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
FOLIA PHILOSOPH ICA 9, 1993
W itold Straw iński
A T O M IS T IC U N IV E R S E S O F IN D IV ID U A L S
Al the beginning o f this p aper I would like to refer to certain selected theses o f B. Russell's philosophical conception. These theses can be presented as a p a rt o f the general „logical atom ism p ro g ram m e". F u rth e r on. an outline and analysis ot N. G o o d m a n 's m odern nom inalistic th eo ry is presented, in the form as it was interpreted by R. Eberle. W ithin this in terp re tatio n the concept o f an „atom istic universe o f individuals" is defined. I consider this concept to be a certain specific realization o f the q u o u ted theses o f R ussell's „logical atom ism p ro g ram m e” .
As an im p o rtan t point o f the atom ism p rogram m e one m ay accept the pluralistic thesis ab o u t the m ultiplicity o f separate an d au to n o m o u s things appearin g in the w orld. Russell writes: „I share the com m on-sense belief that there are m any separate things. I do n o t regard the a p p a re n t m ultiplicity o f the w orld as consisting m erely in phases an d unreal divisions o f a single undivisible R eality” 1.
The second significant idea o f R ussell's conception is, as is know n, the belief concerning the real („ex tern a l” ) existence o f relations, stan d in g in oppo sitio n to Leibniz’s m onadology and B radley’s global m onism .
As the third im p o rtan t thesis o f the p rogram m e one m ay account the opinion ab o u t a relative sim plicity o f ontological objects. These objects can have different properties, and can bear different relations, but the m ajority o f these relations d o no t take part in establishing o f the identity o f an object.
Struggling with the so called axiom o f internal relations Russell assum es a sim plicity o f objects which are related to each other: ,,T he view which I reject holds, if I understand it right, th a t the fact th at an object ,v has certain rela-tion R to an object у implies com plexity in x an d j \ i.e. it implies som ething
in the n atu re o f ,v and у in virtue o f which they are related by the rela-tion R ” 2.
1 will outline now N. G o o d m a n ’s „calculus o f individuals", as it was presented by R. Eberle in his book Nom inalistic S ystem s3. Every nom inalism , including G o o d m a n ’s, proh ib its adm ittan ce o f o th er beings th a n individuals. T his principle needs explication an d . above all. answ ering the question w hat an individual is. T he answ er to th a t question takes on a double form; firstly, the constru ctio n o f a certain form al system and secondly, non-form al rem arks concerning the difference between individuals and classes, and the consequence o f refuting all beings exccpt individuals.
U nfortunately, G o o d m a n ’s explan atio n s concerning the m entioned p ro b -lems are far from being unequivocal. T he nom inalistic decision o f not accepting o th er items but individuals does no t autom atically determ ine w hat kind o f beings could be adm itted as individuals. G o o d m an has nothing against the decision th a t the individuals would be ab stra c t as well as concrete items, singular and collective beings, physical and phenom enal objects. In one o f his articles he writes: „W hatev er can be construed as a class can be indeed co n stru ed as an individual"4. Besides, G o o d m an claim s th at any individual m ay be presented and co n stru ed as a class o r a set. O ne can do that, for instance, by the identification o f a physical object with the class o f its m acroscopic, atom ic or sub-atom ic parts, o r with a certain class o f events which set up the history o f a given object. O ne can also con stru e individuals as classes through the tran slatio n o f all statem ents concerning them into the statem ents ab o u t unit sets, contain in g as the only elem ent the given object („singletons” ). Eberle claim s th a t G o o d m an distinguishes individuals from classes first o f all at the level o f a theory, which m eans th at the theory o f individuals differs from the theory o f classes o r sets. Individuals are distinguished from classes neither by the fact th a t they are m ade from a special kind o f m aterial, nor by their spatio-tem poral character. T here are also no specific epistem ological criteria which w ould let differentiate them . It is no t the case th at individuals could be perceived while classes could not, n o r th at classes are only m ental constru cts while the individuals are „given” . The possibility o f differentiation should be searched at the theoretical level by the analysis o f form al features.
A n im p o rta n t principle for individuals is the „principle o f sum fo rm a tio n ” . This kind o f objects can be pu t together, sum m ed up, aggregated m aking up as
2 B. R u s s e l l , Some Explanations in Reply to Mr. Bradley, Mind 1910, p. 373-374. •' R. A. E b e r l e , Nominalistic Systems, Reidel, Dordrecht 1970.
