Analyticity and metamathematics

Jan Woleński

Analyticity and metamathematics

Acta Universitatis Lodziensis. Folia Philosophica nr 9, 125-131

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

F O L I A P H I L O S O P H I C A 9. 1993

J an W o leň ski

A N A L Y T IC IT Y A N D M E T A M A T H E M A T IC S

T h e d istin c tio n o f a n a ly tic /sy n th e tic w as explicitly s ta te d fo r the first tim e by K a n t w h o referred it to ju d g e m e n ts. O th e r a u th o rs a p p lie d th e d istin c tio n a lso to sen ten ces p ro p o s itio n s a n d statem e n ts; in w h a t follow s. I shall use the fo rm 'se n te n c e ' even if review ed a u th o rs em p lo y ed a n o th e r nam es. B efore K a n t, related id eas c o n ce rn e d w ith th e d istin c tio n s a p rio ri/a p o s te rio ri an d n e ce ssa ry /c o n tin g e n t h a d b een d e v elo p ed m ain ly by H u m e a n d Leibniz. A lth o u g h p rc -K a n tia n s did n o t used th e te rm s 'a n a ly tic ' a n d 's y n th e tic ', it is c o m m o n to reg a rd L e ib n itia n d e fin itio n o f n ecessary tr u t h (as a sen te n c e tru e in all p o ssib le w o rld s) o r H u m e a n tre a tm e n t o f re la tio n s b etw een id eas (as re c o rd e d by ta u to lo g ie s) as im p o r ta n t p ro p o s a ls c o n c e rn in g th e c o n c e p t o f a n aly ticity .

F o r K a n t, th e lin g u istic s tru c tu re ..A is B " is the g en eral fo rm o f sentence. N o w a sen ten ce S is a n a n a ly tic if a n d o n ly if its p re d ic a te A is 'c o n ta in e d ' in its su b ject B; o th erw ise S is a sy n th etic sentence. It follow s fro m K a n t's d e fin itio n t h a t n e g atio n s o f an aly tic sen ten ces a re self-c o n tra d ic to ry . M o re o v er, an aly tic tru th s a re u n in fo rm a tiv e (ta u to lo g o u s ) b ecau se th ey m erely an aly se the re le v a n t subject co n ce p t. F o rm a l logic fo r K a n t c o n sists o f a n a ly tic sentences. O n the o th e r h a n d , s y n th etic sen ten ces co n sist in a sy n th esis o f c o n c e p ts an d p ro v id e a n in fo rm a tio n . All a n a ly tic sen ten ces a re fo r K a n t a p rio ri by d e fin itio n b u t sy n th etic o n es c an be e ith e r a p rio ri o r a p o s te rio ri. T h e c eleb rate d p ro b le m o f K a n t’s p h ilo s o p h y c o n c e rn e d th e p o ssib ility o f sentences w hich w o u ld be a re b o th sy n th etic a n d a p rio ri. K a n t h im se lf w as entirely co n v in ced th a t such sen ten ces exist.

T h e p o s t-K a n tia n p h ilo s o p h e rs p ro p o s e d m a n y d e fin itio n s o f an aly ticity . S everal o f th em a re in clu d ed in th e follow ing lis t1 (a n a ly tic = a n aly tica lly tru e):

1 Sec: B. M a t e s . A n a ly tic s en ten c es, ..P h ilo s o p h ic a l R e v ie w " 1951, N o . 60. p . 525.

126 J a n W o l c ń s k i

( l a ) S is a n a ly tic iff S is tru e in all po ssib le w orlds; ( l b ) S is a n aly tic iff S co u ld n o t be false;

(1c) S is a n a ly tic iff n o t-S is self-c o n tra d ic to ry :

( I d ) S is a n aly tic iff S is tru e by v irtu e o f m ean in g s a n d in d e p e n d en tly o f facts;

( l e) S is a n a ly tic iff e ith e r S is logically tru e o r S c an be tu rn e d in to a logical tru th by p u ttin g sy n o n y m s lo r sy n o n y m s;

(If) S is a n a ly tic iff S co m es o u t tru e u n d e r every state-d esc rip lio n ; ( lg ) S is an aly tic iff S c an be red u ced to logical tru th by d efin itio n ; ( lit) S is a n aly tic in a la n g u a g e L i f f S is tru e a c c o rd in g to th e sem a n tic a l ru les o f L.

