Thomas Benda October 2006
A
BSTRACT
. Set theory, as we know it, even non-classical set theory, presupposes the concept of identity. Therefore, strictly speaking set theory is applicable only to individuals, objects having sharp identities. The following are some largely nontechnical notes on the necessity and possibility of a set-theory of non-individuals. It turns out that macroscopic non-individuals can be dealt with doxastic extensions of set theory. Microscopic non-individuals, like elementary particles in physics, pose a tougher challenge. Attempts to describe them formally using a set theory of nonindividuals, however, have so far not been convincing.
Logic and formal science seem invariably to concern objects with sharp identities.
Objects considered therein are clearly identical or distinct. We call them “individuals”.
All standard logical systems deal with objects that have sharp identities, making the presupposition of sharp identity mostly silently. In some formal languages, the identity symbol is a primitive predicate constant, in other formal languages, it is defined. In the former case, the following axiom are usually part of the logical system.
Self-identity: indiscernability of identicals x = x ; and x = y

( Px

Py )
(Leibniz’ law). with x , y denoting individual variables and P denoting any predicate constant.
The converse of the indiscernability of identicals, that is, the identity of indiscernibles x
 y
 
( Px

Py ) is a theorem of standard set theory by either the extensionality axiom, if the identity symbol is a primitive predicate constant, or by defining identity by extensionality.
Logic with identity is a hugely successful tool. Non-standard versions and extensions notwithstanding, it helps us to get to grips with the central problem of representing states of affairs, namely truth and reasoning. Without an expression of clear identity, there are no substitutivity and no way to formulate truth conditions beyond tautologies and the immediately given. Logic with identity further serves as a foundation for mathematics except geometry. Obviously, without a notion of clear and sharp identity there is nothing to apply identity logic to. As just mentioned, the standard application of logic is mathematics, and we understand at least mathematical objects to have clear identities, whatever we think of their nature.
Any attempt to define identity in a noncircular way does not succeed, yet we understand identity. So we feel encouraged to establish some principles that should apply to every theory that deals with individuals and collections of individuals.
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Sets, collections of objects to aggregates, are intuitively grasped. Let us call sets that are individuals, contain only individuals as elements and are not absolutely unlimited
“extensionally determined”. What is meant by “absolutely unlimited” will become clear further below. Once we have accepted the notion of identity, two judgements very naturally occur to us, judgements which are both unprovable and irrefutable, which do not depend on logic and experience, yet are indispensable presuppositions for logic and representing experience. They are:
(S0) Every set is uniquely determined by its elements and is individuated provided all of its elements are individuated.
(S1) The collection of subsets of any extensionally determined set is extensionally determined.
Judgements (S1) and (S2) are overwhelmingly commonsensical and accepted by virtually every member of the logical community. (S2) allows to form for every set S the set of subsets, the power set of S. Having declared two sets S
1
and S
2
uniquely by their elements according to (S1), a set containing elements of S
1
or S
2
, i.e., the union set of S
1
and S
2
, is readily formed. Sets that are constructed by forming power sets and union sets, starting out from individuals rather than—in the words of David
Lewis—structureless gunk or a bottomless pit of objects are extensionally determined.
With these principles, nothing prevents us from constructing a hierarchy of sets by the following self-referential imperative:
(V0) Consider the empty set (or a set of urelemente).
(V1) Consider the power set of the set considered in the last step and repeat (V1).
(V2) Consider the union of all sets to be considered that way and repeat (V1) and
(V2).
Performing steps (V0) to (V2) builds up the standard set-theoretical hierarchy with levels that build up an order. Every—in an unspecified sense—level of this order is passed through. (V2) demands considering their union, an absolutely unlimited class of everything
V
, which is approached, but never reached. We can therefore add another judgement:
(S2) The class of everything
V
, being absolutely unlimited, is extensionally undetermined, but intensionally determined.
Above quick sketch of constructing the set-theoretical hierarchy would be readily endorsed by platonists. However, no assumption has in fact been made as to the nature of sets, whether they are mental constructs, fictions or real. Only in the case of sets being mental constructs we may leave out step (V2) of above set-theoretical hierarchy and hence not suppose
V
to be the intended end-point, but this will not affect the point being made here. Even without taking a position there, accepting above judgements (S0) and (S1) and allowing for the existence of the empty set or some other set while constructing a formal system results in every model of that formal system comprising a subclass of the standard set-theoretical hierarchy. Models without unlimited descending chains of elementhood, which look natural to us, then simply are subclasses of the standard set-theoretical hierarchy.
