Atomistic universes of individuals

Witold Strawiński

Atomistic universes of individuals

Acta Universitatis Lodziensis. Folia Philosophica nr 9, 97-107

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

W ito ld S tra w iń sk i

ATOMISTIC UNIVERSES OF INDIVIDUALS

At t he b e g in n in g o f this p a p e r I w o u ld like to re fer to c e rta in selected theses o f B. R u ssell’s p h ilo so p h ic al c o n c e p tio n . T h ese theses c a n be p re sen te d as a p a r t o f th e g en eral „lo g ical a to m is m p ro g r a m m e " . F u r th e r on . an o u tlin e a n d a n aly sis o l 'N . G o o d m a n ’s m o d e rn n o m in a listic th e o ry is p re sen te d , in the fo rm as it w as in te rp rete d by R . E b erle. W ith in this in te rp re ta tio n th e c o n ce p t o f a n „ a to m istic universe o f in d iv id u a ls " is defin ed . I c o n sid e r this c o n ce p t to be a c e rta in specific re a liz atio n o f the q u o u tc d theses o f R u ssell’s „lo g ical ato m ism p ro g ra m m e " .

A s an im p o rta n t p o in t o f th e a to m ism p ro g ra m m e o n e m ay acc e p t the p lu ra listic thesis a b o u t th e m u ltip licity o f se p a ra te a n d a u to n o m o u s th in g s a p p e a rin g in the w o rld . R ussell w rites: „ I sh a re th e c o m m o n -se n se b e lie f th a t th e re a rc m an y se p a ra te th ings. I d o n o t reg a rd th e a p p a re n t m u ltip licity o f the w o rld as co n sistin g m erely in p h a se s a n d u n re a l d iv isio n s o f a single un d iv isib le R e a lity " 1.

T h e second sig n ifican t idea o f R u ssell's c o n c e p tio n is. as is k n o w n , the b e lie f c o n ce rn in g th e real („ e x te rn a l" ) ex isten ce o f re la tio n s, sta n d in g in o p p o s itio n to L e ib n iz 's m o n a d o lo g y a n d B ra d le y 's g lo b al m o n ism .

A s the th ird im p o rta n t thesis o f the p ro g ra m m e o n e m ay a c c o u n t the o p in io n a b o u t a relativ e sim p licity o f o n to lo g ic a l ob jects. T h ese ob jects c an h av e d ifferen t p ro p e rties, an d c an b e a r d ifferen t re la tio n s, b u t th e m a jo rity o f these re la tio n s d o n o t ta k e p a rt in e sta b lish in g o f th e id e n tity o f a n o b ject.

S tru g g lin g w ith the so called ax io m o f in te rn a l re la tio n s R ussell assu m es a sim p licity o f o b jects w h ich a re related to e ac h o th e r: „ T h e view w hich 1 reject holds, if 1 u n d e rs ta n d it rig h t, th a t th e fact th a t a n o b je c t .v h as c e rta in re la ­ tio n R to a n o b je c t y im plies c o m p le x ity in .v a n d v, i.e. it im plies so m eth in g

1 ІЗ. R u s s e l l . L o g ic a n d K n o w le d g e. A lle n a n d U n w in . L o n d o n 1971. p . 178.

in th e n a tu re o f .v a n d y in v irtu e o f w hich th ey a re related by the rela­ tio n R '4

1 will o u tlin e no w N . G o o d m a n 's c alcu lu s o f in d iv id u a ls" , as it w as p resen ted by R. E bcrle in his b o o k N o m in a listic S y ste m s 3. Every n o m in alism , in clu d in g G o o d m a n 's , p ro h ib its a d m itta n c e o f o th e r b eings th a n in d iv id u als. T h is p rin cip le needs e x p lic atio n a n d . a b o v e all. a n sw e rin g the q u e stio n w h a t an in d iv id u a l is. T h e a n sw e r to th a t q u e stio n tak es o n a d o u b le form ; firstly, the c o n stru c tio n o f a certa in fo rm al system a n d seco n d ly , n o n -lo rm a l re m a rk s c o n c e rn in g th e d ifferen ce b etw een in d iv id u a ls a n d classes, a n d th e co n seq u en ce o f re fu tin g all beings except in d iv id u als.

U n fo rtu n a te ly . G o o d m a n 's e x p la n a tio n s c o n ce rn in g the m e n tio n e d p ro b ­ lem s a re fa r fro m b ein g u n e q u iv o c al. T h e n o m in a listic d ecisio n o f n o t a cc e p tin g o th e r item s b u t in d iv id u a ls d o es n o t a u to m a tic a lly d e te rm in e w h at kind o f beings co u ld be a d m itte d as in d iv id u a ls. G o o d m a n h as n o th in g a g ain st the d ecisio n th a t the in d iv id u als w o u ld be a b str a c t as well as co n crete item s, sin g u la r a n d collective beings, ph y sical a n d p h e n o m e n al ob jects. In o ne o f his articles he w rites: „ W h a te v e r c an be c o n stru e d as a class c an be indeed c o n stru e d as an in d iv id u a l" 4. Besides, G o o d m a n c la im s th a t a n y ind iv id u al m a y be p re sen te d a n d c o n stru e d as a class o r a set. O n e c an d o th a t, fo r in sta n ce , by the id e n tific a tio n o f a p h ysical o b ject w ith th e class o f its m a c ro sc o p ic , a to m ic o r s u b -a to m ic p a rts , o r w ith a c erta in class o f events w hich set u p the h isto ry o f a given o b ject. O n e c an also c o n stru e in d iv id u a ls as classes th ro u g h the tra n s la tio n o f all s ta te m e n ts c o n ce rn in g th em in to the sta te m e n ts a b o u t u n it sets, c o n ta in in g as th e o n ly e le m en t the given o b ject („ sin g le to n s" ). E bcrle claim s th a t G o o d m a n d istin g u ish es in d iv id u a ls from classes first o f all a t the level o f a th e o ry , w hich m ean s th a t th e th e o ry o f in d iv id u a ls differs fro m the th e o ry o f classes o r sets. In d iv id u als a re d istin g u ish ed from classes n e ith e r by the fact th a t th ey a re m a d e fro m a special k in d o f m a te ria l, n o r by th e ir sp a tio -te m p o ra l c h a ra c te r. T h e re a re also no specific e p istem o lo g ical c rite ria w hich w o u ld let d ifferen tiate th em . It is n o t the case th a t in d iv id u a ls co u ld be perceived w hile classes co u ld n o t. n o r th a t classes a re on ly m en tal c o n stru c ts w hile the in d iv id u a ls a re ..g iv en ” . T h e p o ssib ility o f d ifferen tiatio n sh o u ld be sea rc h e d a t th e th e o re tic al level by the an aly sis o f fo rm al featu res.

