a t { : < a , b > 6 R } . I t i s c l e a r t h a t g s a t i s f i e s condition
26 1 0 =
(D)
and hence i s congruence- d i s t r i b u t i v e . Since A i s f i n i t e , S i s also f i n i t e . The above construction i s i l l u s t r a t e d in Figure 1 , where A = { a , 6 , c , d } and R = { < a , b > , < a , c > , < b , d > , < c , d > , < b , a > , < c , a > , < d , b > , < d , c > l . Figure 1 1
26 2
H. P . SANKAPPANAVAR
L e t us choose Con 8 f o r t h e M~ o f Theorem 3.1 and c o n s i d e r t h e f o l
-
, where x q y i s an a b b r e v i a t i o n f o r t h e f o r m u l a # q &Wz((x = X A z & z = z A q ) * ( z = x oh Z = q)).
l o w i n g formulas i n L J
x
=xA q & X
6 ( x ) -+d e f
Coatom ( x )
-
and
def
P(x , g )
For a € A
6(X) &
6 ( q ) & 3 Z ( l m ~ ( z &)
4 X
2
g).
i t i s c l e a r t h a t {a}" i s a coatom i n Con
since i t
has
two congruence c l a s s e s , namely [ { a } , I ] and L U , { a l * ] . We c l a i m t h a t 6 ( x ) p i c k s o u t p r e c i s e l y t h e congruences o f t h e f o r m { a } " w i t h CLAIM 1.
Con 8 I=
claim, i f and so
a
E A
Con 8
+
a E A.
{a}" f o r some a E A. T o p r o v e t h i s {a}" i s a coatom i n C U M 8 6 ( { a } " ) . Conversely, suppose Con 8 k= S ( Y ) ; then Y i s iff
S(Y)
Y =
then i t i s already noted t h a t
a coatom i n Con 8
. Since
2 i s t h e o n l y s i m p l e PCS, Y
congruence c l a s s e s , n a n e l y [ I ]
and YO]
has c x a c t l y two
(2 i s t h e PCS whose u n i v e r s e
is
{O, J } ) . I t f o l l o w s t h a t [ I ] = [ R , J ] f o r some t E S s i n c e S i s f i n i t e and so Y = 2 . If h E S and tr < .C t h e n h = 0, f o r , if JI # C t h e n d < E < V since < h , . t > E X but @ .t and < O , r , > E 8 ; b u t < O , r , > @ E , t h i s i s i m p o s s i b l e s i n c e 2 i s c o a t o m . Thus i t f o l l o w s t h a t t i s an cLtom i n S. Since, by t h e c o n s t r u c t i o n o f S , atoms i n S a r e p r e c i s e l y o f t h e f o r m { a } w i t h a € A we have .t = {a} f o r some a E A, which proves Claim 1. CLAIM 2.
For
a,b
E A, < a ,b > E R
iff
p r o v e t h i s , f i r s t suppose < a ,b > E P,
{dA A ibl" 2
@;
=
f o r e by
Lemma 2 we g e t
{ a , b}"
in
.
{ a , b}" < (tab)" I n f a c t , s i n c e and
To
=
since
( { a }A ) 8 A ( { b l " )
.
Con S I== p ( { a } " , { b } " ) .
( { < t a {ba,, b } > } U A g )
{a,b l " +
k? {a,b}"
(Rub)" = { a , b}"
i s an atom i n
.
(tab)* Next we c l a i m t h d t
s i n c e t h e o n l y congruences g r e a t e r t h a n
we i n d e e d
have
v ( I < t a b , { a , b l > }UA,)
C O J Z ~ , we c o n c l u d e
that
- irreducible ;
for,
{a,b}" i s A { a , b}"
A
"
a r e (tab)" , a ,b
and
8,
263
PSEUDOCOMPLEMEIITED S E M I LATT I CES
i t i s clear that
c
Can $
p
Y
{ b l " ) . To p r o v e t h e converse, suppose $ J?. It i s { a I AA { b } " a r e precisely
c l e a r t h a t t h e congruence c l a s s e s o f
[{a),
17 n
[{b}, I ] ,
n
and ( S - [ { a } , J ] ) [{a},J]
*],
Let
Y
(S
(S - [ { b } , J ] ) .
(S- [ { a },
(S - I { b } , J : )
have
J] n
-
[{b},
J ] ) , (S -
Since < a , b >
n [ { b } , J ] = r { a , b } , J ] . A l s o w e have
[ { a } ,{b}
n
[{a},
[O, { a , b } * ]
=
a {a}" A { b } "
" 3{a,b}"
i s i m p o s s i b l e ; hence we c l a i m t h a t
Y
2
.
B R,
we
[{a},J] {a}*],
n
see
that
(S - [ { b l , J j ) =
and ( S - [ { a } ,
J
1)
Then i t f o l l o w s t h a t {a}" A { b } " = { a , b } " .
and we w i s h t o show t h a t Y i s n o t A - i r r e d u c i b l e . W e I A l 2 3, {a,b l # 7 . I f Y B = ( { a , b } " ) B t h e n
and s i n c e
E yB 5 Y
u , w E B( $), u #
J ] )n [ { b } , J ] = [ I b } ,
,J ]
[ { a } , J ] ) nL i b }
and so
e ( { a , b}, I ) 5
Y
whence
{ a , b}" = Y
which
W,
YB < ({a, 6})B and consequently t h e r e exist such t h a t < u , W > E ( { a , b l " ) B and < u , w > f Y B . Plow
0
F o r if Y $ @ t h e n f o r some non-closed n, & 8.
.
B = {} U AS which i s c l e a r l y a congruence on S.Since @ 5 { a } ^ and @ 5 { b } " , i t f o l l o w s t h a t 0 5 { a , b } " which i m p l i e s t h a t B 5 Ca,b}". Thus we g e t Y V 6 5 {a,b } " ; b u t V B # { a , b } ^ because (Y V B)B = YE { a , b l " . conf ! Y V B . W e have thus shown t h a t Y < Y V 6 < and s o < U , W > Y 2 @ From Y 3 {a, b}" hence p r o v i n g t h a t tradicting the fact t h a t Let
YJ
.
t h i s i t f o l l o w s t h a t Y i s t h e meet o f maximal elements s i n c e [ a , V] Con ( B ( $ ) ) , i m p l y i n g t h a t 'Y i s n o t A-irreducib1e;hence Con$ #; p ( I a } " , { b } " ) T h i s proves C l a i m 2. From c l a i m s 1 and 2 i t f o l l o w s t h a t i f
+
U
= {{a}" : a E A}
P
and
=
Con $ p({a}", {b}" ) } then < U , P > , hence ( 1 ) o f Theorem 3.1 i s s a t i s f i e d ; w h i l e ( 2 ) o f t h a t theorem i s e a s i l y v e r i
{ :
f i e d . Therefor? Theorem 4.1 i s proved. I n f a c t , we have proved t h e f o l l o w i n g s t r o n g e r r e s u l t once i t i s
-
noted
t h a t t h e PCS S c o n s t r u c t e d i n t h e above p r o o f i s a 2 - Boolean a l g e b r a .
