The Identity of Indiscernibles

First published Wed Jun 4, 2025

The Identity of Indiscernibles is the thesis that there cannot be numerical difference without extra-numerical difference—that is, there cannot be two objects that differ only numerically, solo numero. It is an important issue in metaphysical discussions of identity, haecceitism/non-haecceitism, and theories of properties (since extra-numerical identity is often explained in terms of properties). Although it is most often referred to as a principle, its axiomatic character is not currently a topic of discussion, and the fact that it is referred to as such is simply due to Leibniz’s influence.

1. Brief History of the Identity of Indiscernibles

The Identity of Indiscernibles has long been a topic of philosophical reflection. For instance, the Stoics and Nicholas of Cusa maintained the Identity of Indiscernibles, and Aquinas asserted it of separated intelligences, i.e., non-material minds. But it was Leibniz who made the Identity of Indiscernibles a distinctive topic of philosophical reflection. Indeed, by producing arguments for and from it, and by making it a central element of his influential philosophical system, Leibniz made the Identity of Indiscernibles more salient to his contemporaries and successors (for discussion of the role and place of the Identity of Indiscernibles in Leibniz’s system, and his arguments for and from it, see Rodriguez-Pereyra 2014). After Leibniz, philosophers of the calibre of Kant, Bolzano, Russell, Wittgenstein, and MacTaggart all discussed the Identity of Indiscernibles. And in the mid twentieth century it was Max Black, in a now classic article published in Mind, who, by putting forward a provocative, if not completely original, thought experiment, produced a renewal of interest in the Identity of Indiscernibles and set much of the agenda for discussion of it since the 1950s onwards (see Black 1952). Black’s thought experiment consists in conceiving a universe where there are only two iron spheres, at a certain distance from each other, having the same diameter as each other, the same colour as each other, the same temperature as each other, etc. If such a universe is possible then at least some versions of the Identity of Indiscernibles are not true. The Identity of Indiscernibles has continued to be discussed since then up until the present.

2. Formulation of the Identity of Indiscernibles

As stated above, the Identity of Indiscernibles is the thesis that there cannot be numerical difference without extra-numerical difference. There is numerical difference between any objects when those objects are not one, i.e., when they, taken collectively, are more than one. Thus, the Identity of Indiscernibles means that, necessarily, any two objects differ other than merely numerically. The three elements of this thesis, the modal operator, the domain of the quantifier (objects), and the relation that is said to obtain between any two members of that domain (extra-numerical difference) are open to variation and/or interpretation, and any such variation or interpretation produces a different version of the Identity of Indiscernibles.

Let us start with extra-numerical difference. Two objects differ numerically when they differ with respect to which objects they are, while they differ extra-numerically when they differ with respect to how they are. One can interpret extra-numerical difference in terms of properties, or in terms of similarity. On the first interpretation, two objects differ extra-numerically when they differ with respect to some property. On the second interpretation, two objects differ extra-numerically when they don’t resemble exactly. In the contemporary discussion of the Identity of Indiscernibles extra-numerical difference is invariably understood in the former way. However, it is interesting to note that Leibniz typically formulated the Identity of Indiscernibles in terms of perfect similarity, e.g.: “It is not true that two substances resemble each other entirely and differ solo numero” (Leibniz 1686: §9 [2020: 14]). The former conception of extra-numerical difference is the focus in what follows, given its dominance in the contemporary discussion of the Identity of Indiscernibles.

With extra-numerical difference thus understood, the Identity of Indiscernibles can take a very weak form, namely as stating that, necessarily, any two objects must differ in at least one property. In other words:

(1)
Necessarily, no two objects share all their properties.

But, as has been noted innumerable times, although this thesis is true, it is trivially true (e.g., Adams 1979: 11, Van Cleve 2002: 389–90, to mention just two philosophers who have made the point). For no two objects, \(a\) and \(b\), can share all their properties. For if they did, they would also share their properties of identity (i.e., the properties of being identical with \(a\) and being identical with \(b\)) and their properties of difference (e.g., the properties of being numerically different from \(a\) and being numerically different from \(b\)), and many other similar properties. But if they shared these properties, \(a\) and \(b\) would not be two objects. Therefore, necessarily, any two objects must differ in at least some such properties—but that any two objects differ at least with respect to their properties of identity and difference is a triviality.

But this trivial thesis is not even a version of the Identity of Indiscernibles, since differing with respect to properties of identity, properties of difference, and similar ones, is differing numerically. Indeed, for objects to differ with respect to properties of identity and properties of difference is for them to differ with respect to which objects they are, not to differ with respect to how they are; but what the Identity of Indiscernibles requires is that every two objects differ with respect to how they are, that is, it requires that they differ extra-numerically.

Whether or not (1) should count as a version of the Identity of Indiscernibles, properties of identity and properties of difference are not the only properties that trivialize it. For instance, properties like being a member of {a} and being identical with a and being green also trivialize (1). There is thus the problem of finding a characterization of the properties that trivialize (1). Roughly, a trivializing property is one that differing with respect to it does not require differing extra-numerically, while a non-trivializing property is one that is not trivializing, i.e., a property that differing with respect to it requires differing extra-numerically. So, for objects to differ with respect to a non-trivializing property is for them to differ with respect to how they are and not merely with respect to which objects they are. (For more specific proposals about the characterization of non-trivializing properties see Katz 1983; Rodriguez-Pereyra 2022: 50–51.)

Thus, to formulate a non-trivial version of the Identity of Indiscernibles, or indeed to formulate a version of the Identity of Indiscernibles at all, one needs to either vary or interpret differently the notion of property (and thereby the notion of extra-numerical difference). For instance, some philosophers take properties to be universals that make a causal difference to their possessors (e.g., Armstrong 1997). Now, since on such a conception of properties there are no properties corresponding to predicates like “is identical with \(a\)”, “is numerically different from \(b\)”, etc., on such conception of properties the thesis that, necessarily, no two objects share all their properties is not a triviality.

