A C T A U N I V E R S I T A T I S L O D Z I E N S I S
POLU PHILOSOPHICA 3 , 1905 i
Grzegorz K alinow ski
THE PROBLEM OF DEGREES OF MAXIMALITY (A s u rv e y )
The р з р е г i s a com plete survey o f th e methods f o r p r o v i n g theorem s on deg rees o f m exlm ality and o f th e r e s u l t s o b ta in e d up to 1979 by th e a u th o rs working in th e a re a . In th e f i r s t twô sec-tio n s th e re a d e r w ill f in d th e whole co n ce p tu al and n o ta sec-tio n e l ap-p a ra tu s n e c e ssa ry f o r th e f u r t h e r d is c u s s io n o f th e problem o f s tre n g th e n in g s and, in p a r t i c u l a r , o f d eg rees o f m oxiniality o f s e n t e n t i a l c a l c u l i .
^1. P r e lim in a r ie s . L et L be a s e t1 o f fo rm u las formed by means o f s e n t e n t i a l v a r ia b le s p , q , r , . . . and a f i n i t e number o f con-n e c tiv e s f 1 , f 2 , . . . » f con-n. Then th e a lg e b ra
(1 ) L » (L , f •]»^2* • • • V
i s c a lle d a s e n t e n t i a l la n g u ag e. Endomorphisms e o f L a re c a lle d s u b s t i t u t i o n s (e « End(L) ) . F or every XcL we s e t Sb(X) »' j e a s a e X and е е EndC^)}» By a consequence o p e r a tio n on o r sim ply з con-sequence on L, we u n d erstan d an o p e r a tio n С d e fin e d on 2. (th e po-wer s e t ) o f L such t h a t f o r any X, Y £ L
X E CÍX) - C(C(X)) and X C Y —*-C(X) £ C<Y).
A consequence С i s s a id to be s t r u c t u r a l p ro v id ed t h a t f o r any s u b s t i t u t i o n eeE nd(L), and f o r any X c L , eC(X ) £ C(eX).
Every r e l a t i o n
за
i s r e f e r r e d to as a r u le o f in fe re n c e o f L, o r sim ply a r u l e . When " ' " "" 1 Г... " "r 11 xr
X C L and a e L, in s te a d o f R( X, a) we s h a l l sometimes w rite / a e R, у
and c a l l /tu a seq u en t o f R. Any r u le o f th e form ( 2 ) Sb(X/ a ) » | eX/e o i j e i s a s u b s t i t u t i o n o f L
j
w ill be c a lle d se q u e n tia l* A ll r u le s o f th e form ( 2 ) where X i s th e empty s e t , X - 0 , i . e . r u le s o f th e form S b ( ^ / a ) w i l l be c a l le d a x io m a tic . In th a t c a se th e elem ents o f th e s e t j e a t е е End ( L) J a re c a lle d axioms.
Given a s e t o f r u le s o f in f e re n c e й o f a g iv e n s e n t e n t i a l la n -guage L, l e t us d e f in e an o p e r a tio n Cn^ on 2** as fo llo w s : f o r every X £ l , Cn0( X ) i s th e l e a s t s u p e r s e t o f X c lo s e d under th e ru -le s in 6. Cne i s a consequence o p e ra tio n on L, and, m oreover, f o r every consequence o p e r a tio n on Ł* th e r e e x i s t s a s e t r u le s в , such t h a t С « Cn^, c f . [ 4 ] , Any such в w ill be c a lle d a b a s is f o r C. We a ls o have
LEMMA 1. ( c f , [ 2 6 ] ) , A consequence С on i s s t r u c t u r a l i f and o n ly i f i t has a s e q u e n tia l b a s is .
Given a s e n t e n t i a l language J. • ( L , f1#f 2 , . . . » f n ), any couple M • ( Aj^ Ij^),
, • ' . • ' ' . ' i
' ’ •
i
• \ • • • • • ■ _ •where Ajj, - ( A ^ , f j , f2, **»f f a ) I s an a lg e b ra s im i la r to ^ and I С A..,, i s c a lle d an (e le m e n ta ry ) m atrix co rresp o n d in g to L, and th e elem ents o f 1M a r e sometimes c a lle d th e d is tin g u is h e d v a lu e s o f M, o f . [ 5 ] . In t u r n , l e t ue u se th e symbol HOM(fc,^,) to deno-t e deno-th e c l a s s o f a l l homorphisms o f L in deno-to A.^ - th e elem en ts o f Hom(L, Aj,j) a re c a lle d v a lu a tio n s o f J. in M. F or ev eiy X c L l e t us pu t
( 4 ) Cn^(X) * ^aeL í f o r every h e Hom(i.,AM), h ctcl^ wherever h U ) с
Cn^ i s a s t r u c t u r a l o p e ra tio n on Ł, I f К i s e o la s s o f m a trlc e a co rresp o n d in g to a g iv e n language L, th e n by a consequence opera-t i o n deopera-term ined by К on it« we s h e l l u n d erstan d th e o p e ra tio n Cn^ d e fin e d aa fo llo w s ( o f , [2 в ]){ F o r evexy X c L , 7
I t tu r n s o u t, c f . [ 2 9 ] , t h a t f o r every s t r u c t u r a l consequence С th e r e i s a s e t o f m a tric e s К such t h a t С - Cn^.
•Given a consequence С d e fin e d on L, deno te by M atr (C ) th e s e t o f . a l l m a tric e s co rresp o n d in g to L such t h a t C^Crij,,. We th e n have
LEMMA 2 ( c f . [ 2 8 ] ) . Each s t r u c t u r a l consequence С i s u n iq u ely determ ined by M atr(C ), i . e . f o r any two s t r u c t u r a l consequen-ce o p e ra tio n s С, C*defined on L,C - C ' i f and only i f K atr(C ) « - M a tr iC ) .