A n im p o rta n t p rin cip le for in d iv id u a ls is the „ p rin c ip le o f sum fo rm a tio n " . T h is k in d o f o b jects c an be p u t to g e th e r, su m m e d u p , a g g re g ated m a k in g u p as

2 B. R u s s e l l . S o m e E x p la n a tio n s in R e p ly to M r . B r a d le y , M in d 1910. p . 373 -374.

3 R. A . E b e r l e . N o m in a lis tic S y s te m s . R eid el. D o r d r e c h t 1970.

4 N . G o o d m a n . A W o r ld o f In d ivid u a ls, [in:] T h e P r o b le m s o f (Jniversals. A S y m p o s iu m , N o t r e D a m e U n iv . P re s s . N o tre D a m e . In d .. 1956: re p rin te d in: P h ilo s o p h y o f M a th e m a tic s , ed . P. B e n a c e rra ľ . H . P u tn a m . P rin tic c H a ll. 1964.

A t o m i s t i c U n i v e r s e s o ľ I n d i v i d u a l s 99 a resu lt the o th e r in d iv id u a ls w hich a rc c e rta in w holes. T h e o p e ra tio n o f su m m in g up in d iv id u a ls h as n o p hysical c h a ra c te r, a n d the resu lt o f it d o e s no t need to be a w h o le p re serv in g s p a tio -te m p o ra l c o n tin u ity . It c a n be a n o b ject w ith p a rts d isp erse d in sp ace o r ex istin g in d ifferen t p e rio d s o f tim e. B ecause o f this liberal a n d a lm o st a b s tr a c t c h a ra c te r o f the p rin c ip le o f in d iv id u als s u m m a tio n it becom es q u ite alik e the set-th e o re tic o p e ra tio n o f u n io n fo rm a tio n : „ it seem s th e p rin cip le o f sum fo rm a tio n is q u ite a n a lo g o u s to the ' set-th e o re tic p rin cip le g o v e rn in g u n io n s o f sin g leto n s. F o r e x am p le, th e set o f all red ob jects (th e u n io n o f all sin g le to n s o f red o b jects) h as fo r its c o u n te rp a rt a c e rta in in d iv id u a l, n am ely th e sum o f all red o b je c ts” 5. T h e re fo re , the q u e stio n arises w h a t o th e r b asis c an th e re be fo r th e d iffe re n tia tio n betw een classes a n d in d iv id u als?

A s the m a in c rite rio n d istin g u ish in g in d iv id u a ls fro m classes G o o d m a n su ggests th e „ p rin c ip le o f in d iv id u a tio n " . A s is k n o w n , th e p rin c ip le o f in d iv id u a tio n fo r sets is th e ex te n sio n a lity principle:

A = В <— > ř ( x e A <— ► л- є В ).

T h a t p rin c ip le in th e a b o v e fo rm d o es n o t ap p ly to these o b jects fro m th e d o m a in o f set th e o ry w hich d o n o t have m em b ers (w itli the e x ce p tio n o f th e em p ty set). T o e n su re th e u n iv ersal v a lid ity o f the e x te n sio n a lity p o s tu la te in th e d o m a in o f set-th e o re tica l o b jects o n e can e ith e r assu m e th a t all o b jects have e lem en ts, o r in tro d u c e o n e-p lace p re d ic a te „is a se t" an d relativ ise w ith th e h elp o f it th e p rin c ip le o f e x ten sio n ality .

C o n se q u e n tly , w e id en tify classes an d sets th ro u g h p o in tin g o u t th e co rre la te s o f the re la tio n є w ith re sp ec t to th e given set (class); w h en tw o sets (classes) h a v e the sam e elem en ts, th en th ey a re id en tical. S h o u ld th e sim ilar ru le be a p p lie d in th e c alcu lu s o f in d iv id u als, given th a t o n e in tro d u c e s th e re la tio n „ b e in g a p a r t" in place o f th e re la tio n „ b e in g a m e m b e r" , a n d p o s tu la te s th a t tw o in d iv id u a ls a re id en tical ju s t in case w hen th ey h a v e th e sam e p a rts? B ut sh o u ld o n e ta k e h e re in to c o n sid e ra tio n all a c tu a l a n d po ssib le p a rts o f a given in d iv id u a l o b ject? It seem s th a t in th e c alcu lu s o f in d iv id u a ls such a c rite rio n w o u ld be to o s tro n g , a n d th a t it is n o t n ecessary to p o in t o u t all p a rts to id e n tify tw o in d iv id u a ls w ith each o th e r. G o o d m a n a n d E berle claim th a t fo r in d iv id u a l w h o les it is sufficien t to set fo rth a c o n d itio n re q u irin g th a t o b jects w h ich h av e th e sam e „ u ltim a te c o n s titu e n ts " a re id en tical.