THEOREM 4.2. T h e t h e o h y 0 6 t h e C l a n 4 0 6 c o n g h u e n c e l a t t i c e 4 2 - Boolean a l g e b h a b i b a e c u a n i w e l y i n n e p a a a b l e . COROLLARY 4 . 3 . 1 w i t h any ( o h &)
(1)
06
L e t K be any c h b 0 6 ~@JX&XA which containb l.~~%ceA 0 6 t h e 6oMoWing pope,kCien:
LO d g e b h a i c , dintkLbLLtiwe and atatnic and hub 0 and J ,
264
H . P . SANKAPPAIIAVAR
e u v l y ,Lnte,kud in L i r p b ~ U d O C O m p ~ e m e ~ Cand d, L h.4 ail &anertt buch ,#i& [o, 6; h a cotrip&te atotriic E o o ~ e a n d g c b h a , 6 0 t h e me& 06 aeR coatomn i n 1 and [d , I ] 0 a n dgebhaic n u b l a t t i c e iuhobe cornpad elemem2 60hm a complemented bub4kdXLce. Then T h ( K ) 0 h e c m i v & y imeparrable. (2) (3)
The above c o r o l l a r y i s immediate from t h e f a c t t h a t t h e congruence l a t tices o f 2
- Boolean algebras
have these p r o p e r t i e s (see Sankappanavar 1974).
From t h i s c o r o l l a r y i t f o l l o w s immediately t h a t t h e theory o f Heyting t i c e s i s r e c u r s i v e l y inseparable, which i s an improvement on
lat-
a result of
Ershov and T a i t s l i n 1963 t h a t t h e theory o f d i s t r i b u t i v e l a t t i c e s i s r e c u r s i v e l y inseparable. Ne a l s o o b t a i n t h e h e r e d i t a r y u n d e c i d a b i l i t y o f t h e theory
of
r e s t r i c t e d c l a s s o f f i n i t e d i s t r i b u t i v e l a t t i c c s as given i n the
a
very
following
corollary. COROLLARY 4 . 4
d t h 0 and 1
Let
bUC!t
06 a i d e dina%ibLLt.iue
F be the &n
L
eatticen
ZhaX
L contaim a n dement 6 nuch Rlzat 10, 61 and [d, 1: me Boolean nub&zt.ticw, and (ii) t h e numben. 06 V - h e d u c i b l e elmew2 $ 6 i n L 0 e y u d to .the numbeh 0 6 coatom i n L . Then rh(F1 0 hetleilitatLiey undecidable. (i)
Using t h e same c o n s t r u c t i o n and t h e same p r o o f o f Theorem 4.1, tice of filters of S THEOREM
hence
,in
p(x
,q)
as i n the
F(S),the l a t -
place o f Con S and thus we o b t a i n t h e f o l l o w i n g .
The t h e o h y
4.5
06 a l l
6 ( x ) and
Claims 1 and 2 can a l s o be proved w i t h
06 6 i L t e ~ t&zt.ticw 06 2 - Boolean
d g e h
(and
PCS'b) i n h e C U h b i V d y i n n e p a h a b l e .
5. CONCLUDING
REMARKS
We observe t h a t t h e PCS's S can have d i s t i n c t non
constructed i n t h e p r o o f o f Theorem
- closed elements
Boolean algebre as a neo
- Boolean
which are n o t d i s j o i n t . C a l l
dgebha i f f any two d i s t i n c t
elements i n i t are d i s j o i n t . The above observation leads t o t h e problem which we suspect has an a f f i r m a t i v e s o l u t i o n .
a
4.1 2-
non- closed following
265
PSEUDOCOMPLEME?ITED SEH I L A T T I C E S
PROBLEM. Is t h e theory of the c l a s s o f congruence l a t t i c e s of neo- Boolean algebras d e c i d a b l e ? I t is alsoof interest t o finda lattice-theoretic characterizationofthe congruence l e t t i c e s o f 2 - Boolean algebras.
The author would l i k e t o express h i s g r a t i t u d e t o S. Burris f o r t h e encouragement.
REFERENCES, Burris, S . and H. P . Sankappanavar 1375, L a t t i c e - t h e v h e t i c decinivn p v b l e n i n i~ U n i v e n n d ALgebtran, Algebra Univ. 5, 1 6 3 - 177. Church, A. 1936, A nvte vn t h e e I ~ c ~ I ~ d u i z g n j ~ h vThe b l ~Journal i, of Symbolic Logic 1 , 40 - 41. Ershov, Y . L . , J . A. Lavrov, A. D. Taimanov, and M. A. T a i t s l i n 1965, Ela:iei&mq T / z e v > L u , Russian Math. Surveys, 20, 35 - 105. Ershov, Y. L . , and M. A. T a i t s l i n 1963, Svi;ie uiv,vlvab&e t h e o h i u . ( i n Russian), Algebra i Logi ka, 2 , 37 41.
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Gratzer, 6 . 1971, Lattice Theory, W . H. Freeman and Company, San Francisco. Grzegorczyk, G . 1951, Undecidability 06 nofile t v p v l v g i c d Rhevhiu, Fund. Math. ,38,137-152. Kargapolov, M. I . 1962, U M -tlze eL!w~eiuW~y,#tevhq a 6 .P&Licu 1 , 4 6 - 53.
v6
nubgfivup,Algebra i Logika,
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dinite
Rabin, W. 0. 1964, VecidabiLiZg 06 necvnd - v t d a tlzevnieb and au*vr;iata v n i n d i n i t e & e u , Trans. h e r . Math. S O C . , 141, 1 - 34. 1965, k bblp!A r;i&:vd I v h u ~ u f e & d a m phvv&5 and bVme appfiC&OvLJ,kgic, Methodology and Philosophy of Science, Proceedings
of
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the 1964 I n t e r n a t i o n a l Congress,Ear H i l l e l e d . , Amsterdam (1965) ,5868. Sankappanavar, H. P. 1974, A study of congruence lattices of pseudocomplemented semilattices, Ph. D. T h e s i s , U n i v e r s i t y o f Waterloo, O n t a r i o , Canada. S h o e n f i e l d , J . R. 1967, Mathematical Logic, Addison Wesley, Reading. T a i t s l i n , M. A. 1968a, Elementahy lattice theohies
dolr
ideaeb i n p a l u r z o t i u d h i y o ,
A1 gebra
and Logic, 7, 1 2 7 - 129. 1968b, U n o h p l e d e & i n p a l y n v n i ~h i n g b , Algebra and Logic, 7 , 394-395. 1970, 012 dmenta/ry theohies 0 6 l a t t i c e s 06 oubghoupn, Algebra and Logic, 9 , 285- 290. T a r s k i , A. 1949, UndecidabLLLty 06 t h e theohie, ad .&uXLces and p o j e c t i u e gevmu7uk.4, The Journal o f Symbolic Logic, 14, 7 7 - 78. T a r s k i , A., A. Mostowski and R. M. Robinson 1953, Undecidable Theories, North - Holland, Amsterdam.
l n s t i t u t o de M a t e m i t i c a Universidade Federal da B a h i a Salvador, Bahia, B r a z i l .
Non-Classical Logics, Model Theory and Computability, A . I . Arruda. N.C.A. da Costa and R. Chuaqui (eds.) 0 North-Holland Publishing Company, 1977
POLYNOMIALLY BOUNDED Q U A N T I F I C A T I O N O V E R HIGHER TYPES A N D A NEW HIERARCHY OF THE ELEMENTARYSETS ( * I b y JANOS S I M O N
I t i s w e l l known t h a t nondeterminism and e x i s t e n t i a l q u a n t i f i c a t i o n a r e
r e l a t e d . I n p a r t i c u l a r NP can be o b t a i n e d by p o l y n o m i a l l y bounded q u a n t i f i c a t i o n o v e r p r e d i c a t e s on s t r i n g s , were t h e p r e d i c a t e s a r e i n P. Meyer and Stockmeyersuggested c o n s i d e r i n g t h e analogue o f t h e a r i t h m e t i c h i e r a r c h y , where t h e a l t e r n a t i n g q u a n t i f i e r s a r e a l l p o l y n o m i a l l y bounded. I t i s n o t known whether t h e t h e r e s u l t i n g h i e r a r c h y i s p r o p e r . In t h i s paper we cons i d e r p o l y n o m i a l l y bounded q u a n t i f i c a t i o n o v e r s e t s and h i g h e r t y p e s , show t h a t one o b t a i n s a p r o p e r h i e r a r c h y o f t h e elementary r e c u r s i v e
and lan-
guages. I n p a r t i c u l a r , w i t h a s i n g l e e x i s t e n t i a l s e t q u a n t i f i e r (and p r e d i c a t e s i n P) one o b t a i n s e x a c t l y t h e n o n d e t e r m i n i s t i c e x p o n e n t i a l t i m e
rec-
o g n i z a b l e languages. E x i s t e n t i a l q u a n t i f i c a t i o n o v e r t y p e i corresponds t o
i levels
0s
e x p o n e n t i a t i o n o f t h e t i m e r e q u i r e d t o a c c e p t t h e s e t by a non-
d e t e r m i n i s t i c T u r i n g machine (Tm). The r e s u l t s may be c o n s i d e r e d as a c h a r a c t e r i z a t i o n o f t h e computational power o f t h e p r e d i c a t e ' € ' .
