Concerning Kripke semantics for intermediate predicats logics

Z E S Z Y T Y N A U K O W E W Y Ż S Z E J S Z K O Ł Y P E D A G O G IC Z N E J W B Y D G O S Z C Z Y P r o b le m y M a te m a ty c zn e 1982 z .4 M ir o s ła w S z a tk o w s k i W S P B y d g o s z c z C O N C E R N IN G K R I P K E S E M A N T I C S P O R IN T E R M E D IA T E P R E D IC A T E L O G IC S

In N * g a i in tro d u c ed a n ew ty p e o f se m a n tic s .fo r inten-m ediate p r e d ic a te lo g ic s , w h ie h in

([33

H. O n o and S N a g a i term ed the g e n e r a ł K r ip k e m o d els . In this p a p e r, w e d istin gu ish the g e n e r a ł K r ip k e fra m e s from the K r ip k e m o d e ls on the g e n e r a ł K r ip k e fra m es. B e s id e s d e fin ig the g e n e r a ł K r ip k e fra m es, w e a is o d e fln e s tru c tu ra lly g e n e r a ł K r ip k e fra m e s , an d ex a m in e ce rta in p r o p e r tie s o f th e s e two ty p e s o f s em a n tics.

In I w e e s ta b lis h s y m b o ls and te rm in o lo g y w h ic h a r e u sed throu gh out the p a p er.

In II w e ex a m in e the re la tio n s b e tw e e n the g e n e r a ł K r ip k e fra m e s and the s tru c tu ra lly g e n e r a ł K r ip k e fra m es. W e a ls o exa m in e c e r ta in p r o p e r tie s o f th e s e s em a n tics.

In III w e a p p ly the re s u lts o b tain ed b y the au thor in

C?3

to the in term ed la te p r e d ic a te lo g ic s .

I. W e ll-k n o w n ło g ic a l and s e t - t h e o r e t ic a l n o tio n s and s y m b o ls a r e u s e d in the p a p e r. T h e sy m b o ls : A , V ,—^ , —1 . V , d en o te

the fam ilia r lo g ic a l c o n n e c t iv e s : co n ju n ction , disju n ction , im plication, n e g a tio n , u n iv e r s a l q u a n tifier and e x is te n tia l quantiQ er, r e s p e c t iv e ly . T h e le tte r N d e n o te s the s e t o f n atu ral num bers (fr o m z e r o ) . P o r e a c h n C N , the s y m b o ls : p ^ " \ r^ n \ ... d e n o te n -a r y p r e d ic a te v a r ia b le s . T h e s y m b o l A T d e n o te s a s e t o f atom ie fo rm u la s built in the u su a l w a y b y m ean s o f p r e d ic a te v a r ia b le s and in d iv id u a l v a r la b le s from an c o u n ta b ly in fin ite s e t •? x, y , z, ... r , and the sy m b o l P O R

44

-d e n o te s a s e t o f fo rm u la s built b y m ean s o f c o n n e c t t v e s : A • V • —> , “ t , V , 3 oJid atom ie form u la s. P o r a n y Q C . V ,

, 3 - the s y m b o l P O R ^ d e n o te s a s e t o f form u las w h o s e all c o n n e c t iv e s b e lo n g to Q. S m a li G r e e k le tte r s OC . /3 , w ill s e r v e a s m eta lin gu istic v a r ia b le s ru n n łn g throu gh the s e t P O R , w h e r e a s the s y m b o ls : A\ , V/ , =3>> ,\ = > , , 31 s ta n d fo r co n ju n ction , disju n tion , im p lication , e q u lv a le n c e , n e g a tio n , u n iv e r s a l q u a n tifler and e x is te n tia l q u a n tifier in the m e ta la n g u a g e. C a p ita l g o th ic le tte r s , ^ 5 « • • • d e n o te a lg e b r a s, and the r e s p e c t iv e Capital Latin le t t e r s A , B , . . . the u n h re rs e s o f the a lg e b r a s . T h e p r e s e n t w o rk w ill d is c u s s o n ly the s o c a lle d p s e u d o —B o o ie a n a lg e b r a s ( c f . [ ć 3 ) ,

fu rth e r c a lle d s y m p ly a lg e b r a s . T h e s y m b o ls :

d e n o te r e s p e c t iv e ly i unit elem en t, z e r o elem en t, la ttic e o r d e r o f the a l g e b r a , w h ile the s y m b o ls : A y , , —* jjc , d e n o te p s e u d o -B o o le a n o p e r a tio n s o f that a lg e b r a ( c f . C 6 3 )* In th® a b o v e s y m b o ls the in d e x Ą£ w ill b e omitted w h e n e v e r the p o s s ib ilit y o f c o n fu s io n c a n b e e x d u d e d .

In this w o rk , b y K r ip k e fram e w e s h a ll u n d ersta n d a n y p a rtta iły o r d e r e d s e t, l.e . a p a ir ' W ' - < W , ś: > s u c h that W

a n d 4 i » a p a rtia l o r d e r in W . B y a g e n e r a ł K r ip k e fram e ( g . K . £.) w e s h a ll u n d ersta n d a trip let < , V, > s a t ls fy in g the fo llo w in g co n d itio n s ( c f . ( 3 ^ ) ;

&) Y i s a K r ip k e fram e,

( l i ) V i s a m app in g from W to the p o w e r s e t o f so m e s e t s u ch that V ( a ) f <f> fo r a n y a € W a n d V ( a ) C V ( b ) i ! a £ b, ( i i i ) i s a a lg e b r a in w h ic h th e re e x is t

A

a , a n d V i t

t e t t

e

t ł fo r a n y T , s u c h that T «C K. (W , V ) a n d ł C K . ( v ) , w h e re K ( W , V ) den otes the sm allest Cardinal w hich

i s g re a te r than v ( a ) and ^ fo r an y

a fc W , an d hi.(v ) den otes the sm allest Cardinal which is i s g re a te r than V (o ! ) for a n y a Gr W .

T h e sy m b o l P O R ( v ) i s u s e d to d e n o te the s e t o f fo rm u la s o b ta in ed fTom P O R b y a d d ln g the in d iv id u a l c o n s ta n s ts ^ fo r e a c h elem en t

w e s h a ll u n d ersta n d e v e r y fu n ction v frt>m the C a r te s ia n p rod u ct o f the s e t o f a ll c lo s e d fo rm u la s o f F O R ( V ) and W in to A s a t is fie s the fo llo w in g co n d itio n s ( c f . I 1. fo r an n - a r y p r e d ic a te v a r ia b le v (p ^ n ^ ^5 i * • • "^n* a ) ^ v ( P ^ $ j . • • ^ n. b ) If a 4 b an d < 5 j _ ... 3 n > € V ( a ) n* 2. v ( Ot -A

(3

, a ) - v ( oC , a ) A V ( /3 , a ) . 3. _{v ( o C V /9 • a )} - v ( oC w < < 4. v ( © c~ ^ /3 . a ) - A a ^ b ( v ( oC , b ) —> v

<

ć 5. V ( i OC a ) - A a ^ b ( 1 v ( oC , b ) ) , 6. v (

V

x o t , a ) ■ _{a ^ b}

A

k

A v ( o t S . ;j € v ( b ) _ , b ) . 7. v

(

3 x oC . a )

_-

_V

, . v ( o t ^ . a ) . V ( a )

W e s a y that cC€ P O R is s a tls fie d b y a v a lu a tio n v on the g. K . f. < V . V, I X > , If v ( oC1 . a ) - 3l tor a n y a € W , w h e re oC* i s the u n iv e r s a l c lo s u r e o f ec . If e v e r y a v a lu a tio n v on the g. K . f.

