1. Introduction: The Received View and Its Criticism
Leibniz's two laws that relate identity and indiscernibility are each the converse of the other. The first, the ‘indiscernibility of identicals’, is not contentious either in logic or in relation to classical or quantum physics and states that identicals must be indiscernible or, equivalently, in the more perspicuous contrapositive form, that if a differs from b in some respect, then a and b are not identical. On the other hand, the ‘identity of indiscernibles’ (PII) states that indiscernibles are identical or, equivalently, again in the more perspicuous contrapositive form, that there are no two distinct things that have all their properties in common. Stated in the latter way, the principle is summarized as ‘the dissimilarity of the distinct’ (McTaggart) or, as Leibniz put it, ‘things that are distinct must differ in some way’. Of course, PII is trivial if the way in which the things differ is in being distinct. Hence, the true metaphysical significance of the principle is arrived at if it is true even when restricted to qualitative or non-identity-involving properties, for then it (together with its converse, the indiscernibility of identicals) collapses numerical identity and diversity to qualitative sameness and difference. If this is so, then it will be welcomed by empiricists since arguably the qualitative facts are those that are accessible to experience. So if facts about the identity and diversity of things are reducible to them, then the latter too would be in principle empirically accessible. It is argued below that recent discussion of PII and quantum mechanics has lost sight of the broader philosophical motivation and significance of PII. Before we explain the ‘received view’ of the status of PII in the light of quantum mechanics and the recent criticisms of it, more needs to be said about how PII should be understood.Footnote 1
The fate of PII is often tied to that of the bundle theory of individuation. If PII is true, then the individuality of objects can be grounded in their attributes; if PII is false, then either the objects in question are not individuals (as argued by French and Krause [Reference French and Krause2006]) or if they are individuals they must be individuated either by some primitive thisness, haecceity, bare particularity, or substance or their individuality must be primitive and ungrounded (perhaps being a feature of the structure to which they belong).Footnote 2 In order to consider the validity of PII, we have to be more precise about what can count as an attribute of an object. As mentioned above, on pain of trivializing the problem, we had better not count identity statements as representing attributes such as the property of being identical with particle a. For similar reasons we should exclude all attributes that involve ‘naming’ or ‘labeling’ numerically distinct objects. Obviously it is always theoretically possible to attach different names to numerically distinct objects (if we needed a formal argument for this self-evident claim, we could refer to the ZFC set theory in which it is possible to prove the existence of a bijection from the set of natural numbers {0, 1} to any two-element set), but applying this procedure ought not to be seen as discerning the objects in the sense required for PII. The first approximation to the notion of a qualitative attribute applicable in the context of PII is that of an intrinsic physical property. We will ignore well-known problems besetting proper analysis of the notion of intrinsicness, assuming provisionally that an intrinsic property of an object is a property that it possesses independently of everything else that occurs in the world and that it comes sufficiently close to the notion of nonrelational property.
So formulated, PII is false even for classical particles of the same kind since they possess all their intrinsic properties in common and differ only in their spatiotemporal relations. Suppose that the world is not globally spatially symmetric. Then the classical principle of impenetrability according to which the spatiotemporal trajectories of particles cannot cross implies that classical particles satisfy PII if the scope of ‘attribute’ is taken to include spatiotemporal attributes. This makes it clear that the limitation of attributes to intrinsic properties is too restrictive and that relational properties may fall in the scope of PII. Hence, it is commonly accepted that, for example, Max Black's (Reference Black1952) two qualitatively identical spheres are nevertheless discernible in the presence of a third object placed in a way that breaks the overall symmetry, for example, closer to one sphere than to the other. (Of course we are not making the suggestion that Black's spheres in a totally symmetric universe are discernible solely by virtue of the fact that it is possible to break the symmetry. Discernibility in a possible world does not imply discernibility in the actual world. Dieks and Versteegh [Reference Dieks and Versteegh2008] use the condition of possible relational discernibility only as a criterion of individuality of the objects in question, not of their actual discernibility.) Similarly, two incongruent counterparts, such as a left and a right hand, that are otherwise qualitatively the same should be regarded as satisfying the spirit of PII even though they have all their intrinsic properties in common and differ only by being incongruent. Again, if we consider inserting another hand, then it will be either a left or a right hand and will allow us to discern the original two hands.
It is important to note that the distinction between intrinsic and relational properties does not coincide with that between monadic and polyadic predicates. For example, the monadic predicate ‘is an uncle’ refers to a relational property, and the predicate ‘is a left hand’ also arguably refers to the relational property of being the mirror image of a right hand. Hence, objects that are absolutely discernible in the sense that there is a predicate in one free variable that applies to only one of them need not differ in respect of any intrinsic properties. Another example of absolutely discernible objects is provided by an asymmetric undirected graph in which no node has any intrinsic properties but every node can be described by a unique structure description that cites for each of the nodes it is connected to how many nodes that node is connected to (see Ladyman Reference Ladyman2007). Similarly, it is normally accepted that the natural numbers are absolutely discernible since, for every one of them, n, there is a predicate of the form 
$S_{n}$ that says how many times the successor function has to be applied to get to n from zero and that applies to it and it only; yet it is also often argued that they are individuated solely by the relations they bear to each other.
