LOG-ICAL SYSTEMS
WITH
FINITELY
I.ÎANY TRUTH VALUES
G. ROUSSEAU
T h e s is su b m itte d f o r th e
d e g re e o f D octor o f P h ilo so p h y
a t th e U n iv e r s ity o f L e ic e s te r,
1967
UMI Number: U296357
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'^
3 (.)
7
) /
To th e memory o f
FEUX
BEHEEND
Acknowledgement
The a u th o r w ish es to e x p re ss h i s w arm est th a n k s to
P r o fe s s o r R .L .G o o d stein who supervised, th e p re p a r a tio n
o f t h i s t h e s i s 4,
D uring th e co u rse o f th e r e s e a r c h .
P r o fe s s o r G oodstein h as been a c o n tin u in g so u rc e o f
encouragem ent, and has a s s i s t e d th e a u th o r in many ways
b o th m a th em atical and n o n -m a th e m a tic a l;
he has g iv e n
u n s t i n t i n g l y o f h i s tim e and a d v ic e th ro u g h o u t#
th e s e th in g s th e a u th o r i s d e ep ly g r a t e f u l .
iii
For
C ontents
Acknowledgement
iii
I n tr o d u c tio n
C h ap ter I .
1
Sequents i n many v a lu e d lo g ic *
1.
C alcu lu s o f se q u e n ts f o r c l a s s i c a l n—v a lu e d p r e p o s i t i o n a l c a lc u lu s
2.
C alcu lu s o f se q u e n ts f o r i n t u i t i o n i s t i c n -v a lu e d p r e p o s i t i o n a l c a lc u lu s .
C h ap ter I I .
A lg e b ra ic methods#
5
8
19
33
1.
P o s t a lg e b r a s and p se u d o -P o st a lg e b r a s
34
2.
A p p lic a tio n s to p r e p o s itio n a l c a l c u l i
45
C h ap ter I I I # S h e ffe r fu n c tio n s #
55
1#
S h e ffe r f u n c tio n s in c l a s s i c a l n—v a lu ed lo g ic
57
2#
S h e ffe r fu n c tio n s i n i n t u i t i o n i s t i c lo g ic
64
B ib lio g ra p h y
71
I n tr o d u c tio n
I n c l a s s i c a l and i n t u i t i o n i s t i c l o g i c , th e n o tio n o f v a l i d i t y c an be
d e fin e d s e n a n t i c a l l y
by th e methods of T a rs k i and K ripke r e s p e c t i v e l y .
I f we re p la c e the two tr u th - v a lu e s o c c u rrin g i n th e s e d e f i n i t i o n s by a
s y s ta n o f
n
t r u t h —v a lu e s we o b ta in v h a t may be r e f e r r e d to a s c l a s s i c a l
n -v a lu e d l o ^ c an d i n t u i t i o n i s t i c n -v a lu e d lo g ic r e s p e c t i v e l y .
has b een s tu d ie d by many v ;r ite r s b eg in n in g w ith P o s t [ l l ] ;
n o t seem to have been c o n sid e re d p r e v io u s ly .
The form er
th e l a t t e r does
T h is th e s i s w i l l be concerned
w ith c l a s s i c a l and i n t u i t i o n i s t i c n -v a lu e d p r e p o s it io n a l c a l c u l u s .
I t is
p o s s ib le t o c o n s id e r p re d ic a te c a l c u l i by th e same methods ( o f R ousseau [ I 9 ] ) ,
b u t th e b a s ic p r i n c i p l e s a r e i n e v id e n c e a lre a d y in ihe p r e p o s iti o n a l c a s e .
Gents en [ 4 ] in tro d u c e d two system s
LK
and
L J,
fo r c l a s s i c a l and
i n t u i t i o n i s t i c lo g i c r e s p e c tiv e ly , b ased on the n o tio n o f s e q u e n t.
C h ap ter 1 w i l l be concerned w ith t h e many—v a lu ed an alo g u es o f th e s e c a l c u l i .
We shoviT t h a t f o r each choice o f n -v a lu e d t r u t h —fu n c tio n s th e re e x i s t
c o rre sp o n d in g s e q u e n t c a l c u l i
LKn
and
LJn
tb r the c l a s s i c a l and th e
i n t u i t i o n i s t i c n -v a lu e d p r o p o s itio n a l c a lc u lu s r e s p e c t i v e l y .
The method
o f seq u e n ts seems to have c e r t a i n ad v an tag es f o r t h e stu d y of m any-valued
lo g ic over th e t r a d i t i o n a l methods as developed say by R o sser an d Tui*quette [ I 6 ] :
(i)
as th e above r e s u l t in d ic a te s^ th e method i s co m p letely in d e p en d e n t o f t h e
c h o ice o f p r im itiv e tr u t h - f u n c tio n s ; ( i i )
no a r b i- tr a r y c h o ice o f d e sig n a te d
t r u th - v a lu e s i s n e c e s s a ry s in c e d e s ig n a te d v a lu e s p la y no p a r t ,
On th e o th e r
hand c e r t a i n r e s u l t s on a x io m a tiz a b ility o f m any-valued p r e p o s i t i o n a l c a l c u l i
can be d e riv e d i n th e p r e s e n t c o n te x t .
th e c a l c u l i
LICn an d
G entzen*s c a l c u l i
LJ^
LIC and
The c h ie f m o tiv a tio n f o r s tu d y in g
however i s "fco o b ta in a b e t t e r u n d e rs ta n d in g of
L J,
and th e r e l a t i o n betw een them .
—1 —
I n C hapter I I we s tu d y c l a s s i c a l and i n t u i t i o n i s t i c n -v a lu e d p r e p o s i tio n a l
c a l c u l i from th e a lg e b r a ic s ta n d p o in t.
As is w ell-know n, th e r e p r e s e n ta tio n
th e o ry o f B oolean and pseudo-B oolean a lg e b ra s can be a p p lie d to e s t a b l is h th e
com pleteness o f a x io m a tiz a tio n s o f c l a s s i c a l and i n t u i t i o n i s t i c
p r o p o s itio n a l c a l c u l i .
The same approach can be used to o b ta in
a x io m a tiz a tio n s of c l a s s i c a l and i n t u i t i o n i s t i c n -v a lu e d p r o p o s itio n a l
c a l c u l i ( f o r s u ita b le ch o ice o f p r im itiv e tr u t h - f u n c tio n s and d e sig n a te d
v a lu e ) .
The a p p r o p r ia te a lg e b r a ic system s h e re a re th e P o s t a lg e b ra s
and th e p seu d o -P o st a l g e b r a s .
The fo rm er have b een s tu d ie s by Rosenbloom [ I 5 ] ,
E p s te in [3 ],T ra c z y k [2 9 , 30]5
th e l a t t e r have n o t been c o n sid e re d p r e v io u s ly ,
A stu d y o f th e r e p r e s e n ta tio n th e o r y o f th e s e a lg e b r a s p ro v id e s th e to o ls
n e c e s s a ry to prove th e com pleteness of s u i t a b l e a x io m a tiz a tio n s o f c l a s s i c a l
and i n t u i t i o n i s t i c n -v a lu e d p r o p o s itio n a l c a l c u l i .
I n C hapter I I I we stu d y an a s p e c t of th e p ro b la n o f com pleteness f o r
system s o f tru th -fu n c tL o n s
i n c l a s s i c a l and i n t u i t i o n i s t i c n -v a lu e d l o g i c .
We g iv e a sim ple a lg e b r a ic c h a r a c te r iz a t i o n o f th e S h e ff e r fu n c tio n s o f
c l a s s i c a l n -v a lu e d lo g ic ( c f , Rousseau [ 2 0 ] ) , w hich g e n e r a liz e s a r e s u l t o f
P o s t [ 12 ] f o r th e case
n = 2
and a r e s u l t of W heeler [ 32 ] f o r th e case
n = 3 • Also we o b ta in some p a r t i a l r e s u l t s on S h e ffe r f u n c tio n s in
i n t u i t i o n i s t i c two—v a lu e d l o g i c .
We BOW in tro d u c e th e b a s ic s y n t a c t i c a l and s e m a n tic a l
u sed th ro u g h o u t ih e w o rk ,
n o tio n s to be
F o r the c l a s s i c a l n -v a lu e d p r o p o s itio n a l c a lc u lu s
th e s e n o tio n s o c cu r i n esse n ce a lr e a d y in P o st [ l l ] , w h ile th e s e m a n tic a l
n o tio n s f o r th e i n t u i t i o n i s t i c n -v a lu e d
-
2
-
p r o p o s itio n a l c a lc u lu s a r e
o b ta in e d by g e n e r a liz in g the d e f i n i t i o n s o f K ripke [7] f o r the
case
n = 2,
The p r o p o s itio n a l c a lc u lu s ,
i n th e fo llo w in g m anner.
A p r o p o s itio n a l c a lc u lu s i s d eterm in ed
We s e l e c t a s e t
E = [ e o , © i,
!
of
tr u t h - v a lu e s w hich i s to renm in f ix e d th ro u g h o u t, and a s e t o f t r u t h fu n c tio n s
E : E^
argum ents o f
E,
-> E ,
The i n t e g e r
r(E ) i s th e number o f
and w i l l be d en o ted by
In a d d itio n we s e l e c t a s e t
each t r u t h - f unet i 0n
E
r
when
no a m b ig u ity i s p o s s i b l e .
V o f p r o p o s itio n a l v a r ia b le s and f o r
a co rre sp o n d in g c o n n e c tiv e , a ls o denoted t y
P.
The form ulas a re th e e lem en ts o f th e l e a s t s e t o f e x p re s s io n s
w hich c o n ta in s
V and v ;h ich , f o r each c o n n e c tiv e
P ( a i, . . . , ap)
w henever i t c o n ta in s
c ti,
E , c o n ta in s
« p . The degree o f a
form ula i s th e number o f o c c u rre n c e s o f c o n n e c tiv e s
The s e t of fo rm u las w i l l be denoted by
S
w i l l be denoted by th e l e t t e r s
...
C la s s ic a l S e m a n tic s .
V : V
in i t .
and in d iv id u a l fo rm u las
.
'
By a ( c l a s s i c a l ) v a lu a tio n we mean a map
S which a s s ig n s to each p r o p o s itio n a l v a r ia b le a t r u t h —v a lu e .
The v a lu e v * (a )
o f the form ula
a (w ith r e s p e c t to
d eterm ined by th e re q u ire m e n t t h a t
t h a t f o r each c o n n e c tiv e
v*
a g re e s w ith
v
i s u n iq u e ly
fo r
and
a € Y
E we have
v » (E (% i, . . . , %p)) = E (v * (a i) ,
f o r a r b i t r a r y fo rm u las
v)
a x , . . . , ccp.
- 3 -
v»( op) )
In o th e r w o rd s, i f
considered as" a lg e b r a s w ith o p e ra tio n s
homomorphism w hich extends th e mapping
.
E,
th e n
v : V -» E .
v* : S
S
and
E
E
is a
a re
I n t u i t i o n i s t i c sem a n tic s .
where
A
o f mappings
V
a
, (a )
monotone
w henever À ^
monotone fa m ily [v ^ j^ e
a t th e
E,
a c V
v ^ (a ) ^
a
^ ^
i s an a r b i t r a r y p a r t i a l l y o rd e re d s e t , w i l l be c a l l e d
i f f o r each
Given
A fa m ily
p o in t
Ike v a lu e v ^ * (a )
A f A(w ith r e s p e c t to
d eterm ined by t h e re q u ire m e n t t h a t
[v ^ j^
ç A^
a g re es w ith
o f a fo rm u la
^
v^
u n iq u e ly
fo r
a c 7
and
t h a t f o r each c o n n e c tiv e we have
...,a
f o r a r b i t r a r y form ulas
fo rm ulas
a i,
• , , , «p .
)) = i n f
P(v * ( « i ) ,
E v id e n tly we have fo r a l l
a
Y^**‘(a ) ^
■^f*(a)
7/henever À ^ A' .
—4 -
• • • j V *(ap))
C hapter I : Sequents i n Many V alued Logic
I n t h i s C hapter we c o n s tr u c t se q u e n t c a l c u l i f o r th e c l a s s i c a l and
i n t u i t i o n i s t i c n -v a lu e d p r o p o s itio n a l c a l c u l u s .
S e c tio n 1 d e a ls w ith th e c l a s s i c a l p r o p o s itio n a l c a lc u lu s ,
The seq u e n ts o f ihe fo rm
ro*=> P i
w hich ap p ear
a re re p la c e d by seq u e n ts o f ihe form
Po, P i,
.
Pn- i
Pol P i |
C entaen^s work [ 4 ]
in
|P n -i>
where
o-re f i n i t e s e t s o f fo rm u la s . We f i r s t pro v e a lemma f o r
n -v a lu e d t r u thr-funcuLons r a t h e r analogous to th e c o n ju n c tiv e norm al form
f o r two—v a lu e d tr u t h - f u n c tio n s .
T his e n a b le s u s to fo im u la te in tr o d u c tio n
r u le s in th e manner o f G en tzen ,
C entzen has two
each c o n n e c tiv e , one on th e l e f t o f a seq u e n t
rig h t;
s i m i l a r l y we have
n
in tr o d u c tio n r u le s f o r
Po=> P i
and one on ih e
in tr o d u c tio n r u l e s f o r each c o n n e c tiv e ,
one f o r e a c h p la c e cf th e se q u e n t
P o |P i |
• • • |Pn-i • Each
in tr o d u c tio n
r u l e i s r e v e r s i b l e i n th e se n s e t h a t th e c o n c lu s io n i s v a l i d i f and only
i f th e p rem isses a re v a l i d .
p ro p e rty
The com p leten ess o f th e r u l e s fo llo w s from i h i s
by a sim ple in d u c tiv e argum en t.
v a lu e s we can d e fin e v a l i d i t y f o r fo rm u la s ;
I f we in tro d u c e d e s ig n a te d
th e co m p leten ess of the
c a lc u lu s o f sequents im p lie s th e com p leten ess o f s u it a b le a x io m a tiz a tio n s
o f th e s e t of v a lid fo r m u la s .
W ith th e e x c e p tio n o f Theorem 6 , a l l the
r e s u l t s o f t h i s s e c tio n a p p e a r in
[ l9 ] •
I n S e c tio n 2 we c o n sid e r th e i n t u i t i o n i s t i c p r o p o s itio n a l c a lc u lu s .
The n o tio n o f seq u e n t has to be somewhat r e s t r i c t e d , so t h a t
Pol P i I
o ccu rs,
w ith
jPn-i
in a l l
i > m.
i s a sequent only i f f o r each a
Pi
w ith
i
m or
a
in
Tm
o c cu rs i n a l l
P&
e ith e r
This r e s t r i c t i o n does not a l t e r th e c la s s o f sequents
a
i n th e tv;o—v a lu e d c a s e .
as
i n th e c l a s s i c a l case we prove a lemma
w hich e n ab le s us to fo rm u la te in tr o d u c tio n v a lu e s f o r each c o n n e c tiv e ;
th e r e a r e
2 (n - 1)
such r u l e s f o r each c o n n e c tiv e i n t h i s c a s e .
It
i s e a sy to show th a t th e r u le s p re s e rv e v a l i d i t y , v h il e on th e o th e r
an
hand a system atic a tte m p t to f in d a p ro o f f o r im p ro v a b le se q u e n t le a d s
in
to a monotone fa m ily which e s ta b lis h e s i t s ^ a l i d i t y . As in th e
c l a s s i c a l case we can d e fin e v a l i d i t y o f fo rm u las r e l a t i v e t o a s e t o f
d e s ig n a te d v a lu e s and i n f e r th e com pleteness of s u i t a b l e
a x io m a tiz a tio n s from the co m p le ten e ss o f th e s e q u e n t c a lc u lu s .
We
ap p ly t h i s metliod in th e two—v a lu e d case to a x io m a tize any fra g m e n t o f
th e i n t u i t i o n i s t i c p r o p o s itio n a l c a lc u lu s w hich in c lu d e s im p l ic a ti o n .
Our se q u e n t c a l c u l i a re proved com plete e s s e n t i a l l y by th e
fo llo w in g method:
we a tte m p t i n a s y s te m a tic way to c o n s t r u c t a p ro o f
f o r a giv en se q u e n t II ;
e i t h e r th e a tte m p t su cceed s an d II
i s p ro v a b le ^
o r i t f a i l s , and th e r e s u l t i n g s tr u c tu r e can be t r a n s l a t e d , in to an
i n t e r p r e t a t i o n , in vdiich II
is not s a tis fie d .
In the tw o -v alu ed case
t h i s method has b een e x p lo ite d by many a u th o rs , in c lu d in g
B eth , R anger,K rip k e , Rasiowa and S ik o rsk L ,
S c h u tte ,
E or r e f e r e n c e s , s e e Rasiowa .
and S ik o r s k ix [13 ] and K ripke [ 7 ] •
I t w i l l be c o n v en ie n t i n t h i s C hapter t o i d e n t i f y th e tr u t h - v a l u e s
w ith n a tu r a l numbers:
eo = 0 ,
Gi = 1 , . . . , Gn_i = n-1 .
F i n i t e s e ts o f s ta te m e n ts w i l l b e denoted by th e l e t t e r s
—
6
—
P, A, ...
,
I t i s o f te n co n v en ien t to w r i t e
re s p e c tiv e ly .
An i n t e r v a l of
[m : mi ^ m
nig}
where
a
E
mi , m^
and
m fo r
[a j
and
[mj
i s any s u b s e t of th e form
a re elem en ts o f
E;
in p a r t i c u l a r
th e empiy s e t i s an i n t e r v a l .
The p o s s i b i l i t y o f c o n s tr u c tin g seq u e n t c a l c u l i f o r c l a s s i c a l and
i n t u i t i o n i s t i c n—v a lu e d p r o p o s itio n a l c a lc u lu s r e s t s u l tim a t e ly on th e
fo llo T fing sim ple p r o p o s itio n ;
*L e t
E
be a tr u t h - f u n c t i o n o f
su b set o f
—
■
E.
Then f o r a s u ita b le f i n i t e fa m ily of s e t s
" ' — -----------------------------------------
(i € I; j = 1,
R be any
R Î'. C E
J'—
r ) we have
F(xi,
i f d c p ire d th e s e ts
of
r = r( F ) argum ents and l e t
,.
Xp) e
R^
R <=>
A [xi e E.|'
ic I
V . . . V Xp c Rp ] ;
can be ta k e n to be complements o f i n t e r v a l s
E.
P ro o f.
L et
I
c o n s is t of a l l
F ( k i , . . . , kp) / R,
r-tu p le s
and f o r each such
- 7 -
1 = (k i,
i
le t
R^
. . . , kp) e E*^
such t h a t
= E - {kjj ( j = 1 , .
r)
1.
C alcu lu s o f seq u e n ts f o r c l a s s i c a l
n -v a lu e d
p r o p o s itio n a l c a lc u lu s
A seq u e n t i s an e x p re s s io n o f the form
Po (Pi I
where
•••
Po , Pi , .
I Pn-2 I Pn-i
Pn- i
,
a r e f i n i t e s e t s o f fo rm u la s ,
Sequents w i l l be denoted by th e l e t t e r s
se q u e n t
n
(1 ),
and
Z
p la c e i s
E
th e n t h e
II
Q.
If
i s th e s e t
II
i s th e
II[m] = Pm • I f
a re se q u e n ts th e n we d en o te by II Z th e se q u e n t whosô mth
II[m] u Z[m]
th e n we denote by
mf R
mth p la c e of
11,2,
and
(m » 0 , 1,
| P | t h e
n u l l o th e r w is e .
n - l).
seq u en t whose
I f E.
is a su b set of
mth p la c e i s
P
if
A s e q u e n t i s s a i d to be elem en ta ry i f each
fo rm u la o c c u rrin g i n i t i s a p r o p o s itio n a l v a r i a b l e .
The v a lu a tio n
( 2)
v
m € V (Pm)
i s s a i d to s a t i s f y th e seq u e n t
f o r some
(1 )
if
mc E,
The s e t cf v a lu a tio n s w hich s a t i s f y
II
i s denoted by
s01) •
C le a rly we have
(3)
V € s (|a |g ^ ) <=> v - ( a ) c R
(4 )
sCHi
..« I l k ) = 0 (Hi) U .