T h e d e fin itio n ( l a ) goes b ack to L eibniz, ( l b ) a n d ( l e) a re m e n tio n e d as po ssib le e x p lic atio n s by Q u in e in his very fa m o u s criticism o f a n a ly tic k y 2 (1c) is p ro p o s e d b y S tra w s o n 3, ( I d) re c o rd s a ty p ical po sitiv istic tre a tm e n t o f a n a ly tic k y 4, (If) a n d ( I h ) a re ta k e n fro m C a r n a p 5, a n d ( l g) expresses F reg e's d e fin itio n o f an aly tick y .

V a rio u s g en eral logical term s o c c u r in d e fin itio n s ( l a ) - ( 1 h ). T r u th , logical tru th , d e fin itio n a n d c o n tra d ic tio n a p p e a r explicitly in th e m b u t o th e r, for in sta n ce m o d el, p ro v a b ility o r c o n siste n c y im plicitly via p o ssib le w o rld s (s ta te d e sc rip tio n s), logical tru th a n d c o n tra d ic tio n respectively. W e can rew rite fo r in sta n ce ( l a ) a n d (I f) as

(2) S is an aly tic iff S is tru e in all m odels

a n d ( l g) as (n o te th a t (3) is c lo se r to F reg e 's o rig in al fo rm u la tio n th a n (lg )) (3) S a n a ly tic iff S is p ro v a b le exclusively by logic a n d d e fin itio n s. Im p o r ta n t asp e c ts o f m etalo g ical c o n ce p ts like tr u th , co n sisten cy o r p ro v a b ility a re fo rm ally reg u la te d b y m e ta m a th e m a tic a ! th e o re m s: fo r sim p ­ licity. I assu m e th a t m e ta m a th e m a tic s c o m p rise s m etalo g ic a n d fo rm al sem an tics. So we c an ask w h a t follow s from m e ta m a th e m a tic s fo r the ..p h ilo so p h y o f a n a ly tic k y " . M y a im in this p a p e r is to p u t to g e th e r (w ith som e c o m m e n ts) v a rio u s o b s e rv a tio n s o n a n a ly tic k y w hich h av e been m a d e by several c o n te m p o ra ry lo g ician s fro m th e m e ta m a th e m a tic a l p o in t o f view. I shall c e n te r o n so called lim itativ e th e o re m s, in p a rtic u la r

(4) if X c o n ta in s fo rm alize d P e a n o a rith m e tic , th e n X is in c o m p le te if co n siste n t (th e first G ö d e l in c o m p le te n ess th eo rem );

г S ee: V . v u n Q u i n e . Tm> d o g m a s o f e m p iric ism , „ P h ilo s o p h ic a l R ev iew ” 1951, N o . 60. p. 2 0 -4 3 .

3 Sec: P. S t r a w s o n . In tr o d u c tio n to L o g ic a l T h e o r y . M e tlu ie n . L o n d o n 1952. p. 21. 4 S ec: A . A y e r , L o n g u a g c , T ra ill o tu i L o g ic . P e n g u in B o o k s . H a r m o d w o r d tll 1971. p. 1 0 4 -1 0 6 .

s S ee: R . C a r n a p . L o g ic a l F o u n d a tio n s o f P r o b a b ility . R o u tle d g e a n d K e g y n P a u l. L o n d o n 1962. p . S3 a n d R . C a r n a p . M e a n in g a n d N e c e s s ity . T h e U n iv e r s ity o f C h ic a g o P re s s . C h ic a g o 1956. p. 8. 10.

A n a l y t i c i t y a n d M e i a m a t l i c m a t i c s 1 2 7

(5) if S c o n ta in s fo rm alize d P e a n o a rith m e tic , th e n co n sisten cy o f S is im p ro v ab le in S (th e seco n d G ö d e l in co m p leten ess th eo rem );

(6) P e a n o a rith m e tic a n d first o rd e r logic a re n o t d ecid ab le (th e C h u rc h u n d e cid ab ility th eo rem ).