1, INTRODUCTION
I
The c o n n e c t i o n between n o n d e t e r m i n i s t i c computations and e x i s t e n t i a l
( * ) T h i s r e s e a r c h was s u p p o r t e d i n p a r t by g r a n t 70/755 from de Amparo
2
P e s q u i s a do Estado d e Sao P a u l o
267
(FAPESP) , B r a z i 1 .
Fundasao
268
JANOS SIMON
q u a n t i f i c a t i o n i s w e l l known ( S c o t t 1968). Given a n o n d e t e r m i n i s t i c T ~ T and ' an i n p u t x we add, as a s e p a r a t e i n p u t , t h e s t r i n g o f c h o i c e s y t h a t T ' makes i n an a c c e p t i n g computation. We may t h e n e a s i l y b u i l d a T m T w h i c h o p e r a t e s d e t e r m i n i s t i c a l l y and accepts e x a c t l y t h e same s e t as T ' , i n
t h e same
number o f moves, whenever y i s (an encoding o f ) t h e s e r i e s o f c h o i c e s
that
T I uses. T r e j e c t s i f q does n o t r e p r e s e n t an a c c e p t i n g sequence o f c h o i c e s o f moves. Thus t h e s e t accepted by t h e n o n d e t e r m i n i s t i c machine i s L(T') =
{XI
( 3 y ) T ' ( x , y ) halts and accepts]
T h i s c o n n e c t i o n was e x p l o i t e d i n Meyer and Stockmeyer 1972 and 1973 t o e x h i b i t an analogue o f t h e Kleene h i e r a r c h y f o r P, t h e s e t o f p o l y n o m i a l t i m e r e c o g n i z a b l e languages, by d e f i n i n g
Cp
i+l
= c l a s s o f languages d e f i n a b l e as P
nil
{ y 1 l P x R ( x , Y ) , R in
nPi + l =. c l a s s
o f languages d e f i n a b l e as P
{ y lWPx S ( x , q ) ,
s in
C.
1
where l P x ( W p x ) means t h a t t h e r e i s a polynomial p ( ) such t h a t t h e quant i f i c a t i o n ranges o n l y o v e r s t r i n g s x w i t h 1x1 5 p ( ] y l ) (1x1 d e n o t e s t h e l e n g t h o f x ) . The p o l y n o m i a l i s f i x e d f o r a g i v e n s e t . P
'i+l
= c l a s s o f s e t s d e f i n a b l e by {yl y
is accepted by polynomially ' Tm with a C p oracle)
bounded
i
Then C
P
0
s e t s a r e languages i n P, C
P
1
a r e languages i n NP,
and
Itp
1
s e t s a r e languages whose complement i s i n NP. Many o f t h e p r o p e r t i e s o f t h e Kleene h i e r a r c h y h o l d f o r t h e s e c l a s s e s . I t i s n o t known, however, whether t h e h i e r a r c h y i s proper, s i n c e i t i s n o t
known whether P = NP ; see Stockmeyer 1975 f o r f u r t h e r d e t a i l s . We p r e s e n t i n t h i s paper a p r o p e r h i e r a r c h y o f t h e elementary
recur-
s i v e languages, by a n e x t e n s i o n o f t h e s e d e f i n i t i o n s . T h i s w i l l be done
by
a g e n e r a l i z a t i o n o f t h e p o l y n o m i a l h i e r a r c h y by u s i n g q u a n t i f i e r s o v e r p o l y -
269
P O L Y N O M I A L L Y BOUNDED Q U A N T I F I C A T I O N
n o m i a l l y bounded o b j e c t s o f h i g h e r type, i n a manner analogous t o t h e d e f i t i o n o f the analytical hierarchy generalizing the arithmetical hierarchy. We w i l l q u a n t i f y o v e r s e t s , c l a s s e s o f s e t s , e t c . , where t h e
elements
o f t h e s e t s a r e p o l y n o m i a l l y bounded. E x i s t e n t i a l q u a n t i f i c a t i o n o v e r
type
i w i l l correspond t o i l e v e l s o f n o n d e t e r m i n i s t i c time. We develop now some n o t a t i o n i n o r d e r t o p r e s e n t these r e s u l t s .
A s t r i n g i s an o b j e c t o f t y p e 0 . An o b j e c t o f t y p e i + l i s a c o l l e c t i o n
i > 1 we say t h a t an o b j e c t o f t y p e i i s poLyno-
o f o b j e c t s o f t y p e i. F o r
bounded
nii&y
by i t s parameter y i f f t h e o b j e c t s t h a t i t i s a
collection
o f a r e p o l y n o m i a l l y bounded. The f u n c t i o n s t i ( n )
a r e d e f i n e d by
t i ( n ) stands f o r i l e v e l s o f e x p o n e n t i a t i o n .
Thus
[d(x) ]
NTIME
nondeterministic
denotes t h e language accepted by
T u r i n g machines w i t h i n t i m e
B ( x ) and DTIME [ d ( x ) ] t h e languages accepted
d ( x ) by d e t e r m i n i s t i c T u r i n g machines. L e t
w i t h i n time
NEXPTIME
=
U
p a poLynomiae
NTIME [ Z p ( n ) ]
;
and, i n general
I t i s w e l l known t h a t t h e elementary languages a r e e x a c t l y t h e l a n g u a g e s i n m
U
i=l
DTIME [ t i ( n )
and i t i s easy t o show t h a t
since
3
2 70
JANOS S I M O N
[Zi(n) ] 5
NTIME
Also f o r a l l polynomials
DTIME
[ ti+l( M ) 3 ,
p ( 1, f o r s u f f i c i e n t l y l a r g e n,
We saw t h a t NP ( i . e . U
p a paLynamiae
"TIME [Z,
(ph)] )
can be c h a r a c t e r i z e d as t h e c l a s s o f languages L which c o u l d b e e x p r e s s e d a s
f o r some p r e d i c a t e R L € P . Our
r e s u l t i s an extension o f t h i s characterization t o t h e classes
s. =
'
U NTIME [Zi (p ( M I p a pot ynam i d
namely, t h a t any language L E Si
)
]
may be o b t a i n e d by a s i n g l e
polynomially
bounded e x i s t e n t i a l q u a n t i f i c a t i o n o f a s i m p l e p r e d i c a t e . I n t h e case i = O , t h e q u a n t i f i e r r a n g e s o v e r p o l y n o m i a l l y bounded o b j e c t s o f t y p e 0 and p r e d i i s computable i n d e t e r m i n i s t i c p o l y n o m i a l time, w h i l e i n t h e
cate
case, t o d e s c r i b e a language
general
L € Si, we q u a n t i f y o v e r p o l y n o m i a l l y bounded
o b j e c t s o f t y p e i, and a l l o w a c o n s t a n t number o f polynomiallyboundedquant i f i e r s over objects
o f t y p e a t most i-1. I n a d d i t i o n , t h e m a t r i x ( t h e
q u a n t i f i e r - f r e e p o r t i o n o f t h e f o r m u l a ) becomes a f i x e d Boolean c o m b i n a t i o n o f d e t e r m i n i s t i c p o l y n o m i a l t i m e computable p r e d i c a t e s and membership predicates
-
t h e form t'c
o r , i n o t h e r words, i f we make t h e c o n v e n t i o n t h a t p r e d i c a t e s o f "u € Y"
may be e v a l u a t e d i n u n i t time, t h e m a t r i x i s d e t e r m i n i s -
polynomial t i m e computable i n t h i s new sense. P
3. x t o denote " t h e r e i s a p o l y n o m i a l l y bounded o b j e c t x o f
L e t us use P
t y p e i", and W . x 1
t o d e n o t e u n i v e r s a l q u a n t i f i c a t i o n . We s h a l l w r i t e O_
f o r a n u n s p e c i f i e d q u a n t i f i c a t i o n (i. e. Say t h a t a language L i s
Qy means W
P
i
P
orji).