U odk)
We say t h i t ih e seq u en t
II
ev ery v a l u a t i o n .
M o f se q u e n ts is s a id to be sim u lta n e o u s ly
A set
i s v a lid
(Val II)
#
i f i t i s s a t i s f i e d by
s a ti s fia b le i f
n
n cM
The p r o p o s itio n
fl (lI)
.
.(^) r e f e r r e d to above has th e fo llo w in g more
p r e c is e v e r s io n , a p p lic a b le i n th e c l a s s i c a l c a s e .
- 8 -
LEÎ.ÜÎA. 1 .
of r
F ^ arguments and any t r u t h —v a lu e
F o r any tr u t h - f u n o tlo n
th e r e e x i s t s a fa m ily o f s e t s
( 5)
C E (i e I ;
j = 1,
r)
m,
su ch t h a t
F (x x , .
Xp) = m<?=> A [ XieE.*' v . . . v Xp c r/: ] ,
i€ l
^ =
We oan suppose the number of co n ju n o ts Ï to be a t m ost n^"^ ,
and t h i s bound i s b e s t p o s s i b l e .
P ro o f.
Any s u b s e t
S
of
E*'
can be e x p re sse d as th e uniom o f
c a r t e s i a n p ro d u c ts o f su b se ts o f
E.
In f a c t , i f
r >1
th e n
nT“ ^
S
is
th e u n io n o f a l l th e p ro d u c ts
[m ii X . . . x[nV '-ii X {m :(m i, . . . , nh-i , m) f Sj
such t l i a t (m i, . . . , n h - i) ^
, The f i r s t p a r t of ihe Lemma now
fo llo w s by a n a p p lic a tio n of t h i s rem ark to th e s e t
= E ^ - F“^ (m ),
S = [ (m%, . . .,mp) : m% + . . . + mp 3 O(mod n) ]
On th e o th e r hand th e s e t
h as
S
n! ^ elem en ts b u t in c lu d e s no c a r t e s i a n p ro d u c t w ith
one e lem en t;
L et
from t h i s we e a s i l y deduce i h a t th e bound i s
F
be any t r u t h - f u n c tio n s of
be any t r u t h - v a l u e .
i c i
le t
If
(%%, . . . , ccp
n^((% 1 , . . . , ttp)
more th a n
b e s t p o s s ib le
r = r(F ) argum ents and l e t
a r e any fo rm u la s , th e n f o r each
b® th e seq u en t
By ( 3 ) , (4 ) and ( 5 ) we o b ta in
(6 )
b (|
F(ai , .. . , a p ) Im) =
u s in g th e f a c t t h a t
If
n
v"^; 8
E
A s 0I l («i , . . . , a p ) ) ,
id
i s a homomorphism.
i s an a r b i t r a r y seq u en t th e n we have ^
a (n IF (a 1 , • •
&r)lm ) ~ A
id
- 9 ~
m
(4 )
and
(6)
s (ÜL (i%i , # . , ,ap ) ) ,
z'
whence i t fo llo w s th% t
(7)
V al n |F ( a i , . . . , a p ) | m <==^ A V al II H t (a 1 , , . . , a p ) .
id
There i s a r e l a t i o n of ihe form ( 7 ) fo r each cormeo tiv o o f th e
p r o p o s itio n a l c a lc u lu s and each tr u t h - v a l u e
c a lc u lu s
LK,
m.
we a re le d to c o n s id e r f o r e ac h
By a n alo g y w ith G entzen’ s
F
and
m an
in tr o d u c tio n r u le o f ihe form :
(P^m) :
We say t h a t
s e q u e n ts
( i f I)
II IF (a 1 ^ ***f ^ r ) |in
Q i a an im m ediate consequence by r u l e (F,m) o f th e
(i c l )
ùi
n
has th e form Il|F (a 1, .
if
Q has th e form I I I I i(iz 1 ,
each
se q u e n t o f th e form
( m = 0 , 1,
n-1 )
(I)
«p)
ap)| n,
( i c i ) . An
w h ile
elem en ta ry
is c a ll e d a fundam ental se q u e n t i f ihe
have a t l e a s t one f o r mil a i n common.
Pm
The s e t o f
p ro v a b le seq u e n ts i s th e l e a s t s e t w hich c o n ta in s a l l fundam ental
se q u e n ts and which c o n ta in s a seq u e n t
seq u e n ts
ru le
Oj, ( i
f
l)
of w hich
Q
Q w henever i t
i s an
c o n ta in s
im m ediate consequena® by some
(F ,m) ,
THEOREM 2 ,
P r o o f,
A seq u e n t i s v a l i d i f and o n ly i f i t i s p r o v a b le .
Each fundam ental s e q u e n t i s v a l i d ;
consequence o f v a lid se q u e n ts i s v a l i d ;
by
( 7)
an im m ediate
c o n se q u e n tly e v e ry p ro v ab le
se q u e n t i s v a l i d .
For each seq u e n t
o c c u rrin g i n
Q
Q, l e t
and l e t
o f form ulas o f degree
m Q be th e maximum d eg ree o f fo rm u las
n Q be th e number o f o c c u rre n c e s in
m Q.
L et
Q be a v a lid s e q u e n t.
—10 —
If
Q
nfi = 0
th e n
Q
i s elem en ta ry ;
fu n d am ental s e q u e n ts ;
hence
t h a t a l l v a l i d seq u en ts
(i)
or
S in ce
b u t a l l v a l i d elem en tary se q u e n ts a re
Z
Ü
i s p r o v a b le .
If
m Q > 0,
we suppose
a re p ro v a b le f o r w hich
mZ < mn ,
(ii)
m Z = mQ and
m fi > 0 ,
n Z < n Û•
Ù may be e x p re ss e d in the form
n ( F (a 1 , •«*, ^r)lni
where
F (a i ,
n o t o ccu r i n
a re v a l i d ;
ap )
n[m ] .
i s o f degree
(7)
By
mQ
and
th e seq u en ts
F (cci , . . . /%p )
II lit (ai
b u t fo r each of th e s e seq u e n ts c i t h e r
so by in d u c tiv e h y p o th e sis each is p ro v a b le ;
b ein g an im m ediate consequence by rul@
ap )
(i)
hence
does
or
( i f l)
(ii)
h o ld s ,
Q i s p ro v a b le ,
(F,m) o f p ro v a b le s e q u e n ts .
Thus e v e ry v a lid seq u e n t i s p ro v a b le , and t h i s com pletes the p r o o f .
As a c o r o lla r y o f Theorem 2 v/e have th e fo llo w in g an alo g u e of
G entzen’s H au p tsatz
If
n | a 1^
and
s e q u e n t, p ro v id e d
f o r h is sy ste m
| a | gZ
R
LKî
a re p ro v ab le s e q u e n ts ,
and
th e n
HZ
i s a p ro v a b le
S a re d is jo in t.
T his could a l s o be e s ta b lis h e d p r o o f - t h e o r e t i c a l l y in the manner o f G entzen [4 ] .
A lso a t t h i s sta g e we no te t h a t the com pactness theorem h o ld s in th e
fo llo w in g form ;
THEOREM 3 •
A s e t of se q u e n ts i s sim u lta n e o u s ly s a t i s f i a b l e i f an d o n ly i f
ev ery f i n i t e s u b s e t i s s im u lta n e o u s ly s a t i s f i a b l e .
—1-1 —
P ro o f.
I t i s easy to check t h a t th e s e t s o f the form
b a se fo r the c lo s e d s e t s in ihe space
th e a s s e r t i o n t h a t
b QI)
form a
so th e Theorem am ounts sim ply to
Y
E i s com pact.
To d eterm ine th e in tr o d u c tio n r u l e s fb r a g iv e n co n n ectiv e
F,
we have on ly to o b ta in a s u i t a b l e r e l a t i o n o f the form ( 5 ) f o r
th e c o rre sp o n d in g t r u t h - f u n c tio n .
T his may be done i n a v a r i e t y o f w ays,
and i n p r a c tic e i t is n a t u r a l to choose ihe s im p le s t o r most e l e g a n t .
I n view of Lemma 1
,
th e number o f p rem isses can aLv/ays be l im it e d to
By way o f example v/e c o n s id e r th e th r e e - v a lu e d .Lukasiew icz
c o n n e c tiv e s
C,K,A,N
whose t r u t h - t a b l e s a r e t h e fo llo w in g ;
c
G 1
2
K 0
1
2
A 0
1
2
0
2
2
2
0
0
0
0
0
0
1
2
1
1
2
2
1
0
1
1
1
1 1 2
2
0
1
2
2
0
1
2
2
2
2
N
2
co rresp o n d in g in tr o d u c tio n r u l e s , w r i t t e n i n
TIl (a 1 ,
. .,
%r)
( i ^ I)
I F(ctl , • • • yCCp ) 11
a re as fo llo w s ;
I
|oc
a\a\^
^1
\Cal3\
|g |g |g , ^ |
Kct^
cc\^\^
I
\(3\^
Iw l
-1 2 -
I
k-
|C a ^
IJA I Iw
^1
A
m
a |a |
I
|a , ^ |
|
|%,P
I
I
IA a^l
I
\AoLj3
I
I
I Net I
I
| m
L e t u s s in g le o u t a s u b s e t
D o f d e s ig n a te d t r u t h —v a l u e s .
I t is
th e n p o s s ib le to d e fin e th e v a rio u s s y n t a c t i c a l and s e m a n tic a l notiouA
f o r fo r m u la s .
if
Thus a v a lu a tio n
V'-'(a) c D;
we d en o te by
v
s(a )
i s s a i d to s a t i s f y th e fo rm u la
a
th e s e t o f v a lu a tio n s w hich s a t i s f y
a.
F or th e se q u e n t
I ^ I e -D
we s h a l l w r ite
r ||A ;
C s (y ) =
(8)
A fo rm u la
D
c l e a r l y we have
( ||);
s y
s(s) = s ( | | s ) .
a i s s a id to be v a l i d i f e v e ry v a lu a tio n
i s a n e le m e n t of
s(a );
a
i s s a id to be p ro v a b le i f ihe
seq u e n t | ja i s p ro v a b le .
a set
^
o f form ulas i s s im u lta n e o u s ly
s a tis fia b le i f A
S im ila r ly
s ( a ) ^ p' and
CL €
c o n s is te n t i f f o r no
F C jC i s
r| |
a p ro v a b le s e q u e n t.
The s y n t a c t i c a l and sem antical n o tio n s •sdiich we have d e fin e d a r e
r e l a t e d by th e fo llo w in g th e o re m ,
THEOREM 4 .
A fo rm u la i s v a l i d i f and o n ly i f i t i s p ro v a b le ;
a se t of
fo rm u las i s s im u lta n e o u s ly s a t i s f i a b l e i f and o n ly i f i t is c o n s i s t e n t .
P ro o f,
Theorems
2 and 3 a re e a s i l y s e e n to im ply a more g e n e ra l
p r o p o s itio n , namely t h a t the fo llo w in g two c o n d itio n s a r e e q u iv a le n t:
— 13 “•
(a)
n
Y
s (y )
e u
8(8) ,
S €D
(b)
r
II A
i s p ro v a b le f o r some F e
(4)
The p ro o f u se s
and
AC ^ .
( 8) .
For d ie rem ain d er of i h i s s e c tio n we s h a l l re g a rd the symbols
Fo > .
Fn- i
a p p e a rin g in th e s e q u e n t
o f fo rm u la s r a t h e r th a n s e t s .
(1 )
as d e n o tin g sequences
I t i s easy to check t h a t th e p re c ed in g
r e s u l t s rem ain tr u e i f we add r u l e s p e r m ittin g P e rm u ta tio n , Weakening
and C o n tr a c tio n .
We now show how to c h a r a c te r iz e th e s e t of v a l i d fo rm u las by means o f
axioms
and r u le s o f in f e r e n c e , i n th e manner
Thus v/e suppose t h a t c e r t a i n c o n n e c tiv e s
o f R osser and T u rq u e tte [16] ,
D , Jo ;
J n -i
d e fin a b le i n terras of ihe p r im itiv e c o n n e c tiv e s .
form ula and l e t
II
be th e seq u e n t
sequences o f fo rm u las) .
form ula
ai n
We form a new
Pm th e form ula
seq u e n t has th e form ulas
(1 )
(where nov/ th e
y
Pm
be any
denote
se q u e n t by s u b s t i t u t i n g f o r each
(Jm a D y )
a
L et
a re
(m e
E) .
in t h a t o r d e r ,
I f th e r e s u l t i n g
th e n
II'*' Y
i s d e fin e d as th e form ula
. . . 7 . gk F) Y
in p a rtic u la r i f
F o rg iv e n
F
and
II
i s th e n u l l seq u en t th e n
m le t
Hi ( a i ,
an enum eration of the s e q u e n ts
•
II*--y
i s the f o r m l a
. . . , % r), . . . , II,^ (a i, *. .,g r )
y
-
be
Hi,(cci , . , . ,a p ) ( i e l ) o c c u rrin g i n th e
p re m isse s of the in tr o d u c tio n r u l e
(F,m) .
Y/e c o n s id e r t h e foUovd.ng axiom schem es;
- 14 -
(A1)
a D
D a)
(A2)
a D Os D y ) . D . (a D
(A3)
(A 4)
I« I e " y
^
T i i ( ttx >*
(A3)
Jin (g) D
3 (a 3 y )
îlî
^
* *9
g
ap ) y ^ *
• • • ^ * (g i
f
» • .fg p ) Y ^
THEOREÎ/Î 3 c
F
îli
>
• • » i ? a p ) jfn Y
(m e d) .
Note t h a t th e r e i s a n axiom soheme of "the fo rm
a c o n n ec tiv e
I
and a tr u th - v a lu e
(AZ}-) f o r each ohoioo o f
m«
Ever y v a lid fo rm u la i s d e r iv a b le by means o f modus ponens
from th e ax iom schemes (A1 ) — (A3 ) •
Hence i f th e se axioms a r e v a l i d and
v a l i d i t y i s p re s e rv e d by modus ponens, t h e n a fo rm u la i s v a l i d , i f and
o n ly i f i t is d e r i v a b l e .
P r o o f.
Axioms
(A1)
and
(A2)
y i e l d th e d e d u c tio n theorem.-. I f
a fu ndam ental seq u e n t th e n f o r any fo rm u la
means o f
(A3 ) .
Qi , . .
by
Q i s a n im m ediate consequence o f
r u le
th e n th e fo rm u la
-Ql <Y ^ •. . .
i s d e riv a b le by means o f
th e n the form ula
Q'V
D T tI
(A4 ) • Thus i f
| |a'*'a
n- Y
Q
i s any p ro v a b le s e q u e n t,
i s d e riv a b le f o r a r b i t r a r y
v a l id fo rm u la , th e n by Theorem 4
th e fo rm u la
, "Y
||g
y*
Now i f
a
i s a p ro v ab le s e q u e n t.
i s d e r iv a b le , and so by (A3 ) th e form ula
is a
Hence
g
is
d e riv a b le ,
As a n o th e r a p p lic a tio n o f ihe method u sed i a p ro v in g Theorem 3 ,
we show how to a x io m a tiz e any f r a g iæ n t o f t h e c l a s s i c a l ttvo-valued
p r o p o s itio n a l c a lc u lu s which in c lu d e s i m p lic a tio n .
—13 ~
is
i-S d e riv a b le by
y
S im ila r ly i f
(F ,m ),
Q
T h is was f i r s t
done by Henkin [6 ] u sin g a n o th e r m ethod.
The s e t o f tr u th - v a lu e s i s
th e s e t
E =
and th e s e t
D
o f d e sig n a te d tr u th - v a lu e s c o n s is ts o f th e s in g le elem en t 1 ,
The seq u e n ts o f the c l a s s i c a l tw o-valu ed p r o p o s itio n a l c a lc u lu s a re
th e e x p re ss io n s of th e form
r I A
where
T
and
A a re f i n i t e sequences o f f o r m i l a s ,
We c o n sid e r any system of two—v alu ed tru th -fu n c tL o n s
Tfhich in c lu d e s im p lic a tio n
and suppose
F
has
L = (F , O)
and
r
.
L et
argum ents*
R = (F , 1)
Thus
and i = ( i , .
th e n l o t
ij =
ip ) e ET ,
be one o f th e s e tru th -fu n c tL o n s
We d eterm ine th e in tr o d u c tio n r u le s
by th e method u sed i n ih e o r i g i n a l p ro o f
o f ihe p r o p o s itio n (*'^’) •
Al fai ,
E
if
ai ,
gp
a re ^an y form ulas
T i (gi , . . gp )
gp) be a sequence c o n s is tin g o f th o s e
1r e s p . i j
= 0,
re s p .
aj
Then th e in tr o d u c tio n r u le s fo r
such t h a t
F
w r itte n i n th e f orm
L:
r ,
(g 1 ^ • • •> g p ) I ( g 1 J • • • J g p ) jA ( i
r ,F(oci ,
R:
r
f
(gi J
%
• • *9
F* ( i ) )
gp)IA
gp ) I ( g i ,
gp' ) jA ^ c F*” Co) )
r IF(gi,
gp ) >A
9 • • }
• • • f
• • • 9
— l6
—
may be
I f S I is
and Y
ih e
Y, ,
IA ’"
i-S any fo rm u la , t h e n l e t Q^Y
gi 7 •
■• ■ Z) # g% Z) »(/5i Z) Y^ ^ *
For
any g iv e n t r u t h - f u n c t i o n
le t
I I I , . . . , H jj r e s p ,
^ < F“^ (1 )
• • • Z!) #(^^ Z^ Y) 7 Y •
F fro m ih e sy stem
Fo , Fi , . . . ,
Zi , , , , , Sn be an en u m eratio n o f the se q u e n ts
F I (gl 9 •• *9 gp )
such t h a t
d en o te th e form ula
re sp ,
Z^(g% , • • »9 gp )
± € F*^ (o) .
We c o n s id e r th e fo llo w in g
axiom schemes:
Fj,(y) : I 1i " y ^ *
Rp(Y) Î
. . . 3 . IIijY 3) F (gi , ,
gp) | - Y ,
2 x* t 3 . . , , 3 . Z n " Y 3 |F ( c ti,
g p )- Y»
We o b ta in an a x io m a tiz a tio n of the c l a s s i c a l two—v a lu ed p r o p o s itio n a l
c a lc u lu s by ad d in g to th e s e th e schemes;
A1 :
D a) ,
A2 :
a r>03 n y ).D . ( « 3 ^ ) 3
(cc 3 y ) ,
to g e th e r v /ith th e r u l e modus ponens ,
THEOREM 6 ,
Afo rm u la
o f modusponens
i s v a lid i f and o n ly i f
from the
axiom schemes
i t is d e riv a b le by means
At , A2, L„ (y ) , Rr, (y ) . hr, (y ) .
JQ
i?0
-Cl
Cy ) ^ • • • •
R ro o f,
th e n
L et
Q=Y
y
be an a r b i t r a r y fo rm u la ;
i s d e r iv a b le ;
if
Q i s a fundamen-tal s e q u e n t,
i f a r u le o f th e s e q u e n t c a lc u lu s allow s Q
to be i n f e r r e d from p re m isse s
Q i ‘^ 0 .
Qi , , ,.,Q ^ ,
th e n th e form ula
. . . D Æ * Y 3 n-'Y
-1 7 -
i s d e r iv a b le ;
Q'Y
hence i f
i s d e r iv a b le .
Q
Now i f
i s any v a l i d s e q u e n t, t h e n t h e form ula
a
i s a v a l i d form ula th e n
iX a v a lid s e q u e n t, and c o n se q u e n tly th e fo rm u la
f o r any
y;
ia. d e r iv a b le .
in p a r t i c u l a r
ja'-ia
jg 'Y
ja
i s d e riv a b le
i s d e r iv a b le , and th e r e f o r e
a
T his shows t h a t e v ery v a l i d form ula i s d e r iv a b le ;
on th e o th e r hand modus ponens p re s e rv e s v a l i d i t y , a n d i t i s a
s tr a ig h tf o r w a r d m a tte r to check t h a t each axiom i s v a l i d ;
d e riv a b le form ula i s v a l i d , and t h i s com pletes th e p r o o f .