4 N. G o o d m a n , A World o f Individuals,[in:] The Problems of Unirersals. A Symposium,
N otre D am e Univ. Press, N otre Dam e, Ind., 1956; reprinted in: Philosophy o f Mathematics, ed. P. Benacerraf. H. Putnam. Printice Hall, 1964.
a result the o th er individuals which are certain wholes. T he o p eratio n o f sum m ing up individuals has no physical ch aracter, and the result o f it does not need to be a whole preserving spatio-tem poral continuity. It can be an object with p arts dispersed in space or existing in different periods o f time. Because o f this liberal and alm ost ab stra ct ch a racter o f the principle o f individuals sum m ation it becom es quite alike the set-theoretic o p eratio n o f union form ation: „it seems the principle o f sum fo rm ation is quite analo g o u s to the set-theoretic principle governing unions o f singletons. F o r exam ple, the set o f all red objects (the union o f all singletons o f red objects) has for its c o u n te rp a rt a certain individual, nam ely the sum o f all red o b je c ts" 5. T herefore, the question arises w hat o th e r basis can there be for the differentiation between classes and individuals?
As the m ain criterion distinguishing individuals from classes G o o d m an suggests the „principle o f in d iv id u atio n ” . As is know n, the principle o f individuation for sets is the extensionality principle:
A = В <— • Ý ( .v e А <— • л* 6 В ).
T h at principle in the above form does not apply to these objects from the d o m ain o f set theory which do n o t have m em bers (with the exception o f the em pty set). T o ensure the universal validity o f the extensionality p o stu late in the d om ain o f set-theoretical objects one can either assum e th a t all objects have elem ents, or introduce one-place predicate „is a set" and relativise with the help o f it the principle o f extensionality.
C onsequently, we identify classes and sets through pointing out the correlates o f the relation e with respect to the given set (class); when two sets (classes) have the sam e elem ents, then they are identical. S hould the sim ilar rule be applied in the calculus o f individuals, given th at one introduces the relation „being a p a rt” in place o f the relation „being a m em ber” , and postulates th a t tw o individuals are identical ju st in case when they have the sam e parts? But should one take here into consideration all actual and possible p arts o f a given individual object? It seems th at in the calculus o f individuals such a criterion w ould be to o strong, and th at it is not necessary to p o in t out all p arts to identify tw o individuals with each other. G o o d m an an d Eberle claim th at for individual wholes it is sufficient to set forth a co ndition requiring th at objects which have the sam e „ultim ate co n stitu en ts” are identical.
A n o th er im p o rtan t difference between set theory and the calculus o f individuals is as follows: if we have a given object A, then the transitions A -*■ {A} -> {{A}} -» ... in accordance with the set-theoretical principle o f sets form ation, i.e. from elem ents to sets, sets o f sets etc., lead to objects
non-identical with А: А Ф {А} Ф {{A}} etc. In this m anner, startin g from one object we can o b tain whole infinite w ealth o f objects. This is applied in the reco n stru ctio n of n atural num bers as sets hierarchically founded on the em pty set. In the calculus ol individuals such a situation is excluded. F o r example, from a pair of objects A and В one can build up only one new object . i.e. the whole containing as its im m ediate p ro p e r p a rts only these tw o individuals A and B.
In connection with this m atter Eberle writes: „th e principle o f exten-sionality differentiates classes which have different «im m ediate constituents» relative to m em bership-chains; that is to say, classes which have different m em bers. Individuals which have the sam e co n ten t are to cou n t as identical [...] and having the sam e con ten t is here taken to m ean «having the sam e ultim ate constituents» W hen we look at the question o f „c o n stitu en ts” o f sets from a m ore general p oint o f view not only refering to the relation e itself, but also to the ancestral relation from the m em bership relation then it turns ou t that the extensionality principle differentiates sets which have different „im m ediate co n stitu en ts" with respect to th a t ancestral relation. O n the o th er hand, when individuals are concerned, one m ay acknow ledge as identical those o f them which have the sam e „ c o n te n t". H abing the sam e co n ten t m eans here having the sam e „u ltim ate co n stitu en ts". W hich kind o f constituents would be recognized as ultim ate depends on w hat relations they b ear to each other.
