Non-individuals and Quasi-set Theory

Thomas Benda

Abstract.Quasi-set theory by S.French and D.Krause (2006)has been so far the most promising attempt of a formal theory of non-individuals.It aims to provide a formal elucidation of the intuitive notion of ensembles of non-individuals which nevertheless have finite cardinalities of one or greater.In quasi-set theory,extensional identity is defined with a limited scope to exclude objects that are intended to be non-individuals.However,since every elementary formula of its language is sharply bivalent,a binary relation is obtained which is reflexive and in which members of its equivalence classes are substitutable for each other salva veritate in all formulas.Hence identity of any pair of objects is readily defined in quasi-set theory.On a semantic level,quasi-set theory does not provide an interpretation of its language terms as non-individuals that has explanatory power,and it is not easy to see how such an interpretation can be set up.

1 Introduction

When we speak of objects,at least during a given discourse with a certain context,we naturally and without much ado ascribe to them a capability of being identical with itself and distinct from others,in short,“id-capability”.The circularity of the concept of id-capability seems inevitable and points to the irreducibility of the concept of identity and distinctness,a feature that is usually readily accepted.

Identity is primitive or conferred by constituency,properties or construction.Which properties are to count as bestowing identity is subject of an ongoing debate and identity by function or mental construction is still no well established notion.Nevertheless,the notion of identity itself is clearly and naturally understood.

The expressions of a given formal language are clearly id-capable.So are,within a given formal system,the components of a Tarskian semantic set-up,such as the universe of discourse(often called“domain”),interpretations,structures and models.

Contrasted with identity versus distinctness is indiscernibility versus discernibility.Objects are discernible,if and only if some of their properties differ.(We speak of properties in a wide sense,including relational properties.)Discernibility is classifiable according to what kind of properties count to have it established.Stronger and weaker versions of discernibility have been considered.([27,11])Discernible objects are obviously distinct,as is universally acknowledged.The converse,Leibniz’s principle of the identity of indiscernibles,in short,PII,rests on Leibniz’s peculiar metaphysics and is in the macroscopic world generally accepted,although the issue is not uncontroversial among scholars.([9,16,19];[18],p.6,n.8)

The notion of an individual in contemporary discourse is not unique.Commonly understood,an individual is an object which is a unit rather than an aggregate and singled out as something distinct.So,in the present jargon,an individual is capable of having properties of its own and is id-capable.With that,both sets and everyday artifacts are individuals,whereas proper classes and clouds are not.That is the definition of individuality we adopt.A narrower definition of individuals has been proposed more recently ([26])as objects which are id-capable and are discerned from fellow individuals in a strong sense,that is,by haecceity or discernibility with regard to oneplace,non-relational properties.That sort of discernibility has been labeled“absolute discernibility”.([11])We refer to absolutely discernible individuals as“strong individuals”and stay with the preceding,wider understanding of individuality simpliciter.

Everyday objects are usually strong individuals.But we can think of collections of individuals which are not strong individuals.Each of those is characterized only by relations but no one-place properties,so that a structure with nodes results that share all their one-place properties.Any ensemble of elementary particles of the same kind is an example thereof.Such particles require,for empirical adequacy,a novel statistics which puts Leibniz’s PII seriously into doubt.One may go further and take the notion of entanglement in quantum mechanics as a reason to raise the intriguing possibility that—at least some—elementary particles should not even be viewed as individuals simpliciter,that is,should be denied id-capability.This view,which dates back several decades,is given a strong theoretical underpinning in a widely discussed book([18]).

Thus,in previous work,Krauseet al.([23])inquired(emphasis original):

[I]f we take seriously the view that quantum objects shouldn’t have individuality,that is,that they are to be taken asnon-individualsin a sense,can we present a ‘set theory’ where indistinguishability is introduced right from the start?

Accordingly,French and Krause promise to“offer formal accounts of‘quantum indistinguishability’ in terms of ‘Schrödinger logics’ and ‘quasi-set theory’”.([18],p.3)The authors propose quasi-set theory as a set-theoretical account of non-individuality.Towards the end of the book,they state that,under a sufficiently wide notion of reference,“it simply makes no sense to assert that ‘x=y’ where ‘x’ and ‘y’ ‘refer’to non-individual entities,and of course the result is precisely captured by quasi-set theory”([18],p.384).

Quantum mechanice motivates quasi-set theory,but its authors to not claim that the former compels us to employ the latter or any other theory of non-individuals as a formal account of elementary particles.Indeed,in other work,it has been pointed out that quantum-mechanical indeterminacy cannot be straightly translated into metaphysical indeterminacy,including that of individuals.([12])Furthermore,violating PII is not inevitable.Distinguishability may be weakened to weak discernability,a binary,non-reflexive relation.([27,11])Accordingly,Saunders([31])as well as Muller and Saunders([26])proposed that distinctness of fermions and thereby their individuality,though not their strong individuality,can be established.Even though speaking of weak discernability of fermions may presuppose their individuality([13]),Saunders has given a coherent account of quantum-mechanical objects that does very well without giving up their individuality,that is,their id-capability,and acknowledges their weak discernibility as sufficient for discernibility as required by PII.What is more,even with PII violated,non-individuality is not the only option besides haecceity.([7,15])Therefore,while quantum mechanice provides a powerful incentive to set up a set theory of non-individuals,viewing quantum particles as non-individuals is by no means forced on us.

Nevertheless,the study of quasi-set theory is interesting due to its promise to provide a formal account of non-individuality.