—18 —
th u s e v e ry
C alculus o f seoiien ts f o r i n t u l t l o n i s t l o n~valued p r o p o s itio n a l c a lc u lu s
2m
In t h is S ectio n the n o tio n o f sequent has to be r e s t r ic t e d , so
th e exp ression ( l ) i s a sequent i f and only i f f o r each formula
[m ; a c Tin Î
set
a Ç Pm
i s th e complement o f an in t e r v a l o f
a € Pm’
* * Pm* fo r a l l m* < m or
then e it h e r
E*
th a t
a
th e
Thus i f
m* > m.
fo r a ll
As in S e c tio n 1 seguents w i l l be denoted Ity th e l e t t e r s
Ve observe th a t th e n o tio n o f sequent as here d efin ed c o in c id e s w ith
employed in S e c tio n 1 only in th e case
The
p la ce II[m]
m
in S e c tio n 1#
If
n[m]
Z[m]
Q
II and Z are sequents then there e x i s t s a sequent,
If
p la ce
f o r each
whose
m^^ p la ce i s
P
p a r tic u la r , i f
R i s the s e t
lr|&
(^ esp . |r |^ * ) .
ty
lrln
A monotone fa m ily
fo r each
A e
o f S e c tio n 1 ;
A
v/e w r ite
if
(m = 0 , 1 , . . . , n—1)*
H C Z#
then fo r each
P
If
R i s the
th ere i s a sequent
m e R and n u ll otherw ise#
[0,m]
the mapping vj* : 8E
i.e .
fo r each X e A
A v a lid sequent i s
^ set
E
II[m] U Z[m]
(resp*[m , n - 1 ] )
and
s a tis fie s
th ere e x i s t s
In
then we denote
i s sa id to s a t i s f y th e sequent
a € n[m]
{vp^j^ ^ A*
is
m e E,
Gcmplement o f an in t e r v a l o f
|p|^
n = 2*
o f a sequent II i s d efin ed in th e sameway as
IIZ , whose
denoted by
th a t
II
if
II in th e sen se
a and
m such th a t
vjj^(a) = m .
one which i s s a t i s f i e d l y every monotone fa m ily
M o f sequents i s sim u ltan éoualy s a t i s f i a b l e i f th ere
e x i s t s a monotone fa m ily
(v^}^ ^ ^
which s a t i s f i e s every
II e M .
As in S e c tio n 1 v/e have a p r e c is e v e r sio n o f th e p r o p o sitio n ( * ) ,
as fo llo w s :
- 19 -
LhMïIâ 7*
P
be any t r u t h - f im c t lo n o f
m be any t r u t h - v a l u e .
nj
(i e
arg u m en ts and l e t
Then th e r e e x i s t f a m ili e s o f s e ts
j = 1 ,...,r )
such t h a t each rJ"
r
B.j ( i € l ( + ) ;
i s th e complement of
j = 1 ,...,r ) ,
an I n t e r v a l o f
E,
and su ch
th a t
(9 )
P (x i , . . . , X r ) ^ m
A [x f R^ V . . .
ifl(-)
V
x < R ^],
( 10) P ( x i , . . , , x p ) ^ m
A [x f Rl V . . .
ifl(+ )
V
X e R^] ,
We can suppose th e number of con.iuncts
) r e s p . 5 (+ J to be a t m ost
[&(n'' + i ) ] , and t h i s bound i s b e s t p o s s ib l e .
P roof#
imy s u b se t
[•g’(n‘"+ 1 ) ]
S of
C a r te s ia n p ro d u c ts o f s u b in te r v a ls o f
even th e n
E*"
and i f
is odd th e n
n
can be e x p re ss e d a s th e u n io n o f a t m ost
i s th e sum o f
E.
tv/o—elem ent p ro d u c ts
i s th e sum o f [^(n'^+ 1 ) ]
is
of i n t e r v a l s ,
i n e i t h e r case
one— o r two—elem ent p ro d u c ts o f i n t e r v a l s ;
we o b ta in th e d e s ir e d r e p r e s e n ta tio n o f
of
n
E*^ i s th e sum o f ' ^{rf - I ) two—elem ent p ro d u c ts
of i n t e r v a l s to g e th e r w ith a sinj^le o n e-elem en t s e t ;
E**
In d eed i f
S
by form ing th e i n t e r s e c t i o n s
S w ith each o f th e s e o n e -jo r tw o -elem en t p ro d u c ts o f i n t e r v a l s ,
s in c e each such i n t e r s e c t i o n i s o b v io u sly a p ro d u c t of i n t e r v a l s .
The f i r s t p a r t of th e Lemma nov/ fo llo v /s by an a p p l ic a t io n o f t h i s
rem ark to th e secs
S = { ( x i , . . . , X r ) € E^ ;
S = { ( x i , . , . , X r ) € E*" ;
hand th e s e t
[4 (rJ + 1 )]
S = [( m i,...,m r ) :
r e s p e c tiv e ly #
%% + . . . + mp ^ 0 (mod 2 )j
and
On th e o th e r
has
elem ents b u t in c lu d e s no C a r te s ia n p ro d u c t o f i n t e r v a l s
w hich h a s more th a n one
G&(n^ + 1 )]
P ( x i ,.# .,% r ) <
p ( x i , . . . , X r ) > mj
e lem en t;
is b e s t p o s s ib le .
from t h i s we deduce t h a t th e bound
We n o te t h a t on ly in th e case n = 2
- 20 -
does
t h i s b o u id c o in c id e w ith th e bound
L et
P
o b ta in e d i n Lemma 1©
be any tr u th ^ f u n c tio n o f r = r ( P )
be any t r u t h —value*
If
argum ents and l e t
m
s.re any fo rm u la s , th e n th e seq u en t
• • •
w i l l be d en o ted Ty IT{~ ( a i , . . . , « r )
IIL
IP
if
i C l(-)
o r by
i f l( + ) *
Ey an alo g y w ith G entzen*s c a lc u lu s LJ, we a r e le d to c o n s id e r ,
f o r each
P
and
m, in tr o d u c tio n r u l e s o f th e form
(F,m )~
II
II r
( a i,...,a r )
(i f l(-))
il fP (a i , * • • ,ttr ) I m
(F,m )*
II
n I*" ( g % , . . , , a r )
(i g l(+ ))
II |P (a i,* * » ,& r)lm
The a p p l i c a t i o n of th e r u l e (P,m )* i s r e s t r i c t e d to c a s e s where
(11)
1I[0] ^ I I [ 1 ] 2
We a ls o add a r u l e
in fe r
li *•
D i i[ n - 1 ] .
p e rm ittin g W eakening:
If
II C II ' , th e n from
II
The p ro v a b le s e q u e n ts a r e th o s e o b ta in a b le by i t e r a t e d
a p p lic a tio n of th e s e r u l e s to fu n d am en tal s e q u e n ts, i . e . , se q u e n ts
II
such t h a t
n
n[m]
m eE
/
^
»
We n o te t h a t th e r u l e s (P ,n —1 )"" and (P ,0 )*
c o n c lu s io n s a r e
THEOREM 8 .
P roof#
may be o m itte d s in c e t h e i r
alw ays fu n d a m e n ta l s e q u e n ts .
E very p ro v a b le se q u e n t i s v a l i d .
E v id e n tly e v e iy fun d am en tal se q u e n t i s v a l i d , and an y seq u e n t
o b ta in a b le from a v a l i d se q u e n t
by W eakening i s v a l i d .
- 21 -
Hence i t
s u f f ic e s to show t h a t th e in tr o d u c tio n r u l e s
(P,m )
and
(P,m)*^
p re s e rv e v a l i d i t y .
L et
^ ^
be a n a r b i t r a r y monotone f a m ily , and suppose
th a t
s a t i s f i e s th e p rem isses o f (P,m)*”#
s a tis fy
n,
f o r each
i e l ( - ) , and so by (9 )
th e n
s a tis fie s
If
th e seq u en t
v^
does
not
Hi
F ( v ^ ( a i ) , . , , , v * ( a r )) < m ;
we r e c a l l t h a t
( 12 )
hence v/e
v ? ( F ( a i , . . . , a r ) ) = i n f I '( v * ( « i ) , . . . ) V * ( a r ) ) ;
have
( f (« i , . . . ,ttr ) ) ^ m
so t h a t
s a tis fie s
th a t i f
s a tis fie s
,
|P (a i," * # ,% r)lm
♦ Thus we have proved
th e p rem isses of th e r u l e (P,m )
s a t i s f i e s th e c o n c lu s io n .
th e n i t a l s o
We see th e r e f o r e t h a t th e r u l e p r e s e rv e s
v a lid i t y *
Suppose now t h a t th e p rem ises o f th e r u l e (F,m )*
by a monotone fa m ily
If
v^
^
does noc s a t i s f y
( 13 )
v ^ (a )
II,
f o r each
p ^ A,
A be any elem ent at
A.
then by v ir tu e of th e in e q u a lity
^ v*(oc)
and th e r e s t r i c t i o n ( I I ) on
f o r any
L et
a re s a tis fie d
whenever
II,
A ^ /i ,
we see t h a t
Hence f o r a l l
^ ^ A,
i ( l ( + ) , an d so by (IO )
- 22 -
v*
does n o t s a t i s f y
v* s a t i s f i e s
II
lit** (a i;* * * ;K r)
by ( 12 ) i t fo llo w s t h a t
so t h a t
V*
s a tis fie s
|F ( a i , . . . , a r ) l m * *
th e r u le (P,m )^
i s s a t i s f i e d by
i s s a t i s f i e d by
î''^x^A € A *
Thus
th e c o n c lu s io n of
fo r a rb itra ry
A e A,
and
so i t
have proved th e r e f o r e t h a t i f
a monotone fa m ily s a t i s f i e s th e p re m is se s o f th e r u le (F,m )^ th e n i t
a l s o s a t i s f i e s th e c o n c lu s io n .
Thus th e r u le p re s e rv e s v a l i d i t y ,
and t h i s com pletes tlie proof*
We n o te t h a t th e o n ly p ro p e rty o f
th e p a r t i a l o rd e r
^
w hich was used
i n t h i s p ro o f i s r e f l e x i v i t y ;
t h i s p ro p e rty however i s e s s e n tia l *
TIE OREM 9»
Every v a l i d seq u e n t i s p ro v a b le .
P roof*
If
n
i s an u n p ro v ab le seq u e n t th e n th e r e e x i s t s an im provable
seq u en t
Q*
such tlia t
(14)
0 &n*
and such t h a t f o r
(15)
if
,
each c o n n e c tiv e
P and each tr u t h - v a l u e
m,
|P(ai ,.. .,a p ) | nj~ Ç n*,-then scmelli (ai ,* ..,a r ) Ç fl’’* •
This may be seen-as follows;
ifK ^ i
^
i € l(~ )
th e n because
such t h a t
|p ( a i , * . . , a r ) | in” C 0* ;
* ..
i n a seq u en t
If
Q
i s im p ro v ab le , th e r e e x i s t s
Q II f ’( a i , * . . , a p ) = Q
th e same argum ent t o
n,
Q
Q * w ith r e s p e c t
i s u n p ro v a b le ;
nowa p p ly
t o a d i f f e r e n t seq u en t
c o n tin u in g i n t h i s way we o b ta in a sequence
w hich must te rm in a te a f t e r a f i n i t e number o f s te p s
0*
w ith th e r e q u ir e d p r o p e r ti e s ( I 4 ) and (1$)*
i s a n u n p ro v a b le se q u e n t and
th e r e e x i s t s a n u n provable s e q u e n t
- 23 -
| P( « i »
such t h a t
m”*" C 0 ,
th e n
(16)
i f |o |^
th e n
Qa,
|a |“
(a f S, « f E ) ,
and suiih tîja t
( 17 )
C
f o r some
To see t h i s we argue as fo llo w s;
p la c e i s
and
0 [o ] A . . . A n[&]
I I |P ( c f i , . . . , a r ) l
unprovable;
see th a t
-6 c E ;
i t fo llo w s t h a t
b u t th en si.nee II
II II i^ ( a i, . • • ,ap )
sequent
Let II be th e sequent whose
f o r each
C fi
i e l(+) .
sin c e
Q i s unprovable
I I |F ( a i , . . . , a p ) | m*" i s
s a t i s f i e s the r e s t r i c t i o n ( I I ) , we
i s unprovable f o r some
= II II { ^ ( « i . . , a r )
i c l(+ );
has the re q u ire d
th e
p r o p e rtie s
(16) and ( 17 ) .
Let
II
be a n u n p ro v ab le s e q u e n t.
and a mapping
seq u e n t
A "*■ 11^ which a s s o c ia t e s w ith e a c h node
II. ©
The c o n s tr u c tio n p ro c e ed s by l e v e l s ;
wep la c e a s in g le node
l e v e l and
(J = /j(F,m )
We c o n s tr u c t a " tr e e "
Ao
w ith
|F ( a i , . . . , a r ) | ni** C Ij^
a t th e (k+1
IL = ll* ;
Ao
if
li^
A an u n p ro v ab le
a t th e
0^^ l e v e l
A i s a t th e
th e n we co n n ect
l e v e l and s e t
A
A w ith a
=
node
•
The s e t A
i s p a r t i a l l y o rd e re d in th e obvious way*
By (1A) and ( I 6 )
(1 8 )
No
if
ja ln T
we see t h a t f o r a l l
^
th e n
a e S,
U ln T C
m e E
w henever
i s fu n d am en tal and so i t i s p o s s ib l e , f o r each
to d e fin e
v ^ (a )
Ïv ^ Îa ^ A
a s th e l e a s t rtr such t h a t
monotone
if
T h is h o ld s f o r
% i,...,K r
a e n^[m ]
a ll
a / II^[m ].
m e E,
The fa m ily
th e n
ae S
v ^ (a ) / m
a e V by c o n s tr u c tio n .
and c o n s id e r
a e V,
in view o f (1 8 ) .
We s h a l l prove t h a t f o r
( 19 )
A ^ A*.
(fo ra l l
Suppose ( 19 ) h o ld s f o r
a « P ( c t i ,... , c t r ) *
— 24" ~
A e A ).
If
a c Ii^[m]
th e n e i t h e r
|a I m G
or
|a|m ^ C
I n th e f i r s t c a s e we have by (18)
I ^(*1 >• *• #«r ) I m ^
Hence by ( I 5 ) , f o r a l l
sane
i «■ l ( - ) .
/Li
fo r a l l
^ A we have
/i ^ A*
^
fo r
Thus by in d u c tiv e h y p o th e s is we have f o r each
^
jii ^ A
( * r ) / RH*
We deduce l y ( 9 ) t h a t f o r each
/i ^ A
F ( v* ( % i v ^ ( a r ) )
> m
.
> m
•
Thus by (12) we have
v j ( p ( a i , . , . , a r ))
In th e second
s u ita b le
case
|P ( a i , ♦. • ,Or ) | m*' C
so by (1 ? ) we have f o r
/u ^ A
II
) C
11^f o r some
i e l(+ ) .
Thus by in d u c tiv e h y p o th e sis
V
[ t * ( o i ) / E i . . . . . y * (« r ) / EH .
!(!(+ ) ^
''
Hence by ( 10 ) we
have
) , * '* , v * ( a r ) ) < m,
and so by (12)
In e i t h e r c a se we have
v jj(a) 7^ m,
and t h i s co m p letes th e p ro o f of
(1 9 ).
-
25
-
If
and
n
me
v/ere v a l i d , th e n
E
would
be v a l id ;
hence f o r some a e S
we would have
v ^ ( a ) = m and
w hich c o n tr a d ic t s (19)*
a € I^ [m ]
,
We see th e r e f o r e t h a t e v ery u n p ro v ab le seq u e n t
i s i n v a l i d , w hich was to be shown*
Combining Theorems 8 and 9 we o b ta in
THEOREM 10*
A se q u e n t i s v a l i d i f and onlv i f i t i s p ro v a b le .
As a consequence we may deduce th e fo llo w in g an alo g u e of G e n tz e n 's
H au p b sa tz fo r h i s system LJ ;
L et
Z | a |^
II
and
Z
be s e q u e n ts and suppose
a r e p ro v a b le s e q u e n ts . th e n
&> m*
^
l i j a and
IIZ i s a p ro v a b le s e q u e n t.
T h is could a l s o be e s ta b lis h e d p ro o f—th e o r e t i c a l l y in th e manner of
G entzen [4-]*
\7e a l s o n o te w ith o u t p ro o f t h a t th e i n t u i t i o n i s t i c an alo g u e o f
Theorem 3 i s tru e #
Indeed i t can be shovm t h a t a seq u e n t i s
sim u lta n e o u s ly s a t i s f i a b l e i f and on ly i f i t i s sim u lta n e o u s ly
s a t i s f i a b l e in th e c l a s s i c a l sense#
To d e term in e th e in tr o d u c tio n r u le s f o r a g iv e n c o n n e c tiv e
we have to o b ta in s u ita b le r e l a t i o n s o f th e form
c o rre sp o n d in g t r u t h —fu n c tio n #
In view
and
ru le s
(9 ) and (1 0 ) f o r th e
o f Lemma 7 th e number o f
prem isses can alw ays be lim ite d to [i'(n*"+ 1 )]#
th e r e a r e
E,
For eac h c o n n ec tiv e
F
2(n—1 ) r u l e s i f we a g re e t o ex clu d e th e vacuous r u l e s (P,0)***
(F ,n —1 )”*.
(F ,0 )
The r u l e s
and
(F ,n —1 )
(F ,0 )" ’ and
(F ,n —1
a r e th e
same a s th e
f o r th e c l a s s i c a l seq u e n t c a lc u lu s *
an example v/e c o n s id e r th e th re e -v a lu e d L ukasiew icz c o n n e c tiv e s
— 26 —
As
C,K,A,N
in tro d u c e d i n S e c tio n 1 •
In view o f th e above rem arks i t s u f f i c e s
to g iv e th e r u l e s
and
(P ,1 )
o m ittin g th e p a ra m e te r II,
/9|/?|
( P ,1 ) * ;
w r itt e n i n s k e le to n form ,
th e s e a r e a s fo llo w s :
|a
U la
a \ cl^ \
Co/9 ICc^l
|Cct/9| Ca/9
a,/9 1«,/9|
|a |a
|/?|/9
Ka/9 iKb^l
IKc^ I Ko/9
a a
^ l/? l
|a#/?l
Ao/5 \ho0\
•.a
a
a
Na I Na I
| Na | Na
Suppose we s in g le o u t a s u b s e t
where
D=[mçE;
m ^m ol
D o f d e s ig n a te d t r u t h —v a lu e s ,
f o r some
nvj € E*
[vp^i^ ^ ^ may be s a id to s a t i s f y th e fo rm u la
v * (a ) e D;
fam ily»
a
i s s a id to be v a lid
a
A monotone fa m ily
i f f o r each
A € A
i f i t i s s a t i s f i e d by every monotone
The s e t o f v a l i d fo rm u las can be c h a r a c te r iz e d s y n t a c t i c a l l y ;
in d eed Theorem 10 im p lie s t h a t th e fo rm u la
th e seq u e n t
a
| a |p
a
i s v a l i d i f and o n ly i f
i s p rovable#
I f s u ita b le c o n n e c tiv e s
D,
a r e d e f in a b le i n term s o f
th e p r im itiv e c o n n e c tiv e s , th e n u s in g th e above rem ark we can c h a r a c te r iz e
th e s e t o f v a lid fo rm u las by means o f axioms and r u l e s o f in f e r e n c e ,
u s in g a s im ila r
p ro ced u re t o t h a t u sed i n p ro v in g Theorem 5*
We s h a l l
c o n te n t o u rs e lv e s however w ith an a p p lic a ti o n t o th e o rd in a ry (tw o -v a lu e d )
- 27 -
i n t u i t i o n i s t i c p r e p o s itio n a l c a lc u lu s .
tr u t h - f u n c tio n s
Fq , Fj., • , . ,
For each system o f tr/o—v alu ed
w hich in c lu d e s im p lic a tio n ^ D,
we show
hovf to a x io m a tize th e c o rre sp o n d in g i n t u i t i o n i s t i c p r e p o s i tio n a l c a lc u lu s .
The method i s v e ry s im ila r to t h a t u sed i n p ro v in g Theorem 6 , and we
s h a l l u se th e n o ta tio n in tro d u c e d i n c o n n e c tio n w ith t h a t theorem .