But one could still have a very broad notion of property, on which there are such properties as being identical with \(a\) and being numerically different from \(b\), and decide to restrict the properties referred to in the Identity of Indiscernibles to properties that do not render it trivially true. There are several ways in which one could implement this strategy, but in the contemporary discussion the most common one is to restrict the Identity of Indiscernibles to so-called pure, or purely qualitative, properties. Different philosophers have different conceptions of pure properties, but the general idea is that pure properties are those that “do not mention” any particular object. For instance, being red, being next to a tower, and being a lover are pure properties, while being next to the Eiffel Tower and being a lover of Napoleon are impure properties, since they “mention” particular objects, i.e., the Eiffel Tower and Napoleon (for discussion of pure properties and how to characterize them precisely see, among others, Adams 1979: 7; Cowling 2015a; Khamara 1988; Plate 2022; and Rodriguez-Pereyra 2022: 21–27). Another option consists in restricting the properties referred to in the Identity of Indiscernibles to intrinsic properties—roughly, those properties such that having them does not depend on how their possessors are related to anything external to them. The literature on the distinction between intrinsic and extrinsic properties is huge, but no proposed distinction seems to be entirely satisfactory. Nevertheless, on any plausible account of extrinsic properties, these include many relational properties (e.g., those relational properties that objects have in virtue of relations to external objects, as opposed to relational properties had in virtue of relations objects bear to themselves or their parts). The version of the Identity of Indiscernibles produced by this restriction is usually associated with Leibniz (N.B.: for Leibniz quantitative differences are extrinsic differences and so, in his philosophy, the idea that necessarily, no two objects share all their intrinsic properties entails that, necessarily, no two objects differ in size alone).

But some impure properties do not trivialize (1), for instance properties like being next to the Eiffel tower and being a lover of Napoleon, since objects that differ with respect to them differ with respect to how they are and not merely with respect to which objects they are: in one case they differ with respect to how close they are to Eiffel Tower, and in the other they differ with respect to their romantic feelings about Napoleon. Thus, there is also room for a version of the Identity of Indiscernibles that restricts its properties simply to those that do not trivialize (1). Hence, we have the following three versions of the Identity of Indiscernibles:

(PII)
Necessarily, no two objects share all their non-trivializing properties.
(PIIa)
Necessarily, no two objects share all their pure properties.
(PIIb)
Necessarily, no two objects share all their intrinsic pure properties.

Given the considerations above, the non-trivializing properties consist of all the pure properties and some impure properties. Thus, the class of pure properties is included in the class of non-trivializing properties, and therefore (PIIa) entails (PII)—that is, (PIIa) cannot be true unless (PII) is true. And since the class of intrinsic pure properties is included both in the class of pure properties and the class of non-trivializing properties, (PIIb) entails both (PIIa) and (PII)—that is, (PIIb) cannot be true unless both (PIIa) and (PII) are true. There are no other entailment relations between these theses. Thus (PIIb) is the strongest version of the three, and (PII) is the weakest of the three. Indeed, given that (1) is not a version of the Identity of Indiscernibles, there is no version of the Identity of Indiscernibles weaker than (PII).

Orthogonal to this distinction, there is another distinction between versions of the Identity of Indiscernibles, based on work by Quine in which he distinguishes three different kinds of discernibility: absolute, relative, and weak discernibility (Quine 1960, 1976). Given these distinctions, one can distinguish the three following versions of the Identity of Indiscernibles:

(PIIQa)
Necessarily, no two objects satisfy exactly the same open sentences with one free variable.
(PIIQr)
Necessarily, no two objects satisfy exactly the same open sentences with two free variables in both orders.
(PIIQw)
Necessarily, no two objects satisfying exactly the same open sentences with two free variables are such that each one of them satisfies all of them with respect to itself.

The notion of weak discernibility, the notion featuring in the third version of the Identity of Indiscernibles just distinguished, has been much discussed in recent years, especially in relation to the indiscernibility of physical particles (see, for instance, Saunders 2003, 2006; Muller 2015). Weak discernibility can be explained by saying that two objects are weakly discernible if and only if there is a relation R that they bear to each other but at least one them does not bear to itself. What (PIIQw) demands is that every two objects be at least weakly discernible.

Now, every two objects are weakly discernible, and this is trivially so, since numerical difference is a symmetric but irreflexive relation, which means that any objects \(x\) and \(y\) satisfy (PIIQw) trivially, since every two objects trivially satisfy the open sentence “\(x\) is numerically different from \(y\)”, but, equally trivially, no object satisfies it with respect to itself. Obviously, the intended interpretation of (PIIQw) is that, necessarily, every two objects are non-trivially weakly discernible. But, given that no objects can differ relationally without differing with respect to their relational properties, this is equivalent to demanding that, necessarily, no two objects share all their non-trivializing properties. And since no objects can differ relationally without differing with respect to their relational properties, any objects satisfying the Quinean versions of the Identity of Indiscernibles because they differ in their relations will also satisfy (PIIa) or (PII) because of differing in the corresponding relational properties. Thus (PIIb), (PIIa) and (PII) are no less adequate than the Quinean versions of the Identity of Indiscernibles. From now on the discussion will concentrate on (PIIb), (PIIa) and (PII).

Another element that is open to variation in the Identity of Indiscernibles is the domain of entities it is about. The Identity of Indiscernibles has been formulated as a thesis about objects where an object is understood as anything that can be the value of a first order variable. Objects thus understood include both concrete and abstract objects, and it is possible to restrict the thesis to either concrete or abstract objects (for the distinction between concrete and abstract objects see Falguera et al. 2021 [2022]). It is, indeed, possible to restrict the thesis to any class of objects—though it is likely that the more restricted the thesis, the less interesting it is, especially if it is restricted to a class of arbitrarily specified objects. And it is also possible to make the domain more inclusive and allow in it the values of higher order variables, though this is not discussed in what follows.