T h e first an aly sis o f a n a ly tic ity w ith th e h elp o f m e ta m a th e m a tic s w as given by C a r n a p 0. H e d i s t i n g u i s h e d L a n g u a g e I co n sistin g o f e le m e n ta ry logic to g e th e r w ith th e p o rtio n o f a rith m e tic sufficien t fo r a rith m e tiz a tio n (in the sens o f G ö d e l) a n d L a n g u a g e II w hich c o n ta in s all m ean s w hich a rc needed fo r ex p re ssin g classical m a th e m a tic s in it. N o w a n a ly tic ity in L a n g u a g e I is d efin ed by

(7) S is a n a ly tic in LI iff S is a co n se q u e n ce o f th e null class o f sen ten ces (o r every sentence);

H o w ev er, (7) is to o n a rro w lo r LII b ecau se a rith m e tic is in c o m p le te w h a t cau ses th a t „in every sufficien tly rich system fo r w hich th e m e th o d o f d e riv a tio n is p re scrib ed , sen ten ces c an be c o n stru c te d w hich, th o u g h th ey co n sist o f sy m b o ls o f th e sy stem , a re yet n o t reso lu b le in a cc o rd a n c e w ith the m e th o d o f th e system th a t is to say . a re n e ith e r d e m o n stra b le n o r re fu ta b le in it. A n d . in p a rtic u la r, fo r every sy stem in w hich m a th e m a tic s c a n be fo rm u la ted , sen ten ces c an be c o n stru c te d w hich a re valid in th e sense o f classical m a th e m a tic s b u t n o t d e m o n stra b le w ith in th e s y s te m " 7. So we have sentences w hich a rc n o t c o n se q u e n ce s o f every sentence. T o solve this d ifficu lty . C a r n a p (he w a n ted to have all m a th e m a tic a l tru th s a m o n t a n aly tic sen ten ces) p ro p o se s to a d m it in fin ite sets o f prem ises a n d su p p lem en t ru les o f p r o o f by non -effectiv e o n es, fo r in stan ce « -r u le . C a r n a p 's d e fin itio n o f a n aly ticity fo r L a n g u a g e II is to o co m p lic ated in o rd e r to p re sen t it here in a d e ta iled w ay b u t the g en eral idea is c a p tu re d by

(8) S is a n a ly tic in LII iff S is d e riv a b le fro m an aly tic sen ten ces by rules o f p r o o f w hich a rc a d m issib le in Li I.

A s fa r as I k n o w , G ö d e l a d rc sse d to th e p ro b le m o f a n aly ticity o n ly o n c e in his p u b lish e d w o rk s, n am ely in his p a p e r o n R u ssell's m a th e m a tic a l lo g ic8. A c co rd in g to G ödel

(9) S is a n a ly tic iff A is a special case o f the law o f id en tity in v irtu e o f explicite d efin itio n s o f term s o r ru les o f th e ir e lim in a tio n .

H ow ever. G ö d e l o b serv es t hat (6) im plies n o n -a n a ly tic ity o f a rith m e tic . A d m ittin g sen ten ces o f in fin ite len g th d o es n o t sav e the s itu a tio n becau se to p ro v e th a t so m e im p o r ta n t m a th e m a tic a l th e o re m s (fo r in sta n ce , th e ax io m o f choice) a rc an aly tic, o n e w o u ld have to a ssu m e a n aly ticity o f the w h o le

° See: R . C a r n a p . L o g ic a l S v n la .s o l L a n g u a g e. R o u tle d g e a n d K e g a n P a u l. L o n d o n . ■ /* « /.. p . 100.

B See: К G ö d e l , R u s s e ll's M a lh e m a lic a l L o g ic , [in:] T h e P h ilo so p h y o f B e r lr a n tl R u ssell/, ed . P. S ch ilp p . O p e n C o u r t . L a S alle 1944, p. 123 153.