The essential task which is assigned to the calculus o f individuals is, therefore, the explication o f the concept o f a fundam ental relation between individuals. F o r instance, the m ereology o f S. Leśniewski is interpreted as a theory o f the relation „being a p a rt" . G o o d m an introduces a m ore general concept o f a „generating re la tio n " which is to include both the set-theoretic ancestral relation from the relation e an d the relation „being a p ro p er p a rt" . T hus, at that stage o f the developm ent o f his theory he w ants to cover m ereological as well as set-theoretic concepts7. In the later form ulation G o o d m a n s calculus o f individuals is based on the prim itive term to overlap, i.e. on the concept o f a partial covering o f one individual by an o th er. The principal postulate o f this version o f the calculus looks as follows:
.Y o v v <■> : Ír' ( it’ o v : - » iľ o v x A it’ o v ľ )
w here ov is a sym bol lor the relation o f a partial covering o f individuals or, in o th er w ords, o f having a com m on p a rt8. T he reason, why G o o d m an chooses as a prim itive the sym m etric predicate to overlap, an d no t the b etter know n
6 Ibid., p. 26.
7 G o o d m a n , A W orld o f Individuals...,
predicate „is a p a r t” , is the greater form al sim plicity o f the form er. The relation „being a p a r t” in this version o f the calculus is defined by m eans o f the relation o f overlapping:
x is a part o f у z ( z ov x -* z ov у ).
In words: the individual .v is a part o f the individual y , if and only if, when every individual having a com m on p art with ,v has also a com m on part with y.
A ccording to Eberle, there arc three elem ents m ost im p o rtan t for the concept ot individual introduced by G o o d m an : the concept o f the „generating re la tio n ” , the principle o f sum m ation, an d the principle o f individuation. From a form al point o f view the part-w hole relation should fulfil conditions put forw ard by the follow ing definition:
Def. R is a part-w hole relation, if an d only if the follow ing conditions are satisfied4:
1. R is a partial ordering.
2. 0 (the em pty set) is not a m em ber o f the field o f R. 3. T here exists a set A m eeting the follow ing requirem ents:
a) for every non-em pty subset S o f A, and for all v in A, if .v bears R to s u p k-S’, then .v is in S;
b) the field o f R is equal to the set o f all items x such that for som e non-em pty subset S o f A, ,v = supRS;
c) A is infinite.
Let us note th a t in the q u o u ted w ork Eberle uses the set-theoretic a p p a ra tu s on the m eta-language level describing the nom inalistic calculus o f individuals. T hus, he describes nom inalistic systems in non-nom inalistic language. The condition 2 is, according to him, equivalent to the claim th a t for every non-em pty subset S o f the set A the su p RS exists, and all descriptions form ulated by m eans o f the p art-w hole relation theory term s have definite ch aracter. T he point 3, specifies requirem ents which should be fulfilled by the distinguished subset A o f the do m ain o f the relation R. This subset is interpreted as a set o f R -ato m s (atom s w ith respect to the part-w hole relation), i.e. a set o f m inim al elem ents with respect to R.
N ow , let me underline th a t in the definition o f the part-w hole relation one em ploys a specific idea o f sum m ation o f elem ents. This idea, which is a realization o f the principle o f sum m ation for individuals, uses the concept o f a suprem um o f a set with respect to the relation R (R is a partial ordering). T he concept o f a suprem um can be applied both to the finite and infinite subsets o f the field o f the part-w hole relation, e.g. to the whole distinguished set of atom s A. T he in tro d u ctio n o f a generalized o p eratio n o f sum m ation for
individuals (sup^S) has a dcli nite goal in G o o d m an -E b erle’s theory. Namely, they w ant to stay neutral with respect to the problem o f finiteness or infiniteness o f the relation part-w hole dom ain. T h a t problem sh o u ld n 't be decided at the level o f introducing one or a n o th e r sum m ation operation. The problem o f the possibility o f reconstructing the whole universe o f individuals as the sum of all its constitu en ts also com es in to play here. In this connection, Eberle writes: „C an we be assured th at there arc indeed «ultim ate constituents» in the whole field o f physical objects relative to this relation (between a physical p art and a physical w hole)? Suppose that physical objects tu rn out to be infinitely divisible; should we then be prepared to adm it that physical objects are not individuals? G o o d m an does not preclude, on principle, that there m ay be an infinite n um ber o f least physical c o n stitu e n ts" 10. The nom inalistic stan d p o in t should be form ulated in such a general way that a disagreem ent between it and the con ten t o f a physical theory w ould not be possible.