Quasi-set theory does not introduce non-individuality by vagueness of identity.A centerpiece of quasi-set theory is the capability of each collection of non-individuals to be assigned some cardinality.(In this context,we usually think of finite cardinalities.)The intuition is therefore not an ill-defined scattering of clouds over the sky,but a collection of such-and-such many items none of which allows to be singled out and to have self-identity.Whether such an intuition is tenable has been the subject of an ongoing discussion,which has so far not been conclusive.([22,14,5,10,3])

On the other hand,as above quotation by its authors indicate,quasi-set theory does not merely state the absence of strong individuality of some of its objects.Were that the case,then it would add nothing to Saunders’s ([31])approach mentioned above.

So quasi-set theory speaks of objects which are not id-capable and nevertheless form collections of certain cardinalities.([1])That motive resembles the idea of ontological structuralism in science,according to which structures that are set up by relations enjoy ontological primacy over the respective relata,which remain faceless.([25,17])The weakest structure would be a mere collection with a certain cardinality consisting of featureless objects which are not even required to be id-capable.That is what quasi-set theory aims to describe.The hope that accompanies its construction is to provide a formal underpinning of the idea of collections of determined size comprising non-determined,non-individual elements and to make such a notion clear and comprehensible.

In the following,quasi-set theory is critically reviewed in light of the preceding object.The result is negative:identity can be defined in quasi-set theory,so quasi-set theory is no theory of non-individuals.These findings are brought in the relatively short Section 3.Yet the complexity of quasi-set theory,with a comparatively large number of language elements and axioms,and the great interest taken in quasi-set theory warrant a longer discussion.After a semi-formal characterization of quasi-set theory in Section 2,identity is readily defined in it (Section 3).In comparison,a standard way of defining identity in quasi-set theory is brought in Section 4,which,in the face of the previous section,is somewhat redundant,but instructive,because,due to its syntactic shape,it invites possible objections.Those objections can be overcome,yet,it is hoped,their discussion illustrates how quasi-set theory is supposed to work.Section 5 brings a few general issues with defining identity and the brief Section 6 points out the difficulties to find a model of quasi-set theory that illuminates what the latter talks about.

2 The Quasi-set Theory Q

We briefly give an overview of the quasi-set theory Q and provide,in an appendix,a detailed description of the language and calculus of Q.In the following text,we will frequently refer to the axioms(Q1)to(Q26)found in the appendix.

The theory Q is a set theory with urelemente,whose language we denote byLQ.The theory Q has an intended model,a set-theoretical hierarchy with a basis of urelemente,which fall into two kinds,M-atoms and m-atoms,marked by unary predicate constants U and N,respectively.M-atoms are intended to be individuals(as defined above)and m-atoms are intended to be non-individuals,objects which lack id-capability.From M-atoms and the empty set,sets are formed by the usual set-theoretical operations.Sets are marked by the unary predicate constant Z.Matoms and sets are thought to be things(“dinge”)in the everyday sense of the word.Quasi-sets are generalized from sets,they are formed from the empty set,M-atoms and m-atoms by the usual set-theoretical operations.

Identity is defined as extensional identity,denoted by the symbol"=E"1Straight quotes enclose expressions of LQ(and are often left out),whereas curled quotes enclose expressions of the vernacular metalanguage.,as having common elements for quasi-sets and being elements of common sets for M-atoms.Quasi-sets and M-atoms and only they are the relata of extensional identity.Stating extensional identity of an m-atom with any other object is syntactically inadmissible in the languageLQ(see the remark in the appendix after the definition of=E).That syntactic restriction is intended to keep m-atoms from being individuals.

Q further codifies a concept of indistinguishability,which is introduced by a binary predicate constant"I".Identity and indistinguishability coincide for dinge,but not for m-atoms and those quasi-sets that are not sets.

LQcontains the common binary element symbol"∈".For any pairs of objectsx,y,including those in whichxis an m-atom andx,yare quasi-sets,the expression"x∈y"as well as its negation"¬(x∈y)"are well-formed formulas and syntactically admissible.

Finally,LQcodifies a concept of cardinality as a functional constant,which is applicable to any object,including non-dinge.

Q is furnished with an axiomatic calculus,consisting of the axioms and inference rules of standard first-order logic as well as 28 proper axioms.The latter are roughly and informally paraphrased as follows.

(i)Indistinguishability is reflexive,symmetric and transitive,(Q1)-(Q3).

(ii)Indistinguishable dinge are extensionally identical,(Q11).

(iii)Objects related by extensional identity are generally substitutable for each other(see the appendix for a definition),(Q4).

(iv)Indistinguishable objects are generally substitutable for each other,except in formulas stating elementhood,(Q4),(Q9),(Q11).

(v)Objects are disjointly and exhaustively divided into m-atoms and M-atoms,both of which do not contain elements,and quasi-sets,which contain elements.Quasi-sets that contain only M-atoms or sets are sets,(Q5)-(Q8).

(vi)The axioms of Zermelo’s set theory,except for the extensionality axiom,hold,with indistinguishability substituted for identity in the pair axiom,(Q12)-(Q17).

(vii)Each object is assigned a cardinal number,its quasi-cardinality,which is zero iff the object has no elements.The quasi-cardinality of each ding is its cardinality.Any finite quasi-set has a quasi-cardinality which is larger than that of any of its proper sub-quasi-sets as well as,for any smaller quasi-cardinality,a sub-quasiset having that quasi-cardinality(Q18)-(Q25).

(viii)Any pair of quasi-sets whose elements pairwise match each other in indistinguishability and cardinality has indistinguishable members(Q26).

(ix)Analogues of the Replacement Axiom and the Axiom of Choice hold (Q27,Q28).