< V , V, >. s a t is fie s form uła oC , w e s a y that oC is true in the g . K . f. < V • V- X l > * T h e s e t o f a ll fo rm u la s true in the g . K . U

< , V, (t h e c o n te n ts o f < , V, IX > ) w ill b e d e n o te d b y e ( w . v . ) .

B y a s tru c tu ra lly g e n e r a ł K r ip k e fram e ( s . g. K . f . ) w e s h a ll u n d e r s t a n d a n y trip let < y { , V, > • w h ic h s a t is fie s the fo llo w in g c o n d itio n s :

( i ) W an d a re K r ip k e fra m es,

(U ) V i s a fu n ction from W to the p o w e r s e t o f so m e s e t s u ch that V ( a ) ^ fo r a n y a € W a n d V ( a ) £ V ( b ) i f a ^ j f b . B y K r ip k e m o d el on the s. g. K . f. < Y / , V, ^ w e sh a ll

u n d ersta n d e v e r y fu n ction v w h ic h ta k e s o n e o f v a lu e s { U , <t> J-a s its v J-a lu e fo r J-a trip let < ot , a, w > o f a c lo s e d form u ła ot < P O R ( v ) and an elem en t a ć W and an elem en t w fc W ^ , w h o s e v a lu e s a re determ in ed b y the fo llo w in g c o n d itio n s:

- 46 -1. V ( p ( n ) X x . . . f n, a ; w ) - 3J \ f v t x > x w V b > a v ( p ^ ^ x . . . ? n, b, w 1 ) - al , 2. v (otA jS . a, w ) - U ^ v ( o t , a. w ) - Ji v ( f } , a, w ) - U . 3. v ( a t V ^ , a, w ) -

al

v ( oC

,

a, w )

-

al y v ( ^

,

a, w ) - ai , 4. V (oę,->|3. a, w ) - 314=^ V w Ł > 1 w ^ b a ( v ( o t , b , , W;L) - <t>^v ( fi , b, w Ł ) - 1 ) , 5. v ( y o C . a, w ) - a i- # ^ > - \ ^ b ^ a ‘V [tw 1 > ± w (vfcC, b, w ^

-

0

).

6. v ( V - x . a, w ) - al 4 = ^ - b ) a - \ ^ ^ e v ( b ) ( v ( X ? , b, w ) - al ), 7* v ( ^ * O C , a, w ) - 110 31 ^ € . V ( a ) ( v , a, w ) - ai). W e s a y that form u ła o C e P O R I s s a tis fle d b y a K r ip k e m odel v on the s. g . K . f. <~W T V, '‘W ^ > j ** v , a, w ) - al fo r e v e r y a e W and fo r e v e r y w £ W ^ , w h e r e o C i s the u n iv e r s a l c lo s u r e o f o C

If e v e r y K r ip k e m o d el on the s . g . K . f.^ V / 7 V, ^ s a t is fie s form uła O C . w e s a y t h a iO C is v a lid on the s. g. K . s e t o f a ll form u las v a lid on the s . g. K . f. V, . V i > .< the c o n ten ts o£ < V ^ v - Y i > ) w ill b e d en o te d b y E , V,

B y an e a s y v e r ific a t io n b y m ean s in d u ction with r e s p e c t on the num bre o f lo g ic a l s y m b o ls in form uła o C w e h a v e

-P R O -P O S IT IO N . L e t v b e a K r ip k e m o d el on the s. g. K . v . - \ o . L e t a, b € w , a ^ b and le t w, u £ w ^ . ^ u . T h e n fo r a n y c C ^ P O R , v ( c C * , a, w ) - aj lm p lies that v ( o ć * . b, u ) - al , w h e r e o C * is the u n iv e r s a l c lo s u r e o f o C . T h e s y m b o l C L i s u s e d to d e n o te the s e t o f th eorem s of c la s s ic a l p r e d ic a te c a lc u lu s , w h e r e a s the s y m b o l I N T is u s e d to d en o te the s e t o f th e o rem s o f in tu itio n istic p r e d ic a te c a lc u lu s . W e s a y that a s e t o f fo rm u la s L S= P O R is an in te rm e d ia te p r e d ic a te lo g ie if

lt s a t is fie s the fo llo w in g co n d itio n s ( c f . C 5 j ) : 1. I N T Ł L £. C L

2. L i s c lo s e d u n d e r the su bstltu tion , 3. L is c lo s e d u n d er m odus p o n e n s , 4. L is c lo s e d u n d er the g e n e ra lizn U o n .

IŁ. W e d is c u s s re la tio n s b e tw e e n the g e n e r a ł K r ip k e fra m e s and the s tru c tu ra lly g e n e r a ł K r ip k e fra m e s a n d w e p r o v e o f s o m e p r o p e rtie s o f thls fra m es. P o r this p u rp o s e , w e m odify the k n ow n P ittin g method

(w h ic h a llo w s us to ca p tu re c e r ta in re la tio n s b e tw e e n the K r ip k e fra m es a n d the p s e u d o -B o o le a n a lg e b r a s (c f. £ 2 ] ) ) .

T h e fo llo w in g c o n s tru c tio n a llo w s the c o r r e la tio n with a n y K r ip k e fram e "Y( o f the r e s p e c t iv e a lg e b r a . L e t - < W , 4 > b e K r ip k e fram e. A s u b s e t H O. W is c a lle d a h e r e d ita r y s u b s e t o f fram e

~yf if fo r a n y a , b £ W lt fo llo w s from a £ H a n d a », b that b £ H. T h e s y m b o l D ("W ) d e n o te s the c la s s o f a ll h e re d ita r y s u b s e ts o f fram e "W" • w h e r e a s the s y m b o l A l g (" ^ 0 d e n o te s an a lg e b r a with u n iversu m D ( Y O a n d p s e u d o -B o o le a n o p e ra tio n s d e fln e d a s fo llo w s : fo r a n y H r H 2 6 D ^ ) , A H 2 “ O H 2, H 2 - H 2, H x —ł H 2 - { a ] a 6 W A\ V b £ W ( a 4 b * 4 b ^ b £ H 2 ) } , T H , - It is o b v io u s that In Alg('J»/r) th e re e x is t A H. t f T and V Hh w h e re T , T a r e s e t s o f a n y p o w e r. t £ T 1 T H E O R E M 2.1. P o r a n y s . g. K . f. , V, Y { ± , E ( V ,

v.

-

E ( ' W ' .