We o bserve t h a t a s i n th e c l a s s i c a l c ase we may r e g a rd th e symbols
r o ,...,r n _ i
a p p e a rin g in th e seq u en t ( l ) a s d e n o tin g sequences o f
fo rm u las r a t h e r th an s e t s ;
th e r e s u l t s o f t h i s S e c tio n rem ain tim e
i f we add r u l e s p e rm ittin g p e rm u ta tio n , w eakening and c o n tr a c ti o n .
The n o tio n o f seq u en t f o r th e two—v alu ed i n t u i t i o n i s t i c p r e p o s iti o n a l
c a lc u lu s
r u le s
is
th e
(F ,0 )
and
F o ,F i,,.,
same a s i n th e c l a s s i c a l c a s e .
( F ,l ) ^
f o r a g iv e n c o n n e c tiv e
a r e i d e n t i c a l w ith th e r u l e s
The a p p lic a tio n o f th e r u l e
sequence
A
i s empty.
R
L and
The in tr o d u c tio n
F from th e system
R re s p e c tiv e ly .
i s r e s t r i c t e d to c a s e s w here th e
The a x io m a tiz a tio n o f th e i n t u i t i o n i s t i c
p r e p o s itio n a l c a lc u lu s d i f f e r s from t h a t o f th e c l a s s i c a l p r e p o s i tio n a l
c a lc u lu s on ly in t h a t th e axiom schemes
in s ta n c e s
R ^(y)
a re re p la c e d by t h e i r
R p (F (a i, . . . ,a p ))> o r e q u iv a le n tly by th e schemes
Rp •
* F ( # i , , • , ,0Cp ) ZD#
,,,
Zjij * F ( oc^ , , , , ,o(p ) Z) F ( oci , , , , ,cxp )
In p re p a r a tio n f o r th e p ro o f o f Theorem 11, we observe t h a t , f o r any
tr u t h - f u n c t i o n
F
of
r
argum ents, th e fo rm u la
F ( a i,,.,,a p )
is
i n t u i w i o n i s t i c a l l y e q u iv a le n t to th e c o n ju n c tio n o f th e fo rm u las
i j t l “J
such t h a t
means
i
= (ii,.,,,ip ) e
“J
F“^ ( o ) .
T h is can e a s i l y be deduced by
o f (12) from th e f u l l c o n ju n c tiv e norm al form f o r
- 28 -
F :
P ( x i,.,.,x r )
=
A [x i
p -i(o )
u s in g th e f a c t t h a t th e fo llo w in g
famiOy
(a v /9 )
THEOREM 11*
... V
X p (^ ^ ]
,
i d e n t i t i e s h o ld f o r each monotone
=
v ^ (a )
A
v*(/9)
,
=
v ^ (a )
v
v^(/9)
.
A form ula i s v a l i d i f and on ly i f i t i s d e riv a b le by means
o f modus ponens from th e axiom schemes
If
Q
A1, A2,
i s fu n d a m e n ta l, th e n
fican be i n f e r r e d from p re m isse s
th a n
s in c e th e fo rm u la
*
Qi
$
...
y D.
( y ) , Rp , Lp ( y ) , Rp ,
i s d e r iv a b le f o r any
Suppose
R;
V
^ ^ :
V* (a ^
Proof*
^
D* Qk y
y*
by some r u l e o th e r
DO
*
y
*
*
Oi y ,...,O k y
*
th e n th e same i s tr u e o f 0 y .
i s d e r iv a b le , i t fo llo w s t h a t i f
a r e d e r iv a b le f o r
a rb itra ry
Now suppose t h a t
y,
in f e r r e d from p rem ises
Q i
,
b y
r = y i , • • • fYm ^nd r e c a l l t h a t
a r e d e r iv a b le f o r a r b i t r a r y
Yi T).
means o f ru ü e
A m ust be empty;
y,
R;
if
le t
*
♦
0% y , . . . , 0 ^ Y
th e n th e fo rm u las
. . . D .y mZ) Z k P (# i >• • •
)
(R ~ 1 , . • • , r )
a re d e r iv a b le , and so th e r e f o r e i s
Yi
• • • ^*Ym ^ P(%i*«**f%r) >
t h i s im p lie s t h a t th e form ula
n*y
=
yi D.
i s d e riv a b le f o r a r b i t r a r y
fo llo w s t h a t i f
Q
• • • D.ym ^ ( p ( a i , . . . , a r ) ^ y ) ^ Y
y*
Q
Prom th e above c o n s id e r a tio n s , i t
i s a v a l i d s e q u e n t, th e n th e fo rm u la
- 29 -
0 *y
is
is
d e r iv a b le f o r any
y .
As i n th e p ro o f o f
Theorem 6 ,
we deduce t h a t ev ery v a l i d fo rm u la i s d e riv a b le #
i ’o com plete th e p ro o f o f Theorem 6 , i t i s only
n e c e s s a ry t o prove t h a t th e axiom schemes
I^ (y ) and
a re
( i n t u i t i o n i s t i c a l l y ) v a lid #
F or each
r-tu p le
i = ( lij...,ir ) €
(a j D y )
L et
a
if
and each
j = 1 ,...,r , le t
= 0
ij
be th e c o n ju n c tio n o f a l l fo rm u las
A
lj±=1 “J
such t h a t
V
ij=rO
“J
i = ( i i , . . . , i r ) f F~^(o)#
In H e y tin g 's p r e p o s i tio n a l
c a lc u lu s Y/e have
(21)
. . . 348^ 3 y )
^
i f ET
For any
1
t" Y •
we have
A
aj . 3 . y
a j I- ^
' D.
... 3 . ^
3 Y
,
lj„ o
and so
a |”
•^,,.(^1
i e F "^ (0 )
...D * /9 r
•
By d e f i n i t i o n we have
n r Y ,....i^ Y
I-
. . . ^ . / î^ ^ y )
H ence, u s in g ( 2 1 ) , we deduce
n i y , . . . J I * y , a |- y
S in c e
a
i s i n t u i t ! o n l s t i c a l l y e q u iv a le n t to
fo llo w s t h a t th e axiom scheme
I^ ( y )
_ 30 _
i s v a lid #
P (a i, . . . , a r ),
it
.
Now l e t u s ta k e
th e n f o r some
j
y = % in (2 0 ).
e ith e r
ij = 1
and
If
i e
ij = 0
and
or ij = 0
i* e P “^ ( o ) ,
and i j = 1,
so t h a t
from t h i s we deduce
1“
^Y )
•
;^y d e f i n i t i o n we have
2x Y
f ^ Y
if
A
(/9Z5*
P “^ (0 ) ^
• • • ! > • /5^ Z5 y )
r
•
Hence, u s in g (21) we deduce
Zi y , . . . ,
S in ce
y= a
fo llo w s t h a t
t-y
.
i s i n t u i t i o n i s t ic a l l y e q u iv a le n t t o
th e axiom
t h a t th e a x io m a tiz a tio n s h a re s w ith
H e y tin g 's p r e p o s itio n a l c a lc u lu s th e p ro p e rty :
a
it
scheme Rpi s v a l i d .
I t fo llo w s from Theorem 11
fo rm u la
F ( a i , • . . ,« r )^
f o r e v e ry d e riv a b le
th e r e i s a d e r iv a tio n w hich c o n ta in s , b e s id e s im p lic a tio n ,
o n ly th e c o n n e c tiv e s a p p e a rin g in
a x io m a tiz a tio n of th e
a.
The same p ro p e rty h o ld s f o r th e
c l a s s i c a l p r e p o s iti o n a l c a lc u lu s g iv e n in
Theorem 6 , a s i t does a ls o f o r H e n k in 's a x io m a tiz a tio n [ 6 ] ,
- 31 -
We conclude t h i s C hapter w ith a rem ark on th e re ia tio n c b i-p betw een
th e seq u en t c a l c u l i f o r c l a s s i c a l and i n t u i t i o n i s t i c
n—v a lu ed
p r e p o s itio n a l c a lc u lu s #
The p r i n c i p a l d if f e r e n c e betw een th e c a l c u l i o f S e c tio n s 1 and 2
l i e s i n th e r e s t r i c t i o n w hich i s p la c e d on th e
S e c tio n 2#
c la s s o f se q u e n ts i n
However, i f we p la c e th e same r e s t r i c t i o n on seq u e n ts
in S e c tio n 1
th e n th e r e l a t i o n s h i p becomes much c lo s e r#
th e in tr o d u c tio n r u l e s
(p,m )
and
c h a r a c te r iz e th e v a l i d s e q u e n ts ;
(P,m )*
Indeed
se rv e in b o th c a se s t o
f o r th e c l a s s i c a l
n—v a lu e d
p r e p o s itio n a l c a lc u lu s no l i m i t a t i o n i s p la c e d on th e a p p l ic a t io n o f
th e r u le
(F ,m )*,
w h ile f o r th e i n t u i t i o n i s t i c
n -v a lu e d p r e p o s itio n a l
c a lc u lu s th e a p p lic a tio n o f t h i s r u l e i s lim ite d to c a s e s v h e re (1 1 ) hold s#
F o r 'the i n 't u i t i o n i s t i c c ase t h i s rem ark i s c o n ta in e d i n Theorem 10, v ih ile
f o r th e c l a s s i c a l c a se we need on ly m odify s l i g h t l y th e
In the c a s e
n = 2 , th e s e c h a r a c te r iz a ti o n s o f th e s e t
p ro o f of Theorem 2,
o f v a l i d se q u e n ts
d i f f e r v e ry l i t t l e from th e c h a r a c te r i z a ti o n s w hich G entzen g av e, i n term s
o f h is c a l c u l i IK
and LJ.
Thus we have some i n s i g h t i n t o th e r a th e r
s u r p r is in g r e l a t i o n s h i p w hich e x i s t s be-Ween th e s e two c a l c u l i o f
Gentzen#
- 32 -
C hapter I I ;
A lg e b ra ic Methods
I n t h i s C hapter we stu d y th e r e p r e s e n ta tio n th e o ry of P o s t
a lg e b r a s and pseudo—P o s t a lg e b r a s , and a p p ly i t to o b ta in
a x io m a tiz a tio n s o f c l a s s i c a l and i n t u i t i o n i s t i c n -v a lu e d p r e p o s itio n a l
c a lc u li*
In S e c tio n 1 we in tr o d u c e th e P o st a lg e b r a s and p seu d o -P o st
a lg e b ra s *
P o s t a lg e b r a s w ere f i r s t d e fin e d by Eosenbloom [ 1 5 ] , and
have been s tu d ie d i n r e c e n t y e a rs by E p s te in [ 3 ] and T raczyk [ 2 9 ,30]*
Prom t h e i r work i t fo llo w s t h a t a d i s t r i b u t i v e l a t t i c e w ith
a P o s t a lg e b ra i f and
and a f i n i t e chain*
0,1
is
o n ly i f i t i s th e c o p ro d u ct of a Boolean a lg e b ra
S im ila r ly a d i s t r i b u t i v e l a t t i c e w ith
0,1
w i l l be c a ll e d a pseudo—P o s t a lg e b r a i f i t i s th e co p ro d u ct o f
a pseudo-B oolean a lg e b ra ( i . e . , a r e l a t i v e l y pseudo-com plem ented
l a t t i c e ) and a f i n i t e chain*
A fte r some in tr o d u c to r y rem arks on d i s t r i b u t i v e l a t t i c e s w ith
0 ,1 ,
we d e f in e c o p ro d u c ts i n th e u s u a l c a t e g o r i c a l way an d d e riv e t h e i r
e x is te n c e and u n iq u e n ess from g e n e ra l r e s u l t s i n c a te g o ry th e o ry *
Next we d e fin e P o s t a lg e b r a s and
t h e i r a lg e b r a ic s t r u c t u r e ;
o f o rd e r
n
p seu d o -P o st a lg e b r a s and in v e s t i g a t e
th u s we show t h a t a (pseudo—) P o s t a lg e b ra
may be embedded in th e (n —1 )—fo l d C a r te s ia n power of scane
(pseudo—) Boolean a lg e b r a , and we show t h a t each (p se u d o -) P o s t a lg e b r a
h as th e s t r u c t u r e o f a pseudo-B oolean a lg e b ra *
We th e n g iv e an
e q u a tio n a l c h a r a c te r iz a ti o n o f P o s t a lg e b r a s and p seu d o -P o st a lg e b r a s .
F i n a l l y we prove r e p r e s e n ta tio n theorem s d e sig n e d f o r a p p li c a tio n s t o
many—v a lu e d p r e p o s itio n a l c a l c u l i i n S e c tio n 2*
- 35 -
I n S e c tio n 2 we use th e r e p r e s e n ta tio n th e o ry o f S e c tio n 1 to prove
th e co m p leten ess o f s u i t a b l e a x ic x n a tiz a tio n s of c l a s s i c a l and
i n t u i t i o n i s t i c n—v a lu ed p r e p o s itio n a l c a lc u li#
The a x io m a tiz a tio n s
a r e d e sig n e d so t h a t th e c o rre sp o n d in g Idndenbaum a lg e b r a s a r e P o st
a lg e b r a s and p seu d o -P o st a lg e b r a s r e s p e c t i v e l y , and so we e a s i l y
o b ta in com pleteness theorem s from th e r e s u l t s o f S e c tio n 1*
We
co n clu d e th e C hapter w ith some e le m e n ta ry p r o p e r t ie s o f th e c l a s s i c a l
and i n t u i t i o n i s t i c
n -v a lu e d p r e p o s itio n a l c a lc u li#
The a p p lic a tio n o f a lg e b r a ic methods to c l a s s i c a l and
in tu itio n is tic
n—v a lu e d p r e p o s i t i o n a l c a l c u l i i s su g g este d by th e
s u c c e s s f u l u se of th e s e methods in th e two—v a lu e d c a s e by T a r s k i,
McICinsey, R asiow a, S ih o r s k i, e t . a l #
F o r r e f e r e n c e s , s e e Rasiowa
and S ik o r s k i [1 3 ],
1,
P o s t algjebras and p se u d o -P o st a l/? eb ra s
D i s t r i b u t i v e l a t t i c e s w ith
0 .1 ,
Throughout t h i s C h ap ter
l a t t i c e s a r e supposed d i s t r i b u t i v e w ith
0 ,1 ,
a ll
and a l l homomorphisms,
in c lu d in g th e n a t u r a l i n j e c t i o n o f a s u b l a t t i c e i n t o a l a t t i c e , p re s e rv e
th e e le m e n ts
0,1#
A l a t t i c e i s c a l l e d n o n -d e g e n era te i f i t h a s a t
l e a s t two elem ents#
I f th e l a t t i c e
}x € L ; X A a ^ b}
The two—elem ent l a t t i c e i s d e n o ted by 2#
L
i s su c h t h a t f o r any elem en ts
has a g r e a t e s t elem ent
a pseudo-B oolean a lg e b r a ;
o f a r e l a t i v e to
elem ent
a D 0,
b,
th e e lem en t
and i s d en o ted by
d e n o ted by
—a ,
c
c,
a ,b
we say t h a t
th e s e t
L is
i s c a l l e d th e pseudo—complement
a Z> b#
I n p a r t i c u l a r th e
i s c a ll e d th e pseudo—complement o f
- 34 -
a#
I f f o r e v ery elem en t
a e Lth e r e e x i s t s
a ^ a*^
th e n
L
=0
; a v a* = 1
i s s a id to be a Boolean a lg e b r a .
a lg e b r a i s a pseudo-B oolean a lg e b ra w ith
I n t h i s c a se
-a
a D b
We
a’
such t h a t
,
se e t h a t
ev ery B oolean
a D b = a* v b
an d - a = a * .
i s c a l l e d th e complement o f a r e l a t i v e to b ,
i s c a l l e d th e complement o f
The
a n elem en t
and
a*
f o llo w in g theorem was proved by S to n e [28] ;
THBOIlüM 1 •
licit
le t
filte r.
P be a
I
be an i d e a l i n a non—d e g e n e ra te l a t t i c e
If
a two—v a lu e d homomorphism
I and
F a r e d i s j o i n t , th e n th e r e
h :L
2 w hich i s zero on
I
L and
e x is ts
and u n ity on
T h is theorem im p lie s th e fu n d am en tal r e p r e s e n t a t io n theorem f o r l a t t i c e s :
THh'OriEM 2 ,
E very l a t t i c e i s
C o p ro d u cts,
isom orphic t o a l a t t i c e o f s e t s .
The d i r e c t sum o f m odules, t h e f r e e p ro d u c t o f groups
and th e d i s j o i n t u n io n o f
t o p o l o ^ c a l sp a c e s a r e a l l in s ta n c e s o f th e
c a te g o r ic a l n o tio n o f c o p ro d u c t,
w© s h a l l c o n s id e r t h i s n o tio n i n th e
c a te g o ry o f l a t t i c e s and homomorphisms.
If
[D .}.
A A
A
i s any fa m ily o f l a t t i c e s , th e n a l a t t i c e
s a id t o be th e c o p ro d u c t o f t h i s f a m ily
i^ :
D
(a c a )
in to a l a t t i c e
su ch t h a t
D*,
such t h a t w henever
i f th e r e e x i s t homomorphisms
h^ ;
D*
th e r e e x i s t s a u n iq u e homomorphism
h^ = h 0 ip^ f o r each
A e A#
D.
D*
- 35 -
D is
a r e homomorphisms
h : D
D*
F,
The e x is te n c e and b a s ic p r o p e r ti e s o f th e c o p ro d u c t can be deduced
from g e n e r a l r e s u l t s in u n iv e r s a l a lg e b r a and th e th e o ry o f c a te g o r ie s .
THEOREM *>3*
Any fa m ily o f l a t t i c e s h as a c o p ro d u c t, w hich i s u n ique
up to isom orphism ;
th e c o p ro d u ct i s g e n e ra te d by th e u n io n o f th e
im ages o f th e mappings
lp^(A € A ) , and i f th e l a t t i c e s
a re
n o n -d e g e n e ra te th e n th e s e mappings a r e i n j e c t i v e .
P r o o f#
The c a te g o iy o f l a t t i c e s i s an a lg e b r a i c c a te g o ry \\h ic h has
f r e e a lg e b r a s and i s c lo s e d u n d er th e fo rm a tio n o f homomorphic im ag es;
h ence i t h a s c o p ro d u ctsb y a s l i g h t e x te n s io n o f a theorem o f S ik o r s k i [ 2 6 ] .
A l t e r n a t i v e l y th e c a te g o ry i s c lo s e d u n d e r th e fo rm a tio n o f s u b a lg e b ra s ,
iso m o rp h ic images and c a r t e s i a n p ro d u c ts , and hence i t h a s co p ro d u cts
by a -tiieorem o f Schm idt [25]*
The u n iq u e n ess o f th e c o p ro d u c t, and th e f a c t t h a t i t i s g e n e ra te d
by th e u n io n o f th e im ages o f th e mappings
ip^,
can be proved i n any
a lg e b r a ic c a te g o ry .
To pro v e th e i n j e c t i v i t y o f th e mappings
A e A
^Ao ;
a homomorphism
: Dp^
Dp^
t h i s i s p o s s ib le by Theorem 1.
a homomorphism
h : D
Dp^ such t h a t
so t h a t
ip^,
we choose f o r each
hp^
i s th e i d e n t i t y on
Then we know t h a t th e r e e x i s t s
hoip^ ~ ^A
^
S in ce
hp^
i s i n j e c t i v e , so to o i s
1. .
T h is com pletes th e p ro o f ,
Ao
I n view o f th e p re c e d in g Theorem we may alw ays suppose t h a t i f th e
la ttic e s
Dp^ a r e n o n -d e g e n e ra te
o f th e co p ro d u ct
e q u iv a le n t to t h a t
D.
(A e A ),
th e n eac h
Dp^ i s a s u b l a t t i c e
Thus i n t h i s c a s e th e n o tio n o f co p ro d u ct i s
o f f r e e p ro d u c t ( c f . S ik o r s k i [ 2 6 ] ) ,
w0 s h a l l suppose a l l l a t t i c e s n o n -d e g e n e ra te .
