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 __________________ fOL IA PHILDSüPHICA 7, 1990__________________
Janusz M aciaszek
ON OIFFERENT NOTIONS OF QUANTIFIERS
I . I n tro d u c tio n
Among the most c e n t r a l n o tio n s of lo g ic seem- to be quan t i f i e r and q u a n t i f ic a t io n . Although these terms were in tro d u ce d in the 19th c e n tu ry , the ve ry id ea of q u a n t i f ic a t io n i s o ld e r and i t can be tra c e d to a n t iq u it y . The p re s e n t paper w i l l attem pt an e x p lic a t io n of th a t n o tio n . A r i s t o t l e 's id e a s of q u a n t i f ic a t io n have c lo s e correspondence w ith modern n o tio n of g e n e ra liz e d quan t i f i e r form u lated by Lind strö m in o p p o s itio n to c l a s s i c a l q u a n ti f i e r s of p r e d ic a te c a lc u lu s . MGdern n o tio n of a g e n e ra liz e d quan t i f i e r enabled the development o f p r a c t i c a l a p p lic a t io n s in l i n g u i s t i c s and lo g ic o f in d u c tio n . I t i s obvious th a t the two c l a s s i c a l q u a n t i f ie r s a re u s e le s s in these s u b je c ts .
2. The n o tio n of q u a n t if ic a t io n in A r i s t o t l e 's s y l l o g i s t i c
The e a r l i e s t attem pt to c r e a te a system of form al lo g ic was A r i s t o t l e 's s y l l o g i s t i c A r i s t o t l e reco g n iz e s fo u r c a t e g o r ic a l statem ents and s p e c i f ie s the r u le s of in fe re n c e fo r them. C atego
r i c a l statem ents re p re s e n t r e la t io n s between the i n t e r p r e t a t io n s of s u b je c ts and p r e d ic a t iv e words.
S a P USB - U P H > 0 S e P 11 5 N n II P II = U S i P IISR
n
В P П /9
S о P IIS« - U P II/
9
In s e t - t h e o r e t ic a l terms cop u las " a " , " e " , " i " , "p " can be in te r p r e te d as b in a ry r e la t io n s between the sub sets of a domain of
in t e r p r e t a t io n E , ( i . e . the in t e r p r e t a t io n of "a " in E i s the r e la t io n of in c lu s io n : fo r any X, У £ E holds (X , Y ) e y a It i f f X c Y, where l i a l i denotes the symbol “ a " ) . Making use of Lind- s trö m 's g e n e ra liz e d q u a n t i f ie r the above symbols can be tre a te d as g e n e ra liz e d q u a n t i f ie r symbols c h a r a c t e r i s t i c fo r s y l l o g i s t i c language. These symbols have 3e t - t h e o r e t ic a l d e n o ta tio n s in the m od ified m od el-theory.
On account of the development of m athem atics, modern p r e d ic a te lo g ic w ith the c l a s s i c a l n o tio n of q u a n t i f ie r has been formulated. However, in s p it e of i t s u se fu ln e s s in m athematics and metamathe- m a tic s , A r i s t o t l e 's i n t u i t i o n of q u a n t if ic a t io n has been l o s t .
3. C la s s ic a l p re d ic a te c a lc u lu s and i t s e x te n s io ns
F ir s t - o r d e r p re d ic a te c a lc u lu s i s not a mere ex ten sio n of s y l l o g i s t i c . I t has d if f e r e n t and re a c h e r d ic t io n a r y and . syn tax . There i s a s i g n i f i c a n t d if fe r e n c e of s y n t a c t ic a l r o le s between cop u las in s y l o g i s t i c and c l a s s i c a l q u a n t if ie r s in p re d ic a t e c a lc u lu s . Symbols V and 3 are o p e ra to rs th a t co n v e rt pro- p o s it io n a l fu n c tio n s to p r o p o s itio n s ; in the c l a s s i c a l theory of models they have no e x p l i c i t in t e r p r e t a t io n , they are u s u a lly in t e r p r e t a t e d in the co n tex t of a d e f i n i t i o n of s a t is f y in g a f o r mula b eginning w ith q u a n t i f ie r symbol by a v a lu a t io n a l fu n c tio n .