Two m a tric e s M, N co rresp o n d in g to th e same language a re eq u i-v a l e n t . M~N, p roi-v id ed t h a t th e consequence o p e ra tio n s which th e y d e fin e a re i d e n t i c a l , i . e . i f Cn.^ - CnfJ. I f - la a con-gruence o f the a lg e b ra ^ such t h a t I a I S IM f o r any a e IM, th e n i t i s c a lle d a congruence o f t he m a trix M » ( *Ц, 1{.) . I t t u m b o u t t h a t f o r any m atrix congruence w o f M, th e q u o tie n t m a trix
*M/*^ i s ^ u iv a l ^ n t to M} ~ M . Any couple
(6) S - ( L,C ),
where L i s a s e n t e n t i a l language and С i s a s t r u c t u r a l consequence o p e r a tio n on L, i s c e lle d a s e n t e n t i a l c a lc u lu s . In th e seq u el th e elem ents o f th e c la s s Matr(C ) f o r a g iv e n c a lc u lu s S ■« (l^ C ) w ill be c e l le d S -m a tric e s c f . [ 2 8 ] . >'
D ealing w ith th e c a l c u l i Im p lic a tiv e in th e sen se o f R a-5 1 o w a [1 7] Ccf. a ls o [ 2 8 ] ) , one can improve Lemma 2 re p la c in g th e c l a s s o f S - m a tric e s , M atr(C ), by a c l a s s o f q u o tie n t m a tri-c e s . Assume t h a t S • ( L , C ) i s an im p lic a tiv e c a lc u lu s . L et—►de-n o te th e im p lic a tio et—►de-n co et—►de-n et—►de-n e c tiv e o f L, Then f o r every M « M atr(C ) th e r e l a t i o n d e fin e d on AM as fo llo w s:
( 7 ) а b i f and o n ly i f a —»-b, b —*-a e
i s a congruence o f M, and th e r e f o r e M, ~ M. L e t us p u t
. ■ • H :
(8 ) » M « M a tr(C )|.
The elem ents o f A lg ^ C ) w i l l be c a lle d S -a lg e b ra s ( c f . [ 1 7 ] , [28]). LEMMA 3 ( c f . [.1 7]). A consequence С o f im p lic a tiv e S e n te n tia l -. c a lc u lu s S «' (Ł» C) i s u n iq u e ly determ ined by th e c l a s s
§2. Two k in d s o f s tr e n g th e n ings o f a s e n t e n t i a l c a lc u lu s . The n o tio n o f deg re e o f maxima l l t y v ers u s th e __ no tio n o f degree o f com pleteness. Given a s e n t e n t i a l language t , th e o la s s o f a l l . str u c tu r a l consequence o p e ra tio n s on L, to be denoted h e re au C(L), forms a com plete l a t t i c e w ith th e o r d e r < d e fin e d не f o l
-low s: • ;
( 9 ) G1 < C2 i f and only i f C1 (X ) S C2(X) f o r every X C L,
c f . [ 2 9 Í . In th e p a p e r th e symbols sup (C.j,C2 ) and i n f (C^,CZ ) w ill be used to denote th e supremum and i n f Длил o f C, and C2 , r e s -p e c tiv e ly . In th e case when $ c2 ^ ,C2* C L ) , C2 i s c a lle d a s tre n g th e n in g o f СЦ. I f , m oreover, f o r some X s L, C.j(X)$C2 (X), th e n we say t h a t C2 i s a p ro p e r s tre n g th e n in g o f C.j, and w r ite ' G-j < C2 . In th e seq u el th e symbol L w il l a ls o be used to denote th e s o - c a lle d in c o n s is te n t consequence d e fin e d as fo llo w s:
(Ю ) L(X) * Ĺ f o r every X E L.
O bviously, L i s th e g r e a t e s t elem ent o f C (L). Every consequence o p e ra tio n on L which i s n o t in c o n s is te n t i s c e lle d c o n s is te n t. F in a lly , a consequence С * C(L) w il l be s a id to be maximal p ro v i-ded t h a t i t does n o t have p ro p e r c o n s is te n t stre n g th e n in g s«
Very o f te n th e n o tio n s in tro d u c e d in th e l a s t p arag rap h r e f e r a lso to s e n te n t i a l c a l c u l i th e correspondence between s t r u c t u -r a l consequence o p e -ra tio n s o f a g iv e n language L and s e n te n t ia l c a l c u l i fo rm alize d in t h a t language i s , un d er th e d e f i n i t i o n , q u it e o b v io u s. Thus, g iv e n a c a lc u lu s S ■ (Ł , C ), we s h a l l say t h a t a c a lc u lu s S' - (L , C') i s a s tre n g th e n in g o f S p ro v id ed t h a t С < C‘, and so o n . . . In t h i s aenar a l l n o tio n s which we e re
s t i l l going to in tro d u c e w ill a lso be uaed am biguosly. .
A ccording to Lemma 1 , ev ery s tre n g th e n in g o f a s e n t e n t i a l c a lc u lu s S - (L , С ) can be o b ta in e d from $ by adding to ther s e t o f r u le s o f С some s e t o f s e q u e n tia l r u le s в - t h i s s tre n g th e n in g w ill be denoted as S& - (JL, C®) and i f в • { r } a ls o as sP •
• (L , C*). When a l l r u le s i n ö a r e a x io m a tic , JjP ( ( ^ ) w i l l = b*
c a lle d an axiom atic s tre n g th e n in g o f S ( o f C ). In t h a t ca se th e s e t C(A), where
i s an in v a ria n t systém o f C, i . e . the fo llo w in g holdst C(A) - Sb(C(A)) and C(C(A)) • C(A)
and*Cö can be d e fin e d as fo llo w a i f o r every XC L , C®(X) -• С (X U A).
Given a s e n t e n t i a l c a lc u lu s S - ( L» C ), th e c a r d in a l number o f a l l a x io n a tic s tre n g th e n in g s o f S i s c a l le d th e d eg ree o f comple» te n e s s o f S , d c ( s ) , c f . T a r 8 k i [ 1 в ] . On th e o th e r hand, by th e d egree o f m axlm allty o f S , dm(S), we s h a l l u n d e rs ta n d , f o l l o -wing W ó j c i c k i [2 7] , th e c a r d in a l number o f a l l stren g th e-n ie-n g s o f S, i . e . b o th ax io m atic ae-nd e-n o e-n -ax io m atic. O bviously, dm(S) i s a t l e a s t as g r e a t as th e d eg ree o f com pleteness o f 3 and i t tu r n s o u t t h a t in mony c a se s dm (s) > d c ( s ) .