C p if i t can be expressed as i 1
P
i
271
POLYNOMIALLY BOUNDED QUANTIFICATION
where Qp
Ji
j , < i, t h e q u a n t i f i e r s a r e p o l y n o m i a l l y bounded by 1 x 1 ,
11
and t h e
a r e f i x e d , and R L i s a p r e d i c a t e i n P, except f o r o c c u r r e n c e s o f 'El
T h i s i s t h e o b v i o u s g e n e r a l i z a t i o n o f t h e a n a l y t i c a l and h i g h e r h i e r a r c h i e s f o r polynomial t i m e bounded computations. We c o u l d define .IIP
c l a s s e s and i A
P
j
type
similarly
classes.
Using t h e n o t a t i o n , o u r main theorem i s
P
I n o t h e r words: a language may be d e s c r i b e d by a f o r m u l a o f t y p e
1 i l ob-
( i . e. a s i n g l e e x i s t e n t i a l q u a n t i f i c a t i o n o v e r p o l y n o m i a l l y b o u n d e d j e c t s o f t y p e i, f o l l o w e d by a f i x e d number o f p o l y n o m i a l l y
bounded
t i f i e r s o f l o w e r t y p e s and a d e t e r m i n i s t i c p o l y n o m i a l t i m e matrix) i f
quan-
computable
and o n l y i f i t can be r e c o g n i z e d by a n o n d e t e r m i n i s t i c
Turing
machine i n t i m e bounded by i l e v e l s o f e x p o n e n t i a t i o n o f a p o l y n o m i a l .
I n p a r t i c u l a r we have: LEMMA 1 .
NEXPTIME =
P C . 1 1
I n o r d e r t o p r o v e o u r r e s u l t , we w i l l proceed by i n d u c t i o n . The case
i = 1, t h e s t a r t o f t h e i n d u c t i o n , i s t h e Lemma a b o v e , w h i c h we p r o v e i n the next section.
2, EXPONENTIAL PROOF OF L E M M A 1 : us assume t h a t
TIME Let
p ( ~= )
M
SI
A SINGLE
L E NTIME[Z~(')
-
SET QUANTIFIER, 1,i.e.
L E S ~ .F o r c l a r i t y ,
use t h e f a c t t h a t we use t h i s p a r t i c u l a r p o l y n o m i a l , b u t t h e somewhat c l e a n e r . F o r e v e r y tion o f the
let
i t w i l l become c l e a r t h a t o u r argument does n o t
formulas
are
y € L, we s h a l l d e s c r i b e an a c c e p t i n g computa-
TmM t h a t r e c o g n i z e s y, i. e. o u r p r e d i c a t e w i l l be
L = I y 1 3 X : X represents an accepting computation of M on
input y taking exponential time at most}.
272
JANOS
X w i l l be t h e s e t o f I D ' S o f
- including
M ' s tape
SIMON
M ' s computation ( i . e. a d e s c r i p t i o n o f
t h e square scanned and t h e s t a t e o f t h e f i n i t e
t r o l ) . The o n l y problem i s t h a t t h e t a p e used may be o f e x p o n e n t i a l and o u r q u a n t i f i e r s ( e x c e p t f o r 3 X ) range o v e r s t r i n g s
conlength
o f polynomially
bounded l e n g t h . We overcome t h i s d i f f i c u l t y by r e p r e s e n t i n g each I D as 2' where each element stands f o r a s i n g l e square o f t h e Tm t a p e .
elements, Each
ele-
ment w i l l be a t r i p l e (i.e. a s t r i n g w i t h t h e c o n p o n e n t s s e p a r a t e d markers,
by
$ ) o f t h e form t i m e $ p o s i t i o n $ square. = ( t , p , n ) .
( t , p , 6 ) w i l l represent t h e contents o f t h e
p - t h square o f t h e T m t a p e a t
time
t. We assume t h e T u r i n g machine t o have a s i n g l e i n f i n i t e t a p e and a s i n g l e r e a d / w r i t e head (so t h a t pasLtiun i s w e l l d e f i n e d ) and i f a t t i m e f
t h e T m i s scanning t h e p - t h square i n s t a t e q . we s h a l l r e p r e s e n t t h e c o n t e n t s o f t h a t square as t h e p a i r (u,q) where u i s t h e symbol t h a t t h e s q u a r e c o n t a i n s . I t i s e a s i l y seen t h a t t h e l e n g t h o f t and p w i l l be p o l y n o m i a l l y bounded. I n o r d e r t o ensure t h a t X i s t h e s e t o f I D ' S ,
as claimed,
we
shall
w r i t e a p r e d i c a t e e x p r e s s i n g t h e f a c t t h a t f o r a l l s t r i n g s w E X, a ) w i s o f t h e f o r m (t,p,6) as d e s c r i b e d above (ensured by t i e f o r m u l a A d e f i n e d below); b ) f o r a l l t i m e s we have a complete d e s c r i p t i o n o f M ' s I D ( a n d
only
one), denoted by.8; c ) t h e s e t o f such I D ' S i s an a c c e p t i n g computation o f M on i n p u t y , guaranteed by C. D e t a i l e d d e s c r i p t i o n s o f each o f t h e s e a r e : A = Y Pw E
I'
x 3 't 3'p
r
where
= { M ' s tape alphabet} U {,ti's tape alphabet} x { M ' s state set}.
A c t u a l l y 3'n t i e s f o r b since We
3'6 : w = .t $ p $ n A n E
i s n o t necessary - o n e
r
could simply w r i t e out a l l p o s s i b i l i -
i s f i n i t e - we used i t o n l y f o r conciseness o f e x p o s i t i o n .
s h a l l use t h i s as an a b b r e ' v i a t i o n f o r t h e complete e x p r e s s i o n
which, when expanded, o n l y m u l t i p l i e s t h e l e n g t h o f t h e f o r m u l a constant.
u
n ~ 'r by a
273
POLYNOMIALLY BOUNDED QUANTIFICATION
B = V p t Wpp(t < 2" P
P
P
A p 5 2")
->
36 ~ ' U J[LO = . t $ p $ h A UJ € X ]
A
P
W t W ~ W U J W W ' W6 W O ' [ ~ = t $ p $A u~ J ' = ~ $ ~ $ ~ ' ] - > u J = w ' . C ) An a c c e p t i n g computation i s c h a r a c t e r i z e d by:
c l ) a t t i m e .t = 0 we have t h e i n i t i a l ID o f M w i t h i n p u t y; c 2 ) a t t i m e .t =
P'Mi s
i n an a c c e p t i n g s t a t e ;
c 3 ) i f a c o n f i g u r a t i o n i s assumed a t a t i m e d > O , i t i s a t t a i n e d as t h e r e s u l t o f a sequence o f l e g a l moves f r o m t h e i n i t i a l comfiguration.
