max black - the identity of indiscernibles

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VOL. LXL N O . 242.] [April, 1952

MINDA QUARTERLY REVIEW

OF

PSYCHOLOGY AND PHILOSOPHY

I.—THE IDENTITY OF INDISCERNIBLES

B Y MAX BLACK

A. The principle of the Identity of Indiscemibles seems to me

obviously true. And I don ' t see how we are going to defineidentity or establish the connexion between mathematics and

logic w ithou t using it.B. I t seems to me obviously false. And your troubles as a

mathematical logician are beside the point. If the principle is

false you have no right to use it.

A. You simply say it's false—and even if you said so threetimes that wouldn' t make it so.

B. Well, you haven't done anything more yourself than assertthe principle to be true. As Bradley once said, " assertion can

demand no more than counter-assertion ; and what is affirmedon the one side, we on the other can simply deny ".

A. How will this do for an argument ? If two things, a and b,are given, the first has the property of being identical with a.

Now b cannot have this property, for else 6 would be a, and we

should have only one thing, not two as assumed* Hence a

has at least one property, which 6 does not have, that is to say

the property of being identical with a.

B. This is a roundabout way of saying nothing, for " a has the

property of being identical with a " means no more tha n " a is a ".

W hen you begin to say " a is . . . " I am supposed to know whatthing you are referring to a s ' a ' a n d I expect to be told som ethingabout that thing. But when you end the sentence with the

words " . . . is a " I am left still waiting. The sentence " a is

a " is a useless tautology.

11



1 5 4 MAX BLACK. :

A. Are you as scornful ab ou t difference as abou t ide ntity ?For a also has, and 6 does no t ha ve , th e property of being differentfrom b. This is a second property that the one thing has butnot the other.

B. All you are saying is tha t b is different from o. I thin k th eform of words " a is different from b " does have the advantageover " a is a " th a t it migh t be used to give information. I mightlearn from hearing it used that ' a ' and ' b ' were applied todifferent thing s. Bu t this is no t wha t you wa nt to say, sinceyou are trying to use the names, not mention them. When Ialready know what ' a ' and ' b ' stand for, " a is different fromb " tells me nothing. It , too , is a useless tautolo gy.

A. I w ouldn't have expected you to tre at ' tau to lo gy ' as ater m of abuse. Tautology or no t, th e sentence ha s a philosophicaluse. It expresses the necessary truth that different thingshav e a t least one prop erty not in common. Thu s differentthing s m ust be discernib le; and hence, by con traposition,indiscernible things mu st be identical. Q .E D .

B. W hy obscure ma tte rs b y this old-fashioned language ?By " indiscernible " I suppose you m ean th e same as " hav ingall properties in common " . Do you claim to have proved t h a ttwo things having all their properties in common are identical ?

A. Exactly.B. Then thi s is a poor way of stating your conclusion. If a

an d 6 are identical, there is just one thing having th e two n ames' a' and ' b'; and in that case it is absurd to say that a and 6are two . Conversely, once you ha ve supposed there are twothingB having all their properties in common, you can't withoutcontradicting yourself say that they are " identical" .

A. I c an 't believe you were really misled. I simply mea nt tosay it is logically impossible for two thingB to have all theirprope rties in common. I showed th a t a mu st hav e at least

two properties—the property of being identical with a, and theproperty of being different from b—neither of which can be apro perty of 6. Do esn't this prove th e principle of Id en tity ofIndiscernibles ?

B. Per hap s you have proved something. If BO, the natur e ofyour proof should show us exactly w hat you have proved. Ifyou want to call "being identical with a" a " p r o p e r t y " Isuppose I can 't prevent you. Bu t you m ust then accept theconsequences of thi s way of talk ing . All you mean w hen yousay " a has th e property of being identical with a " is th a t a is a.ATid all you mean when you say " b does not have the propertyof being identical with o " is th a t 6 is no t a. So w hat you have



THE IDENTITY OF INDISCEENIBLES 15 8

" proved " is th at a is a and b is no t a, that is to say, 6 and a aredifferent. Similarly, when you said th a t a, but not b, had the

property of being different from b, you were simply saying thata an d 6 were different. I n fact you are merely redescribing th ehypothesis that a and b are different by calling it a case of" difference of prop erties " . Drop t he misleading descriptionand your famous principle reduces to the truism that differentthings are different. How tru e ! And how uninteresting 1

A. Well, the properties of identity and difference may beuninteresting, but they are prope rties. If I had shown th atgrass was green, I suppose you would say I hadn't shown thatgrass was coloured.