However, we can consider mathematical objects that do not satisfy absolute discernibility but are nonetheless relationally discernible. Formally we can say that two objects a and b are relationally discernible iff there is an n-argument relation R such that it is not the case that R(… a …) is true of exactly the same objects as R(… b …). Consider a one-dimensional space equipped with an antisymmetric ordering. Then every point stands in an asymmetric relation to every other point, and for any two distinct points there will be another point such that one of them stands in that asymmetric relation to it and the other does not. As we are discussing discernibility of physical objects, it is natural to limit relations that can discern to physical ones. However, we have to warn here that even when only relations described by a physical theory are considered, this does not settle the question of whether they are sufficient for physical discernibility. We will return to this issue later.
Given the above general characteristics of absolute and relational discernibility, it is not difficult to see where the dominant view regarding the indiscernibility of indistinguishable quantum particles comes from. By indistinguishable particles we understand particles that are such that when they are in a joint state ψ, permuting any particles within this state produces a state that is physically the same as the original state; in quantum mechanics, physical states are represented by one-dimensional rays, and hence the result of applying the permutation operator to the state ψ must be a state of the form c ψ, where c is a complex number.Footnote 3 As is commonly done, we will recognize two types of indistinguishable particles: bosons, for which c is a positive number, and fermions, for which odd permutations change the sign of the state so that c is a negative number, while even permutations leave it intact. In quantum mechanics, all state-dependent properties are represented with the help of Hermitian operators, and the expectation value of the quantity represented by an operator O in pure state ψ is given by the well-known formula 
$\langle \psi \vert O\vert \psi \rangle $. Now, it is straightforward to prove that in the case of n indistinguishable quantum particles the expectation value of any Hermitian operator is the same for all particles. To see this, let us symbolize by 
$O_{i}$ the operator that is defined as the tensor product of 
$n-1$ identity operators I and one operator O standing in the ith place of the product. The operator 
$O_{i}$ can be interpreted as representing some measurable property of the ith particle.Footnote 4 Let 
$P_{ij}$ symbolize the operator that permutes particle numbers i and j. Obviously, from the definition of indistinguishable particles we have
But 
$P^{-1}_{ij}O_{i}P_{ij}$ is just 
$O_{j}$, and hence we have proven that 
$\langle \psi \vert O_{i}\vert \psi \rangle =\langle \psi \vert O_{j}\vert \psi \rangle $ for all i, j. This result can be interpreted as showing that fermions and bosons are absolutely indiscernible.
The case of relational discernibility is a bit more complicated because there is more than one way in which relations can be represented within the quantum-mechanical formalism. One way of interpreting relational properties of separate particles is with the help of conditional probabilities for obtaining a particular outcome on one particle given that certain outcomes were obtained on the remaining particles (for details, see French and Redhead Reference French and Redhead1988; French and Krause Reference French and Krause2006). But there is a simpler way of speaking about relations in the context of indistinguishable quantum particles that has been proposed by Dieks and Versteegh (Reference Dieks and Versteegh2008). Let us imagine that a new particle has been introduced to the collection of n indistinguishable particles (following Dieks and Versteegh, we can call it a gauge particle). Arguably, all relations between the gauge particle and the remaining particles are representable by Hermitian operators acting in the 
$n+1$ component tensor product of Hilbert spaces. If we choose one of such operators 
$O_{gi}$ that is supposed to denote a physical relation between the ith particle and the gauge particle, then by reasoning analogous to the previous case we can show that the expectation value of 
$O_{gi}$ in any symmetric or antisymmetric state ψ is the same as that of 
$O_{gj}$:
This result, easily extendable for an arbitrary number of gauge particles, can convince us that fermions and bosons should be seen as relationally indiscernible too.
However, recently Saunders (Reference Saunders2006), Muller and Saunders (Reference Muller and Saunders2008), and Muller and Seevinck (Reference Muller and Seevinck2009) have challenged the received view according to which quantum particles violate PII. Originally Saunders argued that fermions are in a weak sense discernible after all. Then he and Muller argued that the same is true of bosons in some but not all states. Finally, Muller and Seevinck argue that bosons and fermions in all states are weakly discernible. The notion of weak discernibility that Saunders and Muller use to support their claims goes back to Quine (Reference Quine1976). Quine suggested that objects can be discerned by way of showing that they stand in an irreflexive relation to each other. In his 2006 paper Saunders applied the concept of weak discernibility to the case of two fermions in the singlet spin state, arguing that they are discernible with the help of the irreflexive and symmetric relation ‘having opposite components of spin’. Muller and Saunders (Reference Muller and Saunders2008) extend this result for an arbitrary number of fermions and arbitrary antisymmetric states (pure and mixed), and they also introduced a notion of probabilistic as opposed to categorical discernibility.