A n o th e r im p o r ta n t d ifferen ce betw een set th e o ry a n d the c alcu lu s o f in d iv id u a ls is as follow s: if w e have a given o b je c t A , th e n th e tra n s itio n s A {A} - {{A}} - ... in a c c o rd a n c e w ith th e set-th c o re tica l p rin c ip le o f sets fo rm a tio n , i.e. fro m elem en ts to sets, sets o f sets e tc., lead to o b jects

n o n -id e n tic a l w ith А: А Ф {А} Φ {{A}} etc. In th is m a n n e r, sta rtin g from o n e o b ject we c an o b ta in w hole in fin ite w ealth o f o b jects. T h is is a p p lie d in the re c o n stru c tio n o f n a tu ra l n u m b e rs as sets h iera rc h ic ally fo u n d e d o n the e m p ty set. In th e calcu lu s o f in d iv id u a ls su ch a situ a tio n is ex clu d ed . F o r ex am p le, fro m a p a ir o f o b jects A a n d В o n e c an b u ild u p o n ly o n e new o b ject . i.e. th e w hole c o n ta in in g as its im m e d ia te p ro p e r p a rts o n ly th ese tw o in d iv id u als A a n d B.

In c o n n e c tio n w ith this m a tte r E b e rle w rites: „ th e p rin cip le o f exten- sio n ality d ifferen tiate s classes w hich have d ifferen t « im m e d ia te c o n stitu en ts» relativ e to m e m b e rsh ip -c h ain s; th a t is to say, classes w hich h av e d ifferen t m em b ers. In d iv id u als w hich h av e th e sam e c o n te n t are to c o u n t as id en tical [...] a n d h a v in g th e sam e c o n te n t is h ere ta k e n to m ean « h a v in g the sam e u ltim a te c o n stitu e n ts » ” 6. W h en we lo o k a t the q u e stio n o f „ c o n s titu e n ts ” o f sets from a m o re g en eral p o in t o f view - n o t on ly re ferin g to th e re la tio n e itself, b u t also to th e a n ce stra l re la tio n fro m th e m e m b e rsh ip rela tio n - th en it tu rn s o u t th a t the ex te n sio n a lity p rin c ip le d ifferen tiate s sets w hich h av e d ifferen t „ im m e d ia te c o n stitu e n ts ” w ith resp ect to th a t a n ce stra l re la tio n . O n th e o th e r h a n d , w hen in d iv id u a ls a re c o n ce rn e d , o n e m a y a ck n o w le d g e as id en tical th o se o f th em w hich h av e th e sam e „ c o n te n t” . H a b in g th e sam e c o n te n t m ean s h ere h av in g th e sam e ..u ltim a te c o n stitu e n ts ” . W hich kin d o f c o n stitu e n ts w o u ld be reco g n ized as u ltim a te d e p en d s o n w h a t re la tio n s th ey b e a r to each o th e r.

T h e essential task w hich is assig n ed to the c alcu lu s o f in d iv id u a ls is, th e re fo re , th e ex p licatio n o f the c o n c e p t o f a fu n d a m e n ta l rela tio n betw een in d iv id u als. F o r in stan ce, th e m e re o lo g y o f S. L eśniew ski is in te rp re te d as a th e o ry o f th e re la tio n ..b ein g a p a r t ” . G o o d m a n in tro d u c e s a m o re g en eral c o n ce p t o f a „ g e n e ra tin g re la tio n ” w hich is to in clu d e b o th the set-th e o re tic an ce stra l re la tio n fro m th e re la tio n e a n d th e re la tio n ..b ein g a p ro p e r p a r t ” . T h u s , a t th a t stag e o f th e d ev elo p m en t o f his th e o ry he w a n ts to c o v er m ereo lo g ical as well as se t-th e o re tic c o n c e p ts 7. In th e la te r fo rm u la tio n G o o d m a n 's calcu lu s o f in d iv id u a ls is b a se d o n the p rim itiv e te rm to overlap. i.e. o n the c o n ce p t o f a p a rtia l c o v erin g o f o n e in d iv id u a l by a n o th e r. T h e p rin c ip a l p o s tu la te o f this v ersio n o f the c alcu lu s lo o k s as follow s:

Л* OV f 1Г o v г -> и* ο ν .Y Л И· ο ν у)

w h ere ov is a sy m b o l fo r th e re la tio n o f a p a rtia l c o v erin g o f in d iv id u a ls o r, in o th e r w o rd s, o f h a v in g a co m m o n p a r t 8. T h e re a so n , w h y G o o d m a n c h o o se s as a p rim itiv e th e sy m m etric p re d ic a te to o verlap. a n d n o t th e b e tte r k n o w n

6 Ib id .. p . 26.

' G o o d m a n . A W o r ld o f In d ivid u a ls...,

A t o m i s t i c U n i v e r s e s o f I n d i v i d u a l s 101 p re d ic a te „is a p a r t" , is th e g re a te r fo rm al sim p licity o f th e fo rm er. T h e relatio n „ b ein g a p a r t" in this v ersio n o f th e c alcu lu s is d efin ed by m e a n s o f the rela tio n o f o v erlap p in g :

,v is a p a rt o f y z ( z ov ,v -► : o v y ).