V,

A l g O ^ ) ) . P R O O P . S u p p o s e that

£.V,

V, V ' 1 > is a s . g. K . f. and v is K r ip k e m odel on the s. g. K . f. < * v t " W \ • W e a fu n ction from the C a r te s ia n p ro d u ct o f the s e t o f a ll c lo s e d form u las o f P O R ( V ) and W in to D ( W ^ ) « « fo llo w s : v ( , a ) -{ w £ W 1 | v ( oC , a, w ) - U . It is e a s y to v e r lfy , b y in du ction

with r e s p e c t o n the num ber o f lo g ic a l s y m b o ls In cx , that is a v o lu a tio n on the g. K . f. £ , V, A l g ( ) > . N o w , s iń c e the unit elem en t o f A l g ( ' ^ ' 1 ) is W Ł its e lf, h e n c e ( OL , a ) - if and

48

-only if fo r i l l « £ W Ł v ( ot , a, w ) - li . T h * p ro o f* of th* remaining * t * p * a r * * a a y an d w ill b * amltted. Q. E , D.

Let <*VY\ V, be a g. K . f. A a u b a *t P o f th* unłvaraum of a lg * b r a 1* c a ll* d th* 01 ter of a i g * b r a X L if P i s non—« mpty and for an y a, b £ A , { » , b } £ P if an d onły if a a b C P . A fil tar

P i s s a ld t* be prima if P C a an d for a n y a, b fc A , if a v b fe P ,

than { a , b \ A p ^ <tf. W * s a y that a fil tar P Is An K (W , V ) -01 ter If lt h a a th* f*ll* w in g preperty; for a n y T 4 R (W , V ) , if ■{ a t 1 t £ t }

£ P , than A a t 6 P . A n d w * s n y that a 01 tar P i * an K . ( v ) -flltar if it h a * tlio^foUcrwing propertyt fo r an y T 4 K .(v ), if V a f

t fc T € P , than {_ a t ) t £ T J P ji j!, w h o r* K. ( V ) den otes th* sm allest Cardinal w hich i s g r * a t * r than V ( a ) fo r a n y a £ W . If K. ( W, V ) 4 than w * idantify ^

(w ,

V )-G it a r* with Gitara. Sim ilarały, if

W (V ) 4 )\ o ‘ 0 ł*n w * idnntify « (V )-G it a r s with prlra* Gitar*.

L E M M A 2.1. ( i ) If R ( W , V ) > X e , than th* H (W , V )-G ita r g * n * r a t * d b y a non-am pty * u b s * t A p of th* univ*rsum of a lg e b r a V[ la th* s * t of all *l*m *n ts a £ A su c h that a > / \ a for s * m *

__ t € T •lam ent* * t £ A ^ , t €. T , and fa r som a T 4 lt

(w ,

V ) ,

( i i ) If K. (W , V ) 4 X o* then **** * ( w « V )-G ita r gen eratad b y a non-ampty a u b s *t A of the univ*raum of a lg * b r a \X is th* sat of *11 •lam ent* a £ A su c h that a ^ a ^ A . . , A a n fo r aem e elem ents a^, . . « .a £ A ,

n o

P R O O P . B y an e a a y variG catien

T h ia foliow a a a a y from Lemma 2.1 that

L E M M A 2.2 (c f. H. R a a io w a and R. S ik orsk i

C

6 j ) . Let a o ba a eloment of urtiveraum of a lg e b r a an d let P b e a K. ( w , V )-G ita r o ł a lg e b ra X t . T h a n [ p , a Q) - - ^ a £ A ) a > a o A c , c ć P } la the R (W , V )-G ita r gan aratad b y the aet { a Q^ P .

L E M M A 2.3. Let P be a H. (W , V )-G ita r in a lg e b r a and au p po ae that ( a —> b ) ^ P . T h on the R

(w ,

V )-G lte r £ P , a ) gen erated b y P and a d o e a not contain b.

P R O O F . S u p p o s e that b £ C P , a ) . T h e n b y Lem m a 2.2 b > a A C fa r sa m e c £ P . S a c A a — b a n d h e n c e a b £ P . Q. E . D. L E M M A 2.4 , L a t P b a a p r a p e r R ( w , V ) - fil te r o f a lg e b r a Vt- a n d a u p p o s e i a 4 T h # n the R ( w , V )- fu t e r C P , a ) g e n e r a ta d b y P a n d a i s p r e p a r . P R O O P . O n th * s tre n g th o f Lem m a 2.3, s in e * ~i a — ( a ~ ? 0 ) . Q. E . D.

LEMMA

2.5. Lat a algebra XC satisfies tha fellawirtg conditions

the unlon aat-thearetical of any Chain of R (w , V )-fil ter a af algebra

Vi

Is a K. (W, V)-filter In

. Let F q be a

K- (W, V ) -TUter of algebra

and a

4

P 0. Then Pft can be *xt*nd*d to a K.(V)-fIlter P such

that a ^ P,

P R O O P . L e t u a nota that i f R (W , V ) v )4 than w e c o n s id a r fllt e r s a n d thus a a c h a lg a b r a s a t is fie s tha co n d itia n t tha u nlon a a U the o r e tleni o f a n y Chain o f fllta rs i s a filta r. T h u s, i f K ( w , V ) 4 X than tha p r o o f o f U m m a i s the sa m e a a in C 2 ], pp . 25—26 . S a n o w w e o n ły h a v a to c o n s id e r tha fo llo w in g c a s o s t 1 ° . R ( W , V ) > X o an d

K ( V ) 4z R ( V ) > X o* W ® om*t th* Pr o ®* ° t Lem m a in c a s a

1 ° , s iń c a it is mada sim p le b y tha p r o o f o f Lem m a in 2 ° . T o p r o v * 2 ° lot u s a ssu m o that P q is a R (W , V )- f ilt e r in XC and a ^ P . L a t K ba the c o ile c t io n o f a ll K (W , V )- filt e r a in not co n ta in in g a. T h e c o ile c t io n K i s n on -em p ty, s iń c e P o £ K . tr a t ^ b a a Chain

in K , £ > an d P - U ( X I X t ^ ) . S in c e a f X fo r a ll X € 'C ,

then a 4 P And on the stre n g th o f assu m p tio n a a f Lem m a P i s a R ( W , V ) - filt e r ln 1y£ . H e n c e , b y K u r n t o w a k l- Z o m Lem m a, K h a s a

m ajdm al e le m e n t P . N o w , a ssu m e that P is n ot a K ( v ) - f i l t e r . T h e n th e re e x is t e le m e n ta a , £ A , t e T a n d T < R ( V ) , s u c h that \ / j * t fc T a, £ P a n d l a , ! t £ T ^ P - <f>. P o r e a c h t £ T , le t P t -[p, a t) b e a K (W , V )- f iit e r g e n e r a te d b y the s e t P o { * t$. H e n c e F t O F fo r e a c h t £ T . L e t u s a s s u m e that a £ P { fa r a a c h t ć T . T h u s b y Lem m a 2.2 th e re e x is t e c t 6 P , t £ T , s o that a > c t /S a { fo r e a c h Ł € T. L e t c - / \ c ,. S in c e P is a R (W , V )- fllt e r

- 50

then c £ P a n d a > c A a fo r e a c h t fc T . H a n c a it fo lla w s that a >

V

( « A a , ) . B e c a u a a c A

V

a -

V

( c a a . ) ( b y p.

t fc T * t ć T t C T

1 3 5 ) a n d c A

V

a € P , than a £ P - a c o n tra d lc tio n . T h a r a fo r a t £ T

thara asdata t £ T s u c h that a f P (. S o P ls n ot m axim al a c o n tr a -d lctien . T h u s P l s K . ( v ) .fU te r. Q. E . D.