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Metaphysics usually is concerned with macroscopic, composite objects. These are the objects we deal with every day and whose nature interests us. They are, however, notoriously ill-defined and suffer from a lack of sharp identities. Related considerations are best illustrated in the well-known problems of macroscopic fuzziness; Theseus’ ship; changing of objects over time; and substantial change.
Talk in daily life does well without sharp identities. Physics, however, owes its success to a mathematization of objects and propositions, which requires setting up of idealized individuals and laws. A precise physical theory cannot do without an underlying logic with identity. It may seem that physics of macroscopic bodies, as it is pursued by the scientific community, is an over-idealization and that, in the face of above problems, an appropriate treatment requires a logical foundation without sharp identity. I propose, however, that a world of macroscopic, composite objects is without over-idealizing yet with full precision described by a doxastic logic with identity.
The cat on the mat, which constantly looses hair and picks up dirt, is extensionally determined by a permanently changing set of cat parts. Each member of a huge collection of sets of cat parts can claim with equal justification to be acknowledged as the cat. In fact, the cat is identified by everyone, that is, by every being equipped with a mind, according to some roughly reasonable criteria. The same happens to Theseus’ ship. Its identity is in the eye of the beholder, and there may be agreements and disagreements among various observers.
The proposed theory has a formal frame which is a standard set theory, e.g.,
Zermelo’s theory. Its vocabulary is extended by adding expression variables, for which we use metavariables x , y , … , a unary predicate constant B, and the quotation symbol
┌ ┐
, with both corners being regarded to form a single symbol. Terms of the language of the theory, L , are individual variables or quotations of strings of symbols of L , for both of which we use metavariables x , y , … , or defined terms. Expression variables themselves are not terms. Formulas of L are elementary formulas of the shape x = y , x
 y , Px
1 x
2
…, with
P being a predicate constant of the formal frame theory, or B x , or are complex formulas constructed inductively in the usual way.
Outside the scope of quantifiers with the same quantification variable, any individual variable is free if not enclosed by a quotation symbol, and any expression variable is free inside and outside of quotation symbols. Free individual as well as expression variables become bound by quantifiers not enclosed by any quotation symbol. The elementary formula B x is read as “ x is believed”. We may furnish the predicate constant B with more places to accommodate contexts and individual believers.
We do not specify axioms, we only mention a closure axiom, contained in all weak modal and doxastic theories, which helps to deal with objects that are ill-defined or hard to individuate. It is
B
┌
A
1

A
2
┐ 
(B
┌
A
1
┐ 
B
┌
A
2
┐
)
“If an implication and its premise are believed, its conclusion is believed.”
Applying this axiom to an example, if some person were in a bad accident hit by a flying rock, he would, presupposing that a volcano had just erupted and that it spewed out that rock, believe to have been the unlucky victim of a volcanic eruption. This belief is not disturbed by a notorious difficulty to individuate events like volcanic
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eruptions. It is now straightforward to make other conclusions from scientific hypotheses in that way, notwithstanding arbitrary individuations.
It may very well be argued that the only physical objects with sharp identities are elementary particles. They would naturally be formalized as individual constants.
However, we point out that the language L does not need to contain individual constants. Macroscopic, composite objects are individuated according to criteria that are closer related to needs and the worldview of the observer than to the collection of particles under observation. Formally, they are created by Russellian designators. A common way of individuating would be by function or significance for the observer.
Indeed, introducing individuals is for the observer the way to make sense of the world and to deal with it. On a more elevated level, criteria of identity follow philosophical considerations. In the framework of the sketched theory, they are expressed in further axioms, appearing only inside quotation marks. Quoted expressions are nonextensional. Thus even beliefs that are outrageous to many will not endanger the consistency of the logical system.
Philosophical arguments about nature and identity of macroscopic, composite objects are logically delegated into the space between quotation marks. Thereby their importance is not diminished. The point is rather that they do not affect the identity logic used to formulate them. Macroscopic objects are individuated as part of beliefs.
Their individuation is undertaken with the help of raw intuition, sensitivity to language and conceptual analysis. All these mental acts, however, do not make macroscopic objects mental objects.
The upshot is, macroscopic, composite objects do not require to abandon identity logic. A doxastic extension of a suitable set-theoretical formal system will sufficiently cover all needs for designating objects and formulating propositions.
Another challenge to proposing that objects always have clear identities are elementary particles, bosons and fermions, in physics. As is well known, bosons and fermions form groups with indistinguishable members. If, for example, the electrons belonging to two atoms of the same kind are swapped, no difference between states before and after the act can be told. Quantum statistics, which relies on the indistinguishability of elementary particles, has been empirically verified to an excellent degree. Therefore, this microscopic challenge to sharp identities deserves to be taken serious.