B. You certainly would not have shown that grass had anycolour other than green.

A. "What it comes to is that you object to the conclusion ofmy argument following from the premise that a and b aredifferent.

B. No, I object to the trivia lity of th e conclusion. If youwant to have an interesting principle to defend, you must inter-pre t " prope rty " more narrowly—enough so, at an y ra te, for" identity " and " difference " not to count as properties.

A. Tour notion of an interesting principle seems to be onewhich I shall have difficulty in establishing. Will you at leastallow me to include among " properties " wh at are sometimescalled " relational characteristics "—like being married to Caesaror being at a distance from London ?

B. Why not ? If you are going to defend the principle, it isfor you to decide wh at version you w ish to defend.

A. In that case, I don't need to count identity and differenceas properties. He re is a different argument th a t seems to mequ ite conclusive. The only way we can discover th at two

different things exist is by finding out that one has a quality notpossessed by the other or else that one has a relational character-istic that the other hasn't .

If both are blue and hard and sweet and so on, and have thesame shape and dimensions and are in the same relations toeverything in the universe, it is logically impossible to tell themap art. The supposition tha t in such a case there might reallybe two things would be unverifiable in principle. Hence itwould be meaningless.

B. You are going too fast for me.A. Think of it this w ay. If the principle were false, the

fact that I can see only two of your hands would be no proof thatyou had just two. And even if every conceivable test agreed with



1 6 6 MAX BLACK :

the supposition tha t you had tw o han ds, you might all the timehav e thre e, four, or any num ber. You might have nine hand s,different from one an oth er and a ll indistinguishable from you rleft hand, and nine more all different from each other but in-distinguish able from your righ t ha nd . And even if you reallydid have just two hands, and no more, neither you nor I noran ybo dy else could ever know th a t fact. This is too much forme to swallow. This is th e kind of abs urdi ty you get into, assoon as you abandon verifiability as a test of meaning.

B. F a r be it from me to aban do n you r sacred cow. Before Igive you a direct answer, let me try to describe a counter-example.

Isn't it logically possible that the universe should havecontained nothing but two exactly «rmilAT spheres ? We mightsuppose that each was made of chemically pure iron, had a dia-m eter of one mile, th at they ha d the sam e tem pera ture, colour, andBO on, an d th at nothing else existed. Then every quality andrelatio nal characteristic of the on e would also be a pro perty of th eoth er. Now if wha t I am describing is logically possible, it isnot impossible for two things to have all their properties incomm on. This seems to m e to refute the Principle.

A. Your supposition, I repeat, isn't verifiable and therefore

ca n' t be regarded as meaningful. B ut supposing you havedescribed a possible world, I still don't see that you have refutedth e principle. Consider one of th e spheres, o, . . .

B. How can I, since there is no way of telling them ap ar t ?Which one do you want me to consider 1

A. This is very foolish. I mean either of th e two spheres,leaving you to decide which one you w ished to consider. If Iwere to say to you " Take any book off the shelf " it would befoolish on your part to reply " Which ? "

B. I t 's a poor analogy. I know how to ta ke a book off ashelf, but I don't know how to identify one of two spheressupposed to be alone in space and so symmetrically placed withrespect to each other that neither has any quality or characterthe other does not also have.

A. All of which goes to show as I said before, the unveri-fiability of your supposition. C an 't you imagine th a t one spherehas been designated as ' o ' ?

B. I can imagine only w ha t is logically possible. Now H islogically possible that somebody should enter the universe I

hav e described, see one of th e spheres on his left h an d and proceedto call it ' a'. I can imagine that all right, if that's enough tosatisfy you.



THE IDENTITY OF INDISCERNIBLES 1 5 7

A. Very well, now let me try to finish what I began to sayabo ut a . . .

B. I still can't let you, because you, in your present situation,have no right to talk about a. All I have conceded is that ifsomething were to happe n to introduce a change into my universe,so that an observer entered and could see the two spheres, oneof the m could the n have a name . Bu t this would be a differentsupposition from the one I wanted to consider. My spheresdo n't yet have names. If an observer were to enter the scene,he could perhaps put a red m ark on one of th e spheres. Youmight just as well say " By ' a ' I mean th e sphere which would

be the first to be marked by a red mark if anyone were to arriveand were to proceed to make a red ma rk ! " You might jus tas well ask me to consider the first daisy in my lawn that wouldbe picked by a child, if a child were to come along and do thepicking. This doesn't now distinguish an y daisy from the others.You are just pretending to use a name.