In w o rd s: th e in d iv id u a l x is a p a rt o f th e in d iv id u a l y , if a n d o n ly if. w hen every in d iv id u a l h a v in g a c o m m o n p a r t w ith x h a s a lso a co m m o n p a rt w ith y .

A c co rd in g to E b erle, th e re a re th re e elem en ts m o st im p o r ta n t fo r the c o n ce p t o f in d iv id u a l in tro d u c e d by G o o d m a n : th e c o n c e p t o f the „ g e n e ra tin g re la tio n ” , the p rin c ip le o f s u m m a tio n , a n d the p rin c ip le o f in d iv id u a tio n . F ro m a fo rm al p o in t o f view th e p a rt-w h o le re la tio n sh o u ld fulfil c o n d itio n s p u t fo rw a rd by th e fo llo w in g d efin itio n :

Dcf. R is a p a rt-w h o le re la tio n , if a n d o n ly if th e fo llo w in g c o n d itio n s are satisfied 4:

1. R is a p a rtia l o rd e rin g .

2. 0 (th e e m p ty set) is n o t a m e m b e r o f the field o f R. 3. T h e re exists a set A m e e tin g th e fo llo w in g re q u ire m e n ts:

a) fo r every n o n -e m p ty su b se t S o f A , a n d fo r all .v in A . i f ,v b e ars R to s u p rS , th en x is in S:

b) the field o f R is eq u al to th e set o f all item s x such th a t fo r som e n o n -e m p ty su b se t S o f A , x = s u p RS;

c) A is infinite.

L et us n o te th a t in th e q u o u te d w o rk E b e rle uses th e se t-th e o re tic a p p a ra tu s on th e m e ta -la n g u a g e level d e sc rib in g th e n o m in a listic c alcu lu s o f in d iv id u als. T h u s , he d escrib es n o m in a listic sy stem s in n o n -n o m in a listic lan g u ag e. T h e c o n d itio n 2 is, a c c o rd in g to h im , e q u iv a le n t to th e c laim th a t fo r every n o n -e m p ty su b se t S o f th e set A th e s u p RS exists, a n d all d e sc rip tio n s fo rm u la te d by m e a n s o f th e p a rt-w h o le re la tio n th e o ry te rm s have d e fin ite c h a ra c te r. T h e p o in t 3, specifies re q u ire m e n ts w hich s h o u ld be fulfilled by the d istin g u ish ed su b se t A o f th e d o m a in o f th e re la tio n R. T h is su b set is in te rp rete d as a set o f R -a to m s (a to m s w ith respect to th e p a rt-w h o le re la tio n ), i.e. a set o f m in im al elem en ts w ith re sp ec t to R.

N o w , let m e u n d e rlin e th a t in th e d e fin itio n o f the p a rt-w h o le re la tio n o ne em p lo y s a specific idea o f s u m m a tio n o f elem en ts. T h is id ea, w hich is a re a liz atio n o f th e p rin c ip le o f s u m m a tio n fo r in d iv id u a ls, uses th e c o n c e p t o f a su p re m u m o f a set w ith re sp ec t to th e re la tio n R (R is a p a rtia l o rd e rin g ). T h e c o n c e p t o f a su p re m u m c a n be a p p lie d b o th to th e finite a n d in finite su b sets o f th e field o f th e p a rt-w h o le re la tio n , e.g. to th e w h o le d istin g u ish ed set o f a to m s A . T h e in tr o d u c tio n o f a g en eralized o p e ra tio n o f s u m m a tio n fo r

102 W i t o l d S t r a w i ń s k i

in d iv id u a ls (s u p RS) h as a d e lin ite g o al in G o o d m a n -E b c rle 's th eo ry . N am ely , th ey w a n t to stay n e u tra l w ith resp ect to th e p ro b le m o f lin iten ess o r in fin iten ess o f the re la tio n p a rt-w h o le d o m a in . T h a t p ro b le m s h o u ld n 't be d ecided a t th e level o f in tr o d u c in g o n e o r a n o th e r s u m m a tio n o p e ra tio n . T h e p ro b le m o f th e po ssib ility o f re c o n stru c tin g th e w h o le u n iv erse o f in d iv id u als a s the sum o f all its c o n stitu e n ts also co m es in to p lay here. In this c o n n ec tio n . E b e rlc w rites: ..C an we be assu re d th a t th e re a rc in deed « u ltim a te c o n stitu en ts» in th e w h o le field o f p h ysical o b jects relativ e to this relatio n (betw een a p h ysical p a rt a n d a p h ysical w hole)? S u p p o se th a t p h ysical ob jects tu rn o u t to be in fin itely divisible; s h o u ld w e th e n be p re p a re d to a d m it th a t physical objects a rc n o t in d iv id u als? G o o d m a n d o e s n o t p reclu d e, o n prin cip le, th a t th e re m a y be a n in finite n u m b e r o f least ph y sical c o n s titu e n ts ” 10. T h e n o m in a listic s ta n d p o in t sh o u ld be fo rm u la te d in su ch a g en eral w ay th a t a d isag ree m e n t betw een it a n d th e c o n te n t o f a p h ysical th e o ry w o u ld n o t be possible.