Although it has traditionally been thought that concrete objects fail to satisfy some version of the Identity of Indiscernibles, presumably (PIIa), while abstract objects satisfy it (see, e.g., Baldwin 1996: 233), in recent years there has been considerable debate about whether there are any abstract objects that violate the corresponding version of the Identity of Indiscernibles. The debate has focused mainly on whether there are numbers and graphs that satisfy some version of the Identity of Indiscernibles (e.g., Keränen 2001; MacBride 2006; Leitgeb & Ladyman 2008; de Clercq 2012; Duguid 2016; Assadian 2024). Numbers and graphs are mathematical objects, and the discussion has taken place mainly in the context of realist mathematical structuralism. But similar debates could be had about whether propositions, concepts, meanings, or any other abstract objects satisfy some version of the Identity of Indiscernibles.

Universals, as usually conceived, are the referents of singular terms (i.e., “wisdom”, “redness”, etc.) and therefore they are values of first-order variables. Consequently, they are objects as the term is understood here. Is the Identity of Indiscernibles true of them? It has traditionally been thought that (PIIa) is true of universals (see, e.g., Armstrong 1997: 50, Heil 2003: 159; cf. Lewis 1986a [1999: 98]). Indeed, some have thought that what distinguishes universals from particulars is that the former can be indiscernible, while the latter cannot (e.g., Ehring 2011: 35, Williams 1986: 3, Wisdom 1934: 208). But recently this line of thought has been questioned, and it has been argued that there is nothing in the concept of a universal that rules out indiscernible universals, and that postulating indiscernible universals can be theoretically useful (Giberman 2016, Rodriguez-Pereyra 2017b; for further discussion see Ou 2024).

The final element in the Identity of Indiscernibles that is open to variation or interpretation is the modal operator. The Identity of Indiscernibles is normally understood as metaphysically necessary. But the modal operator in it could be taken to express other kinds of necessity, e.g., logical necessity or physical necessity. Besides, the Identity of Indiscernibles need not contain any modal operator at all, in which case (PII) states that it is the case that no two objects (of the relevant kind) share all their non-trivializing properties, and (PIIa) and (PIIb) make the corresponding de facto statements. Of course, the Identity of Indiscernibles can also be understood to make a modal claim without making a necessity claim; for instance, it could be understood to say that contingently no two objects (of the relevant kind) share all their properties (of the relevant kind). Indeed, some philosophers have argued that the Identity of Indiscernibles is only contingently true (Casullo 1984; see French 1989 for discussion) and others have argued that in his letter to Clarke Leibniz was trying to establish only the truth of the Identity of Indiscernibles, without making any commitments as to whether it is necessarily or only contingently true (Rodriguez-Pereyra 2014; the modal status of the Identity of Indiscernibles in Leibniz’s philosophy is a traditional topic of Leibnizian scholarship—for further recent discussion see Jauernig 2008; Jorati 2017; Bender 2019; Lin 2025; Ruiz-Gómez 2017).

3. Arguments against the Identity of Indiscernibles

(PIIb) makes a very strong claim and it is plausible that there are actual counterexamples. For instance, there might be two tree leaves or two drops of water that share all their intrinsic pure properties (these cases were considered by Leibniz [Leibniz & Clarke Letters, fourth letter to Clarke, p. 36] and the second case was proposed by Kant in the Critique of Pure Reason; see Kant 1781/87: A 264/B 320; for Kant on the Identity of Indiscernibles in the pre-critical period, especially the Nova Dilucidatio, see Rodriguez-Pereyra 2018). Nevertheless, there is no guarantee that any two tree leaves or water drops share exactly the same intrinsic pure properties, since for all one knows one of the leaves or drops might have one more atom than the other, or there might be another imperceptible difference between them. Indeed, this is Leibniz’s point against this alleged counterexample.

It might be thought that mass produced objects are more likely to constitute a counterexample to (PIIb) than natural objects. Two Pepsi cans, say, at least immediately after being made, are likely to have exactly the same intrinsic pure properties. But even in this case there is no guarantee that they do—for all one knows, one of the cans might have one more atom than the other, or there might be another imperceptible difference between them.

The case of physical particles is different, and indeed some philosophers have argued that there are indiscernible physical particles (Cortes 1976, French 1989, Erdrich 2020). Admittedly, the attribution of properties to particles is justified in terms of explanatory fecundity, and it would be unjustified to attribute to particles properties other than those that are explanatorily necessary. There are two aspects of this argument that make it less cogent than it might seem at first. The first is that from the fact that one is not justified in attributing different properties to particles it does not follow that one is justified in denying of them any other properties. The second, related to the first, is that this argument is contingent on the science of the day, and therefore it is not absolutely conclusive, since science changes continuously, and it might be that the science of tomorrow will assert that any two physical particles have different properties.

Another argument against (PIIb) is based on the so-called Recombination Principle, according to which putting together duplicates of parts of different possible worlds yields another possible world (Lewis 1986b: 87–8; see also van Cleve 2002: 391). If so, if there is a possible world with an object having certain intrinsic pure properties, and there is another possible world with an object having those intrinsic pure properties, then there is a third possible world containing a duplicate of the first object and a duplicate of the second object; this third world contains objects sharing all their intrinsic pure properties and, assuming that there is a possible world with objects sharing all their intrinsic pure properties if and only if it is possible that there are objects sharing all their intrinsic pure properties, this argument, if sound and valid, proves (PIIb) false. As formulated, this argument obviously presupposes the controversial Lewisian ontology of possible worlds, and it is thereby weakened in its persuasive power. This is not, however, an essential presupposition, and the argument could be reformulated using only combinatorial principles without assuming the Lewisian ontology of possible worlds. Nevertheless, it has been argued that the argument cannot be reformulated in an actualist setting and that it is actually invalid, since given that the objects in the first two worlds are duplicates of each other, any duplicate of either is a duplicate of the other, and so there is no guarantee that in the third world there are two objects rather than one object at a distance from itself (Rodriguez-Pereyra 2022: 81–82).