No semantics,in particular,no formal metatheory,is provided by the authors of Q.We assume throughout a bivalent Tarskian framework as well as the soundness of Q.We do not presuppose identity or distinctness of objects in the universe of discourse,but we do presuppose the truth or falsehood of every well-formed formula ofLQas well as the usual truth assignments to complex formulas.Nothing in the setup of Q indicates otherwise.Still,we do not make any assumptions about interpretations of the predicate constants,particularly,the element constant∈,although the axioms involving cardinality(Q18)-(Q25)strongly suggest an interpretation ofx∈yas“xis element ofy”.We further note that,if any formula of the formx∈ywere syntactically forbidden,then so would be some formulasx1⊂x2(see the appendix)hence some instances of the axioms(Q21)to(Q23).But no such restriction is envisaged in the construction of Q.Similarly,soundness of Q requires the truth of all instances of the axioms(Q21)to(Q23)and hence clear and exceptionless bivalence of all formulas of the formx∈y.

We close the section with a note on terminology.We henceforth write“indistinguishable”and“extensionally identical”as mere paraphrases of the expressions"I"and "=E",respectively,in the present vernacular metalanguage and use the nouns“indistinguishability”and“extensional identity”accordingly.We continue to use“identity”in our vernacular metalanguage in its intuitive,only circularly definable sense indicated at the beginning and use“indiscernibility”as defined further below in Section 4,as equivalence regarding all formulas arising from predicate constants.We call properties that contribute to discernibility“qualitative”.We speak of objects as anything that may be the subject of discourse,the referent of a singular term,or the value of an individual variable,regardless of its being an individual,that is,its id-capability.

3 Defining Identity in Q

The languageLQof the theory Q contains no primitive identity symbol,which is natural,since Q is designed to account for non-individuality on a basic level,in particular,for non-individuality of m-atoms.With a primitive identity symbol,only the rather forceful way of syntactic restrictions would ensure that.Instead,the defined symbol of extensional identity=Eis introduced intoLQand made applicable only to non-m-atoms.

In this section,after a brief glance at the apparent lack of individuality of indistinguishable m-atoms and of indistinguishable quasi-sets formed on the basis of m-atoms in Q,we nevertheless,taking advantage of the theorems of Q,find a way to define identity by a predicate which is reflexive and whose places are generally substitutible for each other in every formula of Q,which,in short,codes indiscernibility.

In Q,individual variables denoting extensionally identical objects,those that are places of the defined predicate constant=E,are generally substitutable for each other salva veritate(for a definition,see the appendix),as is common in logical systems due to the universally accepted principle of the indiscernibility of identicals.However,individual variables denoting indistinguishable objects are substitutable for each other salva veritate only in those elementary formulas that are not of the formx∈y([4],p.256).The exception is clearly intended by the authors of Q([18],p.280).Indistinguishable objects shall be substitutable for each other,as far as descriptions of facts are concerned,but,unless they are dinge,their identity or distinctness shall remain open.For some pair of indistinguishable m-atomsm1,m2,the formulam1∈zmay be true,whereasm2∈zis false.Yet there is no obvious way to construct a quasi-setpcontainingm1andm2.Q provides,instead of the regular pair set axiom of Z,only a weak pair axiom(Q12)(see the appendix),so it seems we can only speak of the weak pair set ofm1andm2,the quasi-set having precisely those elements that are indistinguishable fromm1andm2,respectively.Substituting in the putative quasi-setpone m-atom by an indistinguishable partner yields a quasi-set which is,by(Q26),indistinguishable fromp,similarly,higher up,for quasi-sets containingp.Apparently,there is no way in Q to construct sets of objects by enumeration of the latter and so we may conclude that it even makes no sense to speak of substituting indistinguishable m-atoms for each other.They may form quasi-sets which are not extensionally identical,but we seemingly lack the means to characterize those quasi-sets by their elements.That,at least,is the intuition which accompanies the construction of Q.

However,as noted in the introduction,an important motive of Q is,simultaneously,to allow all quasi-sets,even those that contain only m-atoms,to have welldefined cardinalities.The idea of Q is to have a formal framework for collections of such-and-such many objects,without there always being a specification of which objects are contained therein.It makes sense to state that,say,quasi-setp1,containing only m-atoms,is the result of having removed an m-atom from quasi-setp2,so its cardinality,if finite,is one less than that ofp2.Cardinality is taken care of by Axioms(Q18)to(Q25).Accordingly,cardinalities are determined by elementhood.To state,for example,that quasi-setp2has cardinality two necessitates being able to state that some m-atom is element ofp2,but not of some subset ofp2.Forp2having cardinality two implies,by(Q21),p2having a subset of cardinality one,which,by(Q19)and(Q23),is a proper subset.Generally,specification of the cardinalities of all finite quasi-sets necessitates bivalence of elementhood and well-formedness of element formulas throughout,as already mentioned in the previous section.

That is what allows us to proceed to define identity in Q.We first write

sngxfor{z|∨y(z∈y∧x∈y∧qy=E1)}

“the singleton ofx”

The singleton ofxexists,is unique and is a quasi-set,even ifxis an m-atom.For,by the weak pair axiom(Q12),for anyz,∨y1(Qy1∧∧z1(z∈y1↔Iz1z)),so,by(Q1),∨y1(Qy1∧z∈y1).

Then,by the separation scheme(Q13),for any quasi-sety1,

Letz∈y1∧x∈y1,suchy1exists by(Q12),then that is equivalent to ∨y2(Qy2∧∧z(z∈y2↔∨y(Qy∧z∈y∧x∈y∧qy=E1))).The latter is the standard condition for the existence of sngx.

We look at how sngxand∩{z|x∈z}are related.