T h e a b o v e c o n sid e ra tio n is to ju s tify th e c o n d itio n 3c w hich states th a t the d istin g u ish ed set A . rep re sen tin g the class o f a to m s , is infinite. In a d d itio n to th is E berle ju stifie s the a ss u m p tio n a b o u t th e in fin iten ess o f th e set o f a to m s in th e follow ing w ay: „ th e a to m s in q u e stio n a re c o n ce rn e d as possible, ra th e r th a n as a c tu a l ob jects. A n d it d o e s n o t seem c o u n te r-in tu itiv e to re q u ire th a t th e re shall be in fin itely m a n y p o ssib le e n titie s. O n the o th e r h a n d , since un iv erses o f in d iv id u als a re co n ceiv ed as c o m p risin g a c tu a l in d iv id u a ls, we shall re fra in fro m im p o sin g a c o n d itio n o n such u niverses w hich w o u ld im ply th a t every u n iv erse o f a c tu a l th in g s is i n f i n ite " " . Ι Π u n d e rs ta n d this in te n tio n c o rre c tly , o n e sh o u ld defin e the p a rt-w h o le re la tio n in th e m o st g en eral w ay a n d e v en tu ally lim it this g e n erality la te r w hile ap p ly in g th a t rela tio n to specify th e c o n c e p t o f an a c tu a l u n iv erse o f in d iv id u a ls.

T h e p o in t 3 a lso claim s th a t th e set A co n sists o f d iscrete a to m s, i.e. R -m in im al elem en ts h av in g n o c o m m o n p a rts. A n a to m c an be n e ith e r a p a rt o f a n o th e r a to m , n o r a p a rt o f a sum o f tw o o r m o re a to m s. B eside ato m s, in d iv id u a ls a re w holes g en erated fro m a to m s w ith th e help o f th e s u m m a tio n o p e ra tio n .

L et us p ro ceed n o w to th e p rin c ip le o f in d iv id u a tio n w hich is a n o th e r fa c to r c o n stitu tin g , a c c o rd in g to G o o d m a n a n d E b erle. the c o n ce p t o f an in d iv id u a l. T h e p rin cip le o f in d iv id u a tio n to g e th e r w ith the p a rt-w h o le relatio n an d th e s u m m a tio n p rin c ip le c h a ra c te riz e o b jects w hich we w a n t to re ck o n a m o n g in d iv id u als. T h e y d escrib e w h a t c erta in co llectives o f in d iv id u als are. r a th e r th a n w h a t a single in d iv id u a l is. A c c o rd in g to E b erle, betw een such c o llectives a special a tte n tio n d eserv e so -called a to m is tic u n iv erses o f in ­

10 /Л/ŕ/., p . 30. " Ib id .. p . 39.

A t o m i s t i c U n i v e r s e s o ľ I n d i v i d u a l s 1 0 3 div id u als. H e th in k s th a t a d e fin itio n o f such u niverses c o n stitu te a p rin c ip a l e x p licatio n o f th e co n ce p t o f a n in d iv id u a l. In g en eral, a n u n iv erse o f in d iv id u a ls is a su b set o f th e Held o f th e p a rt-w h o le re la tio n in w hich the a p p ro p ria te c o n d itio n s c o n c e rn in g s u m m a tio n o f o b jects a n d th e ir in d iv id u a ­ tio n a re satisfied. A n in d iv id u al is c h a ra c te riz e d in a ro u n d - a b o u t w ay as the e lem en t o f a c e rta in u niverse o f in d iv id u als.

E b erlc tries d ifferen t a lte rn a tiv e v ersio n s o f th e p rin c ip le o f in d iv id u a tio n , a n d finally assu m es th a t the task o f d istin g u ish in g in th e field o f rela tio n R the universes o f in d iv id u a ls is best fulfilled by the fo llo w in g p rinciple: F o r e v e ry .v a n d y b e lo n g in g lo U . .v = y iff f o r e v e ry r , i f г is R -lc a st in U , th e n r R .v ilľ r R y; U sy m bolize h ere a c e rta in 'u n iv erse o f in d iv id u a ls 12.

In o th e r w o rd s, in d iv id u a ls b elo n g in g to U a re id en tical ju st in case, w hen they h av e th e sam e a to m s in re la tio n to U as th e ir p a rts. Such a fo rm u la tio n o f the p rin cip le o f in d iv id u a tio n im p o ses c e rta in re stric tio n s o n th e u n iv erses o f in d iv id u a ls w hich arc n o t im p o sed by th e o th e r, m o re liberal fo rm u la tio n s. F o r in sta n ce , m o re g en eral p rin c ip le o f in d iv id u a tio n , w hich id en tifies in d iv id u als w hen th ey have th e sam e a to m s w ith respect to th e w h o le field o f the rela tio n R. d o es n o t im p o se re stric tio n s 011 th e u n iv erse U; every su b set o f th e field o f the rela tio n R co u ld th e n be a cc e p te d as a n u n iv erse o f in d iv id u a ls U . W h a t reaso n s a re th e re fo r c h o o sin g such a p rin c ip le o f in d iv iduation'? T h is p ro b le m boils d o w n to th e q u e stio n o f th e ro le w hich is p layed by th e un iv erses o f in d iv id u als w ith in the field o f th e p a rt-w h o le re la tio n . L et us rem in d th a t E berlc in te rp re ts th e field o f the p a rt-w h o le re la tio n as th e set o f all ob jects w hich c o u ld be p a rts o r w holes. H e w rites: „ B y c o n tr a s t th e e le m en ts o f a p a rtic u la r u n iverse o f in d iv id u a ls a re re g a rd e d as th o se in d iv id u a ls w hich h a p p en to be actu aliz e d in th a t u n iv erse. T o p ro v id e a suggestive exam ple: su p p o se th a t we conceive o f a n in fin ite class o f item s all o f w hich satisfy a p h y sicist's d e sc rip tio n o f a n a to m . L et a p a rt-w h o le rela tio n be co n ceived betw een these a to m s a n d all p o ssib le eo m p o sities o f the a to m s. A n y selection o f these p o ssib le a to m s o r eo m p o sities m ig h t be a ctu aliz e d in so m e u n iv erse w hich is a « u n iv erse o f in d iv id u a ls» if fo r ev ery c o m p o site o b ject w hich is actu aliz e d in it a sufficien t v ariety o f p a rts a rc also a ctu aliz e d , so th a t d ifferen t actu al e o m p o sities have in the u n iv erse d ifferen t a c tu a l p a rts . It is logically po ssib le th a t the sim plest ph y sical o b jects w hich h a p p e n to be actu aliz e d in such a u niverse a rc m olecules, w hile all p ro p e r p a r ts o f m o lecu les rem ain u n a ctu aliz e d p o ssib le s” 13.