Such d e s c r i p t i o n s o f a c c e p t i n g computations
used i n Meyer and Stockmeyer 1972, and 1973 f o r t h e
were
study
of
1ower bounds. c l ) i s ensured by
CI = O$I$y,
x
... A
qi E X A O $ 2 $ y 2 E X A 0 $ 3 $ g 3 6 X A
0 $ n $ y Y I e X A Wpp(p > M A p 5 2'")->
. .ybl, q,
where t h e i n p u t i s y = y,y,.
O$p$b € X ,
i s t h e i n i t i a l s t a t e o f M, and b denotes
t h e symbol f o r b l a n k i n M ' s tape a l p h a b e t . c2): c 2 = ~ ' U J 3't
3'p
36 [ W € X A UJ = t $ p $ 6 A 6 = u x q 6
where a stands f o r some symbol i n
M ' s t a p e a l p h a b e t and
unique f i n a l s t a t e o f M, such t h a t M upon e n t e r i n g q
q
d
]
9
denotes
the
accepts a n d
stops
c 3 ) : To make t h e e x p r e s s i o n s e a s i e r , l e t us assume t h a t M may
eitner
( i t can be assumed wlog t h a t M has such b e h a v i o r ) .
6
r e w r i t e a square o r move i t s head b u t n o t b o t h . T h i s causes t h e r u n n i n g t i m e t o d o u b l e a t most. Now
(.t,p,h)
i s p a r t o f M's configuration a t time t i f
( 1 ) 6 does c o n t a i n a s t a t e component (6 = u x q ) , a t t h e p r e v i o u s moment M was e i t h e r scanning t h e p - t h square i n s t a t e q ' , r e a d b
on
i t , and one o f t h e v a l i d moves o f M upon r e a d i n g a b i n s t a t e q ' i s
t o r e w r i t e i t as an u and go t o s t a t e q ; o r bl was scanning one adjacent
square, and an a l l o w a b l e t r a n s i t i o n was t o move t h e h e a d t o
t o t h e p - t h square.
274
JAFIOS
SIMON
( 2 ) 6 does n o t c o n t a i n a s t a t e component ( 6 = a ) and a t t h e
previous
move e i t h e r M was scaning t h e p - t h square, r e a d i n g a , a n d
i t s
s t a t e caused t h e head t o move; o r M was scanning a n o t h e r squareand d i d n o t m o v e o n t o t h e p - t h square, which a t t i m e t
-
1 c o n t a i n e d a.
We must, i n a d d i t i o n , r e q u i r e t h a t o n l y one move was made 6(y,a) = {(c,y',O),(a,q",L),(a,q"',R))
( c. 9. i f
t h e n i t i s n o t t h e case t h a t
( t , p , y x a ) E X and more t h a n one o f ( . t + l , p , y ' x c ) , ( t + l ,p - 1 , b x y"), ( t + l p, + l , y " ' x d ) E X . I t can be seen e a s i l y t h a t t h i s r e q u i r e m e n t i s s a t i s f i e d ( g i v e n o u r p r e v i o u s r e s t r i c t i o n s ) by ( 3 ) a t a l l t i m e s e x a c t l y one square has a s t a t e s p e c i f i e d i n i t . The e x p r e s s i o n s t h a t s t a n d f o r t h e f a c t s above a r e : C3(1) = WptWppWA E C X Q ,
0 < t 5 ZCn,
t $ p $ A E X ->
([(t-l$p$bxq')E
[ A = ~ X Q ->
X A(ux~,A)
(b,q')]
E
( s t a t e change o n some square) V (3c,q"
c E C,
q" E
Q, ( q , R )
A (t-l$p-l$cxq")EX V [3d,q"'
E 6(c,q")
A(t-l$p$a)EX]
( o r come f r o m t h e l e f t )
d e C, q"' E Q , ( q , ~ )E 6 ( d , q " ' )
A (t-1$ p + 1 $ d x 4 ' " ) E X A ( t - 1 $ p $ a ) EX])]
( o r come f r o m t h e r i g h t ) ;
C3(2) = WptWppW6 € C, 0 < t 5 ZCM, t $ p $ 6 E X ->
[ b = a ->
[
[ t - l $ p $ a x q ) E X A 9 € Q A [ ( a , g ' , R ) € 6(a,y)
A ( t - l $ p + l $ c ) E X A ( t $ p + l $ c x q ' ) E X] V [ ( u , Q " , L ) 6 6(a,q) A ( t - l $ p - l $ d ) A (t$ p -1 $ d x q " ) E X
V [(t-l$p$a)
]]
E X
(head was h e r e b u t moved)
( a c t u a l l y t h i s s u f f i c e s , we show t h e
E X
possibilities)
A[((t-l$p-l$b)
€ X
V ((t-l$p-l$bxq')
A (t-l$p+l$C) EX, E X
A ( ( t $ p - l $ c x y") E X
V ((t-l$p+l$bxq')EX
b,c E Z)
(head was t o t h e l e f t ) V (t$p-Z$c'xq''')EX))
(head was t o t h e r i g h t )
(didnotmovetop)
275
POLYNOMIALLY BOUNDED Q U A N T I F I C A T I O N
A((Z$p+l$c"
(R$p$a
x
x
4 " ) E X V ( Z $ p + Z $ c " ' x 4"')
q ) E X A ((Z$p' $ A ) E X A p # p'*
]]
E X))]
n E E).
Now t h e language L may be expressed as
L = {y I3'X
A A B A C1 A C2 A C3(1) A C3(2) A C3(3)).
I t i s c l e a r t h a t each o f
A, 8,
...,
C3(3) c o n t a i n s o n l y
predicates
t e s t a b l e i n p o l y n o m i a l t i m e (remember t h a t w € X i s counted as a u n i t operation). Also, o n l y 4 q u a n t i f i e r s were used - i f we c o u n t a l t e r n a t i o n o f
quan-
t i f i e r s , o n l y 3 have been used. ( A s l i g h t l y more c l e v e r encoding would u s e o n l y ~3 p r e f i x e s . Remember t h a t we do n o t c o u n t q u a n t i f i e r s o v e r
constant
s i z e d s e t s . ) T h i s proves t h a t e v e r y language r e c o g n i z a b l e i n NTIME[2P(n)],
p ( ) a p o l y n o m i a l , has t h e c l a i m e d r e p r e s e n t a t i o n . T h i s proves h a l f o f o u r lemma. The o t h e r h a l f f o l l o w s f r o m a
simple
P c o u n t i n g argument: i n a l C 1 formula, we have a p o l y n o m i a l l y bounded s e t
I f t h e bounding p o l y n o m i a l i s
P( ) , t h e s e t has a t most Zp(')
.
elements.
Given t h e i n p u t x, t h e n o n d e t e r m i n i s t i c T u r i n g machine t h a t w i l l accept t h e language denoted by t h e f o r m u l a w r i t e s , on one o f i t s t a p e s a v e c t o r length
of
2 P ( n ) , w i t h a 1 i n p o s i t i o n i i f f iis i n t h e s e t . T h i s takes e x -
p o n e n t i a l time, and t h e r e s t o f t h e s i m u l a t i o n i s s t r a i g h t f o r w a r d : e x i s t e n t i a l q u a n t i f i c a t i o n o v e r s t r i n g s i s t r e a t e d by w r i t i n g
t h e s t r i n g down,
w h i l e u n i v e r s a l q u a n t i f i c a t i o n i s t a k e n c a r e o f , by t r y i n g a l l
strings
bounded l e n g t h . The e v a l u a t i o n o f t h e p r e d i c a t e t a k e s polynomial time, c e p t f o r subexpressions o f t h e t y p e
of ex-
x € X, which a r e d e c i d e d by l o o k i n g up
t h e x-th p o s i t i o n o f t h e vector. A l l o f t h i s takes a t m o s t
exponential
time, so t h a t o u r lemma i s proved.