- 36 -
In th e s e q u e l
A more c o n c re te s e t —th e o r e t i c a l c o n s tr u c tio n f o r th e co p ro d u ct can be
g iv en as fo llo w s ;
THEOREM 4#
Suppose
(a Ç a ) .
to th e
form
i s isom orp h ic to a
Then th e co p ro d u ct
la ttic e
pr~^ (Ap^)
o f th e
D*
o f s u b s e ts o f
w ith
f D»
la ttic e
Dp^ ( a e A)
o f s u b s e ts o f
i s isom orphic
^ p^ Mp^ g e n e ra te d by s e ts o f th e
(A e A)*
That t h i s c o n s tr u c tio n i s g e n e r a lly a p p lic a b le fo llo w s from Theorem 2*
Theorem A can be deduced v e ry e a s i l y from th e fo llo w in g a lg e b r a ic
c h a r a c te r iz a ti o n which was communicated to th e a u th o r by V /.H o lszty n sk i.
The
p ro o f i s
a s tr a ig h tf o r w a r d m o d if ic a tio n o f th e p ro o f of th e
c o rre sp o n d in g r e s u l t f o r Boolean a lg e b r a s
THEOREM 5*
Ala ttic e
D i s th e
(of# S ik o r s k i [ 27 ]
co p ro d u ct of s u b l a t t i c e s
p . 39%#
Dp^ (a € A)
i f and o n ly i f i t i s g e n e ra te d by t h e i r u n io n and f o r any ch o ice o f
d i s t i n c t in d ic e s
A i,* ..,A m € A
and elem en ts
a ^ ,b r e Dp^
( r = 1 ,# ..,m )
we have
a% A
A am :^ b i v
Vbm
im p lie s
P o st a lg e b r a s and pseudo—P o s t a lg e b r a s .
n elem ents (n ^
ap ^ bp f o r some r ( r = 1 , . . . , m )
L et
D
be a f i n i t e c h a in w ith
2 ) ; we suppose th e s e to be th e e lem en ts
0 — Sq < e^ K « • • K e|^_ 1 — ^
If
E
,
i s an a r b i t r a r y (non—d e g e n e ra te d i s t r i b u t i v e ) l a t t i c e (w ith 0,1 )
th e n we d e n o te by [D]n
Boolean we say t h a t
S im ila r ly i f
P o s t a lg e b ra
th e c o p ro d u ct of
[D]n
i s th e P o s t a lg e b ra
D i s pseudo-B oolean, th e n
o f o rd e r
p seu d o -P o st a lg e b ra b u t
n
D and
over
D#
o f o rd e r
If D is
n
o v er
D.
i s c a l l e d th e p seu d o -
C le a rly e v ery P o s t a lg e b r a i s a
n o t c o n v e rs e ly ,
- 37 “
[D]^
E.
Vfe w i l l show l a t e r on t h a t
th e d e f i n i t i o n o f P o s t a lg e b ra g iv e n h e re i s e q u iv a le n t to th o se
a p p e a rin g i n th e l i t e r a t u r e •
The s tr u c tu r e o f th e l a t t i c e [p]n
Theorem 3#
may be v e ry e a s i l y deduced from
A ccording to t h i s Theorem we may suppose t h a t
a r e sub l a t t i c e s o f
[p]p
and t h a t e v e ry elem en t o f
p o lynom ial in th e elem en ts o f D and
E#
[D]p
D and
E
is a la ttic e
T ransform ing t h i s poly n o m ial
i n t o d is ju n c tiv e norm al form by means o f th e d i s t r i b u t i v e law , we see
t h a t each elem ent of
where
[D]p
d i , , . , , d p ^ i e D.
we may assume t h a t
has a r e p r e s e n t a t i o n
F u r th e r , r e p l a c i n g
d]_ ^
^
x =
dx
by
(d [ * e i )
V^]^d^ ( k = 1 , . , . , n - i ) ,
Thus each elem ent o f
[D]p
h as
a r e p r e s e n ta tio n
(1 )
where
X=
(d i A e t ) ,
d%, . , dp_i f D.
by th e d e f in in g
T his r e p r e s e n t a t i o n i s i n f a c t u n iq u e .
p ro p e rty o f
a (u n iq u e ) homomorphism
i d e n t i t y on
[D ]p,
: [D]p
th e r e e x i s t s f o r each
Dsuch t h a t
Dt
F o r,
i ( i= 1 , • • . ,n-1 )
re d u c e s t o t h e
D and
D l(e k ) ■= |o o ^ ^ r î ^ s e
o p e ra tin g on (8 ) w ith
(k = 0 ,1 ,...,n - l) ;
we o b ta in
jDi(x) = d ^
( i — 1 ,# # # ,n —l ) ,
from which th e a s s e r t i o n fo llo w s im m e d ia te ly .
If
x'
i s a n o th e r elem ent
X* =
th e n
(d [ A e t ) ,
D t(x v x*) = d t V dt* and
so t h a t
X V x'
and
w ith th e r e p r e s e n ta tio n
x a x*
d{ >
> dp-x
Di (x a x ' ) = d t ^ d i* ( i = 1 , # . . , n - l ) ,
have th e r e p r e s e n ta t io n s
- 38 -
X V X» =
[(d t
X A x ’ =
|^ ( d i A d i ’ ) A e i j
( ')
V dt')
A e J
1
r
T h is e s ta b lis h e s
THEQEIEM 6 .
The l a t t i c e
[D]p
i s isom orphic t o th e l a t t i c e o f a l l
fo rm a l e x p re s s io n s (1 ) w ith th e com p o sitio rs d e fin e d by ( 2 ) , o r e q u iv a le n tly
to th e sub l a t t i c e of
th a t
c o n s is tin g o f elem en ts
(d ^ , . . . ,d p _ x )
such
d% ^ ## # ^ dp_ X•
fra c z y k [29] has c h a ra c te r:.z e d P o s t a lg e b r a s a s fo llo w s :
a d is trib u tiv e la ttic e
a Boolean sub l a t t i c e
L w ith
0,1
i s a l o s t a lg e b r a i f f i t has
D and a sub c h a in
E
w ith elem en ts
0 — Oq k, ex K m• 9 K ep_ X — 1
such t h a t ev ery elem ent has a
X=
w ith d x ,* ..,d n _ x f D.
u niq u e r e p r e s e n ta ti o n
X
^ ®l)^
dp_ X
Thus i n view of Theorem 6, our d e f i n i t i o n i s
e q u iv a le n t to th o s e a p p e a rin g
in th e l i t e r a t u r e .
Chang and Horn
[I]
d e fin e a g e n e r a liz e d P o s t a lg e b ra a s th e l a t t i c e of a l l c o n tin u o u s
fu n c tio n s on a t o t a l l y d is c o n n e c te d compact H aUsdorff sp ace to an
a rb itra ry
w ith
0,1
c h a in w ith th e d i s c r e t e to p o lo g y .
I f we c o n s id e r o n ly c h a in s
t h i s can be shown to be e q u iv a le n t t o saying t h a t a l a t t i c e
L
i s a g e n e r a liz e d P o st a lg e b r a i f f i t i s th e co p ro d u ct o f a Boolean a lg e b r a
and an a r b i t r a r y c h a in .
The o p e r a ti ohs D x ,...,D p _ x
d e fin e d in any l a t t i c e
and th e c o n s ta n ts
[D ]p.
We nov/ show t h a t i f
©o ^© i> ♦••^ 6n- 1
[D]p
i s a P o s t a lg e b ra
o r p seu d o -P o st a lg e b r a th e n a pseudo—complement — and a r e l a t i v e p seu d o complement P> a r e
a ls o d e fin e d .
The e x is te n c e o f a pseudo—complement
i n P o s t a lg e b r a s was n o te d by E p s te in [ 3 ] .
- 39 -
THEOREM 7*
E very p seu d o -P o st a lg e b ra [D]p
(3 )
Di (~x) = -D x(x)
(a)
Di ( x D y )
in
D and
Proof#
( i = 1, 0. . , n - l )
=
The o p e ra tio n s i n [D]p
I s a pseudo-B oolean a lg e b r a w ith
[D j(x)
D
,
D j(y )]
1 ,...,n - l) .
(i =
a r e e x te n s io n s o f th e c o rre sp o n d in g o p e ra tio n s
E,
S u b s titu ti n g
have to prove t h a t
y = 0
i n (A)
we o b ta in
(3 );
(A) a c t u a l l y d e fin e s an o p e ra tio n i n
o p e ra tio n a g re e s w ith th e r e l a t i v e pseudo—complements
f in a lly th a t fo r
we th e r e f o r e
a ry
(5 ) z ^ x D y
x ,y ,z <? [D]p
<f=>
in
[D ]p,
th a t th is
D and
E,
and
we have
Z A x ^ y
I f we r e p r e s e n t th e ri^^^ht-hand s id e o f (A) by d^
(i = 1 ,...,n - l) ,
th e n f o r
D i(z ) = d^ ( i = 1 , . . • ,nr-1 ) ,
s in c e
th e elem ent
dx
* =
^ ^ n -i>
(d [
a
e^)
we have
hence ( a ) d e fin e s an o p e ra tio n i n [D]p •
We e a s i l y check t h a t (A) h o ld s f o r x ,y e D
and f o r
x ,y e E ,
so th e
d e fin e d o p e ra tio n a g re e s w ith th e r e l a t i v e pseudo-com plem ents in D and E.
I t rem ains t o check ( 5 ) .
If
z (C X D y ,
th e n f o r each
i = 1 , . • • ,n - 1 ,
D [(z ) ^ D i(x D y )
hence
;
we have
D i(z
and so
D i(x ) D D [(y)
za x ^ y*
A
x ) =: D i(x )
D t(x ) ^
A
C onversely i f
D j(z ) A D j(x ) < D j(y )
so t h a t f o r each
D i(y )
z a x ^ y,
( i = 1,...,n - 1 ),
th e n f o r each
,
j ^ i,
D i(z ) < D j(z ) ^ D j(x ) D D j(y )
;
th u s we have
D i(z ) ^ D i(x D y )
(i = 1 ,...,n - l) ,
- 40 -
j
we have
2 ^ x D y*
frcxa w hich we conclude t h a t
T h is com pletes th e p ro o f
of Theorem 7*
Having chosen th e l a t t i c e s
D i , . . . , D n _ i , 6o, e i , # . . , e n _ i
L = [d ] p
D and
\L ,
V,
L
th e o p e ra tio n s
v,
,
i n a P o st a lg e b r a o r p seu d o -P o st a lg e b ra
a r e u n iq u e ly d e te rm in e d .
pseudo—P o s t a lg e b ra
E,
I t i s c o n v en ie n t t o re g a rd a
a s an a lg eb jraic s t r u c t u r e
Af Z)
f f Dx , # . # , D p_ x
•
I f we a g re e to re g a rd th e one—elem en t a lg e b r a a s a P o s t a lg e b r a a l s o ,
th e n we a r e a b le to g iv e a r a t h e r sim p le e q u a tio n a l c h a r a c te r iz a ti o n
o f P o s t a lg e b ra s and p se u d o -P o st a lg e b ra s *
Such a c h a r a c te r iz a ti o n
was g iv e n i n th e case of P o s t a lg e b r a s by T raczyk [ 3 0 ] .
Note t h a t th e
c l a s s o f pseudo-B oolean a lg e b r a s i s e q u a tio n a lly d e f in a b le (s e e e . g .
[ 13 ] P .1 2 A ).
THEOREMS.
In o rd e r t h a t
L=
^ L , v , A , D , - , D x , . . . ,Dp_ x ,
sh o u ld be a pseudo—P o s t a lg e b r a i t i s n e c e s s a ry and
s u f f i c i e n t t h a t ^ L , v , A , D , —^
th e fo llo w in g i d e n t i t i e s h o ld
be a pseudo-B oolean a lg e b r a and t h a t
( i = 1 ,...,n - l)
:
(6 )
Di (x V y ) =
D t(x ) V D i(y )
(7)
Di (x Ay ) =
D i(x ) A D i(y )
(8 )
Di(xDy)
=
A jfx [D j(x ) D D j(y ) ]
( 9)
D i(-x )
=
- Dx (x )
( 10)
D i(D k (x ))
=
D%(x)
( 11)
D i(e k )
=
jo
(12)
X
=
(k = 1 , . . . , n - l )
o th e r i^ s e
(k = 0 , 1 , . . . , n - l )
(D i(x ) A e i )
- 41 -
I f we add th e i d e n t i t y
( 13 )
D i(x ) V - Dx(x) = 1
we o b ta in th e n e c e s s a ry and s u f f i c i e n t c o n d itio n t h a t
L be a P o s t
a lg e b r a .
P ro o f.
In each c a se th e n e c e s s ity o f th e c o n d itio n i s c l e a r .
p ro ve s u f f ic ie n c y we o b serv e f i r s t t h a t by (IO )
( i = 1 , . . . , n —1 )
mapping
have a common image
Dx : L
D say*
S e ttin g
L,
th e n
a sub c h a in
and
and so by ( I I )
of
L.
a Ae i ^ b v e j
]^(12)
w ith
t h a t a p p ly in g
Di
Theorem 4 ,
is
L
hence
D
x = ej i n (12) and a p p ly in g ( I I ) we see
ep_x = 1
0 5/ 1
th e
x = 1
and
x
if L
th e
a re a l l d i s t i n c t ;
ej
th e coproduct [D]p
g e n e ra te s
L;
weo b ta in
to check t h a t th e o p e ra tio n s
D, —, D x,#..,D n_x
a s th o s e in tro d u c e d i n [D]p ;
b u t th e
in
E
is
if
e^ ^ e j ,
D and
Dj,
i n (1 2 )
th u s
or i > j ,
w ith th e a id of ( 1 1 ) .
of
ej_x ^ ej
i s n o t th e o n e-elem ent a lg e b r a
a ,b e D, th e n e i t h e r
a (C b
th a t
= Oq
Bq = 0 ;
DUE
we o b ta in
is
o r a Boolean s u b l a t t i c e i f (13)
( j = 1 , . * . , n —1 ) ; a ls o s u b s t i t u t i n g
r e s p e c tiv e ly
D[
( 9)
By (6 ) -
D i s a pseudo-B oolean homomorphism ;
a pseudo-B oolean s u b l a t t i c e o f
h o ld s .
th e mappings
To
so
Thus by
E . I t only rem ain s
of
[D]p
L
a re th e same
a r e u n iq u e ly
d eterm in ed by th e re q u ire m e n t t h a t ( 6 ) , ( 7 ) , ( I O) , ( I I ) h o ld , and th e
o p e ra tio n s
( 8 ) , ( 9);
D, — i n [D]p
a r e c h a r a c te r iz e d p r e c i s e l y by th e c o n d itio n s
th u s th e p ro o f i s com plete#
V/e o b serv e t h a t th e u se o f
Theorem 4 co u ld be a v o id ed i n fa v o u r o f Theorem 6#
R e p re s e n ta tio n Theory.
we can c o n s tr u c t new
p a r t i a l l y o rd e re d s e t*
B eginning w ith th e s im p le s t P o st a lg e b ra E = [ 2 ] p ,
p seu d o -P o st a lg e b r a s a s f o llo w s .
C o n sid er -the s e t
- 42 -
E^^^ o f a l l
L et
A be any
f a m ilie s
e A ^
^
( 14 )
which s a t i s f y th e fo llo w in g c o n d itio n f o r a l l
a^ ^ a^
w henever
A*/i e A :
A ^ /i
We d e fin e th e o p e ra tio n s on
V
=
la .^
V
=
{a^ A
=
fin f
W x
r> ^ ) i
)l
=
ei
=
iDi(a^ ,)i
( i — 1,# * . , n—1 )
=
N li
( i — 0 , 1 , # * # , n—1)
U sing Theorem 8 we can show t h a t
a lg e b r a :
E^^^
i s in d e e d a pseudo—P o s t
i t i s on ly n e c e s s a ry to v e r i f y t h a t
E^"^^
i s a pseudo-B oolean
a lg e b ra and t h a t th e i d e n t i t i e s (6 ) — (1 2 ) a r e s a t i s f i e d *
most p a r t th e v e r i f i c a t i o n s a r e t r i v i a l ;
u s e f u l to n o te t h a t
I f we d e n o te by
x)
=
o f elem en ts
E^"^^ =
of
i n p ro v in g (8 ) and (9 ) i t i s
D [(x )
f o r any s u b s e t
2^^^ th e s e t o f a l l f a m ili e s i n
c o n d itio n ( 1 4 ) , th e n we see t h a t
th e P o s t a lg e b r a
If
E^
2^
A
of
E.
w hich s a t i s f y
2^”^^ i s j u s t th e pseudo-B oolean a lg e b r a
E^"^^ i n v a r ia n t u n d er each Di;
[2^‘^ ^]jj*
F or th e
a c c o rd in g ly
A
i s i d i s c r e t e l y o rd e re d th e n
of a l l
E -v a lu e d fu n c tio n s on A,
we may w rite
E^"^^ i s sim ply
w ith th e
o p e ra tio n s d e fin e d p o in t-w ise *
I t i s easy to see t h a t i f
D and
th e n any pseudo-B oolean homomorphism
to a p seu d o -P o st homomorphism
h (x ) =
^
D*
a r e pseudo-B oolean a lg e b r a s ,
^ :
D -»
h : [D]n -»|p*]n>
(D i(x )) e i
—43 —
•
h as a u n ique e x te n s io n
g iv e n by th e e q u a tio n
T his rem ark s im p lif ie s th e p ro o f of th e n e x t theorem w hich in th e
c a se o f P o s t a lg e b ra s i s
w ell-know n.
THEOREM 9*
be an elem ent o f a p seu d o -P o st a lg e b r a
L et a / 1
F o r some p a r t i a l l y o rd e re d s e t A th e r e e x i s t s a homomorphism
such t h a t
h (a )
h : L
such t h a t
E
P r o o f.
L et
1»
If
L
1 i f f some component
a e D.
x ç L
D i(x ) i s d i s t i n c t
from 1 , we
a e D.
0 ( a ) / 1 , o r , i n c a se
0 : D -*• 2
such t h a t
D i s B oolean, a Boolean homomorphism
0 (a ) ^ 1,
The p o s s i b i l i t y o f th e second
c o n s tiu c tio n fo llo w s from th e Boolean prim e i d e a l th e o re m ;
i s c a r r i e d o u t a s fo llo w s .
D,
L et
p a r t i a l l y o rd e re d by in c lu s io n .
^ ^ (x ) =
fo r a l l
p D A, X € p
D.
Thus
y e p ;
p
x
l e t us
j € K
iff
i n one d i r e c t i o n t h i s i s t r i v i a l ;
x D y / A
th e n th e f i l t e r g e n e ra te d by
n o t c o n ta in in g
0 i s a pseudo—Boolean homomorphism.
— A4" —
D -> 2^^ ;
to showing t h a t
i s d i s j o i n t frcan th e i d e a l g e n e ra te d by
ex tended to a prim e f i l t e r
i s th e
•
T h is amounts
im p lie s
I f 0^
A
d e f in e
, A
f o r th e o th e r we n o te t h a t i f
A U [x}
A,
(j> i s o b v io u sly a l a t t i c e homomorphism
prove t h a t i t p re s e rv e s
th e f i r s t
A be th e s e t o f a l l prim e f i l t e r s
c h a r a c t e r i s t i c f u n c tio n o f th e s e t
mapping
Thus
A a pseudo—Boolean homomorphism 0 ;• D -> 2^^^
such t h a t
Ih e
is
A ccording t o th e rem ark made ab o v e, we have o n ly t o
c o n s tr u c t, f o r s u i t a b l e
of th e l a t t i c e
E^^^
h (a ) / 1 .
see t h a t i t i s s u f f i c i e n t to prove th e theorem when
suppose
h : L
i s a P o s t a lg e b r a th e r e e x i s t s a homomorphism
L =[p]n •U sing th e f a c t t h a t an elem en t
d i s t i n c t from
L.
y
y
and so may be
by Theorem 1#
F o r any two d i s t i n c t
e lem en ts o f
D "Üiere e x i s t s a prim e f i l t e r
n o t th e o th e r ;
hence
0
i s one—one
A w hich c o n ta in s one b u t
and so in p a r t i c u l a r
0 (a ) 4 1*
T h is com pletes th e p ro o f .
The above a rg u m e n t. i n f a c t y ie ld s th e fo llo w in g r e p r e s e n ta t io n theorem ,
f i r s t proved i n th e case of P o s t a lg e b r a s by Wade [ 3 l ] :
IHiiiOIffiM 1 0 .
th e form
Every P o s t a lg e b r a can be embedded in a P o s t a lg e b r a o f
E"^.