L a t

U

b a a a lg e b r a . B y tha s y m b o l

Pp(7X)

w a d a n o te tha s a t o f a ll H .(W , V ) - f i l t e r o a f a lg a b r a , w h ic h a ra K ( v ) - f l l t e r e . T h a s y m b o l d e n o te s tha K r ip k a fra m e < P p ( \ £ ) , G > .

THE O REM 2.2. ( i ) Lat %£ ba a a lg a b ra satisfies tha c°ndltion: tha unlon sa Utheoredcal of any Chain of K.(W, v )-filte rs ls a K.(W, V )-flltar in , Than E ( <W'. V. I X ) ć: E ^ , V, 3 ^ ( 7 $ ) *

(U ) P o r a n y fln ita a lg a b r a ^ . E ( ”VY\ v . X i ) ■ E ( Y , V,

P R O O P , T o p r o v e ( i ) la t u s a s s u m e that < ' Y f , V, * g. K . f. a n d v i s a v o lu a tla n o n tha g . K . U ć J Y f , V, '1%'?. W e d a fln e a fu n ction fram C a r ta s ia n p ro d u c t o f tha s a t o f a ll c lo s e d fo rm u la s o f P O R ( v ) , o f W a n d P p (7 X ) *nto *L ^ • <t> ł a s fo llo w s : ( cx_ , a, w ) - U lf v ( o l , a ) £ w . In o r d e r ta s h o w t h a t t h u s - d e fln e d fu n ctio n ls K r ip k a m o d el on tha s . g . K . f. O Y . V, & ( 7X ) > w a v e r < jfy O n ly tha c a s a 4 an d

6

, b e c a u s a tha v e r iflc a t io n >f tha ra m aln in g

c a s e s i s a ith a r a a s y , o r sim ila r to 4 o r

6

. L a t v ^ ( o C - ^ /3 , a, w ) - U . T h a n v ( o t -9 / 3 . a )

6

w . S a / \ ( v ( ot , b ) v ( /3 . b ) )

a <c b

t w a n d h a n ca v ( ot, , b )

—9

v (

/3

, b ) e w fa r a n y b > a . If vr^

2

w than l s e le a r that fa r a n y b ^ a , v ( o t , b )

-9

v (

^3

, b ) ć w ^ . T h a r a fo r a o n b a s is o f the d efln itien o f filte r, lf v ( oC ,b ) € w ^ , than v (

/3

, b ) fc w ^ , a n d c o n s a q u a n tly fa r a n y b > a a n d fo r a n y w ^

2

w v ^ ( ot. , b j w ^ ) ■ 4 a r V j ( /3 , b, w ^ ) - U . N a w , c o n s id a r the c a s a w han ( OL —9 /3 , a , w ) - O . P ro m th is a ssu m p tien it fo lla w s that

v ( <3^ f i , a f ) « w a n d c o n s e q u a n tty A ( v ( oC , b ) —» v ( /3 b ) )

J a ^ b .

ę w . T h a r a fo r a thara e x ls t s b Q ^ a s u c h that v ( o t , b Q)

-9

v ( /3 ,bQ) ^ w . T h u s o n tha s tre n g th o f Lem m a 2.3 w a c a n a ssu m e that th e re e x is t s a K (W . V )- f ilt e r w # in a lg a b r a X t s u c h that w q

2

w , v ( o t . b Q)

t w q a n d v ( f i , b Q) £ w ^ , B y a p p lyln g to tha lattar Lemma 2.5 we obtain that W( ca n b a axtan dad to a K (V )-d ilte r au ch that v ( f i , b Q) ^ w ^ . S o thara exłst b Q > a an d w ^ 2 w au ch that v^(eC. > b Q, w ^ ) - K a n d f i , b # , ) ® . In tha c a a a 6 tha fo llo w

-in g a q u iv a le n c a a holdt v ^ ( x eH, , a, w ) - JJ ifl v ( V x d , • )

e

w

iff

A

A

v ( O C ? . a )

e

w Ut \ f b > a > ^ S fc V (b )

b > a 3 ć V ( b )

v ( o L ^f . a ) fe w lff b > a V ( b ) v 1 ( o c ' f , a. w ) - U . T e an d tha p re e f a f ( i ) lt su ffic e a ta nota that lf v ( oi_ , a ) >11 then v ( ot , a ) g w fo r all w £ H a n c a V g ( cC , a, w ) - li .

Canvaraaly, let v ( o L , a ) 1| . T h a n b y Lemma 2.5 wa ca n a s e n d to a K (V )-f llt e r w au ch that v ( ®(. , a ) ^ w, an d conaaquantly v x ( oL , a, w ) - <>.

T a p ra v a ( l i ) o b a a rv a that b y Theoram 2.1, E ( ')\ f , V, S ^ C K T ) ) ) ■ K ( Y f , V,' A l g ( ^ p ( ^ J ) ) ) . S in c e is a fi ni te a lg e b ra , therefere

i * laom orphic with tha a lg e b ra A l g ( P p ( \ 0 ) - tha function fj A H j D ^ d a : ) ) * u c h that f ( a ) - { P | a fc P * P fe , i *

• n laomorphism. T h u s w e con cluda tha p roaf a f The o rem. Q. E . D.

Łat y f x - £ W r 4 1 > an d W g - 4 W g , 4 g >

ha K rip k a fram es; w a aaaum a that ^ W 2 - T h e product of fram es “V f 1 * W 2 i « dafined a s a palr < W 1 W g , ^ ^ >

, w hara 4 -jy ^ • 4 ^ U 4 2- It ca n b e e a s ily noticed

that the condltion r* W g - ^ ls not a ssa n tia l - the product of two K ripka fram as can b e obtalned b y u s ln g thelr isom orphic c o p ia s . In

the c a s e w h en w e c o n s id e r o f the s e t o f K r ip k e fra m es ^ I t i j f w e sh a ll w rite k ( V ( ^ ) | t £ I ) o r j * j in s te a d o f < I ( e 1^,

^

i ^ I

V/

1

>

. w h e re

4 . >< , “V,

-

U { 4

i

\

1 1 •

A n Im portant p r o p e r t y o f s tru c tu ra lly g e n e r a ł K r ip k e fra m e s is. e x r e s s e d in the fo llo w in g :

- 52 -o 1 i ć I } th e re l s a a. g. K . f. < Y^. V, Y { * > a u c h that E ( > Y

. v,

V Y ° )

-

n

e

(

_{vu '}

y f * ) .

I € I 1 1 1 P R O O P . S u p p o a e that la g iv e n of the a e t ot a. g . K . f. ' s V,,

Y f f

| i € 1 ^ • lat

u a c o n a id e r the a . g . K . (, < y r . v , r > . w h a r . w - x v , > y ° - x W i , 1 e i r i i e i

an d V la the fu n ction from W "L W j I i €: I J a u ch that fo r a n y a 6 1 ł ^ i } . V ( a ) - V t ( a ) l f a « W j , W e omlt the p r o o f e f the in d u a lo n E ( W , V, W ° ) £= E ( W . , V., W ? ) s łn c e it is

i € i 1 1 *

c e m p le iiy d ir e c t. T o p r o v e the c o n v e r a e In c lu a ie n ie t ua a ssu m e that cC

4

E (

"W

. V,

y ( ° ) .