A. And I think you are just pretending not to understand me.All I am asking you to do is to think of one of your spheres, nomatter which, so that I may go on to say something about itwhen you give me a chance.

B. You talk as if naming an object and then thinking about itwere th e easiest thing in th e world. B ut it isn 't so easy. SupposeI tell you to n am e an y spider in my garde n : if you can catch onefirst or describe one uniquely you can nam e it easily enough. B utyou can't pick one out, let alone " name " it, by just thinking.You remind me of the mathematicians who thought that talkingabout an Axiom of Choice would really allow them to choose asingle member of a collection when they had no criterion ofchoice.

A. At this rate you will never give me a chance to say an ythin g.Le t me try to make my point witho ut using nam es. Each of thespheres will surely differ from th e other in being at some distancefrom th a t other one, bu t at no distance from itself— that is to say,it will bear at least one relation to itself—being at no distancefrom, or being in the same place as—that it does not bear to theoth er. And this will serve to distinguish it from the o ther.

B. Not at all. Each will have the relational characteristicbeing at a distance of two miles, say, from the centre of a sphere onemile in diameter, etc. And each will have the relational character-

istic (if you wa nt to call it th at ) of being in the same place as itself.The two are alike in this respect as in all others.

A. Bu t look here. Each sphere occupies a different p la ce ;and this at least will distinguish them from one another.

1 1 «



1 5 8 MAX BLACK :

B. This sounds as if you thought the places had some inde-pendent existence, though I don't suppose you really think so.

To say the spheres are in " different places " is jus t to say th a tthe re is a distance between the two spheres ; and we have alre adyseen th a t will not serve to distinguish them . Ea ch is a t adistance—indeed the same distance—from the other.

A. When I said they were at different places I didn't meansimply th at the y were a t a distance from one anoth er. T ha t onesphere is in a certain place does not entail the existence of anyother sphere. So to say th a t one sphere is in its place, and theother in its place, and then to add that these places are differentseems to me different from saying the spheres are at a distance

from one another.B. W ha t does it me an to say " a sphere is in its place " ?

No thing at all, so far a9 I can see. W here else could it be ?All you are saying is th a t th e spheres are in different places.

A. Then my retort is, What does it mean to say " Two spheresare in different places " ? Or, as you so neatly p ut it, " W hereelse could they be ? "

B. You have a point. W hat I should have said was th at you rassertion that the spheres occupied different places said nothingat all, unless you were drawing attention to the necessary truthth a t different physical objects mus t be in different places. Nowif tw o spheres m ust be in different places, as indeed the y must, tosay t h a t th e spheres occupy different places is to say no more th anthey are two spheres.

A. This is like a poin t you made before. You won 't allow m e todeduce any thing from the supposition th at there are two spheres.

B . Le t me pu t it another way. In the two-sphere universe,th e only reason for saying th a t th e places occupied were differentwould be th a t different things occupied them . So in order to

show the places were different you would first have to show, insome oth er way , th a t the spheres were different. You will neverbe able to distinguish the spheres by means of the places theyoccupy.

A. A minute ago, you were willing to allow that somebodymigh t give your spheres different names. Will you let me supposeth a t some traveller has visited your m onotonous " universe "and has named one sphere " Castor " and the other " Pollux " ?

B. All right—provided you don't try to use those namesyourself.

A. W ouldn 't the traveller, a t least, have to recognise tha tbeing at a distance of two miles from Castor was not the same pro-perty as being at a distance of two miles from Pollux f



THE IDENTITY OP INDISCEBNIBLES 1 5 9

B. I do n't see why. If he were to see th a t Castor and Pollu xhad exactly the same properties, he would see that " being at a

distance of two mile3 from Castor " meant exactly the same as" being at a distance of two miles from Pollux ".A. They could n't mean the same. If the y did, " being at a

distance of two miles from Castor and at the same time no t beingat a distance of two miles rom P ollux'' would, be a self-contradictorydescription. B ut plenty of bodies could answer to t hi s de-scription. Again if th e two expressions m eant the same , any -thing which was two miles from Castor would have to be twomiles from Pollux— which is clearly false. So the two expressionsdon't mean the same and the two spheres have at least two

properties not in common.B. Which ?A. Being at a distance of two miles from Castor and being at a

distance of two miles from PoU ux.B. Bu t now you are using the words " Castor " and " Pollux "

as if they really stood, for something. They are just our oldfriends ' o ' and ' 6 ' in disguise.