Trivially,x∈∩{z|x∈z},so∩∅and,by (Q20),q(∩{z|x∈z})E0.Assume 1＜q(∩{z|x∈z}),then,by(Q21),

By definition of∩{z|x∈z},neitherx∈ynorx/∈y,since in both cases ∨y(x∈

With∧t(t∈∩{z|x∈z}↔∧z(x∈z →t∈z))and by(～),we have

hence∩{z|x∈z}⊂sngx.

For the converse,assume

But then,by (～)and by (Q23),1＜y ＜q(sngx)),contrary to the definition of sngx.

Hence we have,keeping in mind that every quasi-set is admissible as a relatum of extensional identity,

Ifxis an m-atom,then,by(Q1),sngxis a strong singleton ofx(see appendix).

Now we write

“xis E-related toy.”

E is reflexive,transitive and,by definition of sngy,symmetric.We further write

“The pair set ofx1andx2.”

Thus sngx={x,x}and{x1,x2}=∪{sngx1,sngx2}.We readily obtain

which is nothing but an analogue of the pair set axiom of Z.So we are able in Q to construct sets by enumeration just as well as in Z.

E-related objects are indistinguishable,while the converse does not always hold.We see that as follows.

First,by(Q1),(Q4)and(Q11),any pair of dinge are indistinguishable iff they are extensionally identical.Furthermore,any pair consisting of an m-atom and a nonm-atom has members that are,by (Q11),distinguishable and,by our definition of E,not E-related.Let nowm1andm2be m-atoms and¬Im1m2.By enumeration,{m1,m2}exists,as we just found.But then,by the separation scheme(Q13),{z|(Ezm1∨Ezm2)∧¬Izm2)}also exists,of whichm1is an element,whereasm2is not.Hence¬Em1m2.For the converse case,if Z has a model,there is a model of the axioms of Q in which there are m-atomsm1andm2with¬Em1m2∧Im1m2,for example,the standard set-theoretical hierarchy with a base extended by a set of indistinguishable m-atoms with quasi-cardinality 2.Hence the equivalence classes of E are subsets of equivalence classes of I.

Finally,we look at general substitutability of individual variables for each other that denote E-related m-atoms.If both places of E denote non-m-atoms,then,by definition of E and(Q4),general substitutability obtains.Now let at least one place of E denote an m-atom.Then,as we saw in the last paragraph,any pair of E-related objects are indistinguishable.Hence,by (Q9),we have N-substitutability and,by(Q3),I-substitutability.So,with Exy,xandyare generally substitutible for each other.

With that and E defining an equivalence relation (of which,as is well known,reflexivity would be enough),it is only natural to interpret E as defined identity that is applicable throughout.If the interpretation succeeds,the purpose of quasi-set theory as a formal theory of non-individuals will have been thwarted.

We will discuss three general problems regarding defining of identity in Section 5,which apply not only to Q.These general issues notwithstanding,nothing prevents us technically from defining identity in Q by E,as done above.Still,it is instructive to look at another way to define identity in Q,which more readily invites objections.Those objections help to illustrate how Q seeks to be a theory of non-individuals.

4 Hilbert-Bernays Identity

A well-known standard way to define identity in first-order theories with a concept of equivalence of formulas was first proposed by Hilbert and Bernays([20])and later by Quine([28,30]).Identity is defined by indiscernibility.That clearly works only if Leibniz’s PII is accepted.In a formal,first-order theory without quotation([8]),indiscernibility ofxandyamounts to coextensiveness ofxandy([29]),the equivalence of formulas which differ from each other by substitutingxandyfor each other.That is the setting we currently consider.In a truth-functional theory whose language has unary predicate constantsP1,...,Pmand binary predicate constantsQ1,...,Qn,indiscernibility ofxandyis then given by

to be extended in an analogous way for predicate constants with more places.Of course,discernibility is language-dependent.A richer language may contain additional formulas in whichxandyare not coextensive anymore.

It is no less than technically straightforward to define identity in Q by an instance of(HB),replacing the different definition of extensional identity found in the setup of Q.Discernible objects are,since we ssuppose PII to hold,distinct.Indiscernible objects are identical,notwithstanding the first objection in the following section.All objects are then individuals,which is clearly at odds with the purpose of Q.

To gain clarity,we specify(HB)for Q.Depending on the theory under consideration,some of the conjunction members of(HB)are redundant due to axioms.In Q,as shown in detail in the appendix,by(Q2)and(Q3),we have symmetry and transitivity of indistinguishablity I.Furthermore,by (Q9)and (Q11),the conjunction members involving unary predicates can be dispensed with.So,in Q,(HB)is equivalently simplified to

With(Q11),(Q4)and(Q6)as well as the definition in Q of extensional identity,(HBQ)is equivalent to

The specifications(HBQ)and(HBQ*)of(HB)are obtained by the axioms of Q.They are true or false,but do not otherwise rely on any assumptions about interpretations of predicate constants,particularly,of the element constant∈.Again,it looks straightforward to employ indiscernibility according to(HBQ*)to define identity in Q.It is not surprising that,for non-m-atoms,thereby defined identity is nothing but extensional identity.For m-atoms—recall that,by(Q9),Nx∧Ixyimplies Ny—we take the second disjunction member of(HBQ*)as a definiens,adopt Leibniz’s PII and are done.Then Q is a theory of individuals throughout with defined identity,just as we found by way of defining of identity by E and such as it would be had extensional identity been made applicable to all objects.

However,defining identity by indiscernibility according to (HB)has never received explicit endorsement,where,of course,its success would undermine the purpose of Q,nor has a thorough technical refutation of that option been put forward in the literature.The following considerations are intended to fill that gap.We have to put ourselves in the shoes of an advocate of Q and propose objections against defining identity by(HB)which are raised in the spirit of Q.

