ON DENUMERABLY MA NY - V A L UE D LOGI CS
ROLF SCHO CK
In t h e paper F = ' O n f init ely many -v alued logic s ' (Logique e t
Analyse, v ol 21, 1964 ('), t he aut hor was not sure about h o w t o deal
wit h to-valued (
2
theory
is developed and used t o characterize w-v alued logics.
)
1.
s ceo-VALUED
m a n t i SEMANTI CS I N E MP TY A N D NO N- E MP TY U N I c sVERSES.
.
I
n
By an co-interpreter, we mean a f unc t ion i such t hat
t
h
(1) t he domain of i = t he set of all indiv idual constants and predie
cates
and
p(2) t here
r
eis a set u such t hat
s
e
n
(a) f o r any indiv idual constant c, eit her i(c) = t he empt y set or,
t
for some m in u, i(c ) = {
p (b)a f o r any positive integer m and m-plac e predicate p, i(p) is a
n
pnon-empty
e
f init e sequence of sets of m-t erm sequences of members of
}
ru; and
,
;
o
(c) f o r any k in the domain of i(I), (i(I ))(k ) = t he set of all t such
n
that,
f or some m i n u, t = ( m m ) .
e
s Given a n co-interpreter i, Ui is t he u under (2) above. Obviously,
u
Theorem 1. I f i is an n-interpreter, t hen i is an co-interpreter.
c
Notice t hat t he converse need not hold.
h
s Givene a n co-interpreter i and assigner i n Ui a, I nt la is understood
m
a f o r n-interpreters i n F; howev er, i n t he case o f at omic
as i t was
nformulas,
t t he div is ion is b y t he greatest member of t he domain o f
ithe sequence
c
wh i c h is t he i-v alue of t he predicate concerned rat her
than by n - 1 (alt hough this objet turns out t o be n - 1 i f i is also an
n-interpreter). Th e clauses f o r f ormulas beginning wi t h A a n d V
remain proper since, f or any v ariable v and f ormula f, the set of all
r such that, f or some m in Ui, I nt ia ( r ( m) ) f = r is finite.
(
ffrin1 i s a metalinguis tic v ariable ra n g in g ov er a l l pos it iv e integers great er
than
) I.
(
W
e2
190
h)
eA
rs
eu
as
du
oa
pl
t,
a(
l1
l)
Given a n co-interpreter i , T i a n d i-t rut h are als o unders t ood as
they were f o r n-interpreters i n F; s imilarly , Toi a n d co-validity are
understood as t hey were i n F, but t his t ime wi t h respect t o co-interpreters. I t can be s hown t hat
Theorem 2. To) = t he set of all r such that, f or some positive integer
m and nat ural number not greater than m k, r = k div ided by m (
3 Also,
).
Theorem 3. There is an co-interpreter i such t hat Ti = To).
Let p = t h e s et o f a l l 1
theorem
- p l a co feset theory, t here is a denumerable set o f mut ually dis joint
t hat p — t he union of q. Hence, let
p rdenumerable
e d i c a sets
t e qs such
.
s be
a
f
unc
t
ion
whic
h
correlates
t he positive integers wi t h q. Als o,
B
y
let at be a f unc t ion whic h assigns t o any positive int eger m a f unc t ion
whic
w h correlates
e
l
l t he
- nat ural numbers wi t h s (m). Finally , l e t x be
ankobject
and
i
be
n
o
w the
n co-interpreter such that
(1) U i { x } ;
(2) f o r any indiv idual constant c, i(c) = { x } ;
(3) f o r a n y 1-plac e predic at e p , pos it iv e int eger m , a n d n a t u r a l
number not greater than m k, i f p = (t (m)) (k ), t hen i(p) = t he mterm sequence u s uc h t hat , f o r any 1 i n t he d o ma i n o f u, eit her
k 0 , I is not greater t han k, and u(I ) = ( ( x ) ) o r not and u(I) - the empt y set;
(4) i(I ) i s a 1
- t ef or rmany n-place predicate p, if p I , then i(p) ( t h e empty set).
(5)
s Hence,
e q uf or
e any indiv idual constant c, 1
in
Ui
p
l
a,
a
c
positive
e
p rinteeger
d i m,
c and
a t nat
e ural number not greater t han m
n c e ;
k,
p
i
f
p
(
,
t
(
m
)
)
(k),
t
hen
I
nt
ia
(cp)
= t he number of members o f
a
n
the
a
set
s
of
s
all
1
i
in
g
the
n
domain
e
r
of
i(p)
such
t hat ( x ) is i n (i(p)) (I ) did
vided b y the greatest member of the domain of i(p) = k div ided by
m. Hence, by t heorem 2 and the def init ion of To), Ti = Toi and the
theorem holds.
Theorems 2 a n d 3 jus t if y t h e pres ent t reat ment o f co-valued
semantics.
Theorem 4. I f f is a f ormula and f is co-valid, t hen f is v alid.
(
A. 3Tars k i's «Investigations i n t o t h e s ent ent ial c alc ulus . ( i n Tars k i's b o o k
Logic,
)
Semantics, Metamathematics, Ox ford, 1956).