T h u s th e re e x ia t K r ip k e m o d el v on the a. g . K . Ł < W , V, Y f ° > , a o e W a n d w o e W ° s u c h that v ( o t ', a o , w o ) - 0 . S in c e W -

U

•(. W j | i 6 i j a n d W * - U { W ° | i € ] ^

then h e r e e x is t i, j € I s u c h that a Q 6 W j a n d w Q g W ° . On the S tren gth o f the c o n s tr u c tie n o f the s . g . K . f. < YY , V. "W ° > it is

o b v ie u s that w e c a n a s s u m e that i m j. C o n a id e r K r ip k e m o d el on the a. g . K . L < Vj, ~W ° > a u c h that /3' , a, w ) - v ( f i ' , a, w ) fa r e a c h (3 € A T , fo r e a c h a f- W j a n d fo r e a c h w « . W ® . T h e r e a d e r c a n e a a ily c h e c k that s o d e fin e d o f the m o d el c a n b e e x te n d e d that: f o r a n y yS g P O R , fo r a n y a g W j a n d fa r a n y w e V/j_ • v j ( w ) - v ( /3' , a, w ) . H e n c e , v ± ( oc' , a Q, wq ) - 0 and a o e t ^ A E ( V Y T ,0 ) . Q. E . D . i t l 1 1 1 T H E O R E M 2.4. P o r a n y a . g . K . f. < ¥ , V , K ° > th e re e x is t s a a e t o f a. g . K / i { < Y f j. VJ( * > | i ć I } s u c h that ( i ) e

( Y T . v .

T Y ° ) - r \ E ( V ... Yi rJ) . l e i ( i i ) P o r a n y i £ I, W ; b a s the le a s t elem en t, ( i i i ) P o r a n y 1 6 I, 'Ylf ° b a s the le a s t elem en t.

P R O O P . L e t u s a s s u m e that < W , V, ) Y ° > i s a s . g . K . f. P o r e a c h a € W , l e t W & - { b | a g b } and l e t b e t h e re s tric tio n o f V to W . S im ila rly , fo r e a c h w € W ° , le t W * m u | w ( uJ.

N o w let u s c o n a id er the se t ef a. g. K . t. ' * { v a . 'W * > I

& 6 W , w e W * It l s e le a r that in o rd e r to p re v e ( i ) - (i ii ) lt su ffices te sh ow : E ( TY , V, >Y ° ) - 0 { e ( T Y *. v a . >Y ° ) I • «

w € W * } , If ot 6 E ( TY , V, 7 / * ) , then for e v e r y K rip k e model v on the s . g. K . f. < W . v . 7/ ° > • o v e ry a e W and for

e v e ry w ć W®, v ( oc‘ , a, w ) « U . T h is g iv e s that for e v e r y a € W and for e v e ry w e W ®, cc e E ( Y f a , V^, 'V Y ^ ) . honee

oC e O { E ( V a . V a . V ^ ) | a G W ,w e W ° } . T h a s there e x is t Kripke

model v on the s . g. K . f. < V , V , X ® > , a _{' O} e W and w _o t W® su c h that v ( ot,1, a Q, w # ) - 4> . This y ie ld s - sim ilarly a s ln the proof of the Theerem 2.3 - that ot i E ( V/ , V V _ ) and

Ot A W

O O o

therefore cC ^ H { e ( Y a . Vft, ^ ® ) | a <S W, w W ° } . Q , E . D.

F ollow in g H. Ono 1^9 J w o s a y that the function fx W t—> W * is a em bedding of TY - < W , ^ > into TY ® • < W , ^ if

and only if it sa tisfie s the follow ing conditions: ( i )

f ( w )

- W®,

( U ) \ f a. b G W ( a 4z. b =$> f ( a ) <£,* f ( b ) ) t

( i i i ) W a G W w G W® ( f ( a ) ^ ° w 3| c e W ( a < c A \ w - f ( c ) ) ) .

If there ls a em bedding of W Into W ° , w e s a y that TY is em beddable in v r ° .

W e s a y that s. g. K . f. , V, ,s om beddable into S. g. K . f. <. y { * , V ° , 'yY ° > if the following conditions hołd:

( l ) th e re e x ists a em bedding f ot TY Into T Y * ,

( l i ) th e re e x is t s a fu n ction g from W ■{, V ( a ) | a 6 W } te

V j { v * ( b ) | b £ W® J s u c h that g ( v ( a ) ) - V * ( f ( a ) ) fo r e a c h

a G W ,

(iii) th e r e e x is t s a em bedding h o f Y f j_ *nto 'W *•

T H E O R E M 2.5. S u p p o s e that < T Y , V, W > and < TY®, V ° , i a1* stru c tu ra lly g e n e r a ł K r ip k e fra m es. U ć. y / , V,

is em b e d d a b le into <. v °> V Y ^ > then E ( T Y , V, T Y ^ )

54

-P R O O F . Let oCo łf. E ( W ° , V ° Y f £ ) . T h en there l s K rip k e model v on the s . g. K . f. < V 0. V ° , ^ > • * 0 €■ W ° and w q ć W®

s u c h that v ( o l0> a o, w ^ ) - <t> . N ow , w e dellne K rip k e model on the s . g . K . U \ ( , V, e s follow s: v 1 ( oC , a, w ) m v ( c*- . f ( s ) , h ( w ) ) fo r a n y oC € A T , for an y a e W and for an y w e W j , w h ere f i s a em bedding of 'W Into W ° and h i s a em bedding of Y/ ^ Into Y f ®. B y lnductien with re s p e c t to the number o f lo g ic a l sy m bo ls ln form uła OC , w e ca n sh o w that is r e a lly

K rip k e model on the s . g. K . f. < ~Y/, V, ^ and that for any

OC € F O R , v x ( oC , a, w ) - v ( oC , f ( a ) , h ( a ) ) . S in c e f ( w ) - W ° an d h ( W Ł ) m W ^ , then there exist b ć W an d u fe su c h that f ( b ) - s o , h ( u ) - w Q. H en ce, v 1 ( oc'o , b, u ) - v ( ©c'0 , s o , w )

-0 • 9*

W e s a y that an a lg e b r a ~^X i s stro n g ly compact lf an d onły lf there eadsts the gre a test element ln s e t A - U j. S u c h an element

(if e x is ta ) w ill b e den eted b y * XC • T h e sym bol X£ /F d en otes the ąuotient a lg e b r a obtalned b y m eans of the relation c o n g ru e n c e determined b y the filter F of a lg e b r a .