Another argument against (PIIb) is based on the idea that all intrinsic pure properties are not only individually shareable but also jointly shareable. Thus, for any set of intrinsic pure properties that an object could have, there could be two objects having the same set of intrinsic pure properties (Rodriguez-Pereyra 2022: 82–86). If this is the case, then (PIIb) is false.

There is a simpler argument against (PIIb), and this is that since we can imaginatively conceive two objects having the same intrinsic pure properties, it is possible that there are objects sharing all their intrinsic pure properties—hence (PIIb) is false. Max Black’s scenario consisting only of two iron spheres, at a certain distance from each other, that have all their pure properties in common (let us call such scenario Black’s world) can be seen as the conclusion of an instance of this argument. Since Black’s spheres are the only two objects in the world or situation in question, they share all their pure properties, both intrinsic and extrinsic; therefore, Black’s spheres constitute a counterexample to both (PIIa) and (PIIb). This is the most influential argument against the Identity of Indiscernibles in the history of philosophy.

One assumption of this argument is that the relevant kind of conceivability entails the relevant kind of possibility. But resisting this assumption is not the only way to resist this kind of argument against (PIIa) and (PIIb). One might also resist it by arguing that the content of our imaginative act leaves open whether what we imagine is two spheres at a distance from each other or merely one sphere at a distance from itself (Hacking 1975: 251; O’Leary-Hawthorne 1995: 195), and thus it is not legitimate to conclude that Black’s world is possible. But there are reasons to be sceptical of this kind of reply since it might be the case, as Calosi and Varzi have argued, that this kind of reply adds nothing to the simple denial that there can be indiscernible objects (Calosi & Varzi 2016). Another reply to the argument, similar in vein in that it also exploits the fact that the content of our imaginative act does not guarantee that we are imagining two spheres at a distance from each other, is that all we might be imagining in such a case is a simple and scattered object occupying two spherical regions (cf. Hawley 2009). This object is not a sphere, nor does it contain any spheres as parts. A fourth response to the argument argues that the spheres are fusions of pure properties and that they share, among their qualitative parts, an unlocated sphere that consists of all the spheres’ pure properties except their location properties. Thus, on this line of thought, Black’s world does not contain two spheres that share all their pure properties but one unlocated sphere and two located ones, and the two located ones differ with respect to their location properties (Shiver 2014: 909–10). Sometimes it is argued that the spheres in Black’s world do not counterexemplify the Identity of Indiscernibles because they are weakly discernible, since although they are at a distance from each other, neither is at a distance from itself; thus, for instance, sphere \(a\) is two miles away from sphere \(b\) but it is not two miles away from itself (Saunders 2006, Muller 2015). But although this point is correct, it does not touch the claim that Black’s spheres constitute a counterexample to (PIIa), which is what Black’s example was intended to counterexemplify.

Bob Adams (1979) has provided a different argument against (PIIa) (and therefore against (PIIb) too). This argument infers the possibility of indiscernible objects from the possibility of almost indiscernible objects—that is, it infers the possibility of objects sharing all their pure properties from the possibility of objects sharing almost all their pure properties. Thus, the argument can be put in this form (the reference to possible worlds is not essential, and the argument can be formulated using modal operators instead):

  1. There is a possible world with objects that share almost all their pure properties.
  2. If there is a possible world with objects that share almost all their pure properties, there is a possible world with objects that share all their pure properties.
  3. Therefore, there is a possible world with objects that share all their pure properties.

For example, if premise 1 is true, there is a possible world \(w_1\) where there are only two iron spheres, at a certain distance from each other, and sharing all their pure properties, with only one exception: whilst one sphere has temperature \(T\), the other one has a slightly different temperature, \(T^*\), e.g., the second sphere is one degree hotter than the first. And if premise 2 is true, then if there is a world with objects sharing almost all their pure properties, there is a world with objects that share all their pure properties, e.g., a world \(w_2\) where the spheres of \(w_1\) are exactly as they are in \(w_1\) except that in \(w_2\) they both have the same temperature, say \(T\). The two premises entail the conclusion. And assuming that there is a possible world with objects that share all their pure properties if and only if it is possible that there are objects that share all their pure properties, the conclusion of the argument is equivalent to the claim that (PIIa) is false. It has been argued that either the second premise lacks adequate support, or almost indiscernible objects are dispensable to establish the conclusion of the argument. If the premise lacks support, the argument does not establish the possibility of indiscernible objects; if almost indiscernible objects are dispensable, the argument is not needed to establish the possibility of indiscernible objects (Rodriguez-Pereyra 2017a).

Finally, a subtraction argument has been advanced against (PIIa) (Rodriguez-Pereyra 2022: 87–100). This argument presupposes the falsity of (PIIb) and it is based on the basic idea that if there is a possible world with objects sharing all their intrinsic pure properties that are spatiotemporally and causally symmetrically and independently related to each other, then there is a possible world where everything else has been “subtracted” and those objects are the only objects, and in such a world those objects share all their pure properties, intrinsic and extrinsic. Assuming that there is a possible world with objects that share all their pure properties if and only if it is possible that there are objects that share all their pure properties, the conclusion of the argument is equivalent to the claim that (PIIa) is false. (Again, the reference to possible worlds is not essential and the argument can be formulated using modal operators instead.)