12 Ib id ., p . 38. 13 Ib id .. p . 3 7 -3 9 .

T h e selected p rin c ip le o f in d iv id u a tio n su g g ests th a t E berle w an ts to re strict the v ariety o f po ssib le item s. In d iv id u als a re to be a ctu aliz e d objects w hich co n sist o f a ctu aliz e d p a rts . A c co rd in g to the p rin c ip le o f in d iv id u a tio n , d ifferen t a c tu a l in d iv id u a ls a rc c o m p o se d in th e last re so rt o f d ifferen t a ctu al a to m s. B eing a c tu a l is co nceived h ere in a specific m a n n e r as a n a tta c h m e n t to c erta in d istin g u ish ed u niverse o f in d iv id u a ls w hich is a su b set o f th e p art-w h o le re la tio n field. A t this p o in t o n e c an raise th e q u e stio n , w h e th e r a rb itra ry g ro u p s o f in d iv id u a ls from the u n iverse c an be p u t to g e th e r by the s u m m a tio n o p e ra tio n , g iving as th e resu lt in each case new in d iv id u a l w holes?

G o o d m a n — as is k n o w n — h as an sw ered this q u e stio n q u ite positively: ..A lth o u g h n o t ev ery in d iv id u a l h as a n e g ate a n d n o t every tw o in d iv id u als have a p ro d u c t, every tw o in d iv id u a ls do h av e a sum . B earin g in m in d th a t o n ly in d iv id u a ls a re values o f o u r v a ria b le s, w e c an a ffirm the u n c o n d itio n a l statem e n t:

S '

Ÿ

as a p o s tu la te o r th e o re m o f o u r c a lc u lu s " 14. T h e n e g atio n a n d p ro d u c t G o o d m a n w rites a b o u t, as well as th e sy m b o l + d e n o tin g th e s u m m a tio n o p e ra tio n , a re term s defin ed in the v ersio n o f the calcu lu s o f in d iv id u als p re sen te d in th e q u o u tc d w o rk .

G o o d m a n w-as o fte n criticized fo r a d o p tin g the a b o v e p rin cip le o f s u m m a tio n , a n d in th is case E b erle jo in s his critics: „w e w o u ld d e p a r t from G o o d m a n 's co n ce p tio n by a d m ittin g o th e r re la tio n s w hich q u alify in tu itiv ely as p a rt-w h o le re la tio n s b u t fail to g e n erate a c tu a l su m s o f a rb itr a r y in ­ d iv id u a ls " 15. E b erle im p o ses th e fo llo w in g w e a k e r c o n d itio n 011 th e o p e ra tio n o f su m m in g th e elem en ts o f th e u n iv erse o f in d iv id u als:

F o r e v e ry x w h ich b e lo n g s lo U , th e re e x is ts a set S c o n s is tin g o f e le m e n ts R -m in im a l in U . s u ch t h a t л = supR S.

In o th e r w o rd s, every in d iv id u a l fro m th e u niverse o f in d iv id u a ls U is a sum o f e lem en ts w hich a re a to m s w ith resp ect to the rela tio n R in U. If any o b ject is a n in d iv id u a l b e lo n g in g to th e u n iv erse U . th e n it m u st have a d e c o m p o sitio n in to a to m ic p a rts w ith in U . H o w ev e r, such a c o n d itio n does n o t assu m e th a t every sum o f a to m s o r a n y o th e r in d iv id u a ls b e lo n g in g to U is a g ain a n elem en t o f U . i.e. a n in d iv id u a l in th is universe. T h a t fo rm u la tio n stresses an aly tic, ra th e r th a n s y n th etic fu n c tio n o f th e in d iv id u a l s u m m a tio n o p e ra tio n .

14 G o o d m a n . T h e S tr u c tu r e o f A p p e a r a n c e ..., p. 30». 15 E b e r l e . N o m in a lis tic S y s te m s .... p . 41.

A t o m i s t i c U n i v e r s e s o f I n d i v i d u a l s 1 0 5 Let us c o m m e n t here on o n e m o re m a tte r. In B. R u ssell's logical a to m ism p ro g ra m m e a n essen tial role plays th e q u o u tc d c o n v ic tio n th a t ..th e w o rld d o es n o t co n sist m erely in p h ases a n d u n re a l d iv isio n s o ľ a single u n d iv isib le R ea lity " . T h u s. Eberle is in a b e tte r a g re em en t w ith R u sse ll's p ro g r a m m e th a n G o o d m a n , since he d o es n o t a ssu m e th a t every po ssib le sum o f in d iv id u a ls is an a c tu a l in d iv id u al. In this w ay he re stra in s h im se lf fro m th e a ss u m p tio n th a t th e w h o le w o rld is o n e m ax im al, g lo b al in d iv id u a l, a n d th a t p o ssib ly all p ro p e rties a n d e x tern al rela tio n s o f o b jects in th e w o rld a re red u cib le to the p ro p e rties a n d in te rn a l re la tio n s o f th e w o rld itself.