4.1 Rejecting PII

A first,rather obvious objection would point out that defining identity by(HBQ)or(HBQ*)presupposes acceptance of Leibniz’s PII.In general,rejecting of PII comes,according to a widely adopted classification ([11]),in two variants.First,one may allow for distinctness of otherwise indiscernible objects by some non-qualitative property,a fundamental thisness,called“haecceity”.Haecceity allows for indiscernible strong individuals.Secondly,as advocated by an approach called“qualitative individuality with indiscernibles”(ibid.),in short,QII,or“contextual ungrounded identity”([24],p.37).Accordingly,objects may be indiscernible,sharing all qualitative properties as expressed by the language under consideration,yet distinct.Thereby obtained distinctness is permitted to be expressed by a negated primitive identity symbol,which does not count as denoting a property,so that it does not appear on the righthand side of(HB).Adopting QII amounts to speaking of indiscernible objects which are capable of being identical and distinct,but where,due to a lack of thisness of objects,there is no way of telling which object is which.In short,we get indiscernible individuals,but not indiscernible strong individuals.

Q stays clear of employing haecceity for m-atoms.Its motive resembles that of the metaphysical position QII.Yet Q aims to speak of m-atoms as indistinguishable non-individuals,whereas QII is about indiscernible individuals.For the purposes of renouncing PII in Q,so as to save non-individuality of m-atoms,we have to beware.Indistinguishability and indiscernibility,with the presently adopted terminology,are not the same.The former is mere I-relatedness,the latter,sharing of all properties,including elementhood and,of course,I-relatedness.In Q,we have collectives of indistinguishables.As said further above,each such collective has a cardinality greater than one only if it has members which undergo different element relations and so are discernible.That is a far cry from disavowing PII in(HB).As long as elementhood counts as a qualitiative property on the right-hand side of(HB),Q fully embraces PII.

Still rejecting PII would lead to an outcome that is rather unhelpful for the purposes of Q.It would be impossible to have quasi-sets containing only indiscernible—in contrast to merely indistinguishable—m-atoms which have a cardinality other than one,counter to the motive of Q,for each quasi set containing only indiscernible matoms and having a cardinality greater than one would,by(Q21),have a sub-quasi-set of cardinality one,which,by virtue of the indiscernibility of its elements according to(HBQ*),could not be a proper sub-quasi-set,so we would have a quasi-set having two cardinalities,contradicting(Q19).

But what if we do not count elementhood as a qualitative property,as the shape of(HBQ)may suggest? Thus the element formulas in the second and third conjunctive members of(HBQ)may be seen as a non-qualitative property.Then qualitative properties become indiscernible by indistinguishablity,that is,I-relatedness,alone.Elementhood plays a mere accounting role,enabling us to set up cardinalities,but is irrelevant for indiscernibility.However,even then it is not irrelevant for identity and distinctness.We recall that formulas stating elementhood,even if interpreted in a non-standard sense,are bivalent for all their places,including m-atoms.So such formulas allow us set up distinct indiscernibles,in a spirit of QII if not outright thisness.Two indistinguishable—and here,indiscernible—objects which do not contain or are not contained in the same quasi-sets are thereby not identical.PII is disavowed,but a definition of identity,barring syntactic restrictions,goes through.

Therefore,rejecting of PII does not help to suppress defining of identity by(HB)in Q.

4.2 Syntactic Restrictions

Secondly,one may resort to a similar move as the designers of Q,syntactical banning of certain expressions in (HBQ)or (HBQ*).Indeed,the last conjunction member in the second disjunction member (HBQ*)reminds us of the definiens of extensional identity of ding-urelemente.Let the expressionsm1,m2denote m-atoms.In Q,the expressionm1=E m2is syntactically forbidden([18],p.276;see also the appendix).So it is not far-fetched syntactically to dismiss the conjunction member∧z(m1∈z ↔m2∈z),as well.Let us formulate the certainly uncontroversial principle

(P1)Any expression having a syntactically forbidden sub-expression is syntactically forbidden.

With(P1),the whole of(HBQ*)would be syntactically forbidden and unsuitable as a definiens of anything.It seems that banning of the expression∧z(m1∈z ↔m2∈z),which is a sub-expression of=E,follows the spirit of Q.

The syntactic restriction works to invalidate(HB),(HBQ)and(HBQ*)as definitions of identity.But,of course,we have already circumvented it in the previous section when we defined identity by E.It is easy to verify that,by(*),Exyand the right-hand side of(HBQ*)are equivalent.Blocking the definition of identity by E via syntactic restriction seemingly requires either a suitable additional ad hoc semantic restriction that is appicable toExyor the following principle:

(P2)Any formula which is equivalent with a syntactically forbidden,but otherwise well formed formula is syntactically forbidden.

However,(P1)in conjunction with (P2)are overly restrictive.It is straightforward to see that then,by creating a disjunction of the syntactically banned expression with its negation,not only tautologies become syntactically forbidden,but even any formulaA,by creating the conjunction of the banned tautology withA.Any syntactic restriction works only if it bans what is desired and no more.The burden of showing that is on the shoulder of its creators.In the present case,disallowing either"=E"or"∧z(m1∈z ↔m2∈z)"for m-atomsm1,m2is,without(P2),easily circumvented and,with(P2),disallows formulas of Q which we wish to keep.