4. Arguments for the Identity of Indiscernibles

Faithful to his style, Leibniz produced a battery of different arguments for the Identity of Indiscernibles. As was pointed out before, Leibniz’s intended version of the Identity of Indiscernibles is a principle stronger than (PIIb), one that entails (PIIb). Here we shall briefly concentrate only on two of Leibniz’s arguments. The first argument, typical from his philosophy in the 1680s and whose locus classicus is Section 9 of the Discourse on Metaphysics, tries to derive the Identity of Indiscernibles from Leibniz’s idea that individual substances have complete individual concepts, concepts that contain absolutely everything that is true of them, past, present, and future (Leibniz 1686 [2020]). The argument, however, is subject to two related objections. Firstly, nowhere does Leibniz argue that numerically different individual substances cannot share an individual complete concept, that is, there is no argument from Leibniz that, necessarily, no two individual substances have one and the same individual complete concept. But if they did, given that such concepts contain all the intrinsic pure properties of the substances having those concepts, there would be indiscernible substances in the sense ruled out by (PIIb), i.e., substances sharing all their intrinsic pure properties. Secondly, even if, necessarily, no two substances share their individual complete concepts, Leibniz has not ruled out the possibility of indiscernible individual complete concepts—that is, Leibniz has not ruled out the possibility of numerically distinct individual complete concepts that specify the same intrinsic pure properties. But if such indiscernible concepts are possible, then two or more individual substances could falsify (PIIb) by virtue of having indiscernible individual complete concepts.

The other Leibnizian argument for the Identity of Indiscernibles to be mentioned here appears in Leibniz’s Correspondence with Samuel Clarke (Leibniz & Clarke Letters, fifth letter to Clarke, page 61). According to this argument, if there were objects sharing all their intrinsic pure properties, God would have had no reason to place them where he did rather than swap them around, since the world would not have been less valuable if the objects had been swapped around and no other change had been made; but since God does not act without a reason, God did not create objects sharing all their intrinsic pure properties, that is, (PIIb) is true. Of course, this argument depends on the assumption that a certain kind of god exists, namely one for whom the only reason to create one world rather than another is that the world to be created instantiates more value than the other. Needless to say, the assumption that such a god, or any god for that matter, exists, is extremely controversial and rests on weak rational grounds. Furthermore, the argument also presupposes that any two objects could have been swapped around, leaving everything else intact. But this presupposition fails in the case of two atomic objects that are the only objects in a world whose space is relational (see Rodriguez-Pereyra 2014: 108–112).

One might try to reformulate Leibniz’s argument without the assumption that God exists and that any two objects could have been swapped around. For instance, one might try the following argument, which has undoubtedly Leibnizian credentials: A sufficiently strong version of the Principle of Sufficient Reason, one according to which the intrinsic pure properties of every object determine its location, is true; if there were any objects sharing all their intrinsic pure properties, their intrinsic pure properties would not have determined their location (assuming, as Leibniz did in the original argument, that no two objects are co-located); therefore no two objects share all their intrinsic pure properties, that is, (PIIb) is true. But this argument, like the previous one, is dubious, since the plausibility of the relevant version of the Principle of Sufficient Reason is inversely proportional to its strength.

Bernard Bolzano, sometimes known as the “Bohemian Leibniz”, rejected some Leibnizian principles, and accepted others, and one of those he accepted was the Identity of Indiscernibles. He argues for (PIIb) and his argument for it is different from any argument for the Identity of Indiscernibles one finds in Leibniz. Bolzano believes that all finite substances are causally connected. On this basis he infers the Identity of Indiscernibles, since he thinks that for two substances to have the same intrinsic pure properties they must not only have been given the same original properties by God, but they must also be surrounded by substances that have exactly the same causal effects on them. But Bolzano thinks that there are infinitely more possible situations in which any two substances are surrounded by substances causally affecting them differently than situations in which they are surrounded by substances causally affecting them equally. Since he takes all these possible situations to be equiprobable, Bolzano infers that (PIIb) is true (Bolzano 1837: 532–33 [2014: volume 1, pp. 384–85]). There are a few problems with this argument, one of which is that the fact that something is improbable, however highly improbable it is, does not entail that it is not true, and another one is that it is not clear that there are infinitely more possible situations in which any two substances are surrounded by substances causally affecting them differently than situations in which they are surrounded by substances causally affecting them equally. Thus, the argument seems to fall short of establishing its intended conclusion.

A. J. Ayer (1953) argued for (PIIa) on the basis that there are reasons to believe in the Bundle Theory and that such a theory entails (PIIa) or something like it. This argument makes the Identity of Indiscernibles even more controversial than the Bundle Theory itself, for it assumes not only the truth of the Bundle Theory, but also that this theory entails the Identity of Indiscernibles, and both assumptions are controversial and therefore they need argumentative support (for more on the Identity of Indiscernibles and the Bundle Theory see the next section).

Michael Della Rocca (2005) has argued, in rationalist spirit, for (PIIa). His argument can be concisely represented as follows: a denial of (PIIa) commits one to a violation of a version of the Principle of Sufficient Reason, since if (PIIa) were not true, the non-identity of indiscernible objects sharing all their pure properties would be primitive and therefore such non-identity would have no explanation. Of course, this requires a commitment to a sufficiently strong version of the Principle of Sufficient Reason, and not everyone would be willing to follow Della Rocca in this (as he himself points out). It might be thought that what the argument supports is (PII) rather than (PIIa), since presumably the non-identity of objects sharing all their pure properties but differing with respect to some non-trivializing impure properties might be explained by their difference with respect to those non-trivializing impure properties—however, a difference only in non-trivializing impure properties requires that identity be primitive, or at least not explainable in terms of pure properties, and therefore the relevant version of the Principle of Sufficient Reason operating in Della Rocca’s argument would still be violated. Thus, the argument, if sound and valid, supports (PIIa), and thereby it also supports (PII).

Although perhaps he was not aware of it, McTaggart offered an argument for (PII), since he was implicitly considering all non-trivializing properties and his argument does not depend on there being any difference in pure properties between substances. His argument requires making the thought experiment of removing in thought all difference in nature from two substances, and he says that when we do that we find that we are contemplating not two substances, but only one. And this, he says, comes from the “recognition of the impossibility of diversity without dissimilarity”. Then he adds, in a passage with Leibnizian reminiscences:

The nature of a substance expresses completely what the substance is. And the same complete expression of what a substance is cannot be true of each of two substances. The substance is made this substance by its nature, and, if the nature is the same, the substance is the same. (McTaggart 1921: 96)

True, if what makes a certain substance the substance it is is its nature, then no other substance can have the same nature—but claiming that what makes it the substance it is is its nature shows that one has already assumed (PII).