A. You surely don't want to say that the arrival of the name-giving traveller creates spatial properties ? Perhaps we caD'tname your spheres and therefore can't name the corresponding

properties ; but the properties must be there.B. What can this mean ? The traveller has not visited the

spheres, an d th e spheres have no names— neither ' C as to r',nor ' Pollux ', nor ' o ', nor ' b', nor any others. Ye t you stillwan t to say the y hav e certain properties which cannot be referredto without using nam es for the spheres. You wa nt to say " theprop erty of being at a distan ce from Castor " thoug h it is logicallyimpossible for you to talk in thi s way. You ca n't speak, b utyou won't be silent.

A. How eloquen t, and how unconvincing!

B ut since you seemto have convinced yourself, at least, perhaps you can explainanother thing that bothers me : I doc't see that you have a rightto talk as you do about places or spatial relations in connexionwith your so-called "u n iv e rs e ". So long as we are ta lkingabout our own universe— the universe—I know what you meanby " distance ", " diam eter ", " place " and so on. Bu t in w hatyou want to call a universe, even though it contains only twoobjects, I do n' t see w hat such words could m ean . So far as Ican see, you are applying these spatial terms in their present

usage to a hypothetical situation which contradicts the pre-suppositions of that usage.

B. What do you mean by " presupposition " ?



1 6 0 MAX BLACK :

A. Well, you spoke of measured distances, for one thing.Now this presupposes some means of m easurem ent. Henceyour " universe " must contain at least a third thing—a ruler or

some other measuring device.B. Are you claiming that a universe must have at least three

things in it ? What is the least number of things required tomake a world ?

A. No, all I am saying is that you cannot describe a configura-tion as spatial unless it includes at least thr ee objects. This ispart of the meaning of " spatial "—and it is no more mysteriousth an saving you ca n't have a game of chess w ithout thereexisting at least thirty-five things (thirty-two pieces, a chess-

board, and two players).B. If this is all that bothers you, I can easily provide for threeor any number of things without changine the force of mycounter-exam ple. The im port ant thing, for my purpose, wasth a t th e configuration of two spheres was sym m etrica l. So longas we preserve this feature of the im agin ary universe, we can nowallow any number of objects to be found in it.

A. Ton mean any even number of objects.B. Quite right. Why not imagine a plane running clear

through space, with everything that happens on one side of it

always exactly duplicated at an equal distance in the other side.A. A kind of cosmic mirror producing real images.B. Yes, except tha t there wouldn 't be any mirror 1 The point

is that in this world we can imag ine any degree of complexity andchange to occur. No reason to exclude rulers, compasses, an dweighing machines. No reason, for th a t m atte r, why th e Ba ttleof Waterloo shouldn't happen.

A. Twice over, you mean—with Napoleon surrendering laterin two different places simultaneously I

B. Provided you wanted t o call bo th of them " Napoleon ".A. So your point is th at everythin g could be duplicated on th e

oth er side of th e non-ex istent Looking Glass. I suppose when-ever a man got married, his identical twin would be marryingthe identical twin of the first man's fiance* ?

B. Exactly.

A. Exc ept th at " identical twins " wo uldn't be numericallyidentical 1

B. You seem to be agreeing with me.A. F ar from it. This is jus t a piece of gratu itous m etaphysics.

If the inhabitants of your world had enough sense to know whatwas sense and what wasn't, they would never suppose all theevents in their world were duplica ted. I t would be much m ore



THE IDENTITY OF LNDISCEKNIBLES 1 61

sensible for them to regard the " second " Napoleon as a meremirror image—and similarly for all the other supposed" duplicates '•'

B. But they could walk through the " mirror " and find waterjust as wet, sugar just as sweet, and grass just as green on theother side.