T
h
191
u
s
,
o
u
r
i
.
t
)
t
This f ollows f rom t heorem 1.
Theorem 5. The set of all (o-valid f ormulas is a proper subset of the
set of all 2-v alid formulas.
This f ollows f rom t heorem 4 and t heorem 2 of F.
Theorem 6. There is a nonzero f ormula whic h is not co-valid.
This f ollows f rom t heorem 4 and t heorem 4 of F. Als o,
Theorem 7. I f i is an co-interpreter and t is a t erm or a f ormula, t hen
there are an n and n-int erpret er j such t hat Uj = U i and, f o r any
assigner in Uj a, I nt ja (t) I n t ia (t).
Assume t he antecedent. Obv ious ly , t here is a non-repeat ing f init e
sequence s such t hat t he range o f s = t he set o f all predicates occ urring i n t. Let d be a f unc t ion such t hat t he domain of d = t h e
domain of s and, f or any k in the domain of d, d(k ) = t he greatest
member of the domain of i(s(k)). Also, let p — t he p such that either
d is empt y and p = 1 or not and p = d(1) mult iplied by m u l t i p l i e d
by d (t he greatest member of the domain of d) a n d let e be a f unc tion such t hat domain of e = t he domain of d and, or any k in t he
domain of e, e(k ) p div ided by d(k ). Finally , let j = t he p + 1 interpreter j such that
(I) f o r any indiv idual constant c, j(c ) i ( c ) ;
(2) j ( I ) is a p-term sequence and the range of j(I ) = t he range of i(I );
(3) f o r any predicate q,
(a) i f q does not occur in t and q * I , t hen every member of the
range o f j(q) is empt y and
(b) i f q occurs in t, then, f or some k and r, q s ( k ) , r is an e(k )
- sequence, t he range o f r = i ( s ( k ) ) ) , and j(q) = r ( I )
term
r(e(R)).
Obviously, U j = U i . Als o, f o r a n y pos it iv e int eger m, m-plac e
predicate whic h occurs i n t q, m-t erm sequence of terms whic h occur
in t u, assigner i n Uj a, nat ural number 1, and positive int eger k, i f
Int la ((u(1)) ' ( q ) ) ' u_t ) = 1 div ided by the greatest member of the
domain o f i(q), s(k) = q , and I nt ja (u(h)) = I n t i a (u((h)) * t h e
empty s et f or any h i n t he domain o f u, t hen I nt ja ((u(1)) ' ( q ) )
= (e(k ) mult iplied by 1) div ided by p = ( ( p div ided by d(k))
mult iplied by I) div ided by p I div ided by d(k) = I div ided by the
greatest member of the domain of i(q). Hence, by an induc t ion among
the members of TF, t he t heorem holds.
But t hen
Theorem 8. Th e set of a l l co-valid f ormulas = t h e set o f a l l v alid
formulas.
192
For assume t hat f is a f ormula and v alid and t hat i is an w-interpreter. By t heorem 7, t here are an n and n-int erpret er j such t hat
f is j-t rue just in case f is i-true. Since f is v alid, f is n-v alid and so
j-true. Hence, f is co-valid and so t he t heorem holds by t heorem 4.
Theorem 9. I f f is a f ormula, t hen f is 2-v alid jus t in case A f is wvalid.
This f ollows f rom t heorem 8 and t heorem 6 of F.
If t is a set of real numbers not s maller than 0 and not greater than
1, t hen Vt = t he set of all functions v such that the domain of y
the set of all f ormulas , t he range of v is inc luded i n t, and, f or any
formulas f and g,
(1) v ( l l f ) = t he z such t hat eit her v(f) is i n { 0 1 } and z 1 o r not
and z 0 ;
(2) v ( Hf ) = t he z such t hat eit her v(f) = 1 and z = 1 o r not and
z 0 ;
(3) v (—f ) = 1—v (f );
(4) v ( f
- g )v ( f A g) = t h e smallest member o f f v (f ) v (g)1;
(5)
= v ( f Vg) = t he greatest member of {v (f ) v (g)); and
(6)
t v(k-->)
(7)
h
= (1—t he greatest member of {v (f ) v (g))) + t he smallest
e member of (v (f ) v(g)1.
s Obviously,
m
a l
Theorem
10. I f f is a f ormula, t hen f is an n-t aut ology jus t i n case
l e
v(f)
s t = 1 f or any v in VTn.
m
Theorem
11. V Tn is a proper subset of \ MD.
e
m We say that a f ormula f is an co-tautology just in case v(f) = 1 f or
b v i n VToo and that f is sententially atomic just i n case t here are
any
e sentential connective c and f ormulas g and h s uc h t hat eit her
no
fr = c g or f = gch.
o
fTheorem 12. I f f is a f ormula and f is an co-tautology, t hen f is (01
valid
and so v alid.
, The proof is s imilar to t hat of theorem 11 of F and via t heorem 4.