L E M M A 2.6. Let < Y( , V, > b e a g. K . f. Let \ C b e a a lg e b ra sa tisfie s the condition: the unlon set-th eo retical of an y Chain of K (W , V )-fllt e r s is K

(w,

V )-fltte »' in U . Let a e A and a ^ U . T h en there e x ls ts k (W , V )-fllt e r F su c h that /F ls a strongly com pact a lg e b r a an d * ^ ^

P R O O F . On the s t r e n g h of Lemma 2.5 w e ca n extend t l i } to H. (V )-filt e r F s u c h that a ^ F . Now , w e will p ro v e that \ t /F

is strongly com pact and X - ^ j p ”

E

a } ] F . B y a dlract argument w e get that l l ^ •* t l f ] F - F It i s e le a r that £ a U F y F, sln c e a ^ F . T o p ro v e that 1} a ] F i s the gre a test element in A/F - - £ f } le t u s s u p p o s e th a t T b D F F . H e n c e b Ą F , and s o b y the proof o f Lemma 2.5 a f e l l F o £ b } ) , w h e re E f o { b } ) is a ( W , V )-fU te r gen erated b y the se t F u { b } . T h e re fo re , on the b a s is of Lemma 2.2 a > c a b fo r som e c ć F . T h u s b a

^ c which re s u lts in b a € F and c on sequ en tly [ b l ] F < ^ j p E c t l V\

T H E O REM 2.6. U U l ł ł n o n g e n t r s i* a lg e b r a then fo r an y g. K . f. < y / • v * ^ ^ «■ * « t a f g. K . f. ' • { < . V ,

> I U l } su ch that

( i ) E ( W . V. U ) - i Q t E ( W , . Vj, U j ) . (i i ) P o r a n y i € I, h a s the le a s t alemant, (iii) P o r a n y i d I, ia stro n g ly compact a lg e b ra .

P R O O P . S u p p o s e that < 7)f, V, U > l » a g. K . f. P o r a a c h a e W , let W - { b I a n d lat V b a ho restrictlen o f V

A A

to W . Łat the sym bol S C denote a set of ali strongly com pact a lg a — b r a s , which a re o f form %t-/P. Łat u s c o n s id a r tha s e t a f g. K . f. ' s { < 'YVa , Va , ^ 5 > | a € W , "73 ć S C } . It i s c ie a r tha in o rd e r to prlve ( i ) -> (l i i ) it s u fB c e s to s h o w E ( ")Y , V, ) - f ') E ( Y Y '., V . 7 3 ) I a Ć W , S C } . T h e in d u a le n from the left

A A 4

ip right i s o b v io u s. T o prove the c o m re rse inclusian let u s a ssu m a

that CL ^ E ( TY” , V, lyt, ) . T h u s h ere ejcist a v a lu a ilo n v on the g. K . t < y f , V, > an d a 6 W su c h -that v ( ot1 . a ) ^ ll . H ence, on the strength of Łemtr.a 2.6 th e re must e x is t a stro n g ly compact a lg e b r a ^yC /P s u c h that “ t v ( <<■' . a)7 l P « C o n s id a r a v a łu a -tion v 1 on the g. K . f. "W ** Va ' Z5' ^ * u ch thAt v ^ ( /^ . b )

- t v ( , b ) ] P s u c h v ^ ( /3 , b ) - C v ( j3 , b ) 3 P ter a a c h ^ d A T o c c u rrin g in an d fo r a a c h b 6 W . W e o mit ihe proof that v , ( ot* , a )_{A .X .}

“ C v ( o t 1, a ) 3 P “ s in c ® h » s q u ite sim p le, th e re fo re , ■cC ^

E ( 7 f a . VfcI " M / P ) . * * E . D.

T H E O R E M 2.7. ( i ) E ( 'W’ , V, "VC ) i s an intarmadiata p re d ic a -te lo gie if and ordy if thara i s an elem en t » ć W s u c h thbt V ( a ) ^ yC^.

( i i ) E ( V ( . V, y ) i s an intermediate predicate lo g ie if an d on ly

if there i s an element a G. W su ch that v ( a )

P R O O P . T o pro ve ( i ) let u s o b s e r u e that b y T h e o rem 3.6 a f C 5 l p. 629, E ( W , V, -C R ) i s an intermediate predicate lo g ie if and

only if there i s a 6. W su c h that V ( a ) > )*( , w h ere the letber

- 56

-n e -n d e g e -n e r a e a lg e b ra , tha-n ia e le a r that ( i ) holds.

T a prove ( l i ) lat u s o b s e r v e that b y T h eorem 2 .2 ( i i ) E ( ^ , v »

CR)

” E ( W , V, ćTp ( 3 R ) ) . N a w , It la o b U o u s that < , V, YY £

ia am baddable Into < ”W , V, ? (C R ) ^ tar a n y K rip k e (ram a

Y { X’ C o n aaqu en U y, b y T haoram a 2.8, E ( "W , V, Y f ^ ) C. E ( Y ^ , V,

J p ( C R ) ) , w hlch g iv a a finalły that ( l i ) h olda. Q. E . D.

m. T h ła aacH an ia a eontinuation ot [ 7 } • W a gtv*» aam a critarian fa r łha incluaion ralatian b a łw a n tha norv-diajunctlve contanta ot two atrueturally g e n e ra ł K rip k e fram es.

b a t 'W ■ < W , b a a K rip k e (trama. P o r a n y a, b e W, w a a a y that b la a u e e e a a o r o f a (a y m b . a < b ) U a ^ b an d a ^ b . W a a a y that b ia a direet a u e e e a a o r e f a (aym b, a -< b ) U a < b

and there d o e s not exist c « W su c h that a < c < b . A K rip k e irame 'Y / ' ^ is strongly atomie ( cf. f i ] ) ii and on ly ii for any a ,,b e .W , ii a < b , then there exists c <5W su ch that a < c ^ b . A Krip­ ke irame aatisiiea the in e re a sin g s e q u e n c e s condition ( cU \_lj ) ii and only ii there d o e s not exist a n iniinite in e re a s in g s e q u e n c e oi elements oi that irame. R can e a s iły be s e e n that a irame sa tisiie s the in e re a sin g s e q u e n c e s condition ii and only ii e a ch ot its non-empty s u b s e ts h as a maximal element,

L E M M A 3.1. Łet b e a stro n gly atomie K rip ke

irame. S u p p o s e that X Ł W ^ contains no maximal element oi the irame and let v b e a model on the s.g.K .i.< lV ^ V ^ su c h that the condltions A T W w e X ; v (o C , a , w )

-min { v ( o C , a , u ) } u i s satisiied. T h en lo r an y iormula o C * 1 F O R ^ - j ^Zj *^le c o n d i t l o n ^ a C W ^ w e x : v(xT,a,w)

-m in -£ v (o C , a , u ) | u J holds.

P R O O F . B y iduction with re s p e c t on the number oi lo g ic a l sym­ bole in o C . F o r o C t a t the Lemma h old s b y assum ption. Łet u s su p­ p o s e for induction that the resu lts ls true tor all ^ e FO R

an d fa r a n y w fc X a n d fo r a n y a ć W : v ( , a, w ) - min { v ( , a, u ) I w x u } an d v ( y , a, w ) - min { v ( ^ , a, u ) | w

^ u ] . S u p p o s e a l M that ot l s a farm u la containing r lo g lc a l

sym bols. Lat ot m /?> , a fe W an d w £ X . T h an v ( oC , a, w ) - v ( /3 A. a, w ) - min i v ( /3 , a, w ) , v ( w , a. w ) } - min { min { v ( / 3 , a . u ) ! w - < 1 u } , min 1 v ( y , a, u ) I w -4 ^ u - mi n { v

( / 3 ^ a , u ) | w A 1 u } - min { v ( ©o , a, u ) ) w -< 1 u } . Lat u s

n a w assu m e that «C - (2 —> , a e W and w 6 X If a, w ) - U , then fa r a n y u > ^ w v ( /3-^ ^ , a, u ) - 11 . S in ce w ls n e t a maxlmal element ln YE , an d ' ł { ^ i s stro n g ly atemic. then