The principle of identity of indiscernibles implies that the existence of several bosons and of several fermions is impossible. But it patently makes sense to say that a nucleus is circled by a certain number of electrons. Speaking of several electrons in common language assumes there are non-identical bosons, so the principle of identity of indiscernibles is violated.
Non-sharp identities of macroscopic, in particular, composite objects, as discussed in the previous subsection, could easily be relegated to epistemology. Identities of their constituents are not put into doubt. In the case of elementary particles, however, one may very well feel that non-individuality occurs at a primitive level. Therefore, for the task of formalizing physical theories, there is some motivation to look for a theory that accommodates non-individuals.
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In a recent book, Steven French and Décio Krause *
suggest an extended set theory, called “quasi-set theory”, which takes into account above-mentioned difficulties.
They claim that “the ontology of quantum physics apparently does not reduce to that of the standard sets”. In their view, the formal language usually employed for a precise description of the physical world unjustifiedly forces the concept of sharp identity on all objects.Their way out of this difficulty is for some objects syntactically to forbid to speak of identity or non-identity, thus avoiding to have to confront the truth or falsehood of the identity of indiscernibles. Quasi-set theory stipulates two species of objects, those with with and without sharp identity.
We briefly sketch quasi-set theory. The non-logical symbols of the vocabulary of quasi-set theory comprise three unary predicate constants m (“is an indistinguishable atom, an i-atom”), a (“is an atom”), z (“is a set”), two binary predicate constants

(“is an element of”), I (“is indistinguishable from”), and a unary function constant q
(“the quasi-cardinality of”). Defined notions are quasi-sets (all objects that are not iatoms or atoms), i-sets (all quasi-sets that contain only i-atoms as elements), sets
(inductively: all quasi-sets that contain only atoms or sets as elements), and extensional identity (containing the same elements).
Thus extensional identity is defined in the same way as in those versions standard set theory where identity is defined. However, a syntactic restriction applies: Only atoms and sets can said to be identical or non-identical. Sets are uniquely determined by their elements. Quasi-sets that are not sets are also determined by their elements since sentences stating objects to be their elements are either true or false, hence assigning objects as elements of a given quasi-set is unique.
Quasi-cardinality is needed as a primitive symbol of the language of quasi-set theory because, in French’s and Krause’s words “quantum particles cannot be ordered or counted, but only aggregated in certain amounts”, so it is not defined in terms of identity, as is usually done in standard set theory. Quasi-cardinality is a function that does nothing but to assign an ordinal number—we usually think of a natural number—to quasi-sets. The quasi-cardinality of sets will be equal to their cardinality.
Quasi-set theory has a calculus which is furnished with 25 axioms in several groups.
Informally paraphrased, the groups of axioms are:
(i) Indistinguishability is reflexive, symmetric and transitive.
(ii) Identicals are indiscernable.
(iii) I-atoms and atoms do not contain elements; no i-atom is an atom and vice versa.
(iv) Indistinguishable objects are substitutable for each other in all elementary formulas, except for membership in quasi-sets.
(v) The axioms of Zermelo’s set theory, except for the extensionality axiom, and the axiom of choice hold, with indistinguishability substituted for identity.
(vi) All quasi-sets have a quasi-cardinality.
Thus the defined relation of extensional identity has all properties of identity in standard set theory, in particular, by (ii), substitutivity of identical objects in all formulas except for membership of sets. With substitutivity of indistinguishable
* French, Steven; Krause, Décio (2006). Identity in Physics: A Historical, Philosophical, and Formal
Analysis . Oxford University Press.
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objects in all formulas, indistinguishability would be the same as identity in standard set theory, so that quasi-set theory would collapse to standard set theory.
We note, however, that membership in quasi-sets is always true or false. As a consequence, principles (S0) and (S1) hold for all quasi-sets, however, somewhat emptily for quasi-sets that are not sets. That is, in quasi-set theory a hierarchy of sets is constructible on the basis of the empty set and atoms. In fact, the intended model of quasi-set theory is an extension of above set-theoretical hierarchy with urelemente.
The basis of urelemente is broadened to include the quasi-set of i-atoms. Quasi-sets constructed of i-atoms have no sharp identities, they are only distinguishable or not.
In quasi-set theory, different orderings of i-atoms and i-sets are indistinguishable, much in accordance with the motive of introducing the theory.
Above axioms constitute a theory which accounts well for the formal structure of quantum physics. Presently, the interesting question is whether quasi-set theory indeed achieves the feat of being a formal theory of non-individuals.
At first glance, quasi-set theory looks like an attractive option to counter the challenge of microscopic non-individuals to setting up a formal theory. Short reflection, however, reveals a couple of open questions.