From the p o in t o f view of l i n g u i s t i c s , p r e d ic a te c a lc u lu s seems a r t i f i c i a l and u n s u f f ic i e n t , because of i t s poor p o s s i b i l i t y to express " q u a n t it a t iv e words" of n a t u r a l language.
The f i r s t step towards the e x te n sio n o f “ e x p re s s iv e power“ of p r e d ic a te c a lc u lu s c o n s is te d in the re fo rm u la tin g of s y l l o g i s t i c in terms of p r e d ic a t e l o g ic . I t was p o s s ib le by in tr o d u c tio n of the symbols o f q u a n t i f ie r s o f lim it e d ran g e 1.
S a P i f f (V B <x>) P(x> * ( V x ) ( S ( x ) — *P (x > ) S i P i f f ( a s <x>) P ( x ) s ( Y x K S ( x ) Л p (x ) S e P i f f ( v s <x>) ~ P.(x) ■ ( V x )C S ( x ) — » P i x ) ) S o P i f f •■<3s0(>) '* P ( x ) e ( V x ) ( S ( x ) Л ~ P ( x ) )
— ... i
* See: Z. S ł u p e c k i , L. B o r k o w s k i , Elem enty l o g i k i matematycznej i t e o r i i mnogości, Warszawa 1984.
ihus the form ulas of p r e d ic a te lo g ic become s h o rte r and t h e ir syntax more adequate fo r n a tu r a l language d e s c r ip t io n .
The next e x te n tio n was the in tr o d u c tio n of n u m erical q u a n ti f i e r s2 . - For e x a c t ly m h o l d s . . . -£<m>, (m > 0 ) (X < o > )x A (x ) “ ~ 3 x A (X )fo r m * 0 ( 2L<m>)xA(x) a 3X1,--X(B ^A( x i . . . Л А (xm) Л Xj t х^ Л . . . Л x. i
* х
m Л . . . А х sn-l .i
x Л V x m м+1. ( А ( х „ . ) ---m+1*
( хra+i, * * Xj v . . . v xm+1 * X )> fo r m > 0 - For e x a c tly n doesn t ' h o l d . . . (n > 0 )ir<n>xA(x) ■ 2 < n> x(- A ( x )) fo r n > 0 - For a t most m h o ld s . .. - 2 [ m ] , m > 0
2ľ [m }xA(x) * 5ľ <o>xA(x) v . . . w *L<m>xA(x) - For a t most n d o e s n 't h o ld . . . - U T (n ], n > 0
7 l[n ]x A (x ) » ЯЧо>хА(х) v . . . v 0Ы п>хА(х)
I t is c le a r th a t both e x te n tio n s are a b b re v ia tio n s o f complex form ulas of p r e d ic a te c a lc u lu s .
The symbols belong to the c la s s of f i r s t - o r d e r q u a n t i f i e r symbols and can be d e fin e d in terms of c l a s s i c a l q u a n t if ie r s v and 3, w ith in f ir s t - o r d e r p re d icate- lo g ic . The f i r s t r e a l e x ten sio n of p r e d ic a te c a lc u lu s is due to M ostow ski3, who in tro d u ced q u a n t if ie r s of o th er c la s s e s .
4. G e n e ra liz e d q u a n t if ie r s of Mostowski
Une of the most im portant step s towards modern n o tio n of quan t i f i e r was made by Mostowski4 . Mostowski p re s e n ts an e x p l i c l t e mo- d e l- t h e o r e tic d e f in it io n of q u a n t if ie r as an in t e r p r e t a t io n o f a q u a n t if ie r symbol.
2
See e .g . A. M o s t o w s k i , On a G e n e r a liz a tio n o f Quan- t i f i e r s , "Fundanenta M athem atics“ 1957, v o l. 44, pp. 12-J6.