A. You do n't und erstand m e. They would not postulate"an othe r s id e" . A man looking a t the "m ir ro r " would beseeing himself, not a dup licate. If he walked in a stra igh t linetoward the " mirror " he would eventually find himself back athis startin g point, not at a duplicate of his star ting point. Thiswould involve their having a different geometry from ours—

but that would be preferable to the logician's nightmare of thereduplicated universe.

B. They might think so—until the twins really began tobehave differently for the first tim e I

A. Now it's you who are tinkering with your supposition.You can't have your universe and change it too.

B. All right, I retract.A. The more I think abo ut yo ur " universe " the q ueerer it

seems. W ha t would hap pen when a ma n crossed your invisible" m i r r o r " ? While he was actually crossing,*his body wouldhave to change shape, in order to preserve th e sym me try. Wouldit gradually shrink to nothing and then expand again ?

B. I confess I hadn't thought of that.A. And here is something that explodes the whole notion.

Would you say that one of the two Napoleons in your universeha d his he art in the right place— literacy, I mean ?

B. Why, of course.A. In th a t case his " mirror-image " twin would have th e

he art on th e opposite side of the body . One Napoleon would

have his heart on the left of his body, and the other would have iton the right of his body.B. It's a good point, though it would still make objects like

spheres indistinguishable. But let me try again. Let meabandon the original idea of a plane of symmetry and to supposeinstead that we have only a centre of symm etry. I mean th ateverything th at happened a t any place would be exactly dup licatedat a place an equal distance on the opposite side of the centre ofsym me try. In short, the universe would be wh at the mathe-maticianfl call " radially sym m etr ica l". And to avoid com-

plications we could suppose that the centre of symmetry itselfwas physically inaccessible, so that it would be impossible foran y ma terial body to pass throug h it. Now in tkis universe,



1 6 2 MAX BLACK :

identical twins •would have to be either both right-handed orboth left-handed.

A. Your universes are beginning to be as plentiful as black-berries. You are too ingenious to see the force of my argu mentabou t verifiability. Can 't you see th a t you r supposed de-scription of a universe in which everything has its " identicaltwin " doe sn't describe any thin g verinably different from a cor-responding universe without such duplication ? This must beso, no matter what kind of symmetry your universe manifested.

B. You are assuming tha t in order to verify th a t there are twothings of a certain kind, it m ust be possible to show th at one ha s apro per ty no t possessed by the othe r. But this is no t so. A pair

of very close but similar magnetic poles produce a characteristicfield of force which assures me that there are two poles, even ifI have no way of exam ining them sepa rately. The presence oftwo exactly similar stars at a great distance might be detected bysome resultant gravitational effect or by optical interference—or in some such similar way—even though we had DO way ofinspecting one in isolation from th e other. D on 't physicists saysomething like this about the electrons inside an atom ? We canverify that there are two, that is to say a certain property of the

whole configuration, even though there is no way of detectingany character that uniquely characterises any element of theconfiguration.

A. Bu t if you w ere to approach your two stars one would h aveto be on your left and one on th e right. And this would dis-tinguish them.

B. I agree. Why sh ou ldn't we say th at the two stars aredistinguishable—m eaning th a t it would be possible for an observerto see one on his left and t he o ther on his right, or more generally,that it would be possible for one sta r to come to have a relation to

a third object that the second star would not have to that thirdobject.

A. So you agree with me after all.B. Not if you mean that the two stars do not have all their

prope rties in common. All I said was th a t it was logicallypossible for the m t o ente r into different relationships with a thi rdobject. B ut this would be a change in the uriverse.

A. If you are right, nothing unobserved would be observ-able. For the presence of an observer would always change it,an d the observation would plways be an observation of some-thing else.

B. I d on 't say th at every observation changes wh at is observed.My point is that there isn't any being to the right or being to the



THE IDENTITY OF INDISCERNIBLES 1 6 3

left in the two-sphere universe until an observer is introduced,that is to say until a real change is made.

A. But the spheres themselves wouldn't have changed.B. Indeed th ey wo uld : the y would hav e acquired newrelational chara cteristics. In the absence of an y asym m etricobserver, I repeat, the spheres would have all their properties incommon (including, if you like, the power to enter into differentrelations with other objects). Hence the principle of Iden tityof Indiscernibles is false.