There are two other recent arguments for (PII). One assumes a relatively weak Humean principle to the effect that, necessarily, no two objects necessarily co-vary with respect to every non-trivializing property—that is, necessarily, for no two objects \(x\) and \(y\) is it the case that, for every non-trivializing property \(F\), necessarily, \(x\) is \(F\) if and only if \(y\) is \(F\). It is plausible that this principle is true. For any objects \(x\) and \(y\) necessarily co-varying with respect to every non-trivializing property would be such that, necessarily, \(x\) is square if and only if \(y\) is square, \(x\) is a left-wing activist if and only if \(y\) is a left-wing activist, \(x\) is exactly in place \(p\) at time \(t\) if and only if \(y\) is exactly in place \(p\) at time \(t\), and so on for every non-trivializing property. But such objects would exhibit a degree of mutual dependence that seems to be incompatible with the degree of ontological autonomy required to be a concrete object. Furthermore, objects violating (PII) would necessarily co-vary with respect to every non-trivializing property. For, necessarily, every object \(x\) has, for every non-trivializing property F, the property of necessarily, having \(F\) if and only if \(x\) has \(F\). Thus, a particular object \(a\) has, for every non-trivializing property \(F\), the property of necessarily, having \(F\) if and only if a has \(F\). But this latter property is, itself, a non-trivializing (impure) property, since differing with respect to it requires differing extra-numerically, since for any two objects to differ with respect to it is for them to differ with respect to whether having \(F\) is necessarily connected to \(a\)’s having \(F\), rather than differing merely with respect to which objects they are. Since, for every property \(F\), the property of necessarily, having \(F\) if and only if a has \(F\) is a non-trivializing property, every object sharing all its trivializing properties with \(a\) will share such properties with \(a\) and, thus, it would necessarily co-vary with \(a\) with respect to every non-trivializing property. Thus, generally, objects violating (PII) would necessarily co-vary with respect to every non-trivializing property—hence, given the Humean principle that, necessarily, no two objects necessarily co-vary with respect to every non-trivializing property, (PII) is true.

The other argument assumes an idea about grounding, namely that for every concrete object \(x\) that exists at a particular time \(t\), \(x\) exists in time because \(x\) exists at \(t\), and for no object \(y\) other than \(x\) is it the case that \(x\) exists in time because \(y\) exists at \(t\). Now, a property like existing in time because \(a\) exists at time \(t\) is a non-trivializing (impure) property, since for objects to differ with respect to it they must differ extra-numerically, that is, they must differ with respect to which object’s existence at a particular time grounds their existence in time, rather than differing merely with respect to which objects they are. So, given any two concrete objects \(a\) and \(b\) that exist at time \(t\), they must differ with respect to some of their non-trivializing properties, since they must differ with respect to the properties of existing in time because \(a\) exists at time t (only \(a\) will have it), and existing in time because \(b\) exists at time t (only \(b\) will have it). Thus, the argument goes, (PII) is true of concrete objects. But the argument can be extended to abstract objects too, since properties like being abstract or concrete because \(b\) is abstract are non-trivializing (impure) properties that cannot be shared, and therefore any two two objects, whether abstract or concrete, must differ in some non-trivializing properties, which is what (PII) demands (For this argment and the previous one see Rodriguez-Pereyra 2022: 106–125).

5. Connections of the Identity of Indiscernibles

The Identity of Indiscernibles interacts, or is thought to interact, in interesting ways with other philosophical theses. One such interaction is by way of the metaphysical consequences of the Identity of Indiscernibles. The stronger the version of the Identity of Indiscernibles, the more metaphysical consequences it has. Thus, some of the substantive metaphysical consequences of (PIIb) are not consequences of the weaker versions of the Identity of Indiscernibles. (PIIb) entails, for instance, that space, time, and spacetime are not absolute, that is, that there are no points or regions of space, time, or spacetime, since they would share all their intrinsic pure properties. Similarly, (PIIb) entails that there are no atoms or physically elementary particles, since these are often conceived as sharing their intrinsic pure properties. Leibniz was aware of these consequences of the stronger versions of the Identity of Indiscernibles, and he also thought that the Identity of Indiscernibles entailed that the mind was not a tabula rasa (since such minds would share all their intrinsic pure properties) and that, consequently, it entailed the existence of innate ideas.

Can a principle of individuation in terms of pure properties be extracted from (PIIa) or (PIIb)? That is, a principle of individuation of objects, according to which what makes a particular object the particular object it is, is that it has the pure properties, or the intrinsic pure properties, it has. No, such a principle of individuation cannot be extracted from them. This is because although (PIIa) and (PIIb) rule out the possibility of objects sharing all their pure properties, neither principle entails that any set of pure properties is necessary and sufficient for being a particular object. That is, both principles forbid the possibility of two or more objects sharing all their pure properties, but neither principle forbids objects that could have had the same pure properties that any actual object has. This is perhaps better seen in terms of possible worlds: although both principles rule out possible worlds in which two or more objects share all their pure properties, neither principle rules out two or more objects having the same pure properties in different possible worlds.

One thesis which is often thought to interact with the Identity of Indiscernibles is Anti-Haecceitism. As is well known, there are many different senses of Haecceitism and Anti-Haecceitism, and we shall here briefly discuss the relations between various of these senses and the Identity of Indiscernibles (see Cowling 2015b [2023] for a discussion of Haecceitism and Anti-Haecceitism).