3 Ibidem .
D e f in it io n t S :
A q u a n t i f i e r lim it e d to the domain of in t e r p r e t a t io n E i s a fu n c tio n Qe which assig n ee one of tru th va lu e s I , 0: to each in t e r p r e t a t io n of one-argument p r o p o s itio n a l fu n c tio n F on E , which s a t i s f i e s the i n v a r i a n c e c o n d i c t i o n: Q (F ) - * Q (F ^ ), where i s a - p e r m u t - a t i o n of C, such th a t fo r a e E holds F p ( $ ( a ) ) = F ( a ) .
By t h is d e f i n i t i o n , fo r every q u a n t i f ie r symbol thetoe i s a fa m ily of sub sets of E th a t " s a t i s f i e s " i t s in t e r p r e t a t io n (a qu a n t i f i e r lim it e d to E ) .
The i n v a r i a n c e c o n d i c t i o n enab les to f o r mu] ate a very in t e r e s t in g n u m b e r - t h e o r e t i c r e p r e s e n t a t i o n o f a q u a t i f i e r . L e t (m, n ) be the sequr- ence of p a ir s of c a r d in a l numbers s a t is f y in g the eq u ation m * n - ~ c a r d ( E ) . For each fu n c tio n T which assig n ee I or 0 to each p a ir we put:
Qt( F ) = T (c a rd [a e E: F ( a ) * l } , c a r d ja e E: F ( A) = o } ) Theorem I 6 :
a ) Qt i s a q u a n t i f ie r lim it e d to E.
b ) For each q u a n t i f ie r Qe lim it e d to E th e re i s a fu n c tio n T such th a t Qr = Qe.
I f Qt = Qe, then T i s a s s o c i a t e d f u n c t i o n of Qe This n o tio n i s very u s e fu l in d e fin in g the s p e c i f i c q u a n ti f i e r s by means of the c o n d itio n s under whách the a s s o c i a t e d f u n c t i o n of a q u a n t i f ie r posesses va lu e 1. Exam ples: 1. C la s s ic a l q u a n t if ie r s B e : 'n i 0 V e : n « 0 2. Num erical q u a n t if ie r s S<m>E: m s- n 2 [ i b ] e : w < m IT <n>E: 11 s n IT [n je : ti < n 5 Ibidem. 6 Ibidem .
3. N o n - firs t- o rd e r q u a n t if ie r s
S e : w < Xo - fo r denumerably many holds
S ~ e : n < Xo v и < X o - fo r f i n i t e l y many holds or fo r f i n i t e l y many d o e s n 't hold A. T r i v i a l q u a n t i f ie r s (d e fin e d in s e t - t h e o r e t ic a l term s) PE : fo r every F hold s P e (F ) = I 0e : fo r every F hold s 0 e ( E ) 3 0 I t i s a ls o p o s s ib le to d e fin e d u a l s and b o o 1 e a n c o m b i n a t i o n s of q u a n t i f ie r s . D e f in i tlo n 27: l e t T(m, ч ) Ь е a s s o c i a t e d f u n c t i o n of Qe. A q u a n t i f ie r Q»e w ith a s s o c i a t e d f u n c t i o n T*0m, n ) * <v T (n , in) i s a d u a l of Qe ( e . g . 3t sV *e) . D e f in it io n J 8 : Qt = ~ Q'e i f f Qe( F ) Q'e ( F ) Qe = Q'e v Q"e i f f Qe( F ) * Q * ( F ) v Q * ( F ) Qe » Q'e Л Q*e i f f Qe( F ) - Q<( F) Л Q '( F ) fo r every F .