The details of their proposal are somewhat technical, but the main idea can be presented in a simplified way as follows. In the case of two fermions a and b, Muller and Saunders construct a Hermitian operator 
$A_{ab}$ with the following properties: if 
$a\neq b$, then any antisymmetric state W is an eigenstate of 
$A_{ab}$ with eigenvalue λ, while for 
$A_{aa}$ it remains true that W is its eigenstate, but with a numerically different eigenvalue 
$\mu \neq \lambda $. From this it follows that if we define a binary relation Rab as ‘a and b are in a state in which the measurable quantity represented by 
$A_{ab}$ takes on value λ', then clearly this relation will be symmetric and irreflexive, and hence a and b are weakly discernible in all available states. Of course the key element of this argument is the construction of the operator 
$A_{ab}$. Muller and Saunders ensure that it possesses all the required features by using a clever combination of projection operators corresponding to some arbitrary physical magnitude. One may note, however, that if we relaxed a bit the stringent requirements for admissible discerning relations that Muller and Saunders insist on, we could achieve the same result in a much simpler way, namely, by introducing the following relation: Sab iff the state that a and b jointly occupy is antisymmetric. Since no single particle can be in an antisymmetric state, Saa is always false; hence S discerns weakly all fermions.Footnote 5
Regarding the case of bosons, as mentioned above, Muller and Saunders distinguish cases of categorical discernibility and probabilistic discernibility. Bosons can be categorically weakly discerned only in certain states; however, they can always be weakly discerned probabilistically. The relation that does the job in this case is defined in terms of expectation values of Hermitian operators. If we choose an operator A that is not dispersion-free in a given state of the system, then it can be shown that the expectation value of the tensor product 
$A\otimes A$ is different from that of the square 
$A^{2}\otimes I$. Again, as in the case of fermions, a much more intuitive example of a weakly discernible relation can be supplied; however, it may be seen as failing to conform to the standards of physically admissible relations adopted by Muller and Saunders. Let us define relation Tab as ‘a's momentum and b's position commute with each other’. Clearly Tab holds for two numerically distinct particles, whether bosons or fermions, but it does not hold for a single particle, since position and momentum of the same physical system are noncommuting. If this is admitted, then even bosons are categorically discernible as argued by Muller and Seevinck (Reference Muller and Seevinck2009).
2. Weak Discernibility, Probabilistic Discernibility, and PII
Having explained the approach adopted by Muller and Saunders, we now turn to the fundamental question of whether the weak discernibility of fermions and bosons should have any consequences for the validity of the PII. On the face of it, it looks like the existence of an irreflexive and symmetric relation joining objects a and b cannot help discern them qualitatively but only ensures that a and b are numerically distinct entities. If we insisted that numerical difference has to be grounded in differences in individual attributes, then perhaps we could argue that weak discernibility indeed implies the existence of some qualitative difference, but this reasoning would beg the question, for it relies on a supposition that is denied by those who argue that PII is violated for quantum particles. Bas van Fraassen and Isabelle Peschard (Reference van Fraassen and Peschard2008) state that the condition of weak discernibility “is a condition which entails that there is more than one entity of the given kind, without implying that there are specifiable differences that distinguish them” (19). They insist that the existence of an irreflexive relation joining two fermions does not vindicate PII but rather explains “how we can assert indiscernible entities to be distinct” (21).Footnote 6
Generally, we agree with van Fraassen and Peschard's position regarding the notion of weak discernibility and its connection with PII. However, Muller and Saunders, Muller and Seevinck, and Dieks and Versteegh clearly do not share this view. In what follows we will try to assess, impartially and charitably, the reasons they can have for claiming that weak discernibility has anything to do with the validity of PII. Muller and Saunders devoted a substantial portion of their long paper to careful analysis of various notions of discernibility and logical relationships among them. Taking their cue from Quine, they treat weak discernibility as a particular instance of the notion of relational discernibility explained above, and the latter seems to be a perfectly legitimate expression of the intuitive notion of discernibility. On the other hand, we argue in the next subsection that their reconstruction of Quine's ‘grades of discriminability’ harbors some confusion, which may have contributed to their view—mistaken in our opinion—that the weak discernibility of indistinguishable particles can save PII in quantum mechanics
2.1. Grades of Discriminability
In order to keep with the way Quine and Muller and Saunders treat the problem of discernibility, we must relativize this notion to a particular language. This move has its disadvantages. For instance, it is customary to distinguish between absolute and relational discernibility on the basis of the distinction between monadic and polyadic predicates. But we should not forget that the fact pointed out above that a predicate is monadic in a given language does not guarantee that the corresponding property is nonrelational (intrinsic). There is nothing that could stop us from counting the term ‘a wife of Henry VIII’ among primitive monadic predicates of our language, although it is dubious to say the least that it denotes an intrinsic property of the six women who were fortunate (or unfortunate) enough to have tied the knot with that famous (or infamous) ruler of England. Thus we should keep in mind that in order to make the following distinctions metaphysically salient, we would have to accept additional assumptions regarding the interpretation of primitive predicates of our language.