How convincing in general is the tool of syntactic restrictions to strip certain objects of individuality? It appears that one has the liberty to restrict definitions as desired,in the present case,to exclude m-atoms from being relata of extensional identity=E.Ideally,an imposed restriction is not arbitrary,but works as a natural part of the theory and is guided and motivated by the remainder of the theory.In the present case,Q without the syntactic restriction of extensional identity to non-m-atoms would be a set theory with two sorts of urelemente,defined extensonal identity and a commonsensical introduction of cardinality.It would be a self-contained theory without the slightest hint at the feasibility of non-individuals.The revolutionary introduction of non-individuals comes about only by said syntactic restriction and is motivated by nothing in the remainder of the theory.To be sure,one cannot demand of a formal theory to provide a complete foundation of an envisaged metaphysical novelty,but one would hope for its clarifying power and some explanatory support,as far as formal aspects are concerned.After all,the apparent motive of the formal system Q is to elucidate the envisaged notion of ensembles of non-individuals with fixed cardinalities.But a mere syntactic restriction,rather than clarifying,entirely relies on what is to be clarified,the object of introducing non-individuals.Furthermore,not allowing,within Q,to speak about individuality of m-atoms does not amount to be able to speak about non-individuals,let alone elucidate what they are.The metaphysical status of m-atoms as individuals or non-individuals is simply left open and any understanding of non-individuals that goes beyond the intuitive from which we started out remains elusive.

4.3 Banning Circular Definitions

It is commonly agreed upon that definitions are not to be circular.So,thirdly,one has to be careful if the statement of indiscernibility(HBQ)or(HBQ*)has to assume the identity of objects referred to by subexpressions thereof.If∈is interpreted in the standard way as“is element of”,we readily find a problem with the subexpression∧z(m1∈z ↔m2∈z).Clearly,withm1andm2being m-atoms,their identity depends on the identity condition of the quasi-sets of whichm1andm2are elements.But those,in turn,according to the first disjunction member of (HBQ*)depend on m-atoms,among themm1andm2.There is circularity in the definition of identity of urelemente,which may be seen as a reason to disallow the subexpression∧z(m1∈z ↔m2∈z)in the definiens of identity.However,we note that the definition of identity of pairs of M-atoms,to which the same consideration applies,proceeds in Q without much ado.

Having identity defined by E entirely avoids the circularity objection.

In conclusion,none of above three objections holds water.Defining of identity in Q by(HB)provokes above objections only by its specific syntactic shape.HavingExybe the definiens of identity ofxandydoes not even invite any of these objections.Still,we have to address a few general issues regarding defining of identity.

5 General issues regarding defining of identity

Defined identity,as contrasted to identity as a primitive,is fraught with several issues,which are not specific to the present definition of identity,but put into doubt the feasibility of defining identity in general.We briefly discuss them in the present context of defining of identity in Q by E,keeping in mind that the same issues concern the original definition of identity in Q by=E.

First,even with Exyobtaining,xandycan be distinct.More precisely,the theory Q may very well have a model-if it has a model at all—in whichxandyare distinct.That distinctness comes about by finer criterion of distinction,which has been applied in such a model,with Q lacking the linguistic resources to express it.The finer distinction is performed by a predicate whose equivalence classes are proper subsets of those of E.In our setting,such a predicate is not contained inLQ.A linguistically richer theory—let us denote it by Q+—may express distinctness ofxandywhile Exystill obtains,where every true sentence of Q remains true in Q+,so that every model of Q+is a model of Q.However,the linguistic poverty of Q does not prejudge on our ability to define identity therein.Identity merely becomes languagedependent.Within a given language,we obtain by defining identity an equivalence relation whose relata are generally substitutable for each other,which is what we aim for.

Enriching the language by introducing additional qualitative properties resembles in spirit rejecting of PII,which we addressed in the previous section,but is not the same.As we saw there,upon dismissing of PII,distinctness of indiscernibles is enabled by non-qualitative properties which do not count as discerning,such as thisness,haecceity,or a fundamental distinctness,QII.Here,however,discernibility becomes finer-grained by a richer language.That issue,of course,has been well known and has not been viewed as prohibitive for defining of identity.If it were,then already definition of extensional identity=Ein the original set-up of Q would have to be dismissed.

Secondly,in a formal definition of identity,identity as a notion may already be required in the definiens.The common set-theoretical definition of extensional identity,in conjunction with the standard interpretation of the element constant∈,provides,as we noted above,an example.To state,as a definiens,that each of a pair of sets contains the same elements presupposes identity and distinctness of objects that are elements referred to in the definiens.In a set-theoretical hierachy,without or with urelemente,that works well if the empty set and—if employed—the urelemente are individuals.But defining identity of urelemente in an analogue way,as done in the definition of extensional identity=Ein Q may pose problems,as already mentioned.

More severe,finally,is an objection according to which a notion of identity is employed in the formal interpretation of the definiens([34],pp.199-201;[21];[32]).First,expressions of the used formal language have to be identified.Then,more importantly,different occurrences of the same expressions have to refer to identical objects,which are the values under a given interpretation.Thus,in establishing of the indiscernibiliity ofxandy—of what the formal language expressions "x" and"y"refer to—by(HB),the identity or distinctness of what places of predicates in the definiens(HB)refer to has to be established for the definition to make sense.However,not only (HB)-indiscernibility is thereby affected,but any formula in general,of which(HB)is just an example.

If there is merit to that objection,then the capability of formulas to be assigned truth values requires identity as a primitive notion and all formal definitions rely on primitive identity.In Q,identity defined both as extensional identity or by E would be a charade,Q would be about nothing but individuals.If the present objection can be overcome,then,other objections to defining of identity notwithstanding,identity in Q is perfectly definable by E and,again,Q is a theory of individuals.

If identity—especially with the last objection in mind—can be defined at all,then E performs this task adequately.Universal applicability of elementhood has allowed to define universally applicable identity in Q.Bivalence of all formulas in the primitive language of Q has established non-identity as clear distinctness.