The p rese n t t h e o r e t ic a l background enables to examine the b o o l e a n a l g e b r a 31 of quantifiers Ignited to d e- n u m e r a b l e s e t E. U i s isonorfic to the p r o d u c t of th re e b o o l e a n a l g e b r a s :
0 3 Ж * X x ( j I . 0 } , -v , v , A )
where x = ( Р ( Ю , * , и , л ) and P (N ) ie a pow erset o f a l l r e a l numbers (in c lu d in g 0 ) . I t occurs th a t q u a n t i f ie r s of l i belong to one of the four c la s s e s : It 0e I , • Se I , I S»t I and I S ~ i I rep re se n te d by the t r i p l e s of 0 r e s p e c t iv e ly :
P ( N ) f i n x". P ( N ) f i n x {q P ( N ) i n f i n x P ( N ) f i n x [ Q P ( N ) i n f in x P ( N ) f i n x j I
P ( N ) i n f i n x P ( N ) f i n x J l }
P ( N ) in f in x P ( N ) in f in x |q|
The c la s s B 0e I is the c la s s of f i r s t - o r d e r qu a n t i f i e r s Other q u a n t ifie r s c a n 't be d efin ed in terms of Ve and 3c o n ly .
C le a r ly , a s s o c i a t e d f u n c t i o n s of Mostowski's g e n e r a l i z e d q u a n t i f i e r s o f U are of the form mRm, tiRn and t h e ir boolean com binations, where R i s a r e la t io n on numbers: "=H, "< ", and n, m < c a r d ( E ) . I t i s p o s s ib le to express o th er in t e r e s t i n g q u a n t if ie r s not considered h e re , e . g. fo r more than h a lf h o ld s . . . : -m > П
Mostowski managed to g iv e an e x p l i c i t d e f i n i t i o n of a q u a n t if ie r as an in t e r p r e t a t io n of a q u a n t i f i e r s y m b o l and p o in ted out how to d e fin e s p e c i f i c q u a n t i f ie r s by means of s e t theory or a s s o c i a t e d f u n c t i o n s . Mo sto w sk i d id n ’ t use terms of t h e o r y o f m o d e l s but h is id e a s were con tinu ed by Lindströro who found t h e i r m odel-the o r e t i c a l form.
5. A form al m o d e l- th e o re tic a l in t e r p r e t a t io n of q u a n t if ie r s
A f i n a l g e n e r a liz a t io n o f a n o tio n of a q u a n t i f ie r was in tr o -9
duced by Lind strö m . Lindström has g ive n a form al m odel-theore t i c a l d e f in it io n of a g e n e r a l i z e d q u a n t i f i e r as an in t e r p r e t a t io n o f a q u a n t i f i e r s y m b o l .
D e f in it io n
A q u a n t i f ie r Q i s a c la s s of r e l a t i o n a l s t r u c t u r e s o f type t e N such th a t Q i s c lo se d under isom or phism.
Choosing a s p e c ia l domain of in t e r p r e t a t io n we get a d e fin itio n o f a g e n e r a l i z e d q u a n t i f i e r l i m i t e d t o E as a r e l a t i o n a l s tr u c t u r e of type t e Nn .
Qe ; <E, R j с P ( E t l ) ... Rn с P ( E t n )>
9 t
P. L i n d s t r ö m , F i r s t Order P r e d ic a te L o g ic w ith Ge n e r a liz e d Q u a n t if ie r s , "T h e o ria " 1966, pp. 186-195.
clo se d under perm utations of E. The type t * < t j , t > (t j > 0) is a sequence of a r i t i e s of r e la t io n s R j , Rn>
I t is obvious th a t M o sto w sk i's g e n e r a l i z e d q u a n t i f i e r s are a s p e c ia l case o f L m ds trb m *s, as they are r e l a t i o n a l s t r u c t u r e s of type <l>.
Qe ■ <E, R £ P (E }>
The copulas of A r i s t o t l e 's s y l l o g i s t i c s are q u a n t if ie r symbols o f type <1. 1>, e .g .