The above consideration is to justify the co ndition 3c which states that the distinguished set A, representing the class o f atom s, is infinite. In add itio n to this Eberle justifies the assum ption ab o u t the infiniteness o f the set o f atom s in the follow ing way: „th e ato m s in question are concerned as possible, rather th a n as ac tu al objects. A nd it does not seem co u nter-intuitive to require that th ere shall be infinitely m any possible entities. On the o th er hand, since universes o f individuals arc conceived as com prising actual individuals, we shall refrain from im posing a co n d itio n on such universes which would imply th a t every universe o f actual things is infinite” 11. If I understan d this intention correctly, one should define the part-w hole relation in the m ost general way and eventually limit this generality later while applying that relation to specify the concept o f an actual universe o f individuals.
The point 3 also claim s that the set A consists o f discrete atom s, i.e. R -m inim al elem ents having no co m m o n parts. An atom can be neither a part of an o th er atom , n o r a part o f a sum o f two o r m ore atom s. Beside atom s, individuals are wholes generated from atom s with the help o f the sum m ation operatio n .
Let us proceed now to the principle o f individuation which is an o th er fa cto r constituting, according to G o o d m an and Eberle. the concept o f an individual. The principle o f individ u atio n to g eth er with the part-w h o le relation and the sum m ation principle characterize objects which we w ant to reckon am o n g individuals. They describe w hat certain collectives o f individuals arc, ra th e r th an w hat a single individual is. A ccording to Eberle, between such collectives a special atten tio n deserve so-called atom istic universes o f
in-10 Ibid., p. .in-10. " Ibid., p. 39.
dividuals. He thinks th at a definition o f such universes co n stitu te a principal explication o f the concept o f an individual. In general, an universe o f individuals is a subset o f the field o f the p art-w hole relation in which the ap p ro p ria te conditions concerning sum m ation o f objects and their individua-tion are satisfied. An individual is characterized in a ro u n d -ab o u t way as the element o f a certain universe o f individuals.
Eberle tries different alternative versions o f the principle o f individuation, and finally assum es th at the task o f distinguishing in the field o f relation R the universes o f individuals is best fulfilled by the follow ing principle:
For every v and у belonging lo U , v v iff for every if г is R-Ieast in U. then : R x iff г R y;
U sym bolize here a certain universe o f in d iv id u als12.
In o th er w ords, individuals belonging to U are identical ju st in case, when they have the sam e ato m s in relation to U as their parts. Such a form ulation o f the principle o f individuation im poses certain restrictions on the universes o f individuals which are not im posed by the other, m ore liberal form ulations. F o r instance, m ore general principle o f individuation, which identifies individuals when they have the sam e atom s with respect to the whole field o f the relation R, does not im pose restrictions on the universe U; every subset o f the field o f the relation R could then be accepted as an universe o f individuals U. W hat reasons are there for choosing such a principle o f individuation? T his problem boils dow n to the question o f the role which is played by the universes o f individuals within the field o f the part-w hole relation. Let us rem ind that Eberle interprets the field o f the p art-w hole relation as the set o f all objects which could be p arts o r wholes. He writes: „By c o n tra st the elem ents o f a p articu lar universe o f individuals are regarded as those individuals which happen to be actualized in th at universe. T o provide a suggestive example: suppose that we conceive o f an infinite class o f item s all o f which satisfy a physicist’s description o f an atom . Let a p art-w hole relation be conceived between these atom s and all possible eom posities o f the atom s. A ny selection o f these possible ato m s o r eom posities m ight be actualized in som e universe which is a «universe o f individuals» if fo r every com posite object which is actualized in it a sufficient variety o f p arts are also actualized, so th a t different actual eom posities have in the universe different actual parts. It is logically possible th at the sim plest physical objects which happen to be actualized in such a universe arc m olecules, while all p ro p er p a rts o f m olecules rem ain unactualized possibles” 13.
12 Ibid.. p. 38. 13 Ibid., p. 37 39.
T he selected principle of' individuation suggests th a t Eberle w ants to restrict the variety o f possible items. Individuals arc to be actualized objects which consist of actualized parts. A ccording to the principle o f individuation, different actual individuals are com posed in the last resort o f different actual atom s. Being actu al is conceived here in a specific m an n er as an attac h m e n t to certain distinguished universe o f individuals which is a subset o f the part-w hole relation field. At this p oint one can raise the question, w hether arb itra ry g roups o f individuals from the universe can be put together by the sum m ation o p eratio n , giving as the result in each case new individual wholes?