Another clarificatory comment is in order here. We have to assume that the language 
in which the notions of discernibility are to be defined does not contain any individual names. This requirement should be self-explanatory in the light of our earlier remarks about the inadmissibility of ‘labeling’ as a way of discerning objects. Clearly, if we had enough individual names at our disposal, we could easily ‘discern’ all objects in the domain (given that the domain is finite, which seems to be a plausible assumption for physical objects). Discernibility should not be so cheap.
Muller and Saunders define absolute discernibility in a given language as follows:
(1) a and b are absolutely discernible in iff there is a monadic predicate M in such that 
$Ma\wedge \neg Mb$ or 
$\neg Ma\wedge Mb$.
Definition (1) roughly corresponds to Quine's notion of strong discriminability, with the exception that Quine speaks about open sentences in one free variable rather than monadic predicates. It is unclear whether by monadic predicates Muller and Saunders mean primitive monadic predicates of or monadic predicates definable in . If the latter is the case and if we are allowed to use any logical constants including quantifiers in the definitions of new predicates, then (1) becomes equivalent to Quine's strong discriminability. But if M is meant to refer to primitive predicates only, the formula (1) defines a much stronger sense of discernibility that implies, but is not implied by, strong discriminability. To see this it suffices to consider the case in which there is a dyadic predicate R such that ∃x Rax ∧ ¬∃x Rbx; however, no monadic primitive predicate discerns a and b. An example can be provided by the variant of two Black spheres in which a third object is placed asymmetrically next to a and b. If all primitive monadic predicates in are selected to denote intrinsic properties of the spheres, then a and b are not discernible in the strongest possible sense. However, when R denotes the relation ‘being d meters apart’ and the third particle happens to be exactly d meters from a and not from b, the spheres are clearly discernible by the open formula in one variable ∃x Rax. In what follows we will assume that predicates in (1) are interpreted in a broader sense as being definable in .
The second notion of discernibility is relational discernibility, which can be characterized as follows:
(2) a and b are relationally discernible in iff there is some (
$n+1$)-ary predicate R (
$n> 0$) in such that 
$\exists x_{1}\,\ldots \,\exists x_{n}$ (
$Rax_{1}\,\ldots \,x_{n}\leftrightarrow \neg Rbx_{1}\,\ldots \,x_{n}$).
As in the previous case we will assume that predicates are definable with the help of any open formula containing an appropriate number of free variables in . Note that the satisfaction of (2) does not imply that there will be an open formula true of a and false of b (the truth of 
$\exists x_{1}\,\ldots \,\exists x_{n}$ 
$Rax_{1}\,\ldots \,x_{n}$ and 
$\exists x_{1}\,\ldots \,\exists x_{n}$ 
$Rbx_{1}\,\ldots \,x_{n}$ is perfectly compatible with 
$\exists x_{1}\,\ldots \,\exists x_{n}$ (
$Rax_{1}\,\ldots \,x_{n}\leftrightarrow \neg Rbx_{1}\,\ldots \,x_{n}$) being true). Definition (2) would imply (1) if every object in the domain had its unique name, for in such a case the discriminating formula would have the form 
$Rac_{1}\,\ldots \,c_{n}$, where 
$c_{1},\,\ldots ,\,c_{n}$ are the names of the objects whose existence is stated in (2). But, as we recall, we have specifically excluded individual names from language .
Now Muller and Saunders introduce two more notions of discernibility: relative and weak discernibility. They suggest that, according to Quine, these new types of discernibility form ‘independent’ categories of relational discernibility (2008, 528). But this is incorrect. As we will see, relative discernibility (in Quine's terminology ‘moderate discriminability’) is logically stronger than (2), while weak discernibility is surprisingly equivalent with (2). But first let us present appropriate definitions of these two concepts.
(3) a and b are relatively discernible iff there is an open formula F in two variables in such that 
$F( a,\,b) \wedge \neg F( b,\,a) $.
(4) a and b are weakly discernible iff there is an open formula F in two variables in such that 
$F( a,\,b) \wedge \neg F( b,\,b) $.
These two notions are often characterized as discernibility with the help of an asymmetric relation (in the case of [3]) or discernibility with the help of an irreflexive relation (the case of [4]). But note that actually we do not need to assume that F defines an asymmetric or irreflexive relation in the whole domain; all we need to discern the two objects a and b is that F is asymmetric or irreflexive in the domain narrowed down to {a, b}, which is precisely what conditions (3) and (4) state.