HallE . < £ , . R l . c f ( f ) , R2 Ł P ( E ) : Rj - R2 = 0>
L in d s trö m 's id eą of q u a n t if ic a t io n seems to be a g e n e r a li z a tio n of a n c ie n t in t u it io n s of A r i s t o t l e , i t w i l l be obvious when examining l o g i c o f i n d u c t i o n based on g e n e- r a 1 i z e d q u a n t i f i e r s . The n o tio n of a t y p e o f a q u a n t i f i e r is bound to s y n t a c t ic a l r o le of a r e s p e c tiv e q u a n t i f i e r s y m b o l 0 of a g e n e r a l i z e d f i r s t - o r d e r p r e d i c a t e l o g i c . I f t = < t j ... t n>, then n ex p resses number of form ulas and t^ number of v a r ia b le s in each form u la, bound by Q, e .g .
♦ • Qxn . . .x u ^... xn l * * xn t ni ^ l *
L e t Mlg be a v a l u a t i o n a l f u n c t i o n (under in t e r p r e t a t io n 1 in a domain E ) . E v e ry fu n c tio n M)e and form ula ♦ f i x a r e l a t i o n a l s t r u c t u r e of a type t .
[ M,E . * ] 1 <E. { ( а ц . "... •. a i t i ^ e Е ' 1; s t s f [Mi§
a l t l /,xt t l ) ł ^1^ } ... í ( a n ľ antn> e £tr>! s t a f £MtÉ ^an l /xln • • • • an tn /xntn • * гД 1 >
where a ^ is an o b je c t of E, such th a t H i e ( x ^ ) = a ^ and
s t s f [ M |E ( . . . ) , ф means th a t M(E . s a t i s f i e s the sub-form ula f Les P j E and Uc be in t e r p r e t a t io n s I in E of a p r e d ic a te symbol PV. of a r i t y m and a q u a n t if ie r Q. G e n e r a l i z e d f i r s t - o r d e r p r e d i c a t e l o g i c is an e x ten sio n of f ir s t - o r d e r p r e d ic a te lo g ic (w ith i d e n t i t y ) , by adding g e n e r a l i z e d q u a n t i f i e r s of v a rio u s typ e s .
The d e f in it io n below is an in d u c tiv e d e f in it io n of s a ty s fy in g a form ula by a v a lu a tlo n a l fu n c tio n Ml£ .
D e f in it io n 5 * * :
) l - 8 t s i K - ... ... М,Е ( ^ ГО)>
€ f V
2. s ts f[M ie , жк с x j } i f f M1£ (x k ) * M(e ( X j ) .
3. s t s f [M(E , } ] where ł begins w ith q u a n t i f i e r s y n - b о 1 Q i f f [ M)E ,♦] e Q(
Examples:
1. W ell known s e n t e n t ia l c o n n e c tiv e s : ~ , v , —» , Л >, are in terms of Lindström q u a n t i f i e r symbols of type <0> and <-0, 0>, as t h e ir in t e r p r e t a t io n s are r e la t io n a l s tr u c t u r e s of a r i t i e s 0. Th is resembles L u k a s ie w ic z 's n o ta tio n of s e n te n tio n a l co n n e c tiv e s.
2. Let W£ be a q u a n t if ie r of type <1, l>, <E, R j , R? > « i l £ i f f card ( R p > c a r d ( R 2) . The q u a n t if ie r denotes: There a re no l e s s . . . t h e n .. .
The examples show the u t i l i t y of L in d s trö m ’Concept of a quan t i f i e r . The next ch ap te r d e a ls w ith the p r a c t ic a l a p p lic a tio n of t h is id e a .
6. P r a c t i c a l a p p lic a t io n s of a modern n o tio n of a g e n e ra liz e d q u a n t if ie r
A cco rd in g ly to A r i s t o t l e 's i n t u i t i o n , the most im portant quan t i f i e r s in the n a tu r a l language in v e s t ig a t io n s are of the type <1, 1> s in c e they f i t fo r the s p e c i f ic syntax of a n a tu r a l la n guage. They were thoroughly examined during la s t ten y e a rs .