A fte r selecting the a p p r o p r i a t e prin cip les o f in d iv id u a tio n a n d s u m m a tio n E berle d efines the cen tra l c o n ce p t o f his re c o n str u c tio n o f G o o d m a n 's calcu lu s a n ..a to m istic u niverse o f in d iv id u a ls " . T h e b o th a b o v e -m e n tio n e d prin cip les a ssu m e th a t in every universe o f in d iv id u a ls exist a to m s; hence, th e e x p ressio n ..u n iv erse o f in d iv id u a ls" is su p p lem en te d w ith the ad je ctiv e ..a to m is tic " . Since, a cc o rd in g to E b erle. the ch o sen p rin cip le o f s u m m a tio n im plies the prin cip le o f in d iv id u a tio n , in th e d e fin itio n o f a n ..a to m is tic u n iv erse o f in d iv id u a ls" o n e can ta k e in to a c c o u n t on ly the fo rm er.

Dcf. U is a n a to m is tic u n iverse o f in d iv id u a ls fo r R iff 1 ) R is a p a rt-w h o le re la tio n .

2) U is in clu d ed in the field o f R.

3) fo r every л- in U . th e re exists a set S su ch th a t all m em b ers o f S a re R -least in U . a n d .v = s u p RS 10.

A fte r p re sen tin g the a b o v e o u tlin e o f G o o d m a n -E b e rle 's th e o ry som e c o m m e n ts suggest them selves. A s 1 have w ritten a t th e b e g in n in g o f th is article, th a t c o n c e p tio n seem s to be a c e rta in re a liz atio n o f selected theses o f R u ssell’s logical a to m ism p ro g ra m m e . It assu m es th a t th e re exist m an y s e p a ra te a n d in d e p e n d en t in d iv id u a ls, a to m s a n d w holes, w hile re stra in in g itse lf fro m co n clu d in g the m a tte r o f existence o f a m a x im a l glo b al in d iv id u a l, iden tical p e rh a p s w ith th e w h o le reality . T h u s, it re p re sen t a s ta n d p o in t o f p lu ra lism . A t the sam e tim e a fu n d a m e n ta l ro le plays h ere th e ..g e n e ra tin g " p a rt-w h o le relatio n w hich is assu m ed in d e p e n d e n tly from in d iv id u a l objects. O n e d o c s n o t a tte m p t to red u ce th a t re la tio n to in te rn a l p ro p e rtie s o f in d iv id u a ls b u t the o th e r w ay ro u n d , th e in tro d u c tio n o f it is c o n stitu tiv e o f th e c o n c e p t o f an ind iv id u al. T h is a p p ro a c h is in a c c o rd a n c e w ith R u sse ll's s ta n d p o in t rejecting in te rn a l rela tio n s ax io m an d p o s tu la tin g e x te rn a l re la tio n s in d e p e n d e n tly from o b je c ts' p ro p e rties. F in ally , th e re is c e rta in k in d o f sim p licity in d o m a in s w hich qu alify as a to m is tic u niverses o f in d iv id u a ls, viz. th e ir m em b ers c an be uniq u ely p re se n te d as relativ ely sim ple w holes c o m p o se d o f e le m en ta ry c o n stitu en ts, a n d such a c o m p o sitio n m u st allo w fo r th e ir c o m p le te id e n ­ tificatio n . T h is a sp ect o f sim plicity c o u ld be e x p ressed by th e s ta te m e n t th a t

a to m istic u niverses o f in d iv id u a ls have th e s tru c tu re p e rta in in g to the relatio n ..sim p le r th a n " , w hich c a n be in te re stin g in the m e a n in g an aly sis o f th e c o n ce p t o f sim p lic ity 17.

T h e fu n d a m e n ta l n o m in a listic claim p o s tu la te s re fu ta tio n o f a b stra c t e n titie s, in p a rtic u la r existence o f classes. G o o d m a n w rites: „ W h a te v e r we are w illing to recognize as an e n tity a t all m a y be c o n stru e d as a n in d iv id u a l [...] we c an c o n stru e a n y th in g as a n in d iv id u a l" 1“. It is ra th e r sem a n tic th a n o n to lo g ic a l a p p ro a c h : th e n o m in a listic thesis co u ld be fo rm u la te d in E b crle's c o n c e p tu a l fram e w o rk as th e sta te m e n t th a t every d o m a in o f ob jects c a n be in te rp re te d as a c e rta in a to m istic u n iverse o f in d iv id u als. L et us co n sid e r the so u n d n ess o f th a t statem e n t.