6 How to Interpret Quasi-set Theory

No formal metatheory of Q has been presented.That would certainly be desirable for precision (see,e.g.,[33]).Besides,and importantly for the object of Q,a precise model of Q is of great help to understand what Q talks about.We recall that Q has been constructed with the hope to illuminate and make precise a rough concept,that of ensembles of such-and-such many objects which cannot be told apart.If Q not only formally embraces non-individuals,but also comes with a well specified model,we will have a coherent and techincally worked out notion of emsembles of indiscernibles.The first antecedent will not come about due to the universal definability of identity in Q,but we should have a brief look at the prospects of the second.

It is not hard to see that a formal metatheory of Q based on ZF fails to express non-individuality of objects,as has been noted ([18],p.273).With ZF being the metatheory of Q,the predicate constants N and I ofLQhave individuals and nothing but individuals as their places.As a putative metatheory of quasi-set theory,ZF at best talks about“non-individuality”,a language expression,but does not use nonindividuality,what the expression is supposed to mean.

Consequently,Arenhart and Krause([4])proposed Q plus a replacement scheme as the metatheory of Q(plus a replacement scheme),similar in spirit to having ZF as the metatheory of ZF.It is assumed implicitly that,if ZF has a model,Q has a model(depicted ibid.on p.258),which looks being straightforward to prove,and a sketch is made of the metatheoretical Q which includes coded language expressions ofLQ.But clearly,having Q be its own metatheory does nothing to understand what Q is about.

Only with such an understanding in place,it will help to elucidate the concept of non-individuals.So far,however,no model of Q or,for that matter,any other putative formal theory of non-individuals with some explanatory power is on the horizon.Perhaps,as Arenhart([2])has proposed,a logical approach to non-individuality will not work,we may have to resort to metaphysical considerations,as in the past.

7 Conclusion

Quasi-set theory Q seeks to account for non-individuality and proceeds to do so by syntactical restricting of the definition of extensional identity to non-m-atoms.Given bivalence of all primitive formulas of Q and without assumptions about interpretations thereof and about individuality of objects in the universe of discourse—notwithstanding the next paragraph—an identity predicate is readily defined in Q.It can be syntactically restricted only by gross ad hoc measures and fares just as well against objections of language dependence as any definition of identity in well known logics of individuals.

We still have seriously to consider the possibility,mentioned in Section 5,that,whenever we employ a semantics with unique references of expressions,individuality of what is referrred to is already presupposed.If so,then our definition of identity in Q is at best coherent with underlying presuppositions of identity.In any case,whether definition of identity by E actually defines something or is merely an ornament,Q turns out to be simply a theory of individuals throughout.

Appendix:Technical overview of Q

In this appendix,we sketch the quasi-set theory Q by French and Krause([18]),while mostly,but not entirely,sticking to the original notation.We add several definitions of our own as well as a brief discussion of the axioms,some redundancies and a few deficiencies.

Q is based on Zermelo’s set theory Z,that is,ZF without the replacement scheme,where identity is defined rather than being primitive.As compared to Z,which has a languageLZwithout identity,the theory Q has a languageLQwith additional predicate constants,the same set of axioms,except for one that has been(insignificantly)weakened,and additional axioms.LQin turn is a pair consisting of a set of symbols,called the vocabularyVQ,and a set of finite sequences of symbols,called formulas.

The symbols of the vocabularyVQare

The logical constants have their usual intended meanings.Metavariables for individual variables are denoted by small letters printed in italics,with or without subscripts.Metavariables for formulas are written in the shapeA(x,y),where at leastxandyoccur freely inA.is written for the formula that arises fromA(x)by replacing all free occurrences ofxbyt,iftis free forxinA,and for the empty expression otherwise.In somewhat sloppy,but common usage,we use the same symbols for metavariables for expressions of the language of Q and for referring in our vernacular metalanguage to their denotations.We refer in our vernacular metalanguage to expressions of the language of Q without having them inserted in quotes.So we say,for example,“two m-atomsn1,n2for which In1n2are intended to be non-individuals”.

Elementary terms are individual variables,withtbeing a term,qtis a term,as well.

Elementary formulas are

We call the fourth of them above“element formula”.WithA,Bbeing formulas,

are formulas,as well.There is one exception([18],p.276):x=E y,to be defined below,is not well-formed if Nxand Ny.As is common,we henceforth leave out outermost parantheses in formulas and apply customary rules for leaving out parantheses according to various strengths of binding of parts of formulas.

The languageLQis extended in a customary manner by defining of further connectives and a universal quantifier within contexts of formulas.Thus we writeA∧Bfor¬(¬A ∨¬B);A →Bfor¬A ∨B;A ↔Bfor (A →B)∧(B →A);and∧x(Ax)for¬∨x(¬Ax).

Furthermore,we writex/∈yfor¬(x∈y);x ⊂yfor¬(Nx ∨Ux ∨Ny ∨Uy)∧∧z(z∈x →z∈y);∨xy(Axy)for ∨x(∨y(Axy));∧xy(Axy)for∧x(∧y(Axy));∨x∈y(Ax)for ∨x(x∈y∧Ax);and∧x∈y(Ax)for∧x(x∈y →Ax).

WithPbeing a predicate constant,we say thatyis“P-substitutable”forxiffyis substitutable forxsalva veritate in every elementary formula in whichPoccurs and we say thatyis“generally substitutable”forxiffyis substitutable forxsalva veritate in every elementary formula.