E very pseudo—
do—P
] o st a lg e b r a can be embedded in a pseudo-r
P o s t a lg e b ra o f th e form
E (A)
U sing known r e p r e s e n ta tio n theorem s f o r Boolean and pseudo—Boolean
a lg e b r a s we can prove r e p r e s e n ta tio n theorem s o f a to p o lo g ic a l c h a r a c te r
f o r P o s t a lg e b r a s and pseudo—P o s t a lg e b r a s .
Such a r e p r e s e n ta tio n
theorem was o b ta in e d by E p s te in [3 ] i n th e
case o f P o s t a lg e b r a s .
Vfe om it th e p ro o fs s in c e we have no need o f such theorem s i n th e s e q u e l.
THEOPuEIvI 1 1 .
Every P o s t a lg e b ra i s iso m o rp h ic to th e P o s t a lg e b r a o f
c o n tin u o u s n-v a lu e d f u n c tio n s on some t o t a l l y d is c o n n e c te d comipact
H au sd o rff sp a c e .
Every n seu d o -P o st a lg e b r a i s iso m o rp h ic to a p seu d o -
P o st a lg e b r a o f low er sem i—con tin u o u s n-v a lu e d fu n c tio n s
on some comyaot
T i- s p a c e .
2.
A p p lic a tio n s to p r o p o s itio n a l c a l c u l i
I n t h i s S e c tio n we c o n s id e r th e p r e p o s it io n a l c a lc u lu s w hich i s
d e term in e d by ta k in g a s p r im itiv e tr u t h - f u n o t io n s th e o p e ra tio n s
V,
*f
o f th e P o s t a lg e b r a
o f th e s in g le elem ent
v a lid i f f
v * (a ) = 1
1,
E.
eo , 6 x , . • • , ep _
The s e t o f d e sig n a te d t r u t h —v a lu e s w i l l c o n s is t
e ^ .^ = 1 .
Thus
f o r every v a lu a tio n
- 45 —
a fo rm u la
t
,
a
is c la s s ic a lly
and i n t u i t ! o n i s t i c a l l y
v a lid i f f
v^(a)=* 1
(A e A)
f o r e v ery monotone fa m ily
^
C le a rly e v e ry i n t u i t i o n i s t i c a l l y v a lid fo rm u la i s c l a s s i c a l l y v a l i d ,
b u t n o t c o n v e rs e ly .
There i s no e s s e n t i a l lo s s of g e n e r a l i t y i n
r e s t r i c t i n g our a t t e n t i o n to th e above system o f t r u t h —fu n c t io n s , s in c e
i t may be shown t h a t a fo rm u la w ith any c o n n e c tiv e s w h a tev e r i s
i n t u i t i o n i s t i c a l l y e q u iv a le n t (a n d , a f o r t i o r i , c l a s s i c a l l y e q u iv a le n t)
to some fo rm u la c o n ta in in g on ly th e
c o n n e c tiv e s
v , At Df etc©
We s h a l l f i r s t g iv e a more a lg e b r a ic fo rm t o th e d e f i n i t i o n s
o f c l a s s i c a l and i n t u i t i o n i s t i c v a l i d i t y .
set
S
o f fo rm u las a s an a lg e b r a
V,
S
D,
1^1 >• • •
1 * ®o >®i >• • • > ® n - i ^
i s i n f a c t th e a b s o lu te ly f r e e a lg e b ra on th e s e t
g e n e r a to r s .
V
Thus we c o n s id e r th e
: V
E
t
V of fre e
There i s a one—one co rresp o n d en ce betw een v a lu a tio n s
and homomorphisms
h ; S -> E
d e term in e d by th e r e l a t i o n
h ( a ) = v * (a )
;
s i m ila r ly th e r e i s a one—one co rresp o n d en ce betw een monotone f a m ili e s
(v ^ )^ ^ ^
and homomorphisms
h ; S -►E^'^^ d e term in ed by th e r e l a t i o n
h(oc) =
^ A
•
Hence th e d e f i n i t i o n s of v a l i d i t y can be r e s t a t e d a s fo llo w s :
(i)
a form ula
h : S -> E
a
c a rrie s
S
i s c l a s s i c a l l y v a l i d I f f each homomorphism
a
i n t o th e u n i t e le m e n t;
(ii)
a fo rm u la
i s i n t u i t i o n i s t i c a l l y v a l i d i f f f o r each p a r t i a l l y o rd e re d s e t
homomorphism
h : 8
E^^^ c a r r i e s
—
a
—
i n t o th e u n i t e le m e n t.
a e S
A
each
U sing th e r e p r e s e n ta tio n th e o iy o f S e c tio n 1 we can c h a r a c te r iz e
c l a s s i c a l and i n t u i t i o n i s t i c v a l i d i t y a s fo llo w s ;
THiiiOEEM 12*
A form ula i s c l a s s i c a l l y (resp * i n t u i t i o n i s t i c a l l y ) v a l i d
i f and o n ly i f i t i s c a r r ie d i n t o th e u n i t elem ent by ev ery homomorphism
from
S
i n t o a P o s t a lg e b r a (resp * pseudo—P o s t a lg e b r a )
P ro o f.
Suppose th e r e e x i s t s a homomorphism
a p seu d o -P o st a lg e b r a
[u]n
e x i s t s a homomorphism
g
such t h a t
from
g (h (a )) / 1#
w ith
f(a )
1,
such t h a t
[u]n
so
a
[u]n
such t h a t
S
in to
% Theorem 9 th e r e
in to some p seu d o -P o st a lg e b ra
i s a homorphism from
S
in to
i s not in tu itio n is tic a lly v a lid .
S im ila r ly i f th e r e e x i s t s a homomorphism
a lg e b r a
from
h (a ) / 1.
Then f = go h
and
h
[D]n*
h (a ) / 1 ,
h
th e n
T his p ro v es th e Theorem i n one d i r e c t i o n ;
from
a
8
in to a P ost
is not c la A c a lly v a lid .
th e o th e r
d ire c tio n i s t r i v i a l .
V/e s h a l l now c h a r a c te r iz e c l a s s i c a l and i n t u i t i o n i s t i c v a l i d i t y
by means o f axioms and r u l e s of in f e r e n c e .
Thus we c o n s id e r th e
fo llo w in g axiom schem es, in which th e e x p re s s io n
a b b r e v ia tio n f o r
th e fo rm u la
(A1)
o
(a D /5 )>
3
Da )
(A2)
a D Os D Y ) 0 . ( a D
(A3)
» 3 Os 3 (« *
(A4-)
(a * /S) D a
(A3)
(a ^ f i ) 0 0
(a
/S))
(a 6 )
a O {a W 0 )
(A7)
/9 D (oc V /?)
(a 8 ) ( a O f ) o ( . ( 0 O r )
(A9)
(A10)
D a)
3 ((a V /9 ) D y ) )
a O (,-aO 0 )
{ a O 0 ) O ((a O - 0 ) O - a )
—47 —
y)
a ®^
:
i s an
(A11
D ;(a
(A12
D l(a A /9 ).S o D i(a ) a D i(^ )
(A13
D i(*
v /9 ).3 .D i(a )
V DiO?)
A jii ( D j ( a ) D D j( ^ ) )
-Di(a)
(A14
(i =
(i
= 1 , # # # , n —1 )
( i,j =
n—1 )
(A15
D i(D j(a ))# a . D j(a)
(A16
D i( e j)
(1 ^ 1 ^ j ^ n -1 )
(A17
-D i(ej)
(O ^ j < i ^ n-1 )
(A18
a . e . V J :J (D i(a ) ^ e i )
D i(a ) V - D i(a )
(A19
We ta k e (A1 ) — (A18)
a s axiom schemes
p r e p o s itio n a l c a lc u lu s , an d (A1) - (A19)
c l a s s i c a l p r e p o s itio n a l c a lc u lu s .
f o r th e i n t u i t i o n i s t i c
a s axiom schemes f o r th e
I n b o th c a s e s th e r u l e s o f in f e r e n c e
a re modus p o n e n s'
a
to g e th e r
a D
w ith th e r u l e
^ n -i (oc)
V/e o bserv e t h a t th e axioms (A1 ) - (AIO)
a r e th e u s u a l axioms
f o r H e y tin g 's p r e p o s itio n a l c a lc u lu s , w h ile axiom s (A11) - (A19)
c o rre sp o n d i n an obvious way to th e e q u a tio n s (6 ) - (1 3 ) of S e c tio n 1*
The p a ssa g e from i n t u i t i o n i s t i c n—v a lu e d lo g ic t o c l a s s i c a l n -v a lu e d l o g i c ,
by th e a d d itio n # f (A19 ) , sh o u ld be compared w ith th e p assag e from
H eyting *s p r e p o s itio n a l c a lc u lu s to th e c l a s s i c a l two—v alu ed p r e p o s it io n a l
c a lc u lu s
th e a d d itio n o f th e law o f ex clu d ed middle*
— 48 —
We now prove th e c o m p leten ess o f Die se a x io m a tiz a tio n s :
THEOREI'd 13*
A fo rm u la i s i n t u i t i o n i s t i c a l l y v a l i d i f f I t i s d e riv a b le
frcan axioms (A1 ) - (A1 8) ;
a form ula i s c l a s s i c a l l y v a lid i f f i t i s
d e r iv a b le frcan axiom s (A1 ) - (A19)*
Proof#
In view o f Theorem 8 , e v e ry p se u d o -P o st a lg e b ra i s a pseudo—
B oolean a lg e b r a
i n w hich th e e q u a tio n s (6 ) - (13) h o ld .
homomorphism from
S
axiom s (A1 ) — (a18)
i n t o a pseudo—P o s t
in to a pseudo—P o st a lg e b r a
i n t o th e u n i t elem ent#
a lg e b ra c a r r i e s
e lem en t, th e n i t a ls o c a r r i e s
i t c a rrie s
th e
a
a
must c a r r y each o f th e
I f a homomorphism from
and
in to th e
a D (3
Hence
S
in t o th e u n i t
u n i t e lem en t;
i n t o th e u n i t e le m e n t, th e n i t must c a r r y
u n i t e le m e n t.
Hence every
s i m ila r ly , i f
D n -i(a )
in to
by Theorem 12 i t fo llo w s t h a t any fo rm u la
d e r iv a b le from (A1 ) - (A18) i s i n t u i t i o n i s t i c a l l y v a lid #
C onsider th e r e l a t i o n
a
and
a ^ /3
/? w henever th e fo rm u la
w hich h o ld s betw een two fo rm u las
(a s /?)
i s d e r iv a b le from (A1 ) — (A 18).
I t i s e a s y t o show t h a t t h i s r e l a t i o n i s a congiruence on
exanç)le, i f
a ^
Dn-i(cc D /9 )
and
h o ld s th e n
Dn_i (/? D a )
t h a t f o r each
i
D i ( a ) '^ D t 0 9 )
(i = 1 ,...,n - l) .
we
th a t
^ D a
a r e d e r iv a b le ;
Di(%) D D[(/9)
b u t th e n by (A13) we see
S in ce t h e r e l a t i o n
8 /^#
a r e d e r iv a b le , so t h a t
a# i s a co n g ru en ce,
The axiom s (A1 ) — (A19) g u a ra n te e
i s a pseudo—P o s t a lg e b r a , i n view o f Theorem 8*
Theorem 12 th e n a t u r a l homomoi^hism
S -> 8 /^
v a lid fo rm u la in to th e u n i t elem ent o f
8/^#
c a r r i e s ev ery i n t u i t i o n i s t i c a l l y ;
From t h i s we i n f e r t h a t ev ery
i n t u i t i o n i s t i c a l l y v a l i d fo rm u la i s d e r iv a b le from ( A1 ) — (A 18).
p ro v es th e
For
a r e d e r iv a b le , so t h a t
and D [(^ ) D D i(« )
may form th e q u o tie n t a lg e b r a
S /^
and
a D^
S.
f i r s t p a r t of our Theorem;
s im ila r#
— 49 "*
T his
th e p ro o f o f th e second p a r t i s
v/e now e s t a b l i s h seme sim ple p r o p e r tie s o f th e c l a s s i c a l and
i n t u i t i o n i s t i c n—v a lu ed p r e p o s i t i o n a l c a lc u li#
a mapping
a -*• a*
F i r s t we o b ta in
o f th e fo rm u la s o f th e n—v a lu e d p r o p o s itio n a l
c a lc u lu s i n t o fo rm u las o f th e two—v a lu ed p r e p o s iti o n a l c a lc u lu s such
th a t
i s c l a s s i c a l l y r e s p . i n t u i t i o n i s t i c a l l y v a lid i f and o n ly
a
if
i s c l a s s i c a l l y re sp # i n t u i t i o n i s t i o a l ] y v a lid ( i n th e o rd in a ry
tw o -v alu ed sen se )#
V/e a rra n g e th e p r e p o s itio n a l v a r i a b l e s o f th e n—v a lu e d p r e p o s iti o n a l
c a lc u lu s i n a sequence
P , %,#**,
p r e p o s itio n a l v a r i a b l e s
and s i m i l a r l y we a rra n g e th e
o f th e two—v a lu e d p r e p o s itio n a l c a lc u lu s i n
a sequence
( 15)
The s e t
Pi,
Sq
•••> P n - i #
Qi
•
of fo rm u la s o f th e tv/o-valued p r o p o s i ti o n a l c a lc u lu s i s
of th e sequence ( 15 ) o f
th e l e a s t s e t w hich c o n ta in s each member
p r e p o s itio n a l v a r ia b le s and
w hich c o n ta in s
-a
and
w henever i t c o n ta in s
V/e d e fin e mappings
a
5 i : S ■* Sq ( i =
ccD /9
1 ,..,,n - l)
and
by in d u ctio n ,
a s fo llo w s ;
& l ( p )
5
l(q)
«
=
P i
A
# , .
A
p i
=
Qi
A
. . .
A
qi
•
•
8 l(a v /9 )
=5 8 1( a ) V 8iC #)
8 i(a A ^)
= 8 1 (a) A 8 i(j9)
8 i(« D j9 )
= A jii( S j(a ) D 8 j(^ ))
8i(-a )
=
~ 81 (a )
8 i( U j( a ) )
=
S j( a )
8 l(®j )
=
I
Q
( j = 1 ,# ..,n - l)
i ^ j
- 50 -
0 ,1 ,# ..,n - l)
Here
and
1
and
0
Pi A —Pi
THEOREM 14®
may be ta k e n to den o te f o r example th e fo rm u las
Pi D pi
re s p e c tiv e ly .
A fo rm u la
a
o f th e n—v alu ed p r o p o s iti o n a l c a lc u lu s i s
c l a s s i c a l l y ( r e s p . i n t u i t i o n i s t i c a l l y ) v a l i d i f and on ly i f th e
co rre sp o n d in g form ula
8 n -i(% ) o f th e two—v a lu e d p r o p o s itio n a l c a lc u lu s
i s c l a s s i c a l l y (resp o i n t u i t i o n i s t i c a l l y ) v a l i d .
P r o o f.
F i r s t we show t h a t i f
B i s any pseudo—B oolean a lg e b r a , th e n
th e r e i s a one-one co rresp o n d en ce betw een homomorphisms
and homomoiphisms
(1 6 )
f ; S -> [B]n >
D i(f(a ))
In d e e ü , i f we a r e
=
g iv e n a
determ ined
by th e i d e n t i t y
(i = 1 ,...,n - l)
h (8 i(a ))
homomorphism
h : Sq -* B
h ; So -> B,
.
th e n we e a s i l y
prove by in d u c tio n t h a t
h (S x (tt)) ^
^ k (8 n -i(o ())
hence th e r e c e r t a i n l y e x i s t s a mapping
5
f ; S -*> [s]n
s a tis fy in g ( l6 ) ,
name]y
f(« )
=
V"-J
h (8 i(a ))e i
;
fu rth e rm o re i t fo llo w s from ( l6 ) t h a t such a mapping i s homomorphic,
C o n v ersely , suppose we a r e
So
g iv e n a homomorphism
f : 8 -* [B]p ;
s in c e
i s th e a b s o lu te ly f r e e a lg e b ra on th e g e n e r a to r s ( 15 ) , th e r e e x i s t s
a homomorphism
h : 8o
B
s a tis fy in g
h ( p i ) = Di f ( p )
(i =
h ( q i) = Di f ( q )
(i
n. * «•
1 ,...,n - l)
= 1 ,..,,n - l)
-
;
by in d u c tio n we e a s i l y v e r i f y t h a t
-
51
h
-
s a t i s f i e s (1 6 ) .
Suppose now t h a t
homomorphism
a e S
h ; So -> B (w here
a homomorphism f : 8
[B]n
f ( a ) = 1 , and so by ( l 6 )
su ch t h a t (16) h o ld s .
5 n - i( a ) i s
F o r any homomorphism
8 n-i(% )
f : 8
[B]n
B i s pseudo—B o o lean ), th e r e e x i s t s a homomorphism
such t h a t ( l é ) h o ld s .
Thus
Thus
Theorem 12
C o n v erse ly , suppose t h a t
in tu itio n is tic a lly v a lid .
a
S ince
F o r any
B i s pseudo—B o o lean ), th e r e e x i s t s
h ( S n - i( a ) ) = 1 .
in tu itio n is tic a lly v a lid .
(where
i s in tu itio n is tic a lly v a lid .
h : So -» B
h ( S n - i ( a ) ) = 1 , i t fo llo w s t h a t
i s i n t u i t i o n i s t i c a l l y v a l i d by Theorem 12.
f(a ) = 1 .
T h is p ro v e s th e p a r t
o f th e Theorem w hich co n cern s i n t u i t i o n i s t i c v a l i d i t y ;
th e o th e r p a r t i s
proved s i m i l a r l y .
Theorem 14 im m ediately g iv e s a s o lu tio n o f th e d e c is io n problem f o r
th e c l a s s i c a l and i n t u i t i o n i s t i c n—v a lu e d p r e p o s itio n a l c a lc u lu s *
As
a n o th e r a p p l i c a t i o n , we e s t a b l i s h an alo g u es o f two w e ll—Icnown theorem s
c o n ce rn in g th e i n t u i t i o n i s t i c (two—v a lu ed ) p r e p o s i tio n a l c a l c u lu s .
(F o r r e f e r e n c e s , s e e Rasiowa and 8 ik o r s k i [ 1 3 ] . )
THEOREM 15 .
If
c a l c u i u s , th e n
a or
a
a,nd ^
a V /9
a r e any fo rm u la s o f th e n-v a lu e d p ro p o s it^ o n a l
i s i n t u i t i o n i s t i c a l l y v a lid i f and o n ly i f e i t h e r
is in tu itio n is tic a lly v a lid .
The form ula
—a
is c la s s ic a lly
v a lid i f and on ly i f i t i s i n t u i t i o n i s t i c a l l y v a l i d .
P ro o f.
If
a V
i s i n t u i t i o n i s t i c a l l y v a l i d , th e n by Theorem 14 th e
form ula
S n ~ i(a ) V 8 n - i = S n _ i(a V /? )(o f th e tw o -v alu ed p r e p o s iti o n a l
c a lc u lu s ) i s i n t u i t i o n i s t i c a l l y v a l i d .
Hence e i t h e r
8 n -i(% )
8p«i(/9)
i s i n t u i t i o n i s t i c a l l y v a l i d , and s o , a g a in u s in g Theorem 1 4 , e i t h e r
is in tu itio n is tic a lly v a lid .
Theorem 14 th e fo rm u la
If
a or
—a i s c l a s s i c a l l y v a l i d , th e n by
- S i ( a ) = S p - i ( —a ) i s c l a s s i c a l l y v a l i d .
- 32 -
Hence
t h i s formula, i s i n t u i t i o n i s t i c a l l y v a l i d , and so
—a
is in tu itio n is tic a lly
v a l i d by Theorem 14»
Our f i n a l r e s u l t co n cern s th o s e fo rm u la s o f th e n -v a lu e d p r e p o s it io n a l
c a lc u lu s w hich a r e a ls o fo rm u la s of th e two—v a lu e d p r e p o s i t i o n a l c a lc u lu s .
Thus l e t
Kp ( r e s p . Jp ) d e n o te th e s e t o f a l l c l a s s i c a l l y (resp *
i n t u i t i o n i s t i c a l l y ) v a lid fo rm u la s o f th e n—v a lu e d p r e p o s it io n a l c a lc u lu s
w hich c o n ta in only th e c o n n e c tiv e s
THEOREM 16.