Notably, Quine introduced his notion of weak discriminability not with the help of definition (4) but rather with (2) (he never used the term ‘relational discriminability’), and only later did he admit (crediting Ivan Fox with the observation) that weak discriminability can be presented equivalently with the help of an irreflexive open formula with two variables. That (2) and (4) are indeed equivalent can be easily shown. First, suppose that 
$F( a,\,b) \wedge \neg F( b,\,b) $ for some a, b. From this it clearly follows that 
$\exists x$ 
$[ F( a,\,x) \wedge \neg F( b,\,x) ] $, which proves that (2) is satisfied. To prove the implication in the opposite direction, let us assume that a and b can be discerned thanks to the following fact: 
$\exists x_{1}\,\ldots \,\exists x_{n}$ (
$Rax_{1}\,\ldots \,x_{n}\leftrightarrow \neg Rbx_{1}\,\ldots \,x_{n}$). Let us abbreviate this formula as 
$F( a,\,b) $. It is obvious that 
$F( b,\,b) $ is a logical contradiction; hence we have proven that 
$F( a,\,b) \wedge \neg F( b,\,b) $.
As for relative discernibility, Quine himself presented an argument showing that (2) is weaker than (3) while (3) is weaker than (1). If our language was limited to those dyadic predicates that are symmetric in a given domain (for instance, if contained identity as the sole predicate), then it still would be possible to discern relationally, for instance, with the help of the formula 
$x=x\wedge \neg x=y$, and yet no formula would be satisfied by pairs of objects in one order only. On the other hand, the fact that (3) is weaker than (1) can be established as follows: all ordinals are relatively discernible, but they cannot be discerned absolutely, for there are not enough open formulas in the language to cover them all. In conclusion, a complete set of logical relations connecting all kinds of discernibility can be presented on the following diagram (we give Quine's alternative terminology in parentheses where available):
-
Absolute discernibility (strong discriminability)
-
⇓
-
Relative discernibility (moderate discriminability)
-
⇓
-
Relational discernibility
-
⇓⇑
-
Weak discernibility (weak discriminability)
2.2. Physical Discernibility
Having sorted out some logical issues related to the notion of discernibility, let us go back to the case of quantum particles. Muller and Saunders observe that indistinguishable quantum particles cannot be discerned absolutely or relatively, for this would imply that there is a formula that is true of some two objects in one order and not in the other and hence would violate the permutation invariance. Thus the only notions left are relational and weak discernibility. Given the equivalence between the two, it looks as though Muller and Saunders's result proves that fermions (and bosons too) are relationally discernible and therefore that PII is not violated by them. Is there any reason this conclusion should be resisted?
We believe there is. The key to our argument is the requirement of physical discernibility that Muller and Saunders insist on. They correctly point out that physical discernibility is the only notion interesting from the point of view of the validity of PII. As we indicated at the beginning of the paper, no amount of playing with names or labels can convince us that two physically indistinguishable particles are in fact discernible. Muller and Saunders make sure that the notion of discernibility they are working with is indeed physical by limiting all extralogical terms in to predicates that can be defined using clearly delineated physical procedures. But is this restriction on the predicates of sufficient to ensure that the discernibility given by condition (2) will indeed conform to our intuitions of what physical discernibility should look like?
Consider again the case of two Black spheres (at best, technical niceties aside, the fermion/boson case is like that of the two spheres). Saunders claims that they are weakly discernible with the help of the relation ‘being 1 mile apart’. Translating this into the language used in definition (2), we can see that this type of discernibility can be expressed in the following statement: there is an object such that one sphere is 1 mile apart from it, while the other one is not. But note that the object that supposedly discerns the spheres via the relation of being 1 mile apart is none other than one of the spheres. But only one of the two spheres can play this role at a time, so we should ask which of these two spheres the required object is. However, there is no way we could answer this question in a noncircular way, thanks to the assumption that the spheres are absolutely indiscernible. If we could introduce names (labels) into the language, there would be no problem. But labeling is not a physical procedure, as Muller and Saunders themselves admit. Hence it may be argued that the discernibility of the two spheres with the help of the irreflexive relation ‘being 1 mile apart’ does not amount to physical discernibility since it in no way allows us to tell which sphere is which.Footnote 7
This is not to say that the entire category of relational discernibility has to be rejected as nonphysical and that the only admissible discernibility is the absolute one. Let us make this clear by considering a perfectly acceptable case of relational discernibility not reducible to the absolute one. Suppose that the Black universe has been expanded by two additional objects 
$a^{\prime }$ and 
$b^{\prime }$ that have been located symmetrically each a mile apart from one of the spheres and that are physically different from each other as well as from the spheres. In this case spheres a and b are discernible with the help of the formula ‘is 1 mile apart from’, for there is an object that is 1 mile apart from one of them, is not 1 mile apart from the other, and moreover is absolutely discernible from any other object in the universe. Hence, although technically we do not have the names of the separate ‘gauge’ particles in our language, we know that there is a legitimate physical procedure that could help us distinguish one sphere from the other. But the situation would be entirely different if the gauge particles were themselves indiscernible. In that case the fact that the spheres stand in different relations to one gauge particle gives us no clue regarding the identity of the spheres, for the relations will be exactly reversed when we take into account the twin gauge particle.