Given a s e n t e n t i a l c a lc u lu s S ■ ( L , С ), from Lemma 2 i t fo l» lows t h a t any s t r u c t u r a l consequence o p e r a tio n С1 > С l a d e te rm i-ned by somo s u b c la s s o f M a tK c ). C o nsequ ently ,
(11) dm(S) < ca rd {k 1 K C Matr(C) ~ } o r , more p r e c i s e ly ,
( 1 2 ) dm(5) - card | Cn^ 1 K S r i a t r ( c V ~ ) «
And, according to Lemma 3, f o r th e ca se o f im p lic a tiv e senten* t l a l c a l c u l i th o l a s t fo rm u las can be improved to
(1 1 1) dm(S) «S c a r d [K 1 КС A18 Í 4 c V ~ }
and *
(12I ) dm(3) - oard { CnK 1 К С AlgR(C)/ ~ ) i
r e s p e c t iv e l y . v.y '• .V V '
F in a l ly , th e c o u n te rp a r ts o f (1 2 ) and (121) f o r th e n o tio n o f d egree o f com pleteness a re th e follow ing*
•Л •
(13)* d o (3 ) . c a rd I 0 0^ (0) t K c , ... JU, ■ i M (1 4 ) d c (S ) * o a r d j C i y í ŕ ) i K C A lgR (c)A } ,
• ■**.
Л
'
-V .. V, Л.. .‘w*- - *r Л •• __* • *. ’ ,v* ‘ •!*•*< *“ía í §3. H le to r lc a l aocount o f p a r tic u la r a tu d l— o f th e problem o j • degrees o f m axlm allty. In th e p resen t s e c tio n e l l e x p lio t c o n tri-■ : : 1 г-" : ■.
г.щ
• \ v ; ,ľ ‘-ľ \ ' 5 »v ... ■b u tio n s to th e to p io ar* l i s t e d I n th e c h ro n o lo g ic a l o r d e r and th e main methods f o r p ro ving theorem s on d eg rees o f m axim ality a re b r ie f ly re p o rte d . A ll u n d efin ed n o ta tio n con cern in g п-v alu e d Lu-kasiew icz s e n t e n t i a l c a l c u l i comes from [ 8 ] .
The paper by W ó j c i c k i , [2 7] , in which th e n o tio n o f degree o f m axim ality was in tro d u c e d was, a t th e same tim e , th e f i r s t c o n tr ib u tio n to th e s tu d ie s on th e problem . The main theorem o f [2 7] says t h a t th e d egree o f m axim ality o f th e th re e -v a lu e d Lu-kasiew icz s e n t e n t i a l c a lc u lu s - ( L , C ^)equals 4 , i . e .
Cl) - dm(Lj) « 4 . ‘ .
The c r u c ia l p o in t o f th e o r i g i n a l method o f p ro o f a p p lie d by R. W ójcicki i s th e re d u c tio n o f th e whole problem to th e problem o f s tre n g th e n in g s o f L^ which can be o b ta in e d by th e use o f r u le s o f in fe re n c e determ ined by seq u en ts o f th e sublanguage o f L g e n e ra te d by a s in g le s e n t e n t i a l v a r ia b le p , * (L v , л , i ) . A ccor-d in g ly , th e f i r s t s te p was to prove th e fo llo w in g a s s e r tio n ;
( p ) F o r ev ery a * L, XCL, a « C3(X) i f and o n ly i f f o r every s u b s t i t u t i o n e t Ł - ^ Ł p o a • C^ieX ).
N ext, u sin g some p r o p e r ti e s o f C^, i t i s p o s s ib le to d e fin e an eq u iv alen c e r e l a t i o n ^ having th e two p r o p e r ti e s : (1) a (3 i f and o n ly i f h(oł) - h ( ( 3 ) f o r ev ery h: L —*A^ Ш ) I f а *^(3 th e n Cj(o») ■ C^((3).
I t tu rn e d o u t t h a t th e q u o tie n t s e t Lp/« ^ had 12 elem ents - in [2 7 ] t h e i r r e p r e s e n ta tiv e s were denoted as (fy, <f2 , . . . , <р12* Sub-s e q u e n tly , from th e p r o p e r ti e s o f i t fo llo w s t h a t every s tre n g th e n in g Lj o f L j by a s e q u e n tia l r u le R » Sb(X/<* ) determ ined by a seq u en t X/ a o f 1^ i s eq ual to some s tre n g th e n in g o f by a r u le o f th e f o r e 3b ( ^ V f y ) . Thus, th e f u l l in v e s t ig a t io n o f th e s e t o f a l l s e q u e n tia l r u le s o f L d eterm in ed by seq u e n ts o f ^ can be re p la c e d by a atudy o f th e s e t 'o f 144 r u l e s o f in f e r e n c e o f th e form 3b ( V<Pj). To do t h i s , th e a u th o r o f [2 7] used some m a trix methods and f i n a l l y reac h ed th e c o n c lu sio n t h a t each o f th e r u le s R o f th e form Sb( V«pj) f a l l « i n t o One o f th e c a te g o r ie s d e fin e d by th e fo llo w in g c o n d itio n s :
( a ) R la a ru le o f L^ thus L^ ■ ( b ) l£ . Ц
( c ) L * . Ц ( d ) » Ł * ,
where L* ■ (L , C*) w ith C* d e fin e d as fo llo w s: C3(X) 1Г C2(X) 4 L L o th e rw is e .
Thus Lj has a t l e a s t fo u r s tre n g th e n in g s : L^» Ł*, Ц - L. The f a c t t h a t th e s e a re th e only p o s s ib le s tre n g th e n in g s o f L^, and th u s t h a t dm(L,) » 4 , fo llo w s e a s i l y from th e fo llo w in g
re»-J \
s u i t s :
” C3 < C3 < C2 < L and t h e r e i o r e * ЬУ th e 'u s e o f ( p ) one can prove t h a t every p ro p e r s tre n g th e n in g o f C„ i^ n o t weaker th a n C*;
e
- Ц i s c o n s is te n t i f and o nly i f ö i s th e s e t o f пие& o f C^j - Ł* i s { ÍÍ } - com plete (Theorem 2 in [ 2 7 ] ) , and th e r e f o r e f o r every С > C * . C(0) ? Cj( 0 )í »
- ł~> i s th e o n ly proper c o n s is te n t s tre n g th e n in g o f L^ (Wajs- b e r g 's theorem on d eg rees o f com pleteness o f Ł* ),
N ext, u sin g th e f a c t t h a t Lukasiewicz s e n te n tia l c a l c u li Ln a re im p lic a tiv e ln the sen se o f h a s i o w a [1? ] , the a u th o r o f th e p r e s e n t review gave ln [1 0] an a lg e b r a ic p ro o f o f ( I ) and, mo-re o v e r, showed t h a t th e d eg ree o f m axim ality o f th e fo u iv v a lu e d Lukasiew icz c a lc u lu s L^ a ls o e q u a ls 4 ,
Both th e method a n l th e r e s u l t s o f [Ю ] were subsequently genera-liz e d In [ 1 1 1 f o r a w ider cla a a o f Lukasiewicz lo g i o s , namely, f o r th o s e c a l c u l i Ln f o r which n-1 i s p rim e. The main r e s u l t o f [11] says t h a t
The o r i g i n o f th e p ro o f o f ( i l l ) g iv e n i n [11] i s th e use o f
( I I ) dm(L^) • 4 .