F o r each
v a l i d fo rm u la s of
Kg , Kq , . . .
n ^ 2,
th e
set
Jp
c o in c id e s w ith th e s e t o f
H eyting *s pro p o s i t i o n a l c a lc u lu s .
The s e t s
form a s t r i c t l y d e c re a s in g c h a in whose f i r s t member i s th e s e t
o f v a lid fo rm u las of th e
Pro o f*
v , * ,1 3 , — *
c l a s s i c a l two—v a lu e d p r e p o s iti o n a l c a lc u lu s ©
The f i r s t a s s e r t i o n fo llo w s from th e f a c t t h a t ev ery pseudo—Boolean
a lg e b r a i s a pseudo-B oolean su b a lg e b ra o f a pseudo—P o s t a lg e b r a o f o rd e r
n,
w h ile , on th e o th e r hand, e v e ry pseudo—P o st a lg e b r a i s i t s e l f pseu d o -B o o lean .
S im ila r ly to show t h a t
Kg D Kg D . . .
each P o s t a lg e b r a o f o rd e r n—1
a lg e b r a o f o rd e r
n.
, i t i s o n ly n e c e s s a ry to o b serv e t h a t
i s a pseudc—Boolean su b a lg e b ra o f a P o st
The s t r i c t n e s s o f th e in c lu s io n s i s o b ta in e d by
c o n s id e rin g th e fo rm u las
Yn =
i t i s e a s y to se e t h a t
V
(pL = P j)
1^ i < j ^ n
(n = 3 , 4 , . . . )
;
Yn ^ Kp-i - Kn (o f* Godel [ 5 ] ) •
We n o te t h a t th e in c lu s io n s rem ain s t r i c t even i f we r e s t r i c t
to fo rm u las c o n ta in in g o n ly im p lic a tio n .
if
a U /?
i s an a b b r e v ia tio n f o r
(a D
F or i f
D /3,
« i j = (p j
P i)
a tte n tio n
and
th e n th e fo llo w in g fo rm u la
( i n w hich th e term s a s s o c ia te to th e l e f t ) i s c l a s s i c a l l y v a l i d i n th e
(n-1 ) -v a lu e d p r e p o s itio n a l c a lc u lu s b u t n o t i n th e n—v a lu e d p r e p o s i t i o n a l
c a lc u lu s :
- 53 -
Yn
— ® i 2 4/ • • • U a 1 p U OC23 V
G-ode3[5] u sed th e f a c t t h a t
• • • U Kgp U
• • • U otp-g n - i ^ (^n-2 n ^ # n - i n
Yn f Kp_i - Kp
t o show t h a t H e y tin g ’ s
p r o p o s itio n a l c a lc u lu s has no f i n i t e c h a r a c t e r i s t i c m a trix ;
u s in g th e f a c t t h a t
Yn * ^ Kp-i - Kp ,
s im ila rly ,
we can v e r i f y a c o n je c tu re o f
P o r t e [ 10 ] , t h a t th e p o s itiv e im p lic a tio n a l p r o p o s itio n a l c a lc u lu s h as
no f i n i t e c h a r a c t e r i s t i c m a trix (s e e ['17])*
- 54 -
C hapter I I I : S h e ff e r F u n c tio n s
I n t h i s C hapter we c h a r a c te r iz e th e S h e ffe r f u n c tio n s o f c l a s s i c a l
n -v a lu e d lo g ic and make a sm a ll b e g in n in g on th e c o rre sp o n d in g problem f o r
in tu itio n is tic lo g ic .
S e c tio n 1 c o n ta in s two r e s u l t s on S h e ff e r f u n c tio n s i n c l a s s i c a l
n -v a lu e d l o g i c .
Theorem 1 s t a t e s t h a t a tr u t h - f u n o tio n
F
i s a S h e ffe r
f u n c tio n in c l a s s i c a l n -v a lu e d lo g ic i f and o n ly i f th e a lg e b r a <E,F>
has no s u b a lg e b ra s , automorphisms of homomorphic images a p a r t from th e
obvious t r i v i a l ones ,
VVheeler [52] (n = 3) •
T his g e n e r a liz e s r e s u l t s o f P o s t [12] (n = 2) and
The p ro o f i s based on a th eo rem o f R osenberg [14]
which c h a r a c te r iz e s com plete system s o f t r u t h —fu n c tio n s i n g e n e r a l;
th e l a t t e r theorem in tu rn g e n e r a liz e s r e s u l t s o f P o s t [12] (n = 2)
[ 33 ]
and Y a b lo n sk ii .(n - 3 ) ' •
Theorem 1 , ï/h io h i s a c o r o lla r y o f Theorem 1 ,
s ta te s th a t
F
i s a S h e ff e r f u n c tio n i f f i t g e n e ra te s a doubly t r a n s i t i v e
subgroup o f the sym m etric group on
E,
T his im proves r e s u l t s o f Salomaa [2^
and S c h o fie ld [22j ,
The problem o f com pleteness f o r system s o f tr u t h - f u n c tio n s i n
i n t u i t i o n i s t i c n -v a lu e d lo g ic has n o t been c o n sid e re d p r e v io u s ly , ev en in
th e c ase n= 2 .
fie ld .
1,
Hence we are a b le to g iv e o n ly v e ry l im ite d r e s u l t s i n t h i s
In S e c tio n 2 we show t h a t tlie re e x i s t S h e ff e r f u n c tio n s in
D r. Roy 0 . D avies k in d ly communicated to th e a u th o r a d e r iv a ti o n o f
W h ee le r’s r e s u l t from th e theorem of Y a b lo n sk ii and rem arked t h a t he had
e s ta b lis h e d Theorem 1 f o r n ^ 8 u s in g R o senberg’s th eo rem ,
5
had n o t y e t p u b lish e d 'die p ro o f o f h is th e o re m ,
- 55 “
Rosenberg
i n t u i t i o n i s t i c tw o-valued l o g i c , th e s im p le s t b ein g a f u n c tio n o f fo u r
v a ria b le s .
Theorem 3 c h a r a c te r iz e s th e S h e ff e r f u n c tio n s o f fo u r
v a ria b le s .
V/e novf d e fin e th e b a s ic n o tio n s
Two fo rm u la s
a
and
^ o f th e
to be u sed i n t h i s C h a p te r,
n -v a lu e d p r o p o s itio n a l c a lc u lu s are
o lo c c ic a lly e q u iv a le n t i f f o r e v e ry v a lu a tio n
= Y * (fi)
V * (a )
The fo rm u la s
a
and
.
,
we have
ç ^
v ’^'p^(a) =
Fo ,
we have
/9 a r e s a id to be i n t u i t i o n i s t i c a l l y e q u iv a le n t
i f f o r e v e ry monotone fa m ily
Let
v
fo r a l l
A e A.
. . . be any sy stem o f n -v a lu e d t r u t h - f u n c t i o n s .
a t r u t h - f u n o tio n F
of r
d e fin a b le in term s of
We s a y t h a t
v a r ia b le s i s c la s s ic a l ly ( i n t u i t i o n i s t i c a l l y )
Fq , Fi , , , ,
i f th e fo rm u la
Fpi
,.,
is
c la s s ic a lly ( i n t u i t i o n i s t i c a l l y ) e q u iv a le n t to some fo rm u la
% ( p i,, . . , & )
c o n ta in in g o n ly th e c o n n e c tiv e s
Fc , Fi ,
Fq , Fi , . , ,
,
The sy sto n
i s s a id to be c l a s s i c a l l y ( i n t u i t i o n i s t i c a l l y ) com plete
i f e v e ry tr u th - f u n c tio n
F i s c l a s s i c a l l y ( i n t u i t i o n i s t i c a l l y ) d e fin a b le i n terms o f
The t r u t h - f u n c t i o n
f u n c tio n
Fq
Fq , F i , , . ,
i s c a l l e d a c l a s s i c a l ( i n t u i t i o n i s t i c ) S h e ffe r
i f th e sy stem c o n s is tin g o f
Fq a lo n e i s c l a s s i c a l l y
( i n t u i t i o n i s t i c a l l y ) co m p le te,
V/e rem ark i n p a ssin g "that th e n -v a lu e d tr u t h - f u n c tio n s
V, A f D f —, D i, , , , , Dp—1 , ®o, e i , , , , , e p—%
in tro d u c e d in C hapter I I form a com plete s e t , b o th c l a s s i c a l l y and
in tu itio n is tic a lly .
—
56
—
.
1.
S h e ff e r fu n c tio n s i n c l a s s i c a l n -v a lu e d l o g i c .
I n t h i s s e c tio n we s h a l l use a lg e b r a ic lan g u ag e t h r o u ^ o u t .
an a lg e b r a <A, (w i ) . ^>
Xc X
i s com plete i f f f o r each ,
r
a l l fiin c tio n s
f ; A*' -►A a r e d e fin a b le i n te rm s o f t h e o p e ra tio n s a i ( i f l ) •
th e sy stem o f n -v a lu e d t r u t h - f u n c tio n s
i f f th e a lg e b ra
<E,%
, . . .>
%
,
We s a y t h a t
O bviously
i s c l a s s i c a l l y com plete
i s c o m p le te .
As m entioned in the in tr o d u c tio n , th e p ro o f of Theorem 1 employs a
g e n e ra l theorem o f R osenberg [12).
d e fin itio n s .
As u s u a l
we l e t
E^ =
(o n E)
i s a su b set
we v /rite
r
v f ill denote th e s e t o f tr u th - v a lu e s
®h 1^
R(xi ,
argum ents and
R if
E
R
x^)
of
®a-ch h ^ n .
when
R is an
( x i , , , , , x^)e
R.
If
h - a r y r e l a t i o n we say t h a t
f(x ^ ,
R(x^^, . , , , x ^ ) ,
An
h -a ry
E ^ , and i s c a l l e d u n iv e r s a l
R ^ f(x ^ ,
An
B efore s t a t i n g t h i s theorem we g iv e some
R (x ^ , , , , ,
xp^
R =E^;
is
a fu n c tio n o f
p re s e rv e s
xj^) a l l h o ld .
X i, , , x^
a re n o t a l l d i s t i n c t , and
p e rm u ta tio n
a
R (x i ,
f
if
w henever
h -a ry r e la tio n is t o t a l l y re fle x iv e i f
o f th e in te g e r s
f
re la tio n
R (x i,
x^)
w henever
t o t a l l y sym m etric i f f o r each
1,
h
v;e have
<=> ^ ^ ^ (j(l)^
• ••»
The c e n tr e o f a t o t a l l y sym m etric h -a ry r e l a t i o n i s the s e t o f elem ents
c
su ch t h a t
R( xi , . , , , x^ ^ ,c )
h o ld s fo r a l l
x%, . , , , x^
f E .
A r e l a t i o n i s s a i d to be c e n tr a l i f i t i s t o t a l l y r e f l e x i v e an d t o t a l l y
sym m etric and has c e n tr e
C, ^C CCE
a re j u s t th e p ro p e r n o n -n u ll s u b s e ts o f
- 57 -
,
Thus th e s in g u l a r l y c e n t r a l r e l a t i o n s
E,
The p r o je c tio n fu n c tio n s
Ej
"*• E
a re d e fin e d by
pi»^(xi, • • • , %m) —
—1 ,
, . m) .
We may now s t a t e th e theorem o f R osenberg a s fo llo w s :
F o r a f i n i t e a lg e b ra
<E,
j.> (|E | = n > 2)
to be com plete
i t i s n e c e s sa ry
and s u f f i c i e n t t h a t i n e a c h o f the fbllowinis: c a se s t h e r e
e x is ts a t l e a s t
one
wi ( i f l )
t h a t does n o t p re s e rv e t h e r e l a t i o n R :-
1° R i s
a p a r t i a l o rd e r on
2° R i s
(th e g ra p h o f) a n o n - id e n tic a l p erm u tatio n of
ÿ
the r e l a t i o n
R is
Xi
w here
n = jf”
and
elem ent has o rd e r
4°
R
E;
i s an a b e li a n group in vdiich each non—zero
p ;
i s a n o n - i d e n t i c a l . n o n -u n iv e rs a l e q u iv a le n c e r e l a t i o n on
an
^
f o r some
R is
th e
I
h- a r y c e n t r a l r e l a t i o n on
h (3 ^ h ^ n )
E (1 ^ h
0 (x i),
have th e same
n) ;
and some s u r.je c tio n
. . , pr^ 0 ( x p i l < h
*6
-8^^
<A,(w
a p e rm u ta tio n o f
0
;
E *> E ^
fo r
4 = 1, .
a t l e a s t two of th e elem en ts
p r o je c tio n in
m
0(x&
0(x^^)
E^ )»
el
^
i s an a r b i t r a r y a l g e b r a , th en a s u b se t of
A,
o r an e q u iv a le n c e on
A
A sub a lg e b ra d i s t i n c t from
A
and
w ( i € l) .
i s s a id to be p ro p e r, as i s an
autom orphism o th e r th a n th e i d e n t i t y , o r a
n o n - u n iv e r s a l, n o n - id e n tic a l
The e x is te n c e o f p ro p e r con g ru en ces on
—
A,
i s c a ll e d a sub a lg e b r a ,
autom orphism o r congruence r e s p e c tiv e ly i f f i t i s p re s e rv e d by each
co n g ru en ce.
E;
h- a r y r e l a t i o n
( t h a t i s f o r each
If
vAth g r e a t e s t an d l e a s t e le m e n t;
+ Xg = X 3 + X *
G- = <E^ + >
5° R i s
(m > o)
E
38
—
A i s e q u iv a le n t t o th e
e x is te n c e o f homomorphic images o f
A n o t iso m o rp h ic to
A
o r to th e
o n e-elem en t a l g e b r a .
THEOREM 1 • A f i n i t e a lg e b r a
< A/ o >
w ith a s in g le o p e r a tio n i s com plete
i f and on ly i f i t has no p ro p e r sub a lg e b r a s , autom orphism s o r congruences .
P ro o f.
I n view of th e above-m entioned r e s u l t o f P o s t we may r e s t r i c t
a t t e n t i o n to th e case A = E , a > 2 , Suppose w i s an o p e r a tio n o f r argum ents.
The
^ n e c e s s ity o f th e c o n d itio n i s d e a r from 2 ° , 4*^, 5° o f R o senberg’s th eo rem .
To prove s u f f ic ie n c y we have to show in c ase s
R
th en
If
w
p re s e rv e s a p a r t i a l o rd e r ^
th e n e i t h e r {uj
f o r some
or
X i,
u = w (x i,
A - [ui
p re s e rv e s
u
Xp eA -[u}
th e n , because
Xp) ^ w (u ,
Suppose
n = p"'
c y c lic groups o f o rd e r
u)
and
p,
w ith g r e a t e s t elem ent
i s a p ro p e r sub a lg e b r a .
so t h a t
Then
F o r, i f
w (u ,
u,
w ( x i , , , , , Xp) = tL
Xi ^ u , . , , , XpV^ u ,
&= < E , + >
each n o n -zero e lem en t has o rd e r p ,
Zp
th a t i f
A has a p ro p e r s u b a lg e b ra , autom orphism o r congruence ,
1°
^
1*^ —
we have
u) = u ,
i s an a b e lia n group in w hich
G,
b e in g th e d i r e c t sum o f
may be i d e n t i f i e d w ith th e sp ace
i s th e re s id u e c la s s f i e l d o f th e in te g e r s modulo
p.
If
(Z ^)^
w
m
where
p re s e rv e s
th e r e l a t i o n
X l + Xg = X3 + X4
,
th e n i t s a t i s f i e s the f u n c tio n a l e q u a tio n
w (x i, . , , , Xp) + w ( y i, , , , , yp) = w (x i + y i ,
and i t
i s n o t d i f f i c u l t to show t h a t th e s o lu tio n s a r e o f th e form
w ( x i,
where
a e (2^)”’
d e t ( l - ZA^,) = 0
X = Xo,
Xp + yp) + w (0 , , . . , 0 )
and
Xp) = a + x i A i +
A i, , , , , Ap
th e n th e e q u a tio n
a re
. . . + XpAp
m x m m a tric e s
0 = x (l -ZA^)
so t h a t
- 59 -
o v er
Z ^, If
has a n o n -zero
s o lu tio n
w ( 35b
i.e.
X
Xq
L et
w
i . e . {xq !
5°
Suppose
Go
be the c e n tre o f
R
u.
x t ^ Co
h -a ry
S in ce
U
R
x = xb ,
^ 0
so t h a t
c e n tr a l r e l a t i o n
le t
v
C
u+i
= C U w ( C* ' ) :
V
y
r= 0 ;
suppose
. , , , x^) e R
we may w rite
XL = &/(% if , .
h ) , so a g a in ( x i ,
R(xi^ ,
R.
%f )
. . . , x^) f &
x^ ),
) .
i s n o n -u n iv e rs a l i t fo llo w s from v h a t has been proved t h a t
C / E;
6°
Xp ) ;
-ZAj,)
T h is i s c le a r f o r
w ith (xL , , , , , x f ) f C^*' ( i = 1 ,
s in c e by in d u c tio n h y p o th e sis
.
i s a p ro p e r s u b a lg e b ra ,
Then ( x i ,
I f no
+ w (xi,
I f d e t(l
and f o r each
Xj^) e
e Cq .
xl
= %
has a s o lu tio n
p re s e rv e s Ihe
C R, f o r each
G. K, (% ,
i f some
Xo + X p )
a = x ( l - ZA[)
Xo) = Xo;
we shovf t h a t
. ..,
i s a p ro p e r autom orphism .
+ X
th e n th e e q u a tio n
w( xq ,
+ X l,
thus V
Suppose
w
G^
i s a proper sub a lg e b r a ,
p re s e rv e s th e
an d , f o r some s u rje c tL o n
h -a ry
re la tio n
0 ; E -> (Ej^)™(m > O) ,
R,
v h ere
3^ h ^ n
R( x i , . , , , x^)
i s e q u iv a le n t
to
|[pr^0(xi),
(a )
If
h'" < n
. . . , pr^ 0 (x j^ )i| < h
fo r
6
= 1,
. . . , m.
th e r e l a t i o n
0 ( x i ) = 0 (x a )
i s a p ro p e r congruence u n le s s
E
has a p ro p e r s u b a lg e b ra .
F o r, t h i s
r e l a t i o n i s a non—id e n t i c a l , n o n -u n iv e rs a l e q u iv a le n c e r e l a t i o n , so i t
s u f f i c i o s t o show t h a t
0(w(xi^,
= 1,
if
0 ( x i ^ ) = 0 (xg^ ) , . , . , 0 ( x i ^ ) = ^ { x j ' )
x i ^ ) ) = 0(w(xg^ , . . . , Xg' ")),
m)
E
F o r some
we have
p r^ (0 (w (x i\
If
Suppose Ihe c o n tr a r y .
th en
X l*'))) / p r^ (0 (w (x a ^ , . . . , X g ^ ) ) ) .
has no p ro p e r s u b a lg e b ra s , th en ih e mapping
— 60 —
w : FT
E
is
s u r j e c t i v e , as a r e 0 ; E -►E ^
X'lf ,
.
( i = 1,
x f (i = 3, .
.
h)
h)
and p r^ i
may be chosen so t h a t
a r e a l l d i s t i n c t , i . e . , R( w( x i ^,
does n o t h o ld .
Since
R (xi^,..., x^ ),
c o n tr a d ic t s our s u p p o s itio n t h a t a
(b)
E ^ -» E ^ .
Suppose
]f = n
b i s e c t i o n we i d e n t i f y
E
and
p r^ (0 (w (x t , . x f ) ) )
. . . , Xi^),
. . . , w (x ^ , . . . , x ^ ))
. . . , R(xi*~ , . . . , x ^ ) h o ld , t h i s
p re s e rv e s
so t h a t 0
Thus elem en ts
R.
is b ije c tiv e .
By means o f t h i s
E ^ , w ith th e consequence t h a t
R( x i , . . . ,
x^)
i s e q u iv a le n t to
lîpr^U i),
C o n sid er th e mappings
. . . , p r^ (x ^ )!| < h
g^î
E^^ (6 = 1 , . . . , m)
(^1 ,
I f some
(of.
g.
th a t
im ply t h a t
~
fmf), ...,(^ 1 ^ ,
s u r j e c t i v e f o r o th e rw ise
.