Muller and Saunders apparently overlooked the fact that there are two elements involved in relational discernibility of objects a and b: the relation itself and the objects that the relation connects with a but not with b. The condition of physicality that they impose on relational discernibility affects only the choice of the admissible relations but is silent about the arguments of the relation that do the work. Of course it does not make sense to demand that the discerning objects be physical as opposed to nonphysical. Rather, we should make sure that there is a proper physical procedure that could in principle identify those objects as distinct from the remainder of the universe that does not discern them. Hence, we suggest that a corrected version of the condition of relational discernibility should look something like that (for simplicity's sake we will consider only the case of dyadic relations):
(5) a and b are relationally physically discernible in iff there is some physically defined relation R such that 
$\exists x$ 
$[ ( Rax\wedge \neg Rbx) \wedge \forall y\,( y\mathrm{Ind}\,x\rightarrow ( Ray\wedge \neg Rbx) ) ] $, where 
$y\mathrm{Ind}\,x$ means that y is absolutely physically indiscernible from x.Footnote 8
Condition (5) ensures that if the discerning (‘gauge’) object has a twin, this twin must stand in the same relations to the discerned objects as the original one. Hence the case of discernibility via the irreflexive relation of being 1 mile apart is obviously excluded, for it is true that 
$Rab\wedge $
$\neg Rbb$, but not that 
$Raa\wedge \neg Rba$, in spite of the fact that a and b are themselves indiscernible.
One may ask why not use the simpler condition that prescribes that discerning objects should not have indiscernible copies (‘twins’). The answer is that this restriction is unnecessarily strong. For instance, it is perfectly acceptable to discern the Black spheres with the help of two qualitatively identical objects that stand in the same relation to one sphere but not to the other. The only situation that is not acceptable is when one discerning object stands in the given relation to one sphere if and only if the other discerning object stands in exactly the same relation to the other sphere, for in that case the overall symmetry precludes any effective discernibility.
But now the question is why we should choose condition (5) rather than (2) as an expression of physical relational discernibility. What makes condition (2) less physical than (5)? We may explain this by asking, what is the ultimate goal of discriminating two objects physically? The answer may be: to ensure the existence of a possible physical procedure that could yield one outcome when applied to one object and a different outcome when applied to the other one. This procedure does not have to be practically executable, but it should be available to us in principle. Speaking metaphorically, we want to be able to ‘mark’ one object and not the other one. Condition (5) affords us such a procedure (if only in the form of a thought experiment). It states that in our domain (which is assumed to be finite) there are objects (possibly more than one) that collectively discern analyzed entities a and b by standing in some relation R to one of them and not to the other one. By assumption, the set of discerning objects is closed under the relation of absolute physical indiscernibility. Thus, there is a (possibly complex) physical property P such that any object c that possesses P discerns objects a and b via relation R. Consequently, a procedure that verifies which objects possess P gives us an algorithm for discerning a from b. But such an algorithm is not available to us when only condition (2) is satisfied. If there are two absolutely indiscernible objects such that only one of them discerns a from b by standing in R to a and not to b, there is no way that we could pick this one by applying any algorithm whatsoever. Thus the discernibility of a and b achieved in this case is no better than the ‘discernibility’ given by reference to hypothetical primitive thisness (see below).Footnote 9
3. Further Reflections
We have argued that the formal condition of weak discernibility, while being sufficient to establish numerical diversity, is not properly thought of as establishing that two objects can be discerned from each other. ‘Weak discernibility’ is a misnomer because it is orthogonal to what is at issue when the status of PII is discussed. Recall that PII is paraphrased by Leibniz as the claim that distinct things must differ in some way. On the contrary, objects that are only weakly discernible do not differ in some way. Black's spheres are such that each is 2 miles from the other and not from itself, but each sphere is the same as the other in respect of being 2 miles from a qualitatively identical sphere. The relation that weakly discerns them does not amount to a difference between them over and above their numerical distinctness. Weak discernibility via relations like ‘is 2 miles from’ or ‘has opposite spin in the z-direction to’ is a qualitative feature, but it is not one by which we can discern the individuals in question. Hence, we conclude that the received view is correct and that PII is violated by quantum particles. However, there are further issues that remain to which we now turn, namely, whether there are important differences between the indiscernibility of quantum particles and that of classical objects such as Max Black's two spheres, the relationship between PII and Pauli's exclusion principle, and finally the question of whether and, if so, how individuals are more than mere objects.