some a lg eb ra ic stru c tu re s corresponding to »-valued Lukasiewicz m a trices, namely s o -c a lle d MVn-a lg eb ra a introduced by R. 3 . O ri- g o lia . Given f i n i t e n > 2 , MVn-a lg eb ra i s a stru ctu re n • . - ( A , 0, 1) o f th e type ( 2 , 2 , 1 , 0 , 0 ) f u l f l l i n g a number o f
eq u ation s, c f . [3] . The primary correspondence between Lukasie- vlo z m atrices and MVn algebras runs аз fo llo w s: For every Luka-siew icz m atrix MQ - U n , - * , V , л , T , {1}) one can d e fin e MVR a l -gebra
Ад * ( An , ♦ , • , - , 0 , 1)
p u ttin g x ♦ у - т х -+ у , x • у ■ i ( x - n y ) and 3c - i x , Moreover, t h is correspondence i s o n e-to -o n e, s in c e con versely: x-*y * 5 ♦ y f xvy - x • у ♦ y , x л у ■ (х + У) - у and, o b v io u sly , nx • x . Se-condly, we have ( c f . [3 ])»
• \ %
(R t) Every MVQ algebra w ith more than one element i s is o -morphic w ith a su b d irect product o f a number o f co p ies o f algebras Ąm, where m^n end »-1 i s a d iv is o r o f i>-1.
T h e re fo re , i t tu r n s o u t t h a t each MVR a lg e b ra n can be considered a s th e a lg e b ra o f th e form rt*- ( А , - * , и , 0 , - , 1 ) , w h e r e и , Л , - его th e n a tu r a l c o u n te rp a r ts o f -+, v , a , i , r e s p e c t iv e l y , defined in th e a p p r o p ria te a lg e b ra s Ад . In th e seq u el we s h a l l u se the symbol' CR to den ote th e consequence Сл * determ in ed on th e language L o f L ukasiew icz s e n t e n t i a l c a l c u l i by th e m a trix (rt*, 0 } ) »
Given n > 2 , l e t * be th e r e la tio n on L m ( L,*+,v,Af -») defined as fo llo w st For every a , (3 « L,
a «* ß i f and on ly i f e С ( 0 ) .
Using th e f a c t t h a t AR co rr ela ted w ith th e Lukasiewicz matrix i s an MVR a lg e b r a f one can e a s il y v e r if y th a t the j^ndenbatmi a l-gebra -/*»n i s en Win all-gebra, Now, r e c a ll th a t th e Lukasiewicz c a lc u lu s LQ - (L»Cn ) i e L a p lic a tlv e . Thus, as a p a r tic u la r ca se Of a g e n e ra l r e s u lt o f R e « 1 o v a o f . ( 1 7 ] , p* 184 we o b ta in th a t - / * n l e a fr e e algebra in th e c la sa AlgR(CQ) o f all" : l^ - a l - gebreb. Using t h is fa c t and th e rep resen ta tio n theorem (R t) oná cen prove th a t f o r a g iv en n > 2 , the d a * * o f a l l Ln-a lg e b r a s co in c id e s w ith th e c la s s ИУ o f e l l MY a lg e b ra s, i . e . th a t
(1 5 ) AIgą (Cn) - MVn.
Now, l e t us assume t h a t n i s a n a tu r a l number such t h a t n-1 I s prim e. I f a o t has o n ly one n o rv -triv ia l su b alg eb ra - A2 . изing t h i s f a c t and ( R t ) one can prove c f . [1 1 3 t h a t f o r any MVn a lg e b ra n , th e consequence o p e ra tio n Cn c o in c id e s w ith one o f th e fo llo w in g consequence o p e ra tio n s :
CV CV “W
cV
where d en o tes th e d i r e c t p ro d u ct o f and Дп » In t u r n , we have L ■ Cn, > Cn. > Cn. „ . > Cn. . - 1 ~2 —2 ^ n ~n And th e r e f o r e ‘ . • Л, , < . w A »V
Á'
* l’ ’ \ ŕ • *.•*
v-:lWd’a n> ■ ( СгУ ' K
- "V.} - {cn4i.
c„v v
C»4J . T h is ends a sk e tc h o f th e p ro o f o f ( I I I ) .REHAHK. T o-give a s im i la r p ro o f o f ( I ) and ( I I ) , ln [1 0 ] th e a u th o r usee th e s o - c e lle d th re e -a n d fo u i> v alu ed Lukasiew icz a l -g eb ras in tro d u ced by M o i s i 1 c f . e .g . [1 3 ]. I n c i d e n t a l y , ' i t can b i proved t h a t in th e ca se s n»3* and r* 4 , b o th th e n o tio n s - t h a t o f n-v alu ed L ukasiew icz ,H oisil a lg e b ra and th a t o f NVn a l-g eb ra - c o in c id e .
*
■
-
■ ’ ' .'rt
-The exam in ation o f th e problem o f d e g re e s o f m axim ality in th e w hcle c l a s s o f n -v alu ed Lukasiew icz c a l c u l i was c a r r i e d o u t ’ by W ó j c i c k i ln [3 1]» where th e fo llo w in g r e s u l t was proved;
dm(Lft) i s f i n i t e f o r every f i n i t e n ^ Z ,
To g e t t h i s r e s u l t , W ójcicki m odified th e a lg e b r a ic te c h n iq u e o f [1 1] and in tro d u c e d a v ery handy n a tio n o f th e c h a r a c t e r i s t i c e le -ment o f an Ln- a lg e b r a . Below, a s k e tc h o f th e method used in [31] i s g iv e n .