G o o d m a n — as is know n has answ ered this q uestion quite positively: „A lth o u g h not every individual has a negate and not every tw o individuals have a prod u ct, every two individuals do have a sum. B earing in m ind that only individuals are values o f o u r variables, we can affirm the unconditional statem ent:
.v (z = .v + v)
as a p o stu late o r theorem of o u r c a lc u lu s" 14. T he negation and product G o o d m an writes ab o u t, as well as the sym bol + d enoting the sum m ation o p eratio n , are term s defined in the version o f the calculus o f individuals presented in the q u o u tcd w ork.
G o o d m an was often criticized for ad o p tin g the above principle o f sum m ation, and in this case Eberle jo in s his critics: „we w ould d ep art from G o o d m a n ’s conception by ad m ittin g o th e r relations which qualify intuitively as part-w hole relations but fail to generate actual sum s o f a rb itra ry in-dividuals” 15. Eberle im poses the follow ing w eaker co ndition on the o peration o f sum m ing the elem ents o f the universe o f individuals:
For every v which belongs to U, there exists a set S consisting o f elem ents R-minimal in U, such that .v = s u p rS,
In o th er w ords, every individual from the universe o f individuals U is a sum o f elem ents which are atom s with respect to the relation R in U. If any object is an individual belonging to the universe U. then it m ust have a decom position into atom ic p a rts w ithin U. However, such a condition docs not assum e th a t every sum o f atom s or any o th er individuals belonging to U is again an elem ent o f U, i.e. an individual in this universe. T h at form ulation stresses analytic, rath er th an synthetic function o f the individual sum m ation operation.
14 G o o d m a n . The Structure o f Appearance..., p. 36. 15 E b e r l e . Nominalistic S ystem s..., p. 41.
Let us com m ent here on one m ore m atter. In B. Russell’s logical atom ism p ro g ram m e an essential role plays the q u o u ted conviction th a t „th e w orld does not consist merely in phases and unreal divisions o f a single undivisible R eality’’. T hus, Eberle is in a better agreem ent with Russell’s p ro g ram m e than G o o d m an , since he does not assum e th a t every possible sum o f individuals is an actual individual. In this way he restrains him self from the assum ption th at the whole world is one m axim al, global individual, an d th a t possibly all properties and external relations o f objects in the w orld are reducible to the properties an d internal relations o f the world itself.
A fter selecting the a p p r o p r ia te principles o f individuation an d sum m ation Eberle defines the central concept o f his reconstruction o f G o o d m a n ’s calculus an „atom istic universe o f individuals” . T he both above-m entioned principles assum e th at in every universe o f individuals exist atom s; hence, the expression „universe o f individuals” is supplem ented with the adjective „ato m istic” . Since, according to Eberle. the chosen principle o f sum m ation implies the principle o f individuation, in the definition o f an „atom istic universe o f individuals” one can take into account only the form er.
Def. U is an atom istic universe o f individuals for R iff 1) R is a part-w hole relation.
2) U is included in the field o f R.
3) for every x in U , there exists a set S such th at all m em bers o f S are R-least in U, and ,v = su p RS 16.
A fter presenting the above outline o f G o o d m an -E b erle's theory some com m ents suggest themselves. As I have w ritten at the beginning o f this article, th at conception seems to be a certain realization o f selected theses o f Russell’s logical atom ism program m e. It assum es that there exist m any separate and independent individuals, atom s an d wholes, while restraining itself from concluding the m atter o f existence o f a m axim al global individual, identical perhaps with the whole reality. T hus, it represent a stan d p o in t o f pluralism . At the sam e time a fundam ental role plays here the „g e n era tin g " part-w hole relation which is assum ed independently from individual objects. One docs not attem pt to reduce that relatio n to internal properties o f individuals bu t the o th er way round, the in tro d u ctio n o f it is constitutive o f the concept o f an individual. This ap p ro ach is in accordance with R ussell's stan d p o in t rejecting internal relations axiom and po stu latin g external relations independently from objects’ properties. Finally, there is certain kind o f sim plicity in dom ains which qualify as atom istic universes o f individuals, viz. their m em bers can be uniquely presented as relatively sim ple w holes com posed o f elem entary constituents, an d such a com position m ust allow fo r their com plete iden-tification. T his aspect o f sim plicity could be expressed by the statem ent that
atom istic universes o f individuals have the stru ctu re p ertaining to the relation „sim pler th a n ” , which can be interesting in the m eaning analysis o f the concept o f sim plicity17.