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m a th e m a tic s in ad v an c e . G ö d e l also co n sid e rs a n o th e r d e fin itio n o l'a n a ly tic ity , n am ely a v ersio n o f ( I d ) b u t lie d o es n o t link it w ith a n y m e ta m a th c m a tic a l fact.

T h e next step in the h isto ry in q u e stio n w as m ad e by C o p i a n d T u r q u e tte 0. C opi ex am in es the fo llo w in g d e fin itio n o f a n aly ticity

(10) S is a n a ly tic iff its tru th o r v a lid ity follow s from the sy n tac tic a l o r g ra m m a tic a l ru les g o v e rn in g a la n g u a g e rules in w hich it is expresed. fro m th e p o in t o f view (4) fo rm u la te d by him as

(11) given a n y re a so n a b ly rich lan g u ag e, th e re is n o n -em p irical. n o n -in d u c tiv e p ro p o s itio n e x p ressib le w ith in it w hich is n o t d ecid ab le o n the b asis o f the sy n tac tic a l rules o f th a t lan g au g e.

T h e n C o p i says th a t (11) leads to

(12) th e re a re a p rio ri lio n -a n a ly tic tru th s,

w hich d e stro y s th e an aly tic th e o ry o f a p rio ri (all a p rio ri sen ten ces are an aly tic).

T u rq u e tte m ak es several o b je c tio n a g a in st C o p i. L et m e m e n tio n tw o. T h e first is gen eral: ..In fact, th e claim th a t th e re a re G ö d e l s y n th etic a p r i o r i tru th s th e n a m o u n ts to n o th in g m o re th a n a re s ta te m e n t in m islead in g p h ilo so p h ic al lan g u ag e o f so m e w ell-estab lish ed logical resu lts, n o ta b ly o f w h a t is u su ally called G ö d e l's seco n d in co m p leten ess t h e o r e m ''10. S eco n d ly . T u r q u e tte o b s e r­ ves th a t u n d c cid ab le s ta te m e n ts co u ld be in te rp re te d as em p irical o r w ell-fo rm ed b u t d e v o id o f m ean in g .

C o p i in his a n s w e r" says th a t his th eses a re n o t deriv ed fro m u n d e cid ab le sentences b u t from the fact th a t „ th e re a re such sta te m e n ts as G ö d e l's w hich a re a p rio ri tru e b u t n o t a n a l y tic " 12. M o re o v e r, he rejects th e em p irical th e o ry o f m a th e m atics a n d o b serv es th a t re g a rd in g u n d e cid ab le sen ten ces as d ev o id o f m e a n in g is u n te n a b le b ecau se we u n d e rs ta n d them .

T u rq u e tte po sitiv e so lu tio n s re q u ire e ith e r a cc e p tin g th a t m a th e m atics is em p irical o r a revision o f logic: b o th p ro p o sa l m u st m eet several w ell-know n ob jectio n s. T u r q u e tte 's g en eral o b je c tio n a g a in st C o p i raises a serio u s m e t­ h o d o lo g ic a l p ro b le m . G ö d e l's th e o re m s (like o th e r lim itativ e resu lts) says n o th in g on a n a ly tic ity o r a p rio rity . So C o p i's fo rm u la tio n o f (4) is in fact its c erta in p h ilo so p h ic al in te rp re ta tio n w hich sh o u ld be s e p a ra ta le ly ju stifie d . M o re o v er. (12) is d eriv ed by C o p i fro m (10) b u t it m ay n o t ho ld u n d e r o th e r d e fin itio n s o f a n aly ticity .

4 See: I. C o p i. M o d e r n L o g ic a m ! th e S y n th e tic tt p rio ri. ..J o u r n a l o f P h ilo s o p h y " 1949, N o . 4 6 . p. 24.1-245: 1. C o p i. G o tici a m i the S y n th e tic и p riori: tt R ejo intier. ..J o u r n a l o f P h ilo s o p h y " 1950. N o . 4 7 . p. 6 1 1 -6 3 6 ; A . T u r q u e t t e , tititle! tintI th e S y n th e tic a p rio ri. ..J o u r n a l o f P h ilo s o p h y " 1950. N o . 4 7 . p. 1 2 5-1 28.