F i r s t , u s in g p r o p e r ti e s o f th a consequence Ш I t i s p o s s ib le t o g iv e a p u re ly l o g i c a l p ro o f o f th e fo llo w in g e q u a lity )
k6 Or*« e»» U tl iim k l
where HSP(Mn) denote« th e v a r ie ty generated by th e »«valued ßiewloz matrix (th e le a s t c la s s o f matrice* con tain in g MJ1 and clo se d under th e op eration s o f fo rain g d ir e c t producta, subal-gebras and homomorphio im ages). N o tice , th a t (16) i s another v e r-ween sim ple MVn algebras AR and m atrices f^ , and the eq u atio- nal d e f in a b ilit y o f MVn a lg eb r a s, i t i s p o s s ib le to tr a n s la te C r ig o lia ’s rep resen ta tio n r e s u lt (R t) in to
Where a e A i s an element o f some algebra-A > (A,-% v , a , t , {1 )) in HSPÍMj^), l e t us denote by [ a ] the subalgebra o f A generated by a , f o r every Ln-a lg e b ra AeSp(Mn ) i t i s p o s sib le to fin d an e le -ment a* in A a " c h a r a c te r istic ele-ment" o f Ą, c f , [31]» with the fo llo w in g two p ro p erties)
Therefore, denoting b / Vr th e s e t o f a l l d ir e c t products o f the form * . . . * Hjj o f p airw ise d iffe r e n t su b o a trices o f Мд ,
i* may p a s s from (1 7) to ,
Jid s in c e Vn i s f i n i t e , dmtł^) i s f i n i t e . This ends the proof o f IV).
Now, we a re going to d iscu as [93» In which the s o lu tio n to ne di*-proble<n f o r some n o n -im p lica tiv e s e n te n tia l c a l c u li was g i - /en* As th e t i t l e o f [ 9 ] makes p la in , the paper concerns the so - - c a lle d dual counterparts Of n-valued Lukasiewicz c a l c u l i , o f . 2 6 ], and [ 1 2 ] . Where L -,(-*, v, л ,т) i s th e language o f Lukasiewicz - a l c u l i » M n > 2 , th e c a lcu lu s dual to L^, dt^ , i s a p a ir
sio n o f (1 5). In tu rn , u sin g th e one-to-ofie correspondence b
et-HüP(Mn) - SP(Hn) . n
Thus, c f , (1 2 * ), f o r every n > 2 (1 7 )
<łŁn - (L , Cn ).
where Čn ie th e consequence o p e r a tio n determ ined by th e m a trix Mn » (Ą ^, An - [1 j ) , c f . [1 2 ]. The f a c t t h a t th e c a l c u l i dL^ a re n o t im p lic a tiv e e a s i l y fo llo w s from th e d e f i n i t i o n o f th e c l a s s S o f R a s i o w a c f . [ 1 7 ] , p . 179. In s p i t e o f t h i s , th e main theorem o f [9] saying t h a t th e d eg ree s o f m axim ality o f a l l ca lc u -l i dL a re f i n i t e , i . e . n '
IV) dm(dLn ) i s f i n i t e f o r ev ery f i n i t e n > 2 ,
was o b ta in e d by th e u se o f m a tric e s analogous to S -a lg e b ra s ( ^ n- - a lg e b r a s ) o f Rasiowa. In th e seq u el th e c l a s s o f a l l such m a tri-ces w ill be denoted as Matr**(dCn ).
Given n > 2 , f o r every x « A n , l e t us p u t 0 i f x • 1 IX ■
. V * .. iy'
1 o th e rw is e .
Using th e c r i t e r i o n i n [1 ^ 3 , i t Í3 easy to v e r if y t h a t 4 i s defL - nab le in Afi. By th e same symbol, n , we s h a ll d enote a s e n t e n t i a l c o n n e c tiv e in L co rresp o n d in g to 1. Where M » (£вд» 1 ^ )« M atr (dC ^), l e t us p u t;
a b i f and o n ly i f ч ( а - * Ь ) , ч ( Ь - * а ) e 1^.
M • v
A i s a congruence o f th e m a trix И and th e r e f o r e M«y
Conse-q u e n tly , th e c l a s s .
MatrR(dCn ) - J M/* s M «M atr(dC ^)J
can be used to re p la c e Matr(dCn > in (1 1 ), and th u s we g e t: (1 8 ) da(dLn ) . c a rd { K i K C M atr^ ídCn )/~ } .
I t tu r n s o u t t h a t a l l m a tric e s in M atr (dCn ) h av e, among o th e r s , a v e ry s p e c ia l p r o p e r ty - In ev ery M«MatrR(dCn ) th e r e i s a s u b s e t V,^ such t h a t th e q u o tie n t s e t M/^ i s th e o n e e le -ment s e t , nam ely, VM/j{ " 1ц end t h * t (A^» 1 ц ) в ИЗР(1^) o f , p . 46. Using t h i s f a c t and th e r e p r e s e n ta tio n theorem f o r HííPÍM^), on», can prove t h a t
(19) M«tx^(dCn ) • HSPÍÍ^),
compare (1 6 ). M oreover, th ere l e • on e-to-on e correspondence between m a tric e s from HSP(Mn ) and th ose from HSPiM^i For every M m (ĄM, IM) e H SPi^) and M^ - ( ^ 1M) and fo r any <*«L, X SL ,
where ч X denotes th e s e t o f formulas r e s u ltin g fr o e X by prece-ding each o f i t s formulas by n . In tu rn , l e t Vn be th e "natural" counterpart o f th e s e t o f product n a tr lc e s used on p . 46. Then, using (1 9 ) and ( ♦ ) one oan prove th a t f o r every M*Mati^4dCn ) th ere I s a matrix Md « ? n such th a t Md ~ M, Consequently, (1 8 ) c a n ’ be improved to
and s in c e Vn i s f i n i t e , «*»(1^> l e f i n i t e .
The method o f [9 ] d escrib ed in the l a s t sequence o f paragraphs was a ls o used to some ex ten t in [ 8 J to g iv e a ch a ra cte riz a tio n o f
w un auperaesJLgnatea l o g ic a l v a lu e si 1 « I , 0 $ I* The main r e s u lt o f [ 8 ] says th a t th e degree o f maximality o f any »-valued L u k a siew icz-lik e s e n t e n t ia l c a lc u lu s i f f i n i t e and equal to the degree o f maximality o f th e corresponding » .valu ed Lukasiewicz c a lc u lu s.