3, THE
MAIN
THEOREM,
F i r s t , we want t o show t h a t a l l t h e languages a c c e p t a b l e by a Tm nondeterministic time
in
276
JANOS
SIMON
( f o r i n p u t o f l e n g t h n ) can be expressed by a p r e d i c a t e .
i
zy
, i. e .
t h a t i s , t h e p r e d i c a t e contains a s i n g l e p o l y n o m i a l l y bounded
existential
q u a n t i f i e r over o b j e c t s o f type i, and a f i x e d f i n i t e number o f polynomiall y bounded q u a n t i f i e r s over o b j e c t s o f type a t most i - 1 , and t h e p r e d i c a t e
preceded by these q u a n t i f i e r s i s computable i n d e t e r m i n i s t i c polynomial time (assuming a l l membership predicates t o take u n i t t i m e ) . We s h a l l proceed by i n d u c t i o n . The case i = l has j u s t been proved. Our approach t o t h e general case w i l l be analogous t o t h e case o f e x ponential time: we s h a l l have an o b j e c t z o f type i, representinganaccepti n g computation o f t h e T m M ( t h a t recognizes t h e language w i t h i n t h e given bound) on i n p u t x . Again, we s h a l l want t o describe an I D as a t r i p l e ( t i m e , p o s i t i o n , square) and w r i t e down the predicates
A-C3(3) that w i l l
ensure
t h a t X indeed represents such a sequence. The problem i s t h a t t h e f i r s t two components now have 1enyth
a n d a l l s t r i n g s must have p o l y n o m i a l l y bounded length. The t r i c k we w i l l use i s t o use encodings f o r t h e numbers, j u s t as we used encodings f o r t h e I D ' S . I n t u i t i v e l y , t h e numbers " s t a y one l e v e l b e low" so t h a t t h e procedure may be c a r r i e d o u t i n d u c t i v e l y . More p r e c i s e l y , we s h a l l prove t h a t w i t h q u a n t i f i e r s o f type may o b t a i n o b j e c t s o f type
i - 1 we
i - 1 t h a t w i l l represent numbers. Also, t h e op-
e r a t i o n s o f forming p a i r s and t r i p l e s o f such o b j e c t s can
be d e s c r i b e d
w i t h i n t h e same formalism, and we may express t h e r e l a t i o n s
a=b a=O a = b+l
a i s t h e f i r s t (second) components o f a p a i r
w = (a,b), a
b, w representing numbers up t o - t i + l ( n ) , using a f i n i t e number o f
quan-
277
POLYNOMIALLY BOUNDED QUANTIFICATION
t i f i e r s of type a t most i - 1 , over polynomial time bounded predicates. Triples can be handled as the r e s u l t of two pairing operations. I t i s only a matter of going over our previous proof ( f o r t h e c a s e i = l )t o check t h a t only the predicates above were used t o write down formulae A - C3(3), so t h a t i f we prove the claim above we shall have proven the theorem. 16 A and €3 ate p o L y n o m i a y bounded objects (pbos) 0 6 t y p e i, x E r, thehe a t e pbos a6 t y p e i ( p e ~ b p w s i t h a kigheh bound) C and U such t h a t LEMMA 2 .
and
1 ) C 4ep4e~enL5 ( A , B ) , 2 ) I) 4ep4ezesenL5 ( A , x ) .
Mufieuuefi, t h e pfiedicate C = ( A , B ) , [D = ( A , x ) ) x = 6A.t I6econd)component 06 C ( U ) ate exp4ennibLe u i n g poLynamidLy bounded quantibiehd 0 6 t y p e s at most i - 1 (except doh i = O When an exintenLLd quanti6iehD needed 604 t h e Lat o n e ) . PROOF: By i n d u c t i o n o n i.
For i = 0, C = A $ B , D = A $ x ; w = A $ B i s deterministic computable, as i s the predicate (w,y) [ L O= x $ y ]
polynomia
time
For i > 0, A = {a,), 8 = { b 1 where a bk a r e objects of type i - 1. J k j' By induction c i k = ( n i , b i ) i s defined, as well as d . = (a.,x).Def ne C =
I, D Jk
{c.
- -
= {d.).Now C = ( A , B ) i f f W W E C , J
A = { aJ. 13bk [ b k E B
1
LO
=
(a. b.) J'J
1
a . E A , bk E 8 and
J
( a j , b j l E C]}.
This lemma j u s t i f i e s the use of t r i p l e s a t a l l l e v e l s . Now l e t us d e f i n e representation f o r numbers. For i = 0 we represent a number bywriting i t down. Since our numbers were bounded by 2' (remember the s i m p l i f y i n g assumption t h a t our polynomials were l i n e a r ) , t h i s could be done in p o l y nomial space. For i = 1, a number will be represented a s a s e t {(position,digit)), with the obvious meaning: p o s i t i o n will be a s t r i n g of polynomial l e n g t h . This enable us t o write - 2 ' positions ( i . e . our number will have length
- 2 ')
and represent values up t o 2 z n
.
I n g e n e r a l , we will assume as inductive hypothesis t h a t the r e p r e -
278
JANOS SIMON
s e n t a t i o n o f numbers up t o using an o b j e c t Ti (pij,digit)
t i + l ( n ) ( i . e. o f l e n g t h t i ( n ) ) c a n b e d o n e
of type i (i2 1) and t h e elements o f Ti a r e o f t h e form
where p . . r e p r e s e n t s *-J
a number ( t h e j - t h p o s i t i o n o f
the
number represented by T i ) and aXgLt i s t h e corresponding d i g i t . M o r e o v e r , we assume t h a t t h e s e t Ti may be defined using a s i n g l e e x i s t e n t i a l q u a n t i f i e r o f type i (and q u a n t i f i e r s o f smaller types). (S'ince we s h a l l use i n a formula which has a q u a n t i f i e r o f type i + l ,t h i s w i l l cause no
Ti prob-
Then we d e f i n e numbers o f l e n g t h ti+l( n ) as s e t s { ( T i , d i g i t )
lems.)
1
where Ti w i l l denote a p o s i t i o n . I n order t o g e t numbers we must a)
ensure t h a t a l l d i g i t s a r e defined;
b)
be a b l e t o deal w i t h these o b j e c t s as numbers 5 ti+,(n) i) ii) iii) iv) v)
t e s t f o r = 0, t e s t f o r = .ti+* (n) - 1 , t e s t f o r e q u a l i t y o f two "numbers", t e s t f o r successor-relationship, test for
_l/n,
290
ROBERT M. SOLOVAY
we have
A.
W(6,h) =
for all
h
2
6
ho,
then
Ci
E Y.
Y
e v e n t u a l l y , make one o f t h e Ai(h)'s w i t h
n
I f t h i s happens we Ci(h)
+
0, equal
w i l l c o n t r a c t , p o s s i b l y t o Y, as we add elements t o
2.2.
To s t u d y t h i s s i t u a t i o n , we f o r m u l a t e an i n f i n i t e
game.
must,
t o Ao.
But
Ai. two
person
A c o n f i g u r a t i o n i n t h i s game i s N - t u p l e o f f i n i t e l y based s e t s : < c 0,...,c ~ > -. The ~ i n i t i a l configuration i s < w 2 ?>.
,...,
P l a y e r I on h i s t u r n p l a y s a c o d e f o r a f i n i t e l y based s e t Y o f measure
.