As mentioned above, Saunders (Reference Saunders2006) originally argued that fermions are weakly discernible in virtue of being in entangled states like the singlet state: consider the relation ‘has opposite spin in direction z to’.Footnote 10 Then Muller and Saunders (Reference Muller and Saunders2008) argued that only elementary bosons in some states are not weakly discernible, and Muller and Seevinck (Reference Muller and Seevinck2009) that both elementary bosons and elementary fermions are weakly discernible in all states. French and Krause (Reference French and Krause2006) and Hawley (Reference Hawley2006) have argued that the appeal to weak discernibility even in the original case is circular. Most recently Hawley (Reference Hawley2009) says that weak discernibility is “grounded in the fact that the objects in question are distinct; weak discernibility cannot itself be the ground of distinctness” (110).Footnote 11 We are not here concerned with the question of whether weak discernibility grounds distinctness but rather with the question of whether the relations employed in its service can really be said to allow the particles to be ‘discerned’. Muller and Seevinck certainly conflate these issues when they say that if particles appear to violate PII when only physical properties are considered, they may nonetheless be “discerned by means that go above and beyond QM, such as by ascribing Scotusian haecceitas to the particles, or by ascribing sibling attributes to them from scholastic and neoscholastic metaphysics” (Reference Muller and Seevinck2009, 180). Of course by the latter they have in mind primitive thisness or bare particularity and the like, but these notions are introduced to provide a principle of individuation for entities that are not discernible and hence for which it is assumed (following the connection that is usually made to which we referred in our opening remarks) that the bundle theory of individuation is not viable. If we suppose that there are haecceities, it is not correct to say that they allow the individuals that have them to be discerned. They are posited only because it is assumed or argued that there can be indiscernible yet distinct entities.
The above critiques of Muller and Saunders all have in common that they apply equally well, if at all, to the claim that Black's two spheres are weakly discernible by the irreflexive spatial relation of being some distance apart. Hence nothing about the quantum nature of the particles is at issue. However, the quantum nature of the properties and relations into which particles enter is of further interest since it considerably complicates matters, as we explain below.
Saunders's appeal to the anticorrelation in spin of fermions in the singlet state to establish weak discernibility might lead one to think that weak discernibility is based on the holding of an irreflexive relation analogous to the classical relation two spheres might have of being oppositely charged. Interesting questions arise about whether two Black spheres of opposite charge should count as absolutely or just relationally discernible depending on whether positive and negative charges are different intrinsically qua qualitative properties or whether swapping positive and negative charges makes no difference. However, either way, such spheres are weakly discernible anyway by their spatial relation if it is granted that Black's original spheres are. Now when we consider fermions (and bosons) because they are quantum particles, they are said to lack the spatiotemporal trajectories that would be sufficient to absolutely discern classical particles (as mentioned above, assuming a sufficiently spatiotemporally asymmetric world). So in most discussions, spatial degrees of freedom are ignored, and instead spin comes to the fore. Hence, Hawley (Reference Hawley2009) says that fermions are weakly discernible because “each has spin opposite to the other” (108), and Lyre (Reference Lyre2009) says that fermions stand in an irreflexive relation when in an entangled state for the relation of having opposite spin (6). (Saunders originally suggested the opposite spin relation and did not mention the spatial degrees of freedom presumably because it is well known that electrons in the singlet state can also be in the very same eigenstate of energy and hence have the same spatial wave function.)
However, saying that they have opposite spin masks the fact that property attribution in quantum mechanics is very contentious. The old orthodox/Copenhagen thing to say would of course be that neither of them has a definite spin in any direction, but that if measurements are made of their spins, then they will never be the same in the same direction. This is what makes many people balk at Saunders's claim that they are indeed discernible because to say that such fermions have opposite spin is to ignore the fact that quantum systems do not have properties like spin ‘in the classical way’ (whatever that means). This line of criticism in a much more sophisticated form underlies the critique of Muller and Saunders due to Dieks and Versteegh (Reference Dieks and Versteegh2008). Our arguments are orthogonal to theirs, but they do make an important claim that we argue is false when they say that after measurement the symmetry is broken and “distinct individuals will arise” (933). There is a temptation to think that fermions in the singlet state are somehow not individuals when they are entangled, perhaps because of the connotations of holism and so on, but that after measurement, since they now can be said to have definite spins, for example, Alice's is up in the z-direction, the situation is just like the classical case of two discernible objects where they really do have opposite spins in a particular direction because of underlying properties possessed by distinct individuals.