A. So perhaps y ou really do have tw enty hand s after all ?B. Not a bit of it. No thing th at I hav e said prevents me from

holding that we can verify that there are exactly two. B ut we

could know that two things existed without there being any wayto distinguish one from the other. The Principle is false.

A. I am not surprised that you ended in this way, since youassumed it in th e description of your fantastic " universe " .Of course, if you began by assuming that the spheres werenumerically different though qualitatively alike, you could endby " proving " what you first assumed.

B. B ut I w asn 't " proving " any thing. I tried to suppo rt mycontention that it is logically possible for two things to have alltheir properties in common by giving an illustrative description.(Similarly, if I had to show it is logically possible for nothing atall to be seen I would ask you to imagine a universe in whicheverybody was blind.) It was for you to show th at my de-scription concealed some hidden contradiction. And you ha ve n'tdone so.

A. All the same I am not convinced.B. Well, then, you ought to be.

1

Cornell University.

1 The following notes and references might be helpful to anybody wishingto make up his own mind on the questions raised :

The Definition of Identity: See Principia Mathematica, vol. i, definition13.01. The theory of types required Whitehead and Russell to sayx and y are identical if and only if the same predicative functions aresatisfied by both. For a similar definition see W. V. Quine, MathematicalLogic (1940), definition D 10 (p. 136). See also G. Frege, The Foundationsof Arithmetic (Oxford, 1940, p. 76).

Self-evidence of the Principle : " I think it is obvious that the principleof Identity of Indiscernibles is not true " (Q. E. Moore, PhilosophicalStadia, 1922, p. 307). " Leibniz's ' principles of indiscernibles ' is all

nonsense. No doubt, all things differ ; but there is no logical necessityfor it" (C. S. Peirce, Collected Papers, 4.311). " Russell's definition of' = ' won't do ; because according to it one cannot say that two objectshave all their properties in common (even if this proposition is never true,



1 6 4 MAX BLACK : THE IDENTITY OF INDISCERNIBLES

H is nevertheless significant)." (L. W ittgenstein, Tractalus Logico-Philo-sophious, 5.5302). See also C. H. Longford, " Otherness and dissimilarity "in MIND, vol. m i l (1930), pp. 454-461.

Bradley's rtmark: Not made in connexion with this topic. See EthicalStudies, 2nd edn., 1927, p. 165. Bradley affirmed wha t he called " th eAxiom of the Identity of Indiscernibles " (The Principles of Logic, 2nded., 1928, p. 288).

Identity as o relational property: " . . . numerical identity, which isa dyadic relation of a subject to itself of which nothing but an existentindividual is capable " (C. S. Peirce, CoUecied Papers, 1.461).

The proof of the principle by treating identity as a property : " I t shouldbe observed that by ' indiscernibles ' he [Leibniz] cannot have meant twoobjects which agree as to all their properties, for one of the properties ofi is to be identical with x, and therefore this property would necessarily

belongto y if i and y

agreedin all

their properties. Some limitationof

the common properties necessary to make things indiscernible is thereforeimplied by the necessity of an axiom " (Prindpia Mathematica, vol. i,p. 51). Cf. K. Grelling, " Identitas indiscernibilium " in Erkenntnis, vol.vi (1936), pp. 252-259.

Counter-examples : C. D. Broad tries to refute McTaggart's form of th eprinciple (" The dissimilarity of the diverse ") by th e example of a universeconsisting of two minds, without bodies, th at are exactly alike u> all respec ts.(Examination of McTaggarfs Philosophy, vol. i, p.' 176). Broad holdshowever, t h a t " either spatial or temporal separation involves dissimilarity "(op. dt., p. 173).

The argument from verifiability : " To say that B and C are ' real ly '

two, although they seem one, is to say something which, if B and C aretotally indistinguishable, seems wholly devoid of meaning " (B. Russell,An Inquiry into Mtaning and Truth, 1940, p . 127).

The distinguishability of asymmetric bodies and their mirror images :There U a famous discussion of this in Kant 's Critique of Pure Reason.See, for instance, H. Vaihinger, Kommentar zur Kant's Kritik der rtinenVernunfl, 1922, vol. ii, pp. 618-532 ("AnhaDg-Das Paradoxon dersymmetrischen Ogenst&nde " ).

max black - the identity of indiscernibles

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