Quasi-sets are syntactically forbidden to be individuals. Yet it is straightforward to extend quasi-set theory to an equivalent theory which allows certain quasi-sets to be individuals. Call the equivalence classes of i-sets under the relation of indistinguishability “p-sets”. Let each p-set be identical to itself as well as to each pset that is indistinguishable therefrom, and non-identical to each set as well as to each p-set that is distinguishable therefrom. Every p-set has a quasi-cardinality, which is usually a natural number n . For the intended physical application, we can think of each p-set as being placed at n spots in space simultaneously. In known quantum physics, there would be no more than two quasi-sets, that of bosons and that of fermions with quasi-cardinalities, in fact quite large natural numbers. We may even allow their quasi-cardinalities to change over time to allow for effects of quantum field theory. The p-sets of bosons and fermions are individuals in our extended theory, yet quasi-set theory and our extended theory are equivalent. In other words, standard set theory very well describes all the physics that is needed, provided such a description has been given within quasi-set theory. It is now a small step in the spirit of Ockham’s razor to get rid of quasi-atoms and quasi-sets as well as unnecessary axioms to yield a still equivalent theory. Describing quantum physics, in spite of its seeming embracing of non-individuals, therefore does not need a set theory of nonindividuals.
If we are not content with such an elimination of non-individuals through the back door, it is because we attach some meaning to i-sets and quasi-cardinality. To evaluate what our reduction has modified, we need to become clear about those meanings.
As mentioned above, quasi-cardinality has been defined without reference to identity, simply as an ordinal number assigned to quasi-sets. In a similar way, membership is strangely disconnected from identity, lending respective definitions a strong ad-hoc flavor.
As the last three paragraphs show, we need to attach meaning to the sentences of quasi-set theory to justify employing it. This requires setting up of a semantics of
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quasi-set theory which provides interpretations to expressions of its language and is formulated withn a metatheory of quasi-set theory.
Any metatheory of quasi-set theory obviously employs its own language, in terms of which we express the meaning of language expressions of quasi-set theory. What would that metalanguage be? Three possibilities readily offer themselves. First, the metalanguage can be the language of standard set theory, as metalanguages belonging to various formal theories often are. This choice, however, would render the whole project of quasi-set theory useless, since what we mean by its expressions would all be extensional, and all talk of non-individuals on the object level would be no more than speaking of symbols. Identity would be firmly restored. As a second possibility, the metatheory may employ the same language as quasi-set theory. "I-atoms" would mean what i-atoms as expressions of the metalanguage mean. If we do not know that, a meta-metalanguage steps in, and so on. For understanding what notions like i-atoms, i-sets and quasi-cardinality mean in terms we understand, this approach is obviously not helpful.
A third possibility is to use natural language, imprecise but close to our original intuition. We would then, for example, say that two indistinguishable i-atoms are interpreted as two objects that cannot be distinguished, that is, they can be swapped without us noticing any difference. (This sounds tautological, but is not.
“Indistinguishable i-atoms” is the paraphrase for a language expression of quasi-set theory.) Indistinguishability notwithstanding, the metalanguage makes a clear difference between one atom and two, even if they cannot be distinguished. Common sense strongly suggests that “two” in the metalanguage is nothing but “an object and another object not identical with the first”. Indistinguishability then has to be expressed by other means in the metalangauge. A hint at what French and Krause are thinking of is their saying “we cannot have, say, a decision procedure (even for finite sets) for checking whether a certain element belongs to a quasi-set or not” (2006, Sec.
7.4). So it seems the authors mean indistinguishability to be epistemic. In that case, the metalanguage is best made more precise by adding an epistemic operator to any metatheoretical description of two objects that cannot be distinguished, calling them
“two objects which no one knows to tell apart”. Then, non-individuals as ontological objects have vanished again, and quasi-set theory is no more than an epistemic extension of standard set theory, simlar to the proposal in the last section that considered composite nopn-individuals.
Even if some convincing interpretation of quasi-set theory as a genuine theory of nonindividuals came up, we have to contend that the principle of identity of indiscernibles is still violated. In that instance, quasi-set theory offers no progress over standard set theory. We briefly mention that for both approaches a way around that objection is to allow different spatial positions to count as different properties. This contravenes the spirit of Leibniz and requires some presuppositions about the nature of space which we do not discuss here. In both cases, the terms of quasi-set theory do not mean objects without clear identity.
Above remarks concern only the quasi-set theory constructed by French and Krause, not possible and future other attempts. Nevertheless, these considerations ought to show that setting up a logical system of non-individuals is less straightforward than one may hope for.
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