(V I) d a ( L ^ ) . d«(Ln ) every f i n i t e n > 2 , a v eiy I&An, v 1 « Z and О й l .
In h i s a b str a c t [ 6 ] , M a d u c h reproted b r ie f l y some re-s u l t re-s o f re-s tu d ie re-s on pure liq p U eation al re-s e n t e n t ia l c a lc u li o f toko* s ie w lc z . Given a f i n i t e n > 2 , th e »-valued Lukasiewicz L aplica- t io n a l c a lc u lu s i s th e p a ir (fc, C* ) c o n s is tin g / o f th e pu-re im p lic a tio n s! s e n te n tia l language ),* • <L*~+) and C* - Cr^#, where 1« th e I x p lic a t lo n a l reduct o f th e Lukasiewicz 1 matrix Mn. Among o th e r s , one can fin d in { é 3 th e fo llo w in g theorem: : ;
(♦ )
a e C i ^ i X ) i f and on ly i f •» a « Ci^ ( чХ) ч а е Cn^ ( *» X) i f end o n ly I f a e Cr^ ( X ),
( VI I ) dm(L*) - dc(L*) • n (Гог every f i n i t e n > 2 ) ,
The o r i g i n a l method o f p ro o f iß based on some r e s u l t s conoex*- nlng Lindenbaum'3 a lg e b ra s determ ined by th e s o - c a lle d ir r e d u c ib le th e o r ie s o f L*. U n fo rtu n a te ly , th e p ro o f c o n ta in s some gaps n o t easy to remove. The r e s u l t , how ever, i s c e r t a i n l y v a lie d - a v e ry sim ple p ro o f o f (VI I) w i l l be g iv e n in S e c tio n 4.
M. T o k a r z i n [2 0] examined th e problem o f s t r u c t u r a l s tre n g th e n in g s o f th e Donsequence o p e r a tio n s C® and C® determ in ed by th e s o - c a lle d S u glhara m a tric e s and M®, r e s p e c tiv e ly (m | •
• ( { - 1 , 0 , 1 } , - , v , A , - I , { o ,l} ) and mJ - ( { - 2 , - 1 , 1 , 2 ) , -*,V,A0 ,{ 1 ,2 ft
where ix ■ - x , x v y » m ax (x ,y ), x a у - m in (x ,y ), x - * y • - x v у i f
x ^ y and x-+y m - х л у o th e rw is e ) . The main r e s u l t o f [ 2 0 ] say s t h a t th e deg rees o f m axim ality o f C® and a re b o th eq u al to 4 , ( V I I I ) d m (c |) - dm( cj ) - 4.
The d e t c i i s o f p ro o f o f ( V I I I ) a re s tr o n g ly based on p a r t i c u -l a r p r o p e r ti e s o f S ugihara m a tric e s . The method, however, seen s to be more u n iv e r s a l - i t has some p o in ts o f c o n ta c t w ith Wójo lc - k i ’ s method o f p ro o f o f ( I ) . A cco rd in g ly , th e p ro o f f o r th e case o f c | can be sk etch ed so as to c o n s is t o f th e fo llo w in g fo u r s te p s :
Step 1. I f c®< C, where С i s a c o n s is te n t s t r u c t u r a l conse-quence o p e r a tio n , th e n C í Cg (Cg being th e c l a s s i c a l consequence o p e ra tio n - Lemma 2 in [ 2 0 ] ) .
Step 2. I f i s used t o denote t h e ,s t x u t u r a l l y com plete con-sequence c f . [1 5 ] f o r which
c|*(0)
- C j( 0 ) , th e n f o r ev ery C, C j< C<C2 we have C < C® * (a p a r t i o u l a r case o f th e g e n e ra l r f ts u lt concern in g s t r u c t u r a l co m p leten ess, c f . S e c tio n 4 , p . 521S tep 3 . L et u s now assume a g a in t h a t C^< C< Cg, Then, i f f o r some a « L, XC L,cx • c(X ) anda*C® (X ), th e n by Lenaa 4 i n [20 ] we g e t CgiX) 4 L and t h i s to g e th e r w ith Lemma 8 i n [ 2 0 ] im p lie s t h a t f o r some s e n t e n t i a l v a r ia b le s p0 and p1# ? Qe С (p . n P l) , F in a l ly , th e s tre n g th e n in g ) o f c j by th e r u le £ - { p1 p1/ p J i s s tr u c -t u r a l l y co m p le-te, and -th u s - c | * (Lemma 7 o f (°20j>. So,
Stor- A. Steps 2 and 3 to g e th e r Imply t h a t C^* i s th e o nly stru ctu r a l consequence o p e r a tio n betv/een and C2 . On th e o th e r han-1, C'2 i s maximal and th c re f o r o Cy- and C’2 a re a l l c o n s is te n t stren gth en in gs o f c j . Thus, dra(Cj) «■ 4 .
As e a r ly as in [2 7 ] W ó j c i c k i posed th e fo llo w in g con-jectu res
(H ) The degree o f m axim ality o f any s tro n g ly f i n i t e s e n t e n t i a l c a lcu lu s i s f i n i t e .
R e ca ll, o f. [3 0] , t h a t a c a lc u lu s S - ( L , C) i s s tr o n g ly f i n i t e i f th ere i s a f i n i t e s e t o f f i n i t e m a tric e s К s tr o n g ly adequate to C, i . e . such t h a t С • Cn^. I n c id e n ta l ly , i t was proved in [ 2 5 ] th a t the degree o f co m pleteness o f a s tr o n g ly f i n i t e s e n t e n t i a l ca lc u lu s i s always f i n i t e .