10 T u r q u e t t e . G ö d e l.... p. 126. 11 See: C o p i . Gotiel... 12 Ih itl.. p. 6.14.

A n a l y l i c i t y a n d M e t a i m t h c m a t i c s 129 K e m e n y 1·’ arg u e s th a t th e c o n ce p t o f in te n d e d m o d el (in te r p re ta tio n ) fo rm s a n a d e q u a te c o n ce p tu a l b ase fo r fo rm al sem an tics. A ssu m e th a t we define a n aly tic p ro p o s itio n s as th o se w hich a rc u n iv ersally valid , i.e. h o ld in all m odels. T h is d e fin itio n is to o n a rro w in v irtu e o f in co m p leten ess o f a rith m e tic . K em en v arg u e s th a t a m o re sa tisfa c to ry a c c o u n t o f a n a ly tic k y is to be o b ta in e d w ith the help o f th e c o n c e p t o f in te n d e d m odel.

Let T be a th e o ry , i.e. a set o f sen ten ces closed u n d e r th e co n se q u e n ce o p e ra tio n . N ow all in te n d e d m o d els o f T h av e ex actly th e sam e universes. M o re o v er, if M a n d M ' a rc in te n d e d m odels, th en b o th c an m u tu a lly d iffer o n ly w ith resp ect to v a lu a tio n o f e x tra lo g ic al c o n sta n ts ; K e m en y c o n sid e rs a rith m e tic a l c o n sta n ts as logical. T h e n

(13) S is an aly tic in L iff A is tru e in all L -in le rp re ta tio n s , i.e. L in te n d e d m o d els o f L.

A ssu m e th a t S is a n a ly tic in L. A th e o ry T is c o m p le te (K e m e n y say s th a t it is th e m o st n a tu ra l c o n ce p t o f c o m p le ten e ss) if an d o n ly if

(14) S b elongs to C n (T ) if A is a n a ly tic in T.

If T is c o m p le te, then its an aly tic tru th s c an be defin ed as valid in all m odels. B ut if'T is in co m p lete, th is d e fin itio n m u st be rep laced by (13) b ecau se fo r in sta n ce we have a rith m e tic a l tru th s w hich a re n o t valid in all m o d els o f arith m e tic .

K e m en y 's a p p ro a c h raises so m e d o u b ts . L et S be a n u n d e cid ab lc fo rm u la in its in te n d e d m ean in g . C o n sid e r its n e g atio n not-S . W e c an easy defin e a set o f m o d els in w hich S holds. O n e c an even claim th a t m o d els o f n o t-S (n o t th o se o f S) a rc in te n d e d . T h is m e a n s th a t no t-S is a n a ly tic o n th is c laim . So we o b ta in th a t tw o m u tu a lly c o n lia d ic t o iy sen ten ces a re a n aly tic. T h is re a so n in g sh o w s th a t the c o n ce p t o f a n a ly tic k y via L - in tc rp e la tio n s is ra th e r p ra g m a tic a n d relativ ised th a n sem a n tic a n d a b so lu te.

B o rk o w sk i14 c o n sid e rs tw o d e fin itio n s o f a n a ly tic k y , n am ely

(15) S is a n a ly tic in th e sy n tac tic sense iff S is p ro v a b le exclusively by logic; (16) S is an aly tic in th e sem a n tic sense iff S is tru e in all m odels. A c co rd in g to B o rk o w sk i, the first G ö d e l th e o re m im plies th a t n o t every sen ten ce se m a n tic a lly an aly tic is also sy n tac tic a lly a n aly tic. H o w ev er, this th esis is d u b io u s. I f sen ten ce S is tru e in all m o d els, it is (by co m p le ten e ss th e o re m ) p ro v a b le exclusively by logic. T h is m ean s th a t b o th classes o f an aly tic sen ten ces m u tu a lly co in cid e.