T h in g s an d m a te ria l o b jects d o n o t seem to fulfil th e n o m in a listic prin cip le o f in d iv id u a tio n . F ro m th e sam e th in g s -p a rts wc c an c o n stru c t d ifferen t w holes in d ifferen t m o m e n ts o f tim e; a little ch ild d o es th a t w hile p la y in g w ith b u ild in g blocks. M a te ria l ob jects are n o t in d iv id u a ls in G o d d m a n -E b e rle ’s sense, in o rd e r to a tta in this s ta tu s the tim e d im e n sio n sh o u ld be ta k e n in to acc o u n t. T h u s, lo r in stan ce, a ta b le is n o t a n in d iv id u a l b u t th e ta b le -h o u r, ta b ­ le-m in u te. a n d tab le-seco n d are. Sixty la b le -m in u te s su m m ed up to g e th e r give as a result a n in d iv id u a l w hich is o n e ta b le h o u r. T h e q u e stio n a rise, w h at w o u ld in th is case a to m s be. T h e sam e c o m m o n -se n se tab le, ta k e n in to c o n sid e ra tio n fo r th e p e rio d o f o n e m in u te y e ste rd a y a n d to d a y , c o n sists o f tw o co m p letely d ifferen t n o m in a listic , in d iv id u a ls - tw o s e p a ra te la b ­ le-m inutes. T h e y a re o n ly co n n ec te d by th e o th e r in te rm e d ia te ta b le -m in u tc s w hich ad jo in to each o th e r o r succeed o n e a fte r a n o th e r. T h e id e n tity o f tw o ta b le -m in u tc s s e p a ra te d in tim e d o cs n o t c o m e in to p lay , a lth o u g h th ey c an be p a rts o f th e sam e tab le-w eek o r ta b lc -m o n th . since th e y co n sist o f co m p letely d ifferen t le t's say - p artic lc -sec o n d s (th e m in im al d is tin g u ish ed space-tim e regions). T h e only k in d o f id en tity w hich c an o c cu r b etw een tw o in d iv id u als s e p a ra te d in sp ace o r tim e is the g e n id e n tily . w hich h a s n o t m u ch to d o w ith th e id en tity in a n o m in a listic sense. It is also n o t difficu lt to sec th a t lan g u ag e e x p re ssio n s d o n o t fulfil th e n o m in a listic p rin c ip le o f in d iv id u a tio n eith er; from th e sam e signs we u su ally m a y b u ilt up d ifferen t e x p ressio n s.

O ne c a n o b v io u sly c o n stru c t d o m a in s w hich w o u ld be a to m istic universes o f in d iv id u als; o n e co u ld also d o th a t w ith th e h e lp o f set-th e o re tic c o n cep ts. F o r ex am p le, the p o w er set o f so m e n o n -e m p ty set Z (th e em p ty set ex clu d ed ) w ith the o p e ra tio n o f u n io n a n d the in clu sio n re la tio n , i.e. th e relatio n s tru c tu re < 2 z, и , о . is a n a to m is tic u n iverse o f in d iv id u als. T h e a to m s here a re th e u n it sets fo rm ed fro m th e e le m en ts o f the set Z . If the em p ty set w as

17 S t r a w i n s k i . / ! F o rm a i D e fin itio n o f th e C o n cep t o f S im p lic ity . [in:] P o lish E s s a y s in the

P h ilo so p h y o f th e N a tu r a l S c ie n c e , ed . W . K ra je w s k i. R e id e l, D o r d r e c h t і 982 . p. 195 197.

A t o m i s t i c U n i v e r s e s o ľ I n d i v i d u a l s 1 0 7 in clu d ed , th en it w o u ld have b een the on ly a to m in th is u n iv erse w hich, how ever, w o u ld n o t h av e been a b le to g e n erate o th e r elem en ts. N ev erth eless, the c o n stru c tio n o ľ such d o m a in s seem s to be a ra th e r w eak ju s tific a tio n of' G o o d m a n 's c o n v ic tio n th a t „w e c an c o n stru e a n y th in g as an in d iv id u a l" .

It a p p e a rs th a t we d o n o t m eet a to m is tic u n iv erses o f in d iv id u a ls to o often . T h in g s a n d m a te ria l ob jects d o n o t seem to be in d iv id u a ls in this sense. It is ra th e r th e e n titie s o f cven tistic o n to lo g y , co n siste d o f sp a tio -te m p o ra l events, w hich satisfy the c o n d itio n s re q u ire d fro m in d iv id u a ls by G o o d m a n an d E berle.

W a rs a w U n iv e rs ity P o la n d

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A T O M I S T Y C Z N E U N I W E R S A I N D Y W I D U Ó W

A u t o r p rz e d s ta w ia k ry ty c z n y a n a liz ę p e w n y c h idei n o m in a lis ly c z n e j te o rii N . G o o d m a n a w in te rp r e ta c ji R . E b e rle g o . T e o r ia la . czyli „ r a c h u n e k i n d y w id u ó w ” , o p a r ta je s t n a trz e c h p o d s ta w o w y c h p o ję c ia c h : z a s a d z ie s u m o w a n ia , relacji ..c z ę ś ć -c a ło ś ć ” i z a s a d z ie in d y w id u a liz a c ji. O w e p o ję c ia w z ię te ra z e m c h a r a k te r y z u ją p rz e d m io ty , k tó re c h c e m y za lic zy ć d o in d y w id u ó w , p rzy czy m c h a r a k te r y s ty k a t a o k re ś la raczej cz y m je s t p ew ie n z e s p ó l in d y w id u ó w , n iż to cz y m je s t p o je d y n c z e in d y w id u u m .

W e d łu g E b e rle g o n a specjalni} u w a g ę z a s łu g u ją tzw . a to m is ty c z n e u n iw e rs a in d y w id u ó w . D e fin ic ja ta k ic h u n iw e rs ó w m a s ta n o w ić w ła ś n ie o k re ś le n ie te g o , cz y m są in d y w id u a . W o g ó ln o ś c i ..a to m is ty c z n e u n iw e rs u m in d y w id u ó w ” to p o d z b ió r p o la re la cji „ c z ę ś ć -c a ło ś ć " . w k tó ry m są sp e łn io n e o d p o w ie d n ie w 'a ru n k i d o ty c z ą c e s u m o w a n ia i in d y w id u a liz a c ji p rz e d m io tó w . P o ję cie to a u t o r w'iąże /. a to m iz m e m lo g iczn y m B. R u s s e lla o r a z ro z w a ż a j e g o m o żliw e z a s to s o w a n ia .