The s tu d ie s were s ta r te d by Barw ise and Cooper12, who t r e a t a q u a n t if ie r as a n o u n p h r a s e -NP:
Ibidem. 12
j . B a r w i s e , R. C o o p e r , G e n e ra liz e d Q u an tiT T ie r s ąnd N atu ra l Language, " L in g u is t ic s and P h ilo s o p h y " 1981 No. 4, pp. 159-219.
Q ( q u a n t i f i e r ) = NP
\
0 (d e te rm in e r) Set ex p resio n Exam ples:
I E ve ry man 1 = [ x : D mani с X }
I Two boys I = { X : c a rd ( II boy 1 I) X ) = г ]
I most women II * | X : c a rd ( II woman It Л x ) > c a rd (ll woeanll - X )]. I! E ve ry ( t h i n g ) I = { X : X » e } .
where e v e ry , two, most e t c . are d e t e r m i n e r s . In terms of B a rw ise and Cooper in t e r p r e t a t io n e v e ry q u a n t i f ie r i s o f type <1> and i s c lo s e ly r e la t e d to the q u a n t i f ie r o f l i m i t e d r a n- 9
e-The main goal of B a rw ise and C o o p e r's paper was to fo rm u late a form al language LGQ cap ab le of fo rm a liz in g a fragm ent of En g lis h . The LGQ syntax i s c lo s e ly r e la t e d to the E n g lis h s yn ta x , e .g . the s p e c ia l fu n c tio n in LGQ i s assig n ed to a term " t h in g " which w i l l m an ifo ld compound e x p re s s io n s l i k e " e v e r y t h in g " , " n o t h in g " , "so m eth in g ".
Other papers co n ce rn in g q u a n t i f ie r s in n a t u r a l lan g uag e17 deal w ith a q u a n t i f ie r i d e n t i f i e d w ith a d e te rm in e r ( i t i s of type <1. l>. As q u a n t i f ie r s are b in a ry r e l a t i o n s on su b sets of E I t is p o s s ib le to impose some c o n d itio n s on them. F i r s t c o n d itio n i s a u n i v e r s a l one, i t i s s a t i s f i e d by a l l q u a n t i f i e r s .
CQNSERV. ( c o n s e r v e t i v i t y )
For every E and e v e ry A, B c E h o ld s : QgAB i f f 0Е А (А П В).
T h is c o n d itio n im p lie s fo r example th a t e .g . some men are run ning i f f some men are running men, Second c o n d itio n i s not a u n i v e r s a l one. I t i s not s a t i s f i e d by some s p e c ia l q u a n t i f ie r s e .g . many, some, few, a few.
QUANT, ( q u a n t it y )
For e v e ry E* E ‘ , e v e ry b is e c t io n Ф : í — * E ' and e v e ry A, B C E h o ld s: QeAB i f f Q£* ( A ) * ( B ) .
-13 See: J . van B e n t h e m, Q u estion s about Q u a n t if ie r s , "The Jo u rn a l of Sym b olic L o g ic " 1984, v o l. 49, pp. 443-446; i d e m , Esseys In L o g ic a l Sem a n tics, Amsterdam 1985; 0. W e s t e r s t S h l , Some Remarks on Q u a n t if ie r s , Göteborg 1982,
This c o n d itio n I s c lo s e ly r e la te d to M o sto w sk i's and L in d s trö m 's c o n d itio n of in v a r ia n c e .
I t i s a ls o p o s s ib le to fo rm u la te another c o n d itio n s a t i s f i e d o n ly by s p e c ia l groups of q u a n t i f ie r s . The m o n o t o n i c i t y c o n d itio n s of q u a n t if ie r s a re fundam ental in d e fin in g the c la s s of f i r s t - o r d e r q u a n t i f i e r s of type <1, l>. For q u a n t if ie r s th a t s a t i s f y CQNSERV. and QUANT, th e re e x is t s
possi-14
b i l i t y to in trod u ce n u m b er- te o retica l re p re s e n ta tio n , th a t ena b le s to examin t h e ir r e la t io n a l p ro p e rtie s .