The con jectu re (H ) appeared t p be n o t t r u e - T o k a r z sue- ceded [2 3] in c o n s tr u c tin g a s tr o n g ly f i n i t e l o g i c , whose degree o f mfcximallty i s i n f i n i t e ^ more e x a c tly , he showed t h a t th e con-sequence o p e r a tio n determ ined by th e fourw valued im p lic a tio n a l- -n e g a tio n a l S ugihara m atrix (INSA) has i n f i n i t e l y many s t r u c t u r a l stren g th en in g s. The b a s io T o k a rz 's id e a was su b se q u en tly m odified by W r o ń s k i , who i n [3 2] have a s i m i l a r counterexam ple to (H) by th e use o f a th re e -e le m e n t m a trix . From th e two exam ples, Wronski’ s i s much e a s i e r to d e s c r ib e . I t runs as fo llo w s;
Let A ■ ( { o ,1 ,2 } ,* ) ' be an a lg e b ra o f ty p e ( 2 ) , whose b in a ry operation • i s d e fin e d by th e c o n d itio n s : 0*0 = 2*2 * 2 , 1*1 * t . and x*y • 0 o th e rw is e . I t i s easy to see t h a t B - ( { o ,2 } ,* ) i s a . subalgebra o f A. In th e se q u e l we s h a ll c o n s id e r th e two m a tric e s
A * CA*{o}) and В . ( B ,{ o } ) .
For every n - 1 , 2 , . . . d e fin e a p e t o f s e q u e n tia l r u le s
Rn * { í * » 01)* (A) x i s a f i n i t e s e t o f fo rm u las b u i l t up from a t most n s e n t e n t i a l v a r ia b le s p^, . . . , pn , ( U ) СПд(Х) * L}.
Subsequently, l e t us p u t Cn » Cn^nC^ i s th e s tre n g th e n in g o f Cn^ by th e s e t o f r u le s K^). One can p rove t h a t f o r any n •> 1 ,
CnA ** e n ^ £n+1*
c f . [3 2] - Lemma 1 .1 . T his im m ediately Im p lies t h a t Спл has i n f i -n i t e l y ma-ny s t r u c t u r a l s tre -n g th e -n i-n g s .
§4. Some r e l a t e d to p lc ą . In th e lo g i c a l l i t e r a t u r e th e re i s a number o f r e s u l t s which p ro vide из w ith very co n v en ien t methods of e s ta b lis h in g deg rees o f m axim ality o f some s p e c ia l s e n t e n t i a l c a l-c u l i . E s p e c ia lly im p o rtan t a re th e r e s u l t s concerning such no-ti o n s as m ax im ality, alm ost m axim ality and s t r u c t u r a l com plete-ness f o r th e extended d is c u s s io n o f t h i s s o r t o f th in g s see e , g . [ 2 1 ].
M axim ality. I f S > (L , С ) i s a maximal s e n t e n t i a l c a lc u lu s , i . e . i f С does n o t have p ro p e r s t r u c t u r a l s tre n g th e n in g s except L, th e n o b v io u sly dm(S) m 2 . In [ 2 2 ] one con f in d a v ery u s e fu l m a trix c r i t e r i o n o f m axim ality o f consequence o p e r a tio n .
M1 ( c f . [19] ) , I f every c o n s ta n t f u n c tio n o f A., i s d e fin a b lekvl i n th e m a trix M * th e n Cr^ i s maximal,
A very i l l u s t r a t i v e example o f th e u se o f th e c r i t e r i o n (M1) i s a p r e t t y s h o r t p ro o f o f m axim ality o f th e c l a s s i c a l s e n t e n t i a l c a lc u lu s Lg ■ (L , C2 ), A cco rd in g ly , we have t h a t th e matrLx A2 r-* ( { 0 , 1 ) »~ л » i s s tr o n g ly adequate f o r C2 , Cn4 * C2 and bo th th e c o n s ta n ts : 0 and 1 a r e d e fin a b le in M2 , e , g . as 0 *> • T ( x - * x ) and 1 » x -»x .
A lm ost-m axlm allty. Given a s e n t e n t i a l language L l e t us pu t
(19)
0 i f X - 0 L o th e rw is e .
H A* a ( s t r u c t u r a l ) consequence o p e r a tio n on L and i t i s c a l l e d ,
c f . [ 2 2 ] , a lm o s t- in c o n s is te n t. In t u r n , a consequence С on L w ill be c e lle d alm ost-m axlroal w henever f o r every s t r u c t u r a l consequence С ', C< С * im p lie s t h a t C* • L^ o r C ' • L, c f . [ 2 2 ] , From th e de-f i n i t i o n i t im m ediately de-fo llo w s t h a t th e d eg ree o de-f m axim ality o de-f any alm ost-m axim al s e n t e n t i a l c a lc u lu s S *> ( L , C ) eq u a ls 3 o r 2, The fo llo w in g m a trix c r i t e r i o n o f alm o st-m ax im ality can - be found
AM-)• I f M • ( Лм, { a} ), where ®*АМ 1* a m a trix *uch t h a t f o r every b « A H th e r e i s a fu n c tio n f b d e f in a b le in ĄM such t h a t f b ( a ) в b , th e n Cn^ i s alm ost-m axim al.
A nother c r i t e r i o n , w hich, o r i g i n a l l y , was s t a t e d e a r l i e r by WdJ- c io k i and Wro iís k i, oan be t r e a t e d as a c o r o lla r y to AM^j
AM2 ( Wójc ic k i-W ro ria k i, u n p u b lish e d ). L et A^ have no p ro p e r su b alg eb ra and l e t a e A M. Then la alm ost-m axim al.
N otice t h a t i f a consequence o p e ra tio n С i s almost-maximal and C ( 0 ) i 0 , th e n С i s maximal. Thus, any maximal s e n t e n t i a l ca'lcu- lu s can se rv e as an example o f alm ost-m axim al c a lc u lu s w ith th e d eg ree o f m axim ality 2 . F in a l ly , th e fo llo w in g theorem seems to be o f some i n t e r e s t :
•
THEOREM 1 . (W roński, unpublished)« I f M i s a tw o-elem ent ma-t r i x , ma-th en Cn^ i s alm osma-t-m axim al.
A p ro o f o f Theorem 1 can be foiled in [ 2 1 ] .
S tr u c t u r a l com p leteness. Given a s e n t e n t i a l language L, a ru-l e o f In fe re n c e R o f L i s c a ru-lru-le d s t r u c t u r a ru-l i f and o n ru-ly i f f o r
v y “ --- aV '
every seq u en t л/а » A/c< e R im p lie s t h a t eA/ e a e R f o r every sub-s t i t u t i o n e e E n d ( L ) . Where С i s a consequenoe o p e r a tio n on L, R i s c a lle d p e rm is s ib le ln C(0) i f and o n ly i f f o r every X/oi e R, a e c ( 0 ) whenever X с C (0 ). A s e n t e n t i a l c a lc u lu s S - ( L , С ) i s s t r u c t u r a l l y com plete, c f . [ 1 8 ] , i f and o n ly i f ev ery s t w c t u r a l r u le which i s p e rm is s ib le in C(0) i s a r u le o f C.