The Identity of Indiscernibles is a supervenience thesis, namely the thesis that numerical identity supervenes upon indiscernibility. And the different versions of the Identity of Indiscernibles, (PII), (PIIa), and (PIIb), make numerical identity supervene on different kinds of indiscernibility: indiscernibility with respect to intrinsic pure properties (PIIb), indiscernibility with respect to pure properties, whether intrinsic or extrinsic (PPIIa), and indiscernibility with respect to non-trivializing properties, whether pure or impure (PII). Since pure properties are purely qualitative properties, (PIIa) and (PIIb) make numerical identity supervene upon the purely qualitative. That is, according to these two principles, once you fix the purely qualitative properties of objects, you thereby fix the facts about their numerical identity and difference. That is, once the purely qualitative properties of objects are fixed, the facts about how many objects there are in the world are fixed (assuming objects having different qualitative properties are numerically different, an uncontroversial assumption). Since fixing all the purely qualitative facts of the world includes fixing the purely qualitative properties of the objects, if (PIIa) or (PIIb) are true, fixing all the purely qualitative facts of the world fixes the facts about how many objects there are in the world.

In one sense of the term, Anti-Haecceitism is the view that fixing all purely qualitative facts fixes all facts. That is, necessarily, any purely qualitative description of reality entails all facts. This version of Anti-Haecceitism rules out the possibility of merely non-qualitative differences; for instance, it rules out the possibility that I could have had all the pure properties of Napoleon Bonaparte, or that someone who does not actually exist could have had all the pure properties of Indira Ghandi (Skow 2008 and Scarpati 2023 argue that Anti-Haecceitism should be understood along these lines). This version of Anti-Haecceitism entails neither (PIIa) nor (PIIb), since nothing in it requires that a purely qualitative description of reality entails that there are no objects having the same pure properties, or even the same intrinsic pure properties—indeed, that there are two objects sharing all their pure properties is a purely qualitative description, and therefore it might be part of a purely qualitative description of reality. And nor does such a version of Anti-Haecceitism entail (PII) either, since the fact that, necessarily, any purely qualitative description of reality entails all facts does not entail that no two objects can share all their non-trivializing properties, pure and impure. However, Alexander Roberts has argued that Anti-Haecceitism entails (PIIa) on the basis of some recombination assumptions—for instance, that it is metaphysically possible for any concrete object to be either the tallest, the smallest, the most massive, or the least massive of its kind, or that for any intrinsic pure property that neither of two objects has, it is possible that at least one of those objects has it while the other one does not have it. In effect, what Roberts argues is that, necessarily, there are no objects that, necessarily, share all their pure properties. Some of these assumptions will not be accepted by everyone, and indeed Roberts himself has noted that some of them do not hold up in certain ontologies (Roberts 2024: 95, fn. 4), so there is some degree of conditionality to Roberts’ conclusion.

Do any versions of the Identity of Indiscernibles here discussed entail Anti-Haecceitism? No. (PIIa), (PIIb), and (PII) all require that it is impossible that numerically distinct objects share all their properties of a certain kind. But no matter which kinds of properties objects cannot share, that does not rule out the possibility of any object other than Napoleon having had all his pure properties. Indeed, while (PIIa), (PIIb), and (PII) are theses that rule out possibilities which include distinct objects sharing all their properties of a certain kind, Anti-Haecceitism rules out distinct possibilities in which distinct objects share all their pure properties. Another way of putting this point is that while (PIIa), (PIIb), and (PII) are “intra-world” theses, Anti-Haecceitism is a “cross-worlds” thesis.

And, indeed, often Anti-Haecceitism is presented as a thesis about possible worlds. In this sense Anti-Haecceitism is the thesis that no distinct possible worlds are purely qualitatively indiscernible (though see Skow 2008 for an argument that Haecceitism and Anti-Haecceitism should not be defined in terms of possible worlds). In so far as (PIIa) and (PIIb) quantify over all objects of all kinds, they entail Anti-Haecceitism understood in terms of possible worlds, for those two principles entail that any two possible worlds differ in some of their pure properties. (PII), on the other hand, does not entail this version of Anti-Haecceitism, since (PII) does not require objects (possible worlds included) to differ purely qualitatively. Anti-Haecceitism understood in this way does not entail any of the three versions of the Identity of Indiscernibles here distinguished, since Anti-Haecceitism thus understood is a thesis that rules out the qualitative indiscernibility of certain objects, namely possible worlds, while (PIIa) and (PIIb) rule out the qualitative indiscernibility of all types of objects, and (PII) rules out a different kind of indiscernibility of all types of objects.

In a less common sense, Anti-Haecceitism is the thesis that, necessarily, there are no haecceities, where haecceities are what we have called properties of identity, i.e., properties like being identical with Napoleon. All three principles (PIIa), (PIIb), and (PII), are compatible with the existence of haecceities, and therefore they do not entail Anti-Haecceitism in this sense. Nor does Anti-Haecceitism thus understood entail any of those three principles, since the non-existence of haecceities is compatible with difference solo numero: to differ solo numero two objects must differ with respect to which objects they are without differing with respect to how they are, intrinsically or extrinsically. But to differ with respect to which objects they are, objects need not have any haecceities: Castor and Pollux, let us suppose, may differ only in that one of them is Castor and the other is Pollux without thereby having to have the properties of being identical with Castor and being identical with Pollux.

In yet another sense Anti-Haecceitism is the thesis that the identity of objects is reducible to their pure properties. One minimal condition of the identity of objects’ being reducible to their pure properties is that, for every possible object, having certain pure properties is a necessarily sufficient condition for being that particular object. But neither (PIIa) nor (PIIb) entail that having any pure properties are necessarily sufficient for being any particular object. All they entail is that, necessarily, no objects can share all their intrinsic pure properties (PIIb) or all their pure properties (PIIa). But this does not mean that having any pure properties is necessarily sufficient for being a certain object in particular, i.e., that there are pure properties such that only one object could have had them. Thus, neither (PIIa) nor (PIIb) entail Anti-Haecceitism in this sense (and it should be even clearer that (PII) does not entail it either). On the other hand, Anti-Haecceitism in the present sense entails (PIIa) (and therefore it also entails (PII)), since if, for every possible object, having certain pure properties is a necessarily sufficient condition for being that particular object, then, necessarily, no two objects can share all their pure properties. Similarly, a version of Anti-Haecceitism such that, for every possible object, having certain intrinsic pure properties is a necessarily sufficient condition for being that particular object, entails (PIIb) (and therefore it also entails (PIIa) and (PII)).