Lo g ic of in d u c tio n is another domain of a p p lic a t io n of genera liz e d q u a n t if ie r s . A q u a n t if ie r as a " r e la t io n on r e la t io n s " is ve ry u se fu l to express v a rio u s c o r r e la t io n s searched by e m p iri c a l s c ie n c ie s . L e t us look a t A r i s t o t l e 's scheme of in d u c tio n :
V i s P
Sj i s P
Sn is P
S a P (e v e ry S is P )
A l l oth er schemes of in d u c tio n have the same form but as other q u a n t if ie r s are used i t is p o s s ib le to express v a rio u s c o r- r e l a t i o n s of in t e r e s t .
The p r a c t ic a l u t i l i t y of g e n e ra liz e d in d u c tio n was p o s s ib le thanks to computers. The group of Czech s c ie n t is t s have inven ted s p e c ia l computer methods of o b ta in in g in d u c tio n h yp o th e sis , c a lle d GUHA methods - G eneral Unary H ypothesis Automaton . The GUHA methods are based on n um ber-theoretic re p re s e n ta tio n of genera liz e d q u a n t if ie r s . Whith support of a s s o c i a t e d f u n c t i o n s of q u a n t if ie r s i t i s p o s s ib le fo r a computer to search in t e r e s t in g c o r r e la t io n s in e m p iric a l d a ta . Very im portant s o rt of g e n e ra liz e d q u a n t if ie r s used in GUHA methods are s. t a t i s- t i с a 1 q u a n t i f i e r s. T h e ir a s so cia te d fu n c tio n s ex
i t '• • "■'■■■' .
See: A. M o s t o w s k i , op. c i t ,
15 See: P . H a ] e . k , T. H a v r á n в к , M echanizing Hy p o th e s is Form ation, S p rin g e r V e rla g , H e id e lb e rg 1977.
p ress w e ll known s t a t i s t i c a l t e s t s . S t a t i s t i c a l q u a n t i f ie r s were a p p lic a te d in s o c io lo g y , m ed icine, l i n g u i s t i c s and even in d u s tr y .
The g iven examples prove th a t modern n o tio n of a q u a n t i f ie r due to Lindström i s a ve ry im portant is su e of same branches of contem porary lo g ic .
U n iv e r s it y of Łódź Poland
Janusz M aciaszek
0 RÚÍNYCH POJfCIACH KWANTYFIKATQRА
Jednym z podstawowych pojęć lo g ik i j e s t p o ję c ie k w a n t y f ik a c ji i związane z nim p o ję c ie k w a n ty f ik a t o r a . P ie rw sz e problemy związane z k w a n ty fik a c ją p o ja w iły s ię wraz z powstaniem s y l o g is t y k i A r y s to t e le s a , ch o ciaż n ie I s t n i a ł o wówczas p o ję c ie k w a n ty fik a to r a . Po w sta n ie rachunku predykatów I rzędu oraz wprowadzenie k la s y c z n y c h kw a n tyfik a to ró w V i 3 z m ie n iło z u p e łn ie sens p o ję c ia k w a n ty fik a c j i .
P ra c e Mostowskiego i lin d s trö m e poświęcone kw an tyfikatorom u- ogólnionym w prow adziły z u p e łn ie nowe rozum ienie k w a n t y f ik a c j i, k tó re zaowocowało wieloma p raktycznym i zastosowaniam i w lin g w is ty c e lo g ic z n e j i lo g ic e in d u k c ji.
Głównym celem p racy j e s t w ykazan ie, że koncepcja k w a n ty fik a to rów uogólnionych Lind strö m a n aw iązuje bezpośrednio do i n t u i c j i A r y t o t e le s a , zaś s p ó jk i " a " , “ e " , " i " , "o " w ystę p u jące w języku s y l o g is t y k i mogą być uznane za symbole k w a n ty fik a to ró w u o g ó ln io nych typu <1, l>.