A v e ry u s e f u l c r i t e r i o n o f s t r u c t u r a l - com pleteness was g iv e n by D. Makinson:
SC ( c f . [ 7 ] ) . S - (L , C) i s s t r u c t u r a l l y com plete i f and o n ly i f f o r ev ery s t r u c t u r a l consequence o p e r a tio n C* on L, С Ч 0 ) •
m c ( 0 ) im p lie s t h a t c ' < C* •
The p ro p e rty m entioned i n SC can sometimes be used f o r e s t a b l i s -h ig th e d egree o f m axim ality - i n th e s e q u e l t h i s problem w ill be d is c u s s e d i n two exam ples, namely f o r re v a lu e d G ödel’« c a l c u l i end f o r n -v alued p u re im p lic a tio n a l Lukasiew icz c a lc u li*
Given a f i n i t e n > 2 , th e n -v alu e d m a trix o f Gödel i s defined- as fo llo w s c f . e . g , [ 2 ] t gn » ( { l , 2 , . # . , n } , - ^ , $ , A » 4»» {1}) , where f o r every x ,y « { l , . . , , n j
I
n i f x<n f 1 ♦ x - ^ y - j 1 i f x - n { у i f x>y у i f x<yand xvy • max(x, y), x k y • mi n( x, y) . Where L i s th e a p p ro p ria te s e n te n t ia l lan g u ag e,
d e fin e d as th e p a i r
s e n te n t ia l la n g u ag e, th e n-v alu ed G ö d e l's c a lc u lu s , Gn , can be
S„ - C b,o„)
w ith G ■ Cn , c f . [30] p. 65. I t i s im m ediately seen t h a t f o r
n e n
every n > 2 ,
( . ) Gn < ^ Og ^ • L
and, in p a r t i c u l a r , a lso
( • •) G^i 0 ) <* . < . ф Ggi 0 )?G^ (0 ) •> L.
Secondly, A n d e r s o n proved i n [1 ] th o t th e d egree o f com-p le te n e s s Of Gn eq u als n , dc(Gn ) ■ n , what to g e th e r w ith ( . . )
im p lie s t h e t th e c a l c u l i G^ w ith k < n a re th e o n ly axiom atic s tre n g th e n in g s o f Gn. On th e o th e r hand, a l l G ödel' s c a l c u l i Gn ore s t r u c t u r a l l y com plete, c f , [ 2 ] , a n d -th u s , acco rd in g to_(SC) th e number o f s t r u c t u r a l s tre n g th e n in g s o f G^ i s n o t g r e a t e r th a n
n. F in a lly , u sin g ( • ) we o b ta in
(IX) dm(Gn ) - dc(Gn ) • n f o r every f i n i t e n > 2 ,
An extended v e rs io n o f th e method a p p lie d f o r th e p ro o f o f ( I X) can a lso be used f o r p ro v in g Maduch’s r e s u l t ( VI I ) , p . 49. Now, we s h a l l a lso u se some r e s u l t by P. W ojtylak, namely, th e fo llo w in g :
THEOREM 2 ( c f , [ 2 4 ] ) . Let M be a m a trix co rresp o n d in g t o » g iv e n s e n t e n t i a l language L. Then, f o r every XCL, ".
СПц(ЗЬ(Х)) r O [ c n jj( 0 ) : M i& n submatrix o f M and X S Сп^(0)}. Given o f i n i t e n > 2 , one can c o s ily v e r if y th a t the purely i m p l i c a t i o n Lukasiewicz matrix M* has the fo llo w in g submatriceej M*. M*- 1 , « . . , kJ*, ln turn, one can a lso v e r if y that
Cn < cr£*1 < • • • < **2.. < C1 “ L
c n(^ )
9
cn-1(r>)9 ••• 9 cz ^ ) 9
m
L,Making u se o f Theorem 2 , from th e l a t t e r in c lu s io n we o b ta in t h a t any in v a r i a n t system o f Cn must be equal to one o f th e fo llo w in g Cn( 0 ) , • • • » L. Thus, d c ( L * ) » n . F in a lly , a l l Lns a re s t r u c t u r a l l y com p lete, c f . [1 6 ], s o , by th e same ty p e o f argument as f o r Gödels c a l c u l i we o b ta in dmO.,*)» n.
N£t£> The n o tio n o f s t r u c t u r a l com pleteness o f a s e n t e n t i a l c a lc u lu s can be u n d erstan d i n two waya, acco rd in g to how th e no-tio n s o f a r u le o f in fe re n c e i s d e fin e d . F i r s t , i f we r e s t r i c t th e n o tio n o f a r u le to i t s f i n i t e r y sen se assuming th e r u le to be a s u b s e t o f Ln x L f o r some f i n i t e n , th e n we o b ta in th e n o tio n o f s t r u c t u r a l com pleteness i n th e f l n i t a r y s e n s e . And, i f th e r u le i s assumed as in o u r p a p e r, as a s u b s e t o f 2L, we o b ta in th e noti o n o f s t r u c t u r a l com pleteness in th e i n f i n i t a r y se n se . An e x te n -ded d is c u s s io n o f t h i s d i s t i n c t i o n can be found e . g . in [ 7 ] . I would l i k e to s t r e s s t h a t th e n o tio n o f s t r u c t u r a l com pleteness was used h e re in th e l a t t e r s e n s e . I n c id e n ta l ly , one can show by an easy argument t h a t th e two n o tio n « c o in c id e in th e c ase o f s tro n g ly f i n i t e s e n t e n t i a l c a l c u l i - J u s t f o r t h i s re a so n we could u se th e r e s u l t s o f Dzik-Wrońaki and P ru cn al concerning f i n i t a r y s t r u c t u r a l com pleteness o f C ödel' s and Lukasiew ioz c a l c u l i .
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C hair o f Logic and Methodology o f S lencea Łódź U n iv e rs ity
Grzegorz Malinowski
ZAGADNIENIE STOPNI MAKSYMALNOáci (P r z e g lą d )
A rtyk uł j e s t celnym przeglądem metod dowodzenia tw ie rd z e ń o sto p n ia c h maksymainoéci 1 re z u lta tó w uzyskanych w t e j d z ie d z in ie do 1979 r .