14 See: J . K e m e n y . .t N c »■ A p p r o a c h 10 S e m a n tic s, ..J o u r n a l o l'S y m b o tic L o g ic " 1956. N o . 21. P a n I. p. I 27: P a ri 2. p. 149 161.

14 L. В о г к о w s k i . D e d u ctive ľ o u n d a tio n tou! A n a ly tic P ro p o sitio n s. ..S tu d ia L o g ic a " 1966. N o . 19. p. 59 72; t.. B o r k o w s k i , l.o g ik a fo r m a ln a . P W N . W a rs z a w a 197t); L . B o r k o w s k i . W p ro w a d zen ie do lo g ik i i te o rii m n o g o ś c i, T o w a rz y s tw o N a u k o w e K a to lic k ie g o U n iw e rs y te tu L u b e ls k ie g o . L u b lin 1990.

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D e L o n g 15 a rg u e s th a t (th e fo rm u la C o n (A r) m e a n s „ a r ith m e tic is co n sis­ te n t" ).

(16) T h e sen ten ce C o n (A r) u n d e r its in te n d e d in te rp re ta tio n is sy n th etic

a priori.

T h e fo rm u la e x p re ssin g co n sisten cy o f a rith e m ic is sy n th etic b ecau se it is n o t p ro v a b le exclusively by g en eral logic a n d d efin itio n s a n d a p rio ri b ecau se if a rith m e tic is c o n siste n t, it is necessarily so.

N o w a ssu m e th a t C o n (A r) is n ecessary tru e . L et M is the s ta n d a rd m odel o f A r. So N e c (C o n (A r)) is tru e iff an d o n ly if C o n (A r) is tru e in all m o d els accessible fro m M . H o w ev e r, these m o d e ls a re n o t d e te rm in e d a p rio ri b u t w ith resp ect to p ra g m a tic c rite ria o f s ta n d a rd e n e ss. T o o b ta in (16) o n e h a s to show th a t C o n (A r) h o ld s in all m o d els in w h ich P e a n o a x io m s h o ld b u t it w o u ld be in c o n siste n t w ith u n d e cid ab ility o f C o n (A r).

C a s to n g u a y 16 claim s th a t C h u rc h ’s th e o re m (to g e th e r w ith C h u rc h 's thesis) im plies th a t m a th e m a tic a l k n o w led g e is s y n th etic a p rio ri. H o w ev e r, th is is to o stro n g claim b ecau se (6) im plies o n ly th a t m a th e m a tic a l k n o w led g e is n o r red u cib le to p u re ly a lg o rith m ic p ro c e d u re s. C a s to n g u a y seem s to assu m e

(17) if X is a set o f an aly tic sentences, th a n X is decid ab le. B ut this s u p p o sitio n is b y n o m ean s o b v io u s.

T h e re is n o t sy ste m a tic tre a tm e n t o f a n aly ticity fro m th e p o in t o f view m e ta m a th e m a tic s . O n the o th e r h a n d , m e ta m a th e m a tic a l seem to be o f a fu n d a m e n ta l im p o rta n c e fo r a n y a n aly sis o f a n aly ticity . L et m e finish this su rv ery w ith so m e very g en eral o b s e rv a tio n s 1’ . M e ta m a th e m a tic s su ggests tw o d iv isio n s o f a n a ly tic sentences: (I) in to sy n tac tic , sem a n tic a n d p ra g m a tic (n o te h o w ev er th a t m y p ro p o s a ls in th is re sp ec t c o n sid e ra b ly d iffer fro m th o se o f B o rk o w sk i18), a n d (II) in to a b so lu te a n d relativ e. T h e p ro p o s e d d e fin itio n s are as follow s:

(17) S is a n a b so lu te sem an tic a n a ly tic sen ten ce iff S is un iv ersally valid; (IB) S is a n a b so lu te sy n tac tic an aly tic sen ten ces iff S is an a b s o ­ lu te sem a n tic an aly tic sen ten ce a n d S b elo n g s to a d e cid ab le set o f logical tru th s;

(19) S is a relativ e sem a n tic an aly tic sen ten ce in a th e o ry T iff S is tru e in all m o d els o f T;

(20) S is a relativ e sy n tac tic a n a ly tic sen ten ce in a th e o ry T iff S is a relativ e sy n tac tic a n a ly tic sen ten ce in a th e o ry T a n d S b elo n g s to a d e cid ab le set o f tru th s o f T;

15 See: H . D e L o n g , A P ro file o f M a th e m a tic a l L o g ic . A d d is o n -W e s le y . R e a d in g , M as s . 1970.