{ u ( w - < A u } therefore min 1 v ( f i - * ^ , a, u ) | w u j

- U . N ew , c o n a id e r -the c a s e w hen v ( fi , a. w ) m <b . From

this assum ptlon tt fo llo w s ihat fo r som e z > ^ w and fo r som e b > ej v ( f i . b, z ) - U an d v ( y , b, z ) - © . If z - w , th e n a c c o rd in g

to the induction h yp o th e sis th e re e x ls ts u^, w ^ an d v ( yS, b ,

ux ) - U and v ( , b, - ©. T h u s v ( jf , b, u ^ - <J> and

brom h ere v (/ 3 a, u ^ ) - © which re s u lts in min { v ( Q ^ , a, u )

I w Ł u } - ® . Let u s now su p p o s e ihat z ^ w, T h e n w < 3 z,

s o b y the aasum ptlen that is stro n g ly atemic, there e x lsts u q

su c h that w ^ uq 4 . z. S in ce u q ^ ^ z, hen v(y3>^, b, u# ) -© and v ( ^ ^ - , a, u q ) - ©, whlle siń c e w ^ u q , then min {

a, u ) | w ^ u } - <t>. F o r oC ■ ~l |3 , the re a so n ln g is simlllar.

Let u s su p p o s e that oC -

V*/3

, a € W an d w € X . E viden tly & I w -< Ł u } y J*. If v ( V *

(3

, a, w ) • U , then for a n y b ^ a

and fo r a n y ^ Ł V ( b ) , v ( ( S § ,b ,w ) - 3( . H e n c e b y the in du ction h yp o ­ t h e s is fo r a n y b ^ - a and fo r a n y i ę e v ( b ) t min-£ v ((J § , b , u ) ) w ^ u ^ - l l , and s o m in ^ v ( V x ^ »* »u ) I w " ^ i u3 “ ^ * In 016 ° P P ° 3łte w a y , if

v ( V * fi ,a ,w ) - ^>, then th e re is b ^ a and ) § € v ( b ) su ch that

v ( P § ,b, w ) - and h e n c e v (& ^ ,a, w ) - T h e n on the stren gth o f the in du ction h ip o th e s is m i n / v ( ^ ' ^ , a t u ) | u 3" ” s o m in - £ v ( "Y ,a, u ) | ^ u ^ - (j). F in a lly s u p p o s e that o C - 3 x fi • a e w and w G X . S im ila rly a s a b o v e •{ u | w ^ 3 u 3 ^ v ( 3 x ^ a' w ) ” 3** then th e re is s o m e § € v ( a ) s u ch that ,a ,w ) - 3i . T h e r e fo r e b y

58

-min v ( 3 * fi . « . u ) | w u } - 31 . If v ( 3 * ^ • a»w ) ” • then fo r a n y S e v ( a ) » v t y Ś »*• w ) ■ I • S o b y the in d u ction h ip o th e a is fo r a n y ^ € - v ( a ) , min { v ( 0 - ^ ,a, u ) | w u } - \ , th e re fo re

min * { v ( 3 x {3 ,a, u ) | w u ^ “ Q-E *D«

L E M M A 3.2. L e t ' Y f i “ ^ w i » ^ i ^ ^ a a tro n S iy a tom ie K r ip k e tram ę s a tis fy in g the in e r e a s in g s e q u e n c e s co n d itio n . b e t C b e a s u b s e t c o n ta in in g a ll m axim al e lem en ts o f the iram e

T h e n e v e r y m o d el v q o n the s. g. k . t. < - y { , v . - y o > c a n b e e x te n d e d to a m o d el v o n the s . g. k . u v . y / ± > s o

that fo r a n y form u ła OC £ P O P r fo r a n y a ^

<A. ^ , -l . V .3 V

W an d o r a n y w £ W , v ( c < , a , w ) - v q(c>C . a, w ) .

P R O O F . L e t the a ssu m p tio n s o f the Lem m a b e s a t is fie d and le t Vq b e a m o d el o n the s . g . K . f. , V, ~tyf0 ^ • F o r anY

o C £ A T w e put v ( c £ , a , w ) - v q(c*C , a, w) l f w £ W q ) an d v fc jC , a, w ) - min "{ v ( a C , a , u ) | w x u J l f w £ W ^ - W q . S u c h a d efin itio n is c o r r e c t b e c a u s e , If w € - W ^ - W ©, then w is n ot the

m axim al elem en t o f the fram e thus o n the stre n g th o f s tro n g a tom icity o f the fram e j the s e t -^u | w ^ i u ca n n o t b e em pty.

W e s h a ll flr s t p r o v e that the fu n ction v h a s b e en , In the a b o v e w a y , d e fin e d fo r e v e r y trip le t from the s e t A T >< W X W ^. L e t u s s u p p o s e

that Y — W 1 i s s u c h that u £ Y i f an d o n ly if the fu n ction v is d e fin e d fo r a ll trip let from the s e t A T / - W / 'tu ^ J ’ . It Is e le a r that

£ Y a n d fo r a n y w £ W 1 - i f { u | u ] j , then w £ Y . L e t u s a a su m e to the c o n tr a r y , i.e . that Y C- W ^. T h e n - Y

thus th e re e x is t s a m axim al elem en t in the s e t W - Y b e c a u s e the

Ą 1

in e r e a s in g s e q u e n c e s c o n d itio n is s a tis fie d on the s tre n g th o f the a ssu m p tion . L e t w b e a m axim al elem en t in the s e t W , - Y . T h e n

O _ . j

w q W ^ - W q an d u | w q ^ u ^ ^ Y b e c a u s e w Q Y . T h e r e ­ fo r e th e re e x is t s u s u c h that w jC . u , u Y , thus w . c o n tr a r y

O O ^ J. O O ł o

to the defin ition , is n ot m axim al in s e t - Y . W e h a v e d em on stra ted that the fu n ction v is d e fin e d fo r e v e r y trip le t from the s e t A T X W X W ^. It is o b lv io u s that v must b e a m o d el on the

N o w , it re m o in s to p r o v e that F O R

{ A .

1. ¥ . 3 }

w ill b e a form u ła s u c h that fo r o om e w 6. W and som e a £ W ,

o o '

v ( c ^ 0,a 0,w ) ^ v o ^ c^ o ,a o ,W^ * W e k n ow n that a form u ła t>C0 with thls p r o p e r t y c a n n o t b e a atom ie fo rm u ła . I<et u s s u p p o s e that the fu n ction v is łn a g r e e m e n t with the fu n ctlon v q fo r

a n y a € W , fa r a n y w € W a n d t o r a n y formuła o o n ta ln in g ła s a than

a C łotzłoal „w m b o łs. Su o d o m that w ^ W la & m adm al element

contradictłon. If ^ # - fi v ( 0 ^ Q. a o , w# ) - Jl a n d v o (oC0 > a Q, w o ) m <J>, then there eadata a ^ ^ a Q a n d a 6 W q au o h that z ^

w0, v 0 ( ^ , a r * ) - Jl a n d v 0 ( ^ , a Ł, z ) - T h ia i * not

p o a a łb łe fo r z - w o b e o a u a e then v ( fi , a ^ , z ) - v ( , a 1# w # ) ~