The fundam ental nom inalistic claim postulates refutation o f abstract entities, in particu lar existence o f classes. G o o d m a n writes: „W hatev er we are willing to recognize as an entity at all m ay be construed as an individual [...] we can con stru e anything as an individual” 18. It is ra th e r sem antic than ontological approach: the nom inalistic thesis could be form ulated in h berle's conceptual fram ew ork as the statem ent th a t every d om ain o f objects can be interpreted as a certain atom istic universe o f individuals. Let us consider the soundness o f that statem ent.
Things an d m aterial objects do no t seem to fulfil the nom inalistic principle o f individuation. F ro m the sam e things-parts we can constru ct different wholes in different m om ents o f time; a little child does th a t while playing with building blocks. M aterial objects are not individuals in G o d d m an -E b erle’s sense, in o rd e r to attain this status the tim e dim ension should be taken into account. T hus, for instance, a table is not an individual but the tablehour, ta b -le-m inute, and table-second arc. Sixty tab-le-m inutcs sum m ed up together give as a result an individual which is one table hour. T he questio n arise, w hat would in this case atom s be. T he sam e com m on-sense table, taken into consideration for the period o f one m inute yesterday and today, consists o f tw o com pletely different nom inalistic, individuals two separate tab- le-m inutes. They arc only connected by the o th er interm ediate table-m inutcs which adjoin to each o th er or succeed one after an o th er. T h e identity o f two table-m inutcs separated in tim e docs not com c into play, although they can be parts ol the sam e table-w eek or table-m onth, since they consist o f com pletely different let's say particle-seconds (the m inim al distinguished space-tim e regions). T he only kind o f identity which can occur betw een two individuals separated in space or time is the genidentity, which has not m uch to do with the identity in a nom inalistic sense. It is also not difficult to sec th a t language expressions do no t fulfil the nom inalistic principle o f individuation either; from the sam e signs we usually m ay built up different expressions.
O ne can obviously construct dom ains which would be atom istic universes o f individuals; one could also do th at with the help o f set-theoretic concepts. F o r exam ple, the pow er set o f som e non-em pty set Z (the em pty set excluded) with the o p eration o f union and the inclusion relation, i.e. the relation stru ctu re < 2 Z, u , c > , is an atom istic universe o f individuals. T he atom s here
are the unit sets form ed from the elem ents o f the set Z. If the em pty set was
17 S t r a w i ń s k i , A Formal Definition o j the Concept o f Sim plicity, [in:] Polish Essays in the Philosophy o f the Natural Science, ed. W. Krajewski. Reidel. Dordrecht 1982, p. 195 197.
included, then it w ould have been the only a to m in this universe which, how ever, would not have been able to generate o th e r elem ents. Nevertheless! the co n stru c tio n o f such dom ains seems to be a ra th e r weak justification o f G o o d m a n ’s conviction that „we can co n stru e an y th in g as an individual” .
It ap p e ars th at we do not m eet atom istic universes o f individuals to o often. Things and m aterial objects do not seem to be individuals in this sense. It is rath er the entities o f event ist ic ontology, consisted o f spatio-tem poral events, which satisfy the conditions required from individuals by G o o d m an and Eberle.
Warsaw University Poland
Witold Strawiński
A T O M IST Y C Z N E U N IW E R S A IN D Y W ID U Ó W
Autor przedstawia krytyczny analizę pewnych idei nominalistyeznej teorii N. Cioodmana w interpretacji R. Eberlego. Teoria ta. czyli „rachunek indywiduów", oparta jest na trzech podstawowych pojęciach, zasadzie sum owania, relacji „część-całość" i zasadzie indywidualizacji. Owe pojęcia wzięte razem charakteryzują przedmioty, które chcem y zaliczyć do indywiduów, przy-czyni charakterystyka ta określa raczej czym jest pewien zespól indywiduów , niż to czym jest pojedyncze indywiduum.
W edług Eberlego na specjalni} uwagę zasługują tzw. atom istyczne uniwersa indywiduów. Definicja takich uniwersów ma stanowić właśnie określenie tego, czym sq indywidua. W ogólności „atom istyczne uniwersum indywiduów" to podzbiór pola relacji „ c z ę ś ć - c a l o ś c w którym są spełnione odpowiednie warunki dotyczące sum owania i indywidualizacji przedmiotów. Pojęcie to autor wiąże z aiontizmein logicznym B. Russella oraz rozważa jego możliwe zastosow ania.