>- I
If
n+1 Y, f o r some Ci
prevent
I
Y n Ci = with Y
0
for all
i < N, I w i n s .
n Ci # 0, and
from winning
I f not, I 1 r e p l a c e s Ci by
I 1 wins i f he can
t h e game c o n t i n u e s .
as d e s c r i b e d above
f o r the entire,
infinitely
l o n g game. The game d i f f e r s s l i g h t l y f r o m t h e i n t u i t i v e m o t i v a t i o n o f 2.1 i n t h a t
2 1 / n by
we have r e p l a c e d
1/n
at
h =
00,
>,&.
The reason i s t h a t a s e t which has measure
m i g h t have measure 1 > -). )I + 1
< 1/n f o r f i n i t e
(but w i l l
b
eventu-
a l l y have measure
THEOREM, ( M a r t i n ) L t l N = -2h . ( n + 1 ) . Then pPage4 11 h a a h e c u h h i v e w i n n i n g bLJmtegy ,504 t h e game ju&t d u d b e d .
2.3.
The p r o o f i s based on a p r o p e r t y o f c o n f i g u r a t i o n s , which we dub
p r o p e r t y M. I t i s e f f e c t i v e l y checkable whether a c o n f i g u r a t i o n erty
M.
has prop-
Moreover, t h e i n i t i a l c o n f i g u r a t i o n has p r o p e r t y M, a n d
c o n f i g u r a t i o n has p r o p e r t y M , t h e n f o r any move o f I ,
if
a
I 1 has a t l e a s t one
r e p l y t h a t r e s u l t s i n a c o n f i g u r a t i o n w i t h p r o p e r t y M. A r e c u r s i v e s t r a t e g y f o r I1 i s t o p i c k
Ci
minimal so t h a t
Ci n Y # 0 and t h e r e s u l t i n g con-
f i g u r a t i o n has p r o p e r t y M. (Another r e c u r s i v e w i n n i n g s t r a t e g y
i s easily
e x t r a c t e d f r o m o u r p r o o f t h a t p r o p e r t y M can be p r e s e r v e d . )
2.4.
We l e t T = { < i , j > :1
configuration.
A bijection
as a t r i a n g u l a r a r r a y :
Z.
J-,j
sj
< i < d.Let
l-
be
a
h : T -> N determines an arrangement o f t h e Y's
. = Yh ( i ,j )
.
( C f . Figure 1.)
291
ON RANDOM R . E. SETS
Zl,l
z2,i
z2,2
... ... ... ... ...
Figure 1 .
...,Y N > h a p h o p e h t y M i6 d o h some LhhnglLean_ m y h a the6oi%wingp~op-
DEFINITION, The con6igutdtion < Y 1 , b i j e d o n h: T ->N, ehtq: L e i 1 s i, 5 i, 5
2.5. LEMMA,
.the huu&ing
... 5 ik 5
j ; then
The following l e m m a i s now evident. 1 ) The i n i z i a l c o n 6 i g u h a t i o n <w2,...,
..,YN >
2 ) 1eL < Y1,.
have p h O p e h t y M .
@2> h a phopehty M .
Suppose YiS
Y l C- w2.
Then
ha4 phopehty M.
3 ) W e can t e l l ed~ectiucLyiuhetheh n n c q u e n c e 0 6 i n t e g e h s < x,, ,x N > -io a code a can6igiguhation with p m p e h t y M .
...
2.6. The following l e m m a will complete the proof o f Theorem 2.2. LEMMA, L e t
Y be a g i n i t e l y b a e d subsei
i 5 N such t h a t Yi Yi by Y O
fl Y
e a c o n 6 i g u h a t i o n w i t h phopehty M . L e t 06
+ 8,
m h a phopehty M .
w2, w i t h
p(Y) >
n+l . Then thehe
0
an
and t h e condigiguhation obtained by heplacing
292
ROBERT M. SOLOVAY
Yi's i n a t r i a n g u l a r a r r a y , Z . . , with the $9 J p r o p e r t y guaranteed by t h e Y ' s h a v i n g p r o p e r t y M. Then t h e u n i o n o f t h e Z ' s
PROOF: We a r r a n g e t h e
>
i n t h e b o t t o n row has measure tersection with Y. Let
n.
n+ 1
be minimal such t h a t some Z .
j
,
JIk
Permuting t h e
jth
Yh(j,j)
has nonempty i n -
n
Y
Z. ;
f 1J
# 0. We show
by Y t h e r e s u l t i n g c o n f i g u r a t i o n h a s p r o p e r t y M.
j > 1 , we p u t Zk,P. = Z k , &
I f j = 1, t h i s i s e v i d e n t . I f Z'.
n,j
has non-empty i n t e r s e c t i o n w i t h Y .
row, i f necessary, we may assume
t h a t i f we r e p l a c e
o r j - 1 . Put
Z
Whence, some
, unless k = j
= Z. , f o r P. 5 j - 1 , and Z ' . = Z . for!L<j-1. J,P. JIP. j-199" = Y. We show t h a t t h e t r i a n g u l a r a r r a y Z ' e s t a b l i s h e s
j-1,P-
Finally, put
Z'. J.J t h e new c o n f i g u r a t i o n has p r o p e r t y M.
that
I
What rows c o u l d cause t r o u b l e ? The o n l y rows t h a t do n o t a p p e a r i n t h e array are the
j - l s t and j t h . The
i s 0 . k . The o n l y way t h e
. We
let
Ai(b)
r.e. s e t s ,
be t h o s e elements p l a c e d i n t o Ai b e f o r e stage
6 . (We w i l l a r r a n g e m a t t e r s so t h a t A i ( b )
ZA.)
L e t Ci(n) =
I d : Wn(n E
(1)
A i ( ~ ) -> W i ( 6 , ~ ) ( n )= 1).
We w i l l a r r a n g e o u r c o n s t r u c t i o n s o t h a t ,
. ..
for all A ,
the
array
Lemma has p r o p e r t y M. 2.5 1 ) . I f i t h o l d s a t 6 , and we add no new elements t o any Ai a t stage b , i t w i l l hold a t
( T h i s i s t r u e a t 6 = 0, by
n + l by Lemma 2.5 2 ) . )
A t each s t e p b , we
f i r s t check
whether o r n o t A i s opportune f o r a d d i n g new elements t o the opportune, we w i l l add elements t o t h e Ails such t h a t
Ails.
If b i s
i n an a t t e m p t t o pursue those
p ( 6 ) 2 l / n . We use Lemma 2.7 i n d o i n g t h i s so t h a t
property
6
M
i s preserved.
3.2. We now g i v e t h e p r e c i s e c o n s t r u c t i o n . R e c a l l t h a t o u r c o n s t r u c t i o n
294
ROBERT M . SOLOVAY
e,
n , and k. Here k < n + 1 . L e t I k = ( a k , b k ) be 1 = - n ( n + l ) . To s t a r t t h i n g s o f f , l e t ' s agree t h a t 2 A i ( 0 ) = 0, i < n.
depends on t h e parameters
N
as i n S e c t i o n 1.9. L e t -1 i s opportune and t h a t
We now d e f i n e what i t means f o r a stage, 6 , t o be oppohtune. L e t
t be
t h e l a r g e s t number < 6 t h a t i s opportune. L e t m = (o),,II = ( A ) ~(where,
as
usual, ( x ) , i s t h e exponent o f p . i n t h e p r i m e power f a c t o r i z a t i o n . o f x,and 1
(0)i = 0 by c o n v e n t i o n ) . Then
Let hl,
2)
p(hi,6)
...,hh
be t h o s e b i n a r y sequences o f l e n g t h R s u c h t h a t
1 1 - -. n m
2
6 i s opportune i f
m
A i ( 6 ) E t + l < R.
i d : W(d,n)lR
2 . By
=
hi}.