Jl an d v ( ^ , a t , * ) - v ( ^ , a ^ a ) - <j>, thua v(qC e . a ^ w e ) - <J>

contr a r y to the aaaumptton. T h e re fo re there muat b e z > _{^ 1} w ._O

H e n c e v ( o £ Q, a ^ , a ) - v o ( ° ^ o ' &o* z ^’ k>#cau* e w 0 to maxtmal łn the se t w e W # 1 v ( o < Q, a ^ , w ) -f v ft( cC q , a o , w ) ^ . H en ce

it folłow s that v Q( o < 0» « 0i a ) - Jl - a contradictłon. Let u s now a u p p o s e that ofT 0 - fi v ( c < 0 . a o , w o ) - <|> an d v q ( c < 0. a o,

■*v0 ) - Jl . T h e n there exista a ^ €. W a n d z ^ su c h that a ^ ^ a Q

« n d a ^ Ł w o , v ( fi , a .^ a ) - Jl , v ( ^ , a ^ a ) - <|>. Let a ^ be

a majdmal element in the set z & j x ^ l w o* v ( ^ • * i »

- 60

-v ( 3f * * 1 * *o > “ h®nc® v * ( c<-0. ® !» w 0 ) " Q an d s® v ( oL0 . *<,.

w ) <■ 0 - a contradłction. If * e W , - W , then, on the strength

€> O 1 O

©f Łemma 3.1, v ( ot o , a ^ xft) - min { v ( ^ o, « 0, « ) I * 0 ^

■ <t>. T h e n there eadsts u su o h that z „ u an d v ( oC a „ ,

O 0 X 0 O X

uq ) - ® . H e n c e there must a 2 fe W a n d z ć s u c h that a 2 ^ a^

®n d * 1 uo* • a 2* * ) “ 1 a n d v ( • a2 * * ) “ ° * T h u * x > ^ * o , -which contradicts the assum ptlon that z Q i s majdmaL P o r

oC 0 « 1 /3, the e z p e c te d eontradletfon c a n b e obtaln ed b y a slm iłar

re a so n in g . bet oC 0 - V * , v ( oC c , a o . » # ) - 11 a n d v o( ^ 0,

c o , w o ) - 0 . T h e n fo r a n y a Ł > a o a n d fo r a n y V ( a Ł ) , v ( | g § .

a , , w ) - 11 , a n d there e z is ts a . > a „ a n d ^ _ e V (a „ ) su c h

x O X o O X

that v o ( (3 ^ ś 0, w # ) - O . T h u s on the strength of induction

h lp oth esls v o ( f i g o , a ^ w

q)

- v ( £ a 1> w o ) - 0 , from this v ( cX.o> a o , wo ) m 0 , w hich contradicts the assum ptien. It l s e le a r

that the e xp e e te d contradłction c a n b e obtaln ed b y a sim ilar re a so n in g w h en oC 0 - V x jS . v ( oCQ, a o . w ft) - 0 a n d v # ( cX0,

w o ) - ll . Pin ally, let u s a ssu m e that oL 0 “

3

x|3

, v ( oL 0, a Q,

w 0 ) - 11 a n d v 0 ( ot Q, a Q, w Q) - 0 . T h e n for a n y fc V ( a Q) ,

v © ( / 3 ^ , a o , w o ) - 0 an d there e z łs t s ^ o e v ( a 0 ) s u c h that

v ( { 3 ^ 0 . a 0 , wo ) - U . S o b y the Induction h lp oth esls fo r *50

* V ( a 0 ) , v ( f:l a o , w o ) . v ( f i ^ 0. a 0, w

q)

. U , thus v ( o t0, a o , wq ) - v q ( oL Q, a o , w # ) - U - a contradłction. W e o mit the

p ro o f of the rem aining part s iń c e it ls eom pletły sim ilar. Q. E . D.

T H E O H E M 3.1. Let ± m £ W r 4 ± > b e a stro n gly atomie Kripk frame satisfyin g the in e r e a s in g s e q u e n c e s condition. Let Y - ^ W ®, Z. a b e a K rip k e frame. S u p p o s e that there exista a function h: W ° |-^ s u c h that:

( i ) h is o n e -to -o n e ,

h ( w 2) •

( U l ) for e a c h U W j Ib a maximal element of then there

e xists u W ° su c h that h ( u ) «.

T h e n E

npoR{ A . n . Y . B ^ •

P R O O P . It fo U o w s from Lem m a 3.2.Q .e.D.

T H E O R E M 3.2. S u p p o s e that the co n d itlo n s o f T h e o re m 3.1 hołd. L e t b e a em b ed d in g o f '" W - W ^ I n t o A ^ 0

-and g b e a fu n ction from U - ^ _ v ( a ) | a C W "Jj t o U ' ^ V ° ( b ) | b £ W ° J s u ch that g ( v ( a ) ) - V ° ( f ( a ) ) fo r e a c h a € w . T h e n E ("W^. V

For{A .- » . i . v . 3 i ( ŁE(^ 0 ' v° '^i> °

3y

- 62

-R E F E -R E N C E S

jJlU P .C r a w le y a n d R .P.D ilw orth, A lg a b r o lc theory of lattiees, Prentice— Mail, IN C . E n g lr r a e c t Cliffs, N a w J e r s e y 1973.

M.C»Fitting, Intuittaniatlc L e g ie, M e d a l T h e o r y a n d F o rcin g , N orth - H e lland, Amatardam 1969.

£ 3 ] S .N a g a i an d H.O ne, A p plication s of g e n e ra ł K rip k a m odele to intermedlate lo g ic e , Journal o f T s u d a C o lle g e , 6 ( l 9 7 4 ) , pp. 9-21.

S . N a g a i, O n a eem anties for n o n -e la e s ie a l lo g ic e , P ro c e e d ln g s Japan A c a d e m y , 4 9 (1 9 7 3 ), pp. 337-340.

Z

53 H.Ono, A etudy o f lntarmadlata pro di c a ta lo gic e , Publlcatlon e o f the R e s e a r c h Institute fo r M athem atical S c ie n c e s , 6 (1 9 7 2 /7 3 ), pp. 619-649.

t « 3 H J ia e lo w a an d R .Stkorski, T h e Metamathematiea o f Mathematios, P W N , W a r s z a w a 1970.

U 73 M .Sxatko w ski, O n fragm ent* o f M a d v e d e v 's lo gie . Studia L o gi c a , 4 0 (1 9 8 1 ), pp. 3^*54.

C sH A .W roń ek i, Intermedia te lo g ic e an d the disjunctlon pro perty, R ep orts on M athem atical L o gic, 1 (1 9 7 3 ), pp. 39-51.

^ 9 ] H*Ono, K rip k a m odele a n d intermedlate lo g ic e , Publlcatlon e of the R e s e a r c h Institute for Mathem atical S c ie n c e s , 6 (1 9 7 0 -7 1 ), pp. 461-476.

S E M A N T Y K I T Y P U K R I P K E 'G O D L A P O Ś R E D N IC H S K W A N T YF1 K O ­

W A N Y C H L O G IK

S T R E S Z C Z E N I E

W p r a c y w y r ó ż n ia m y u o g ó ln io n e K r ip k e stru k tu ry o r a z stru­ ktu ra ln ie u o g ó ln io n e K r ip k e struktury, n a s tę p n ie b a d a m y p e w n e w ła s ­ n o ś c i ty c h d w ó c h ty p ó w sem antyk .