However, it is important to consider what really happens according to standard quantum mechanics if a projective measurement is made on a pair of fermions in the singlet state. Certainly their spin state becomes a product state; however, according to the Spin-Statistics theorem, fermions must always be in antisymmetric states. Saunders insists that the account of discernibility should respect the symmetries of the physical theory. That is why he distinguishes elementary bosons and elementary fermions. Similarly, Muller and Saunders (Reference Muller and Saunders2008) cite Weyl referring to the ‘Leibniz-Pauli’ exclusion principle and point out that their view explains why Weyl, who is otherwise so philosophically astute, could deny that elementary particles violate PII in the face of the obvious fact of the indistinguishability postulate and quantum statistics as the received view would have it. It is worth noting that on Muller and Seevinck's (Reference Muller and Seevinck2009) view, there is no connection between the Pauli exclusion principle and discernibility even if we grant that the particles are discerned. On the Muller and Seevinck view, the difference between bosons and fermions is not respected since even the former can be weakly discerned.
It is worth examining more closely the relationship between the exclusion principle and PII. On a superficial level one might reason thus. The exclusion principle says that no two fermions can have all the same quantum numbers. This is not true for bosons. Hence the former can be said to satisfy PII since any two will be found to have different physical properties. To be concrete, consider a pair of electrons in the first orbital of an atom that are in the same energy eigenstate. They cannot both have the same spin in a given direction. So they are discernible because they have different spins. (However, of course, their joint state is not one that quantum mechanics says can be interpreted as one having spin up and the other spin down.) This reasoning is quite sensible. But now consider how things stand if we think of the difference between bosons and fermions not in terms of the above form of the exclusion principle but in terms of symmetric and antisymmetric states. With regard to the singlet state and what the state is after Alice's projective measurement, intuitively they will be in the state corresponding to Alice's up and Bob's down, say. However, the quantum state that we write down may not be such a state because that is a product state in the full tensor product Hilbert state and no antisymmetric states in the latter are product states. To write down the state of the electrons we must antisymmetrize by adding terms corresponding to swapping which particle is on the left and which on the right so the state is actually on with two terms.Footnote 12
So in a sense, antisymmetry demands not that no two fermions be in the same state but that they all be in the same state. Not in the sense that if we make localized measurements of them in different regions where they give the same values for all observables we can measure—they will not; hence the original form of the exclusion principle has a validity if interpreted as referring to measurement outcomes—but in the sense that after we have made such measurements, their states are still antisymmetrized and include terms incorporating the superposition with respect to which fermion was where. Now of course it is fair to argue that this fact shows that we should eschew the excess structure and use a Fock space formalism; however, neither fermions nor bosons are even weakly discernible according to the latter because there is not even the linguistic means to putatively refer to or label individual particles. In any case, for the moment we are assessing standard QM, and there is certainly something to the idea that all fermions are in the same state according to the state we write down, since it is true of it that whatever is predicated of one particle is predicated of the other up to a phase factor of −1. It is not true that one particle belongs to Alice and is spin up, because it matters that the quantum state of Alice's and Bob's fermions is antisymmetric.Footnote 13
As we mentioned above, appeal to that fact that all fermionic states are antisymmetric provides a trivial proof that they are all weakly discernible since the relation of being antisymmetrized with is irreflexive. However, the original motivation for PII involves the grounding of individuality in qualitative features of the world. The forms of weak discernibility appealed to by Muller and Saunders involve mathematical features of the quantum mechanical representation of the world, and it is not always clear when we should say that they represent real properties and when they are mere artifacts. Even granting that weak discernibility by a qualitative relation holds, we argued that it need not have anything to do with the original idea of discerning, that is, telling apart, two objects. Consider two bosons in exactly the same state. Muller and Seevinck prove that they are discernible. But they are obviously not discernible. So the mathematical results appealed to by them cannot have the significance they are alleged to have. There is something to this argument. It may be debatable whether Black's spheres can be said to be discerned by the spatial separation, but it is definitely an abuse of language to say that two bosons in the same state can be told apart. The received view has in its favor that the field theoretic Fock space formalism disposes with particle labels altogether, and interpretations of field theory and particle physics in terms of individual particles are not taken seriously. In the case of nonrelativistic many-particle quantum mechanics, both fermions and bosons are completely indiscernible.
Finally, we return to the relationship between individuals and objects. Recall that French and Krause argue that quantum particles are nonindividual objects because they violate PII and they reject a nonqualitative principle of individuation. Muller and Saunders argue that even though fermions satisfy PII, they are ‘relationals’, not individuals, since they regard the latter as necessarily absolutely discernible. Other plausible criteria for individuals over and above objects include the obtaining of identity over time and/or determinate transworld identity. It is not clear that these have anything to do with absolute discernibility. On the other hand, perhaps it is better to think of the terms ‘individual’ and ‘object’ as having the same meaning. The most minimal view is that anything that is the value of a first-order variable is an individual. There is much further work to be done on these matters.