Comprehension terms of the form{x|Ax}are defined as usual.We write potxfor{z|z ⊂x},the power set ofx;∩tfor{x|∨y(y∈t)∧∧y∈t(x∈y)},the intersection ofx;∪tfor{x|∨y∈t(x∈y)},the union ofx;x∪yfor{z|z∈x ∨z∈y)},the union ofxandy;andxyfor{z|z∈x∧z/∈y)},the difference ofxandy.The existence of the sets denoted in this paragraph is yet to be ensured by axioms of Q.

The following definitions are specific to quasi-set theory.We write

The definition of extensional identity([18],p.277),as it appears above,implies,without further qualification,∀x1x2(Nx1∧Nx2→¬(x1=E x2)),which is,given the motive of Q,a little awkward if we read it as“the members of any pair of m-atoms are extensionally distinct”.That would apply to the pair〈x1,x1〉.Then the members of any pair of m-atoms would be discernible in the sense of Hilbert-Bernays.That outcome is avoided by a syntactic rule.The rule’s motive is that“expressions likex=yare not well-formed ifxandydenote m-atoms”.([18],p.276)So,it is stated,there can be no primitive identity symbol.Instead,a symbol=E,“extensional identity”is to be defined.If we reasonably assume that“expressions likex=y”include extensional identityx=E y,then the latter is,by way of syntactic restriction,not applicable to m-atoms.With that,we write ∅for{x|D,the empty set,for which we also write 0.The first conjunction member within the definiens excludes non-dinge,which,given the syntactic rule just mentioned,is redundant,but makes the definiens always well-formed.

A strong singleton([18],p.292)ofxis any quasi-set having quasi-cardinality 1 and containing some object that is indistinguishable fromx.Given the setup of Q,which avoids speaking of identity and distinctness of m-atoms,we are prima facie not able to state whether the strong singleton of a given m-atom is unique.The axioms below imply that,for dinge,weak and strong singletons are the same.

We recall that the vocabulary ofLZis a proper subset of the vocabulary ofLQFor the purpose of defining cardinals further below,we introduce the notion of ding reduction of formulas ofLZinductively:The ding reduction ofx∈yis Dx∧Dy∧x∈y;of¬Axis Dx∧¬Ax;ofAx ∨Bxis Dx∧Dy∧Ax ∨Bx;of ∨x(Ax)is Dx∧∨x(Ax);and,by definition of=E,ofx=E yis Dx∧Dy∧x=E y.

The quasi-set theory Q has a calculus which is furnished with 28 proper axioms beyond logical axioms:

We pause here to introduce cardinals.First,we note,by checking one by one,that the ding reductions of all axioms of Z,save the pair axiom,are theorems of Q.Furthermore,ding reductions of all inference rules of Z are inference rules of Q.Now we define natural numbers within Z in the common von-Neumann way.For natural numbersx,y,x ＜yindicates thatxis smaller thanyandx ≤yis written forx=E y ∨x ＜y.We further write,as is common,x+yfor the arithmetic sum ofxandyand 2xfor thex-th power of 2.Therein,we use common set-theoretical definitions for arithmetic operators.Finally,we write Finxforxbeing finite;Carxforxbeing a—so far,finite—cardinal in the sense of standard set theory;carxfor the cardinality ofx,ifxis a set and 0 otherwise;and 2carxfor the cardinality of the set of all functions from carxto{0,1};and quoIxfor the quotient set ofxunder I,if I is reflexive,symmetric and transitive,and 0 otherwise.Above definitions and requisite theorems derived in Z are ding-reduced.Then,as noted at the beginning of this paragraph,they are well formed in Q and are theorems of Q,respectively.Thus finite cardinals are introduced into Q.French and Krause([18],p.285)take the theory ZFU to define cardinals and,in a brief sketch,apply the definition by the same device on Q,which allows for the definition of finite and infinite cardinals.Infinite cardinals are referred to in Axioms(Q21),(Q22)and(Q24).For their set-up,an analogue of the Replacement Axiom is needed,which appears only later(p.291).In contrast,we do not elaborate on defining infinite cardinals,since finite cardinals suffice for our discussion of Q and since Q is motivated by quantum theory which,as a standard,speaks of finitely many particles.

We leave out two final axioms stated by French and Krause([18],pp.291,297),which are analogues of the Replacement Axiom and the Axiom of Choice,since,as mentioned above,what characterizes Q over standard set theory does not require the discussion of sets beyond the first infinite level.

Axiom (Q9)implies∧xy(x=E y∧Qx →Qy).As in Zermelo’s set theory Z,the axioms of Q,particularly(Q12)-(Q17)entail the existence of the comprehension terms defined above.Following a venerable tradition in set theory,above axioms are not independent from each other.(Q10),stating the existence of the empty set,is obtained from the separation scheme (Q13).The second and third conjunction members of (Q9)are implied by (Q4).So Axiom (Q4)could be weakened tokeeping(Q9)in place,with the same set of theorems resulting.In particular,substitutibility of objects related by extensional identity is for the respective first occurrences in formulas of the formx∈yensured by the weakened(Q4),for second occurrences in the case of ding-urelemente by definition and(Q9),for formulas of the form Nxtrivially given,for formulas of the form Uxand Zxby(Q9),for formulas of the form Ixy,for quasi-sets by(Q26)and otherwise by a short proof at the end of Section 3 below,and,given the above,for complex formulas by truth-functionality.

Two axioms should be added to the calculus of Q,since not all comprehension terms{x|Ax}denote sets,only those do for which ∨y(∧x(x∈y ↔Ax))is a theorem.Zermelo’s axioms,even with above modifications,ensure that comprehensions are sets in the case of enumerations,unions and power sets,but not for the classes of urelemente and n-urelemente.To fill the lacuna,we state the axioms

We further note that urelemente are not comprehensions.([6])