16 See: C h . C a s t o n g u a y , C h u rch 's T h e o re m a n d th e A n a ly tic - s y n th e tic D is tin c tio n in M a th e m a tic s . „ P h i l o s o p h i c a " 1976, N o . 18. p . 7 7 -8 9 .

17 See: J. W o l e ń s k i . M e ta m a te m a ty k a І ep istem o lo g ia , P W N , W a rs z a w a (f o rth c o m in g ). IB See: B o r k o w s k i , D e d u ctive F o u n d a tio n ...

A n a l y t i c i t y a n d M e i a m a t h e m a t i c s 131

(21) S is a p ra g m a tic a n a ly tic sen te n c e in a th e o ry T iff S is tru e in all s ta n d a rd m odels o f T.

O b v io u sly we have,

(22) a b so lu te sy n tac tic a n a ly tic sen ten ces £ a b so lu te sem a n tic a n a ly tic sen ten ces E p ra g m a tic a n a ly tic sen ten ces (th e sam e h o ld s i f ‘a b so lu te ’ will be rep laced by ‘re la tiv e ’).

So sy n tac tic a n a ly tic sen ten ces a rc th o se w hich can be reso lv ed by a lg o rith m ic m e th o d s. M o re o v e r o n ly logic c o n sists o f a b so lu te an aly tic sentences. T h ese co n se q u e n ce s a re c o n sis te n t w ith m an y tra d itio n a l a c c o u n ts c o n c e rn in g a n a ly tic sentences. In s titu te o f P h ilo s o p h y J a g te llo n ia n U n iv e rs ity P o la n d J a n W oieľtski A N A L I T Y C Z N O Ś Ć I Μ Ε Τ Α Μ Λ Τ Ε Μ Α Τ Υ Κ .Α

C h o c ia ż r o z ró ż n ie n ie s ą d ó w a n a lity c z n y c h i s y n te ty c z n y c h p o ja w iło się p o r a z p ierw szy u K a n ta , to p o k re w n e p o ję c ia m o ż n a o d n a le ź ć j u ż u H u m e 'a i L e ib n iz a . A u t o r z e s ta w ia i a n a liz u je ró ż n e d e fin ic je i c h a r a k te r y s ty k i p o ję c ia a n a lity c z n o ś c i, j a k ie p ro p o n o w a li m . in .: K a n i. p o z y ty w iś c i. F re g e . C a r n a p . S tr a w s o n i Q u in e . W s k a z u je się. że w b a d a n ia c h n a d z a g a d n ie n ie m a n a lity c z n o ś c i cz y sto o d w o ły w a n o się d o ta k ic h p o ję ć n ie la lo g ic z n y e h . ja k : p ra w d z iw o ś ć , n ie s p rz e e z n o ś ć . czy d o w ic d ln o ś ć . a te z ko lei z o s ta ły s c h a ra k te ry z o w a n e n a g ru n c ie m e ta m a te m u - tyki p rz e z tzw . tw ie rd z e n ia lim ita c y jn e , w’ s z c zeg ó ln o śc i p rz e z tw ie rd z e n ia G o d ła o n ie z u p e ln o ś c i i tw ie rd z e n ie C h u r c h a o n ie ro z s trz y g a łn o ś c i. W z w ią z k u z tv m re fe r o w a n o d y s k u s ję n a d zw ią z k ie m \vw. tw ie rd z e ń z z a g a d n ie n ie m ro z s tr z y g a ln o ś e i p r o w a d z o n ą p rz e z s a m e g o G ö d la . a ta k ż e p rz e z T u r q u e t te 'a . C o p ie g o . K e m e n y 'e g o . B o rk o w s k ie g o i in.