(
1
Theorem
13. I f f is a f ormula and f is an co-tautology, t hen f is a
—
tautology.
v This f ollows f rom theorems 10 and 11. Also,
(
0
193
)
+
v
(
g
)
1
;
Theorem 14. I f v is i n VT(0 and f is a f ormula, t hen t here are an n
and a w in V Tn such that w(f ) = v (f ).
Assume t he antecedent. Obv ious ly , t here i s a non-repeat ing a n d
non-empty f init e sequence s s uc h t hat t he range of s = t h e set of
all sententially at omic f ormulas oc c urring i n f . Let d be a f unc t ion
such that the domain of d = t he domain of s and, f or any k i n t he
domain of d, d(k ) = t he smallest b such that, f o r some a, v (s (k )) =
a div ided b y b. Also, let p = d(1) mult iplied by m u l t i p l i e d b y
d (the greatest member of the domain of d) and let e be a f unc t ion
such that the domain of e = t he domain of d and, f or any k i n the
domain o f e, e(k ) p div ided by d(k ). Finally , let w = t h e w i n
V Tp +1 s uc h that, f or any sententially at omic f ormula g,
(1) i f g goes not occur in f, then w(g) 0 div ided by p and
(2) i f g occurs in f, then, f or some k in the domain of s, g = s(k) and
w(g) = (e(k ) div ided by p) mult iplied b y (t he a such t hat y (g) =
a div ided b y d(k )).
Hence, f o r any sententially at omic f ormula wh i c h occurs i n f g
and k i n the domain of s, i f g = s(k), t hen w(g) = (e(k ) div ided by
p) mult iplied by (the a such t hat v(g) = a div ided by d(k)) ( 1 divided b y d(k )) mult iplied by (t he a such t hat v (g) = a div ided by
d(k)) — v (g).
No w let UG = t he set of all pairs t, g i n TF such that, i f g is a
f ormula whic h occurs i n f . t hen w(g) = v (g). I f t he pairs t, g and
u, h are in UG and c is a sentential connective, t hen w(c g) = v(cg)
if cg is a f ormula oc c urring i n f and w(gc h) = v (gc h) i f g c h is a
f ormula occuring i n f. Hence, ' I T is inc luded i n UG and so, since f
occurs i n f, t he t heorem holds. I t f ollows t hat
Theorem 15. The set of a l l f ormulas whic h are oktautologies — t he
set of all f ormulas whic h are tautologies 4
. For assume t hat f is a f ormula wh i c h is a t aut ology and t hat
is i n VTto. By t heorem 14, there are an n and a w i n Tn such t hat
w(f ) = v (f ). Sinc e f is a n n-tautology, w( f ) = 1 a n d s o f i s a n
(0-tautology. Hence, by t heorem 13, t he t heorem holds.
2. co-VALUED LOGI CS
By an okvalued logic, we mean a deductive system d such t hat t he
set of all d-provable f ormulas = t he set of all okvalid f ormulas .
4 This is a generaliz ation o f t he f ourt h p a rt o f t heorem 17 o f t he papet
cited i n note 3.
194
Theorem 16. I f d is an w-v alued logic and f is a f ormula, t hen f is
d-provable jus t i n case f is e-provable f or any n-v alued logic e.
This f ollows f r o m t heorem 8.
Lw a n d co-provability are understood as t hey were i n F, but t his
time wi t h respect t o w-tautologies. By t heorem 13,
Theorem 17. I f f is a f ormula and f is w-provable, t hen f is provable.
Also, b y theorems 12 a n d 8 a n d theorems 7 t hrough 11, 15, 16,
and 18 t hrough 24 of F, i t f ollows t hat
Theorem 18. I f f is a f ormula, t hen f is w-prov able o n l y i f f i s
wv
a l Because of theorems 18 and 8 a n d theorems 25 a n d 32 t hrough
i d of F, it seems lik ely that Lw is an w-valued logic; nevertheless, the
40
.
proof
cannot be giv en i n quit e t he us ual wa y f or t he reason giv en
for t he case of the systems Ln wi t h n great er t han 2 i n F.
The w-v alued logic s are more adequate t han t he f init ely ma n y valued ones i n t h a t t hey c ont ain t h e f i n i t e l y many -v alued ones
in t he sense o f t heorem 16. Nevertheless, except wi t h respect t o t he
problem of the number of t rut h values t o be employed, the co-valued
logics are less adequate t han t he 2-v alued ones i n a l l t he way s i n
whic h t he higher-v alued logics are.
Stockholm
Rolf SCHOCK
ERRATA FOR O N FI NI TELY MANY-VALUED LOGICS»
of t he same aut hor, print ed i n t he 25-26 n., A p r i l 1964
p. 54, l i n e 4, replace t he f irs t «( » b y a (».
p. 55, line 10, replace t he f irs t (cv» b y g A "•
p. 56, line 9 f rom bottom, replace the phrase: t h e n x is n-consistent
just i n case x does not n-imply —4» by the phrase: «. Hence,
x does not n-imply g.0
p. 57, bot t om line: replace v o l 4, 1963» b y ' v ol. 5, 1964. »
195