By 1 ) o f t h e d e f i n i t i o n o f
. 2 ) ' o f t h e same d e f i n i t i o n , p ( Y i ) >
n+1
opportune,
- . Also n+l
the Yi's
a r e c l e a r l y p a i r w i s e a i s j o i n t . We now a p p l y Lemma 2.7. T h i s g i v e s d i s t i n c t
...,ih< N
i n t e g e r s il,
, such
that: 1)
Yj
n
C . ( 6 1 2 0 ; 2) i f we r e p l a c e
"j
each C . ( 6 ) by Y . and l e a v e t h e o t h e r Ci's unchanged, t h e r e s u l t i n g "j J f i g u r a t i o n , say < C & . ,Ci(r-l > , has p r o p e r t y M.
con-
..
Let
Bi
hi(q)= 1 ) . From 1 ) and t h e f a c t t h a t sup A i ( 6 ) < R,we Bi. We now s e t A i , ( 6 + 1 ) = Bi. F o r i n t e g e r s i < N n o t o f t h e
= (yew:
A i , ( o ) C_ J f o r m i . , we l e t
get
J
Ai(6
+
J 1 ) = A . ( o ) . Note t h a t we have C i 5 Ci(6 + 1 ) . S o prop-
"
e r t y M i s preserved. Note a l s o t h a t o u r p r o o f o f Lemma 2.7 g i v e s f e c t i v e procedure f o r d e t e r m i n i n g il,..., ihand we s h o u l d
an
follow
efthis
procedure so t h a t o u r c o n s t r u c t i o n w i l l be e f f e c t i v e . We have now completed s t e p 6 ; t h e c o n s t r u c t i o n now proceeds t o s t e p 6 + 1 . The d e s c r i p t i o n o f o u r c o n s t r u c t i o n i s now complete. Our n e x t t a s k
i s
295
ON RANDOM R. E. SETS
t o show t h a t if k i s c o r r e c t l y chosen ( c f . S e c t i o n l . l O ) ,
the
construction
succeeds.
4, THE 4.1.
CONSTRUCTION
We l e t e,n,N
WORKS,
be as i n t h e p r e c e d i n g
now assume t h a t , w i t h n o t a t i o n s as i n S e c t i o n 1.10, t i o n , f o r t h i s 12, y i e l d s a sequence o f s e t s t h a t p ( g ) 2 1/n. L e t
B
= {PI: g ( n ) =
paragraph, but
we
A E l k . Our c o n s t r u c -
Ao,...,AN-l.
L e t gEWZ
11. We must prove t h a t f o r
such
some
i,
B = A . . 1
Our p r o o f w i l l proceed as f o l l o w s :
( a ) We show t h a t i n f i n i t e l y
stages a r e opportune. T h i s w i l l f o l l o w f r o m t h e c o n t i n u i t y AElk.
S e c t i o n 1 . 5 and t h e f a c t t h a t
many
properties
of
( b ) The n e x t s t e p i s t o a n a l y z e what
happens a t a l a r g e opportune stage. Our method f o r d o i n g t h i s i s t o t a k e a n i n c r e a s i n g sequence o f opportune stages and c o n t i n u a l l y s i m p l i f y t h e s i t u a t i o n by p a s s i n g t o a subsequence. The f i n a l u p s h o t i s t h a t f o r a sequence
o f elements o f
gl,. ..,g, 1
P(9i) 2
2,
Ii=,P(gi)
3)
certain
we have:
y
1)
h
?!,
2 a& *
F o r any i n t e g e r m, t h e r e a r e i n f i n i t e l y many stages o a t which i s t h e c h a r a c t e r i s t i c f u n c t i o n o f some A , ( & )
gm li
J
I t i s c r u c i a l , o f course, t h a t 5 b e among t h e
A 2 P(g)+
I
h
i =1
P(gi) 2
+
1
' bk'
n m. But, i f n o t ,
5i's.
This contradicts A E T k .
argument now shows t h a t f o r some i < N,
( c ) An easy
A . = B.
4.1. We now commence p a r t ( a ) o f t h e p r o o f . Towards
a
contradiction,
assume t h a t t h e r e i s a l a r g e s t opportune stage, say t. ( R e c a l l
that
i s a t l e a s t one opportune stage, a t l e a s t f o r m a l l y , namely t = -1.)
Our
...,n4.
By t h e
2 ZM0 3n3 Sn4
i s op-
argument w i l l c o n s t r u c t a s e r i e s o f v e r y l a r g e i n t e g e r s , no, t i m e we a r e done, i t w i l l be c l e a r t h a t some i n t e g e r
there
p o r t u n e . T h i s w i l l complete p a r t ( a ) o f t h e p r o o f .
4.3. To s t a r t t h i n g s o f f , l e t g,l
...,g,
be t h o s e elements a t w2
such
296
ROBERT M. SOLOVAY
p(gi) 2
that
1 7 . Since
a t most n t l
f u n c t i o n s g have p ( g ) 2
1
1
, we n + l
can
1 ) no 2 t + 2 ;
.e p i c k no so l a r g e t h a t p(g) < 7 , p(g) < 7-EW 2) no > n 2 + n ; 3 ) 1 < E ; 4 ) t h e f u n c t i o n s gllnO, ...,ghl no
are d i s t i n c t ;
5)
f i n d an E > O so t h a t i f
0
LEMMA,
as
1 7 - E.
n l > n O hU that wheneveh h E m2, 1 h f gilm, t h e n P ( ~ ) < T- E
and
m 2 n 1’
p ( g i ) = l i m p ( g i l m). An i n s p e c t i o n o f t h e d e f i n i n+p ( c f . S e c t i o n 1.5) shows t h a t p ( g i l m ) i s monotone n o n i n c r e a s i n g
By S e c t i o n 1.5,
i i o n of i n m.
>
T h e h e 0 an
hln, = gilnl, PROOF:
p(gilnO)
Pick
n1
2
no so l a r g e t h a t p ( g i l n l )
i n t h e statement o f t h e
Id:
W(6)1n1 = g i l n l
lemma, and W ( d )
1
-
p(gi) < M _ - E . Then i f h i s
p ( h ) i s t h e measure o f a subset o f
.
# gil
,By o u r c h o i c e o f n1 t h i s l a t t e r s e t has measure
p ( g l m ) , and
.
by 1 )
Then f o r i s u f f i -
hilm = glm.
ON RANDOM R .
29 7
E. SETS
Contradiction! 4.5. LEMMA, Thehe 0 an
then h = gilm
60h
n2 2 n1
i5
name
that i d
6uch
m 2 n2, and p ( h l m ) 2
1
M - E ~
h.
PROOF: D e n y t h e l e m m a . L e t be a sequence o f f i n i t e b i n a r y sequences whose l e n g t h t e n d t o m such t h a t i f mi = length ( h i ) , # 1 Apply Lemma 4.4. F o r some g , a f t e r g j l m i , any j 5 h , and p(hj) 2 ;-E. 1 and p a s s i n g t o a s u b s e q u e n c e , we h a v e p ( g ) 2 T - E , (2) Thus
liin hi(") = g ( m ) .
i +m
large, tradicts
1
Lemma 4.3,
(since p(g)>,-E).
( C f . S e c t i o n 4.3.)
hi #
by ( 2 ) . B u t by assumption
h . ] n = gilnl, J
I
some i 5 h
g =gi,
1
p(h.) 2 J
since
9.1". I J
If
. This
j
i s con-
;-E
4.6. By o u r c h o i c e o f k ,
Let
n3 > n2
11.1 Next s e l e c t (3)
h
p(h)