E has a p ro p e r s u b a lg e b ra .
depends e s s e n t i a l l y on more th a n one o f i t s argum ents th e n
[ 2 ] , ^ 3 ], [53])
E^ such
g^
. . . , m.
d e fin e d b y
5 ••• 5 ^ 1 f
= p r^ (w (fe iS
V/e may suppose each
fo r 6 = 1 ,
(o
we can f i n d
g^ . assumes
h
does n o t p re s e rv e
(h-1 ) -e le m e n t s u b s e ts
v a lu e s on Hi x
R. Thus
H i,
. . . x H^r;
each
H^r
of
b u t t h i s w ould
= 1 , . . . , m) depends
e s s e n t i a l l y on o n ly one argum ent, so we may w rite
g^C^i,
•••,
f o r s u ita b le mappings
o’, r .
i n v a r i a n t under th e
) (6- = 1,
n o n -n u ll s u b se t I
mapping
~ ^ f . . . , m)
E v id e n tly , we have
pr^ w ( 3^ , . . . ,
I f some p ro p e r
(gj)
. . . , <ani*")=
o f tie i n d ic ie s
r (r(l) C l ) ,
—
61
—
. . , m) •
1,
th e n the r e l a t i o n
m is
p r ^ ( x i) = pr^Cxg)
i s a p ro p e r c o n g ru en ce.
1,
Set
.
.
f o r eac h
6 c I
I n th e c o n tr a r y case
I
i s a p e rm u ta tio n cf
m c o n s is tin g o f a s in g le c y c le o f le n g th
a ( x ) = w( x,
m.
. . x) ,
so t h a t
p r^ a ( x ) =
By in d u c tio n T/e have f o r
W
U = “I ,
m) .
k > 0
p r^ «*< (x) =
Y.r(«) • • • \ _ i (8) P V (8)
(« = 1 , . . . , m) ,
SO, i n p a r t i c u l a r ,
pr^a^(x) =
. . . , m).
(^ = U
From t h i s we v e r i f y t h a t
(wCx' , . . . . xT)) = w ( a ' " ( x ‘ ) ,
If
a'’’
i s n o t th e i d e n t i t y we c o n s id e r th r e e c a se s : ( i )
i s c o n s ta n t = Xo ,
w(xo,
so t h a t [xoi
«'“ ( x ! ') ) .
th e n
Xo)
if
or
=w (a"'(xo),
. . . ,
i s a p ro p e r su h a lg e b ra ;
a p ro p e r autom orphism ; ( i i i )
if
0^ ( x o ) )
(ii)
= a"* ( ûj ( x q , . . . , X o ) )
if
= Xq,
a"’ i s one-one th e n
a”’
i s n o t c o n s ta n t and n o t one-one th e n
th e r e l a t i o n
a"’ ( x i ) = a'" (xa)
i s a p ro p e r c o n g ru e n c e ,
If
i s th e i d e n t i t y th e n
i s th e i d e n t i t y on
(6 = 1 ,
YgY^(^) • * •
m) .
^ (6)
Thus f o r any c h o ic e o f
we can s a t i s f y th e e q u a tio n s
( « = 1 ,
- 62 -
e
) ;
f
is
th e n i f
Xq = ( ^ i ,
we have
w(xo, .
%o) = %o
so t h a t [xo!
i s a p ro p e r s u b a lg e b ra .
The p ro o f o f Theorem 1 i s now c o m p le te .
THEOEEUi 2 ,
A f i n i t e a lg e b r a
< A,w >
w ith more th a n two elem en ts i s com plete
i f and o n ly i f th e s e t of fu n c tio n s d e fin a b le from
t r a n s i t i v e subgroup of th e sym m etric group on
P ro o f.
If
A
w
in c lu d e s a doubly
A.
i s n o t com plete th e n by Theorem 1 i t has a p ro p e r s u b a lg e b ra ,
autom orphism o r congruence, w hich must be p re s e rv e d by a l l fu n c tio n s
d e f in a b le i n term s of
R (ai^ , . . . , a ^ )
w.
B ut i n g e n e r a l i f
does n o t, where
c o n s is t o f
h e le m e n ts , th e n
c a r r i e s a%
to
R
{ai,
. . . , a^}
. , , , a^)
and
[ai^,
h o ld s w h ile
.
a^ }
each
i s n o t p re s e rv e d by a r y p e rm u ta tio n w hich
ai^ , . . . , a^ to a^^ ,
A pplying t h i s i n th e case o f a p ro p e r
s u b a lg e b ra , automorphism o r congruence, vfhere
fo llo w s t h a t
R(ai,
h
does n o t exceed 2 , i t
Ù) can n o t g e n e ra te a doubly t r a n s i t i v e group o f p e rm u ta tio n s .
— 63 —
2•
S h e ff e r fu n c tio n s in i n t u i t i o n i s t i c l o g i c .
In t l i i s s e c t i o n we w i l l be concerned s o l e l y w ith o rd in a ry i n t u i t i o n i s t i c
lo g ic ;
th u s
n = 2
th ro u g h o u t.
I n m ost c o n te x ts i t i s c o n v e n ie n t to om it
th e a d je c tiv e " i n t u i t i o n i s t i c " and r e f e r sim ply to S h e ffe r f u n c tio n s e t c .
I f fo rm u las
a,/3
a re ( i n t u i t i o n i s t i c a l l y ) e q u iv a le n t we w r ite
a ~ p .
B efore p ro c e ed in g we m ention a co n cep t in tro d u c e d by Kuznecov [ 8 ] ,
w hich is c l o s e l y r e l a t e d t o Ihe concept of S h e f f e r f u n c tio n .
a(pi,
. . . , Pp)
a S h e ffe r form ula i f e v e ry fo rm u la i s e q u iv a le n t to some
member o f the l e a s t s e t
w hich i s su ch t h a t i f
F
L e t us c a l l
which c o n ta in s a l l p r o p o s itio n a l v a r ia b le s and
U
yi;
•••> Yr
i s a S h e ffe r fu n c tio n i f f F p i
th e n
^
. . . . pp
a(Yi,
. . . , Yr) ^ ^
•
C le a rly
i s a S h e ff e r fo rm u la , b u t t h e r e
e x i s t S h e ffe r fo rm u las w hich do n o t corresp o n d to any S h e ffe r fu n c tio n ^ ,
Kuznecov has shown th a t th e r e e x i s t S h e ff e r form ulas o f th r e e v a r ia b le s b u t
none o f two v a r ia b le s .
E very tr u t h - f u n c tio n i s d e fin a b le in te r n s o f
In d e e d , by re d u c in g th e (tw o -v alu ed ) tr u t h - f u n c t io n ^
a
v,
,
F
D ,
-
.
to i t s p r i n c i p a l
conjUDative norm al f or m,
F(xi,
we se e t h a t th e fo rm u la
. . . , Xp) =
F pi
...
pp
A
K ( i i , *•
(x^^i ^v
...
V
^k) “ C
i s e q u iv a le n t to a c o n ju n c tio n o f
fo rm u las o f the form
2.
T h is phenomenon o f c o u rse does n o t o ccu r in c l a s s i c a l tw o -v alu ed l o g i c .
— 64 —
(1)
p ,
A
...
A
D
p,
p
v/here
ky . . . , kp
0 ^ 6 ^
V
j
6+1
6
...
V
p
iCp
i s a p e rm u ta tio n o f th e in te g e r s
A, V, D , -, a r e d e f in a b le i n term s o f
a Dj S
i t fo llo w s t h a t
F
F.
F
In view o f ihe e q u iv a le n c e s
F
v , s , -, a re d e f i n a b le .
vhose norm al for m is
^ ( p A q A r A s) A ( p D q v r v s ) A
We have ih e e q u iv a le n c e s n .p
q
~
FlO pq,
f o u r v a r ia b le s .
ps
q
In f a c t ,
i s a S h e ffe r f u n c tio n
(aD/3),
i s a S h e ffe r f u n c tio n i f f
C onsider the t r u t h - f u n c t i o n
V
F.
.~ .(a.v/3)3/3,
a AjS . ~ . a s
p
and
r . ° T h is c o n ju n c tio n w i l l be c a l l e d ihe norm al form o f
The e x is te n c e o f such a normal for m im p lie s t h a t
iff
1, . . . r
~
~
F
Kpppp,
FppOO.
1
Hence
( q D p v r v s ) .
~
E p p p -, p , 0
F
~
-, 1 ,
i s a S h e ffe r f u n c tio n o f
may be re g a rd e d as th e s im p le s t example o f a
S h e ff e r f u n c tio n , s in c e e v e ry S h e ffe r f u n c t io n has a t l e a s t fo u r v a r ia b le s and
a t l e a s t th r e e c o n ju n cts i n i t s norm al f o r m.
method o f
T his w i l l be proved by th e
McKinsey [ 9 ] .
C onsider th e d i s t r i b u t i v e l a t t i c e
L re p re se n te d in F ig u re 1 .
0 ^
la
< / \o d
V b
1.
Fig. 1
3.
In th e extrem e cases
(p
xCi
V
...
V
p
Kp
)
6 = 0
and -(p
Kj
an d
aI
.. .. ..
6 = r , ( l ) d a io te s th e fo rm u las
a
A
— 65 —
p, )
Kp
re s p e c tiv e ly .
By s e t t i n g
-, x = sup[z ; z a x = Oj
v/e c o n v e rt
L
and
x D y = sup [z : z a x ^ y i ,
in to a pseudo-B oolean a lg e b r a .
d i f f e r e n t v a lu e s of
0
b
c
d
a
1
If
F
i s a S h e ffe r
v/hich i s c lo s e d u n d e r
th e s u b s e ts [1]
under
F,
and
D
0
b
c
d
a
1
0
b
c
d
a
1
1 1
0 1
0 d
0 c
0 b
0 b
1
1
1
c
c
c
1
1
d
1
d
d
1
1
1
1
1
a
1
1
1
1
1
1
1
0
0
0
0
0
V
and
F
we can d eterm in e i t s v a lu e in
f u n c tio n o f
F
r
v a r ia b l e s , th e n any s u b s e t o f
must a ls o be c lo se d u n d e r
[O]
are n o t c lo s e d under
a
v, D, -
,
th e v a lu e
a
.
S ince
L
-
-(p i
[a]
^ th e y a r e n o t d o s e d
F
th e v a lu e s
re s p e c tiv e ly .
a
F
v,
Fpi
d i s t i n c t from
a,
...
pp
a.
Some
and i f t h i s is
Thus the a n te c e d e n t c o n ta in s a
and th e co n seq u en t c o n ta in s
a t l e a s t tv/o p r o p o s itio n a l v a r ia b le s o f w hich one ta k e s the v a lu e
c
o th e r
d . I n a d d itio n th e r e must be y e t a n o th e r v a r ia b le which ta k e s
v a lu e
0 , o th e rw ise th e c o n ju n c t
- (p i
would ta k e th e v a lu e
has a t l e a s t f o u r v a r i a b l e s .
t alces
and th e a n te c e d e n t and c o n seq u en t ta k e
p r o p o s itio n a l v a r ia b le w hich ta k e s th e v a lu e 1 ,
F p i . . . Pp,
must
b u t n o t the fo rm u la
must ta k e th e v a lu e
0 < 6 < r,
and
Pp)
a
o f p i , . . . , pp
o f th e form ( 1 ) , th e n
1
. . .
i s n o t c lo se d un d er
f o r c e r t a i n v a lu e s
c o n ju n c t i n th e norm al form o f
a
L
Since
,
and from t h i s we e a s i l y deduce t h a t th e norm al f or m o f
. . . V Pp)
L
o f v a lu e s to t h e v a r i a b le s .
c o n ta in a s a c o n ju n c t th e fo rm u la
(p i
-, x
a re sh ovm in th e
x ,y
From th e normal form o f a tr u t h - f u n c t i o n
f o r any g iv e n a ssig n m e n t
The v a lu e s o f
0.
Since
—
a
. . .
Pp),
—
th e
and hence a l s o
We have th e r e f o r e shov/n t h a t
L - [b]
66
a
and th e
i s n o t c lo s e d un d er
F
a.
F p iy . . y P p
takes th e v a lu e
d i s t i n c t from
b.
b
f o r c e r t a i n v a lu e s o f
F
.
A c o n ju n c t of th e form (1) can n o t ta k e ihe
f o r v a lu e s o f ih e v a r ia b le s d i s t i n c t from
of
p i,
b.
-
two o th e r c o n ju n c ts o f w hich one ta k e s th e value
F
v a lu e b
Hence th e norm al for m
must c o n ta in , i n a d d itio n to th e c o n ju n c t
t h a t i s the norm al form of
pp
(p i a . . . a p p ) ,
cand th e o th e r
d;
has a t l e a s t th r e e c o n ju n c ts .
An e la b o r a tio n o f th e above argum ent e n a b le s us to c h a r a c te r iz e th e
S h e ff e r f u n c tio n s o f fo u r v a r ia b le s .
THEOREM 3 • L e t
In o rd e r t h a t
F
F
be a c l a s s i c a l S h e ffe r f u n c tio n o f fo u r v a r ia b le s .
sh o u ld be an i n t u i t i o n i s t i c S h e f f e r f u n c tio n i t i s
n e c e s s a ry and s u f f i c i e n t t h a t ev ery s u b s e t o f
F
sh o u ld be a pseudo-B oolean s u h a lg e b ra o f
Proof.
L.
The n e c e s s ity o f th e c o n d itio n i s o b v io u s .
th e s u f f ic ie n c y , l e t us suppose t h a t
F
L w hich i s c lo se d u n d er
F
In o rd e r to prove
s a t i s f i e s t h e c o n d itio n .
i s a c l a s s i c a l S h e ff e r f u n c tio n , th e fo rm u la
Since
- ( p * q A r A s) ap p ears
a s a c o n ju n c t in i t s norm al form b u t n o t th e fo rm u la ( p v q v r v s ) .
Hence
Fpppp
i s e q u iv a le n t to
e q u iv a le n t to
-»p.
- , 0, 1
Since
F
is c la s s ic a lly
0 , and s o , by a w ell-know n theorem o f G liv en k o ,
i n t u i t i o n i s t i c a l l y e q u iv a le n t to
th a t
Some fo rm u la i n
0.
From th e s e c o n s id e ra tio n s i t fo llo w s
a r e d e fin a b le i n te rm s of
L - [a]
i s n o t c lo s e d un d er
u sed above t h a t th e norm al form of
F
F.
v,
we deduce by th e argum ent
c o n ta in s a c o n ju n c t o f th e fbrm (1)
such t h a t , f o r a s u i t a b l e assig n m en t o f v a lu e s to t h e v a r ia b l e s ;
(i)
th e a n te c e d e n t c o n ta in s a v a r i a b l e ,
-
6 7
-
p
s a y , w hich ta k e s th e v a lu e 1 ;
(ii)
th e consequent c o n ta in s two v a r i a b l e s ,
ta k e th e v a lu e s
c
and
d
r e s p e c tiv e ly ;
0
q
(iii)
v a ria b le , s
s a y , takes th e v a lu e
c o n se q u e n t.
We conclude t h a t the norm al for m o f
an d
r say, which
th e rem ain in g
and so m ust a p p ea r i n th e
F
c o n ta in s th e
c o n ju n c t
p D q
V
r
V
s•
On th e o th e r hand, th e fo llo w in g fo rm u las c an n o t a p p e a r as c o n ju n cts
s i n c e , u n d er the a ssig n m e n t o f v a lu e s w hich we a re c o n s id e r in g , Ihe y
ta k e v a lu e s l e s s th a n
a;
p A q D r v s ;
p Ar D q
v s;
p A q Ar 3 s .
U sing t h i s in fo rm a tio n we s e e t h a t th e form ula
e q u iv a le n t to
to
p—q
q v r.
o r to
q D p v r v s
A lso , th e form ula
pD q
L ~ [b]
o f th e form (1)
re s p e c tiv e ly .
v
F
a
,
F.
we deduce as b e fo re t h a t
and
X i D Yi
v a r i a b l e s , ta k e th e v a lu es c
A ll o th e r c o n ju n c ts ta k e v a lu e s g r e a te r
takes th e v a lu e
^ d
In
Xg D Yg
w h ich , f o r a s u i t a b l e assig n m e n t o f v a lu e s
th e c o n ju n c t
and
and
and
th a n
- (p A q A r A s ) ,
0; upon s u ita b le renam ing of
may suppose t h i s v a r ia b le to be
v a lu e s
i s e q u iv a le n t
a re d e f in a b le i n term s o f
c o n ta in s two c o n ju n c ts
owing to th e p re sen c e o f
c
and D
i s n o t c lo se d un d er
( d i s t i n c t from b) to th e
th e v a lu e s
is
i s p r e s e n t as a c o n ju n c t i n th e norm al f o r m.
th e norm al form of
v a r ia b le
F p q 0 0
r 0
depending on v/h e t h e r o r n o t th e form ula
e i t h e r c ase i t fo llo w s t h a t
Since
F 1 q
a o
The co n seq u en ts
d
b . A gain,
some
th e v a r ia b le s we
Yi
and
Yg
d
r e s p e c tiv e ly ;
th e a n te c e d e n ts
^ c
re s p e c tiv e ly .
Hence th e r e a r e v a r i a b l e s ,
—
68
—
X%
and
Xg
ta k e
ta k e
p
and
q s a y , w hioh ta k e th e v a lu e s
such t h a t
Y i.
q
occurs i n
The v a r ia b le
s
Xi
and
c
Yg
and
and p
d
r e s p e c tiv e ly and
occurs i n
n u s t ap p ear in b o th Y% and
Yg.
c a se s a cc o rd in g to th e p o s itio n o f th e r e n a in in g v a r ia b le
(i)
v a lu e
If
r
o c cu rs i n b o th
Yi
and
Y g,
th e n
3 r
A q
is
r .
m ust ta k e th e
V s
c an n o t a p p e a r as a c o n ju n c t i n the norm al for m o f F ,
(ii)
We d is tin g u is h
0 , so th e f o m u la
p
FpqOO
r
Xg and
i s e q u iv a le n t to
If
r
Thus
th e form ula
pa q.
o ccu rs i n
X% o r
Xg,
th e n th e v a lu e ta k e n by
r
non—z e r o , and so th e fo rm u la
p Aq Ar 3 8
c an n o t a p p e a r a s a c o n ju n c t in th e normal form of
(a)
FpqqO
If
r o ccu rs i n
i s e q u iv a le n t to
(b)
If
i s e q u iv a le n t to
(c)
If
i s e q u iv a le n t to
X% an d
F.
Yg,
th e n th e form ula
p= q.
r o ccu rs in
Yi
and
X»,
th e n th e
fo rm u la FpqpO
Xi
and
Xg,
th e n th e
form ula FpqtO
p s q,
r o ccurs in
p a q
o r to
ps
q , depending on v/hether o r n o t
th e fo rm u la
r 3 p
V
q
V
s
a p p ea rs a s a c o n ju n c t in the norm al form of
th a t a
or A
i s d e f in a b le in term s o f
—
69
—
F.
F.
In a l l c ases vre see
Combining t h e r e s u l t s of th e p re c ed in g p a ra g ra p h s , we deduce t h a t
F
i s a S h e ffe r f u n c tio n , a s was to be shown.
Vfe observe t h a t an a n a ly s is o f the p ro o f o f Theorem 3 would r e a d i l y
g ive a c h a r a c te r iz a ti o n o f th e S h e ff e r fu n c tio n s o f fo u r v a r ia b le s i n
term s o f t h e i r t r u t h - t a b l e s .
I t would be o f i n t e r e s t to Imov/ w hether theorem s of th e above ty p e
h o ld f o r fu n c tio n s of more th a n fo u r v a r i a b l e s .
th a t th e l a t t i c e
r e s u lt [ 5]
L
We rem ark f i n a l l y
can be u sed to g iv e a s h o r t p ro o f o f McKinsey’s
on th e independence o f
a , v , D ,
In d e e d , f o r each
o f Ih ese c o n n e c tiv e s th e r e i s a fiv e -e le m e n t s u b s e t o f
L
vshich i s
n o t c lo s e d u n d er t h a t c o n n e c tiv e b u t w hich i s c lo s e d u n d e r the rem ain in g
th re e .
—
~/0
—
B ib lio g ra p h y
[1 ]
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[3 ]
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