Another interesting interaction is that between the Identity of Indiscernibles and the Bundle Theory. It is often thought that the Bundle Theory entails a version of the Identity of Indiscernibles, namely (PIIa) and/or (PIIb) (Armstrong 1978: 91; Loux 1998: 107). There are different versions of the Bundle Theory, but one version of it says that properties are universals, and individual objects are entirely constituted by the universals they instantiate (thus individuals are “bundles” of universals, hence the theory’s name). Given this, the thought is, there cannot be two objects instantiating exactly the same universals, since objects that instantiated exactly the same universals would be entirely constituted by the same entities (namely the same universals), and no two objects, it is assumed, could have exactly the same constituents. Since universals are supposed to be pure properties, the idea is that the Bundle Theory entails either (PIIa) or (PIIb).

There are several possible reactions to this. One is to accept the entailment and, on the premise that the relevant versions of the Identity of Indiscernibles (PIIa and PIIb) are false, reject the Bundle Theory (Armstrong 1978: 91). Another one is, of course, to accept the relevant versions of the Identity of Indiscernibles on the premise that the Bundle Theory is true (Ayer 1953; see section 4 above). A third option is to defend the Bundle Theory from the falsity of the Identity of Indiscernibles by holding that the relevant versions of it have not been proved to be false (O’Leary-Hawthorne 1995). Yet another option is to reject the assumption that no two objects can have exactly the same constituents, and thereby reject the entailment from the Bundle Theory to the Identity of Indiscernibles, which allows one to simultaneously uphold the Bundle Theory and reject the relevant version of the Identity of Indiscernibles, i.e., (PIIa) and/or (PIIb) (Rodriguez-Pereyra 2004).

There is a further option: to reject the assumption that properties are universals and take them to be tropes, and take substances or objects to be entirely constituted by their tropes (for such theories see Keinänen & Hakkarainen 2024). Tropes have the peculiarity that, while they cannot be shared by objects, they can resemble each other exactly. This option requires accepting the entailment from the Bundle Theory (interpreted as a Trope Bundle Theory) to the Identity of Indiscernibles, and yet it does not jeopardize the Bundle Theory since on this view the relevant versions of the Identity of Indiscernibles are true, given that tropes cannot be shared by any objects. However, it might be argued that only the letter, but not the spirit, of the Identity of Indiscernibles is preserved under this strategy. For although no two objects can share tropes, since tropes can resemble each other exactly, two objects can resemble each other exactly if they have exactly resembling tropes. And it is plausible that the point of (PIIa) and (PIIb) is to rule out perfect resemblance between objects.

The Identity of Indiscernibles claims a connection between numerical identity and indiscernibility, namely that there cannot be indiscernibility without numerical identity (from now on the qualifier ‘numerical’ will be omitted). Given the centrality of grounding in the contemporary metaphysical discussion, many philosophers have inquired after the grounds of identity, and the Identity of Indiscernibles might be thought to offer an obvious candidate, namely indiscernibility—that is, what grounds the fact that \(a = b\) is that \(a\) and \(b\) are indiscernible. But although the connection between identity and indiscernibility established by the Identity of Indiscernibles might suggest that indiscernibility grounds identity, the Identity of Indiscernibles is not a principle about grounding. That is, the Identity of Indiscernibles, in its various forms, does not claim that indiscernibility grounds identity—instead, it asserts a supervenience relation between identity and indiscernibility, or between identity and a given kind of indiscernibility, namely that \(a = b\) supervenes upon (the right kind of) indiscernibility. But that it is not a principle about ground does not mean that the Identity of Indiscernibles is not an interesting principle—the question whether indiscernibility (of whatever kind) can obtain without identity is an interesting question in itself.

Nevertheless, though the truth of the Identity of Indiscernibles is a question independent from any questions about ground, the question whether indiscernibility grounds identity is also an interesting one. The idea that indiscernibility grounds identity has been thought to be problematic—in particular, it has been thought to be circular, on the grounds that the indiscernibility between \(a\) and \(b\) is the fact that every property F is such that \(a\) has it if and only \(b\) has it, and the fact that \(a\) has property F (for any \(F\) \(a\) has) is one of the grounds of this biconditional fact. But given \(a = a\), one of the properties \(a\) has is the property of being identical with a, and so (assuming the transitivity of grounding), \(a\)’s having the property of being identical with a partially grounds the fact that \(a = a\). But the fact that \(a\) has the property of being identical with a is the fact that \(a = a\); thus, the fact that \(a = a\) partially grounds itself, something many have found inadmissible (Elgin forthcoming).

Obviously, this puzzle does not affect those who think identity is ungrounded, fundamental. But even those who accept that identity is grounded have resources to avoid the puzzle. One obvious way out of the puzzle is to reject the transitivity of grounding, which some have found reason to reject (e.g., Schaffer 2012). But there are other options. One of them is to restrict the properties whose sharing grounds identity to purely qualitative ones. That, of course, assumes the truth of (PIIa), which is very controversial, to say the least. Alternatively, one can attempt to ground identity in necessary indiscernibility, as Thornton proposes (Thornton forthcoming). According to this proposal, what grounds that \(a = a\) is that \(a\) is necessarily indiscernible from \(a\), and what grounds that \(a\) is numerically different from \(b\) is that \(a\) and \(b\) are possibly discernible from each other. On this proposal indiscernibility is restricted to purely qualitative properties. The proposal presupposes that there cannot be numerically distinct entities that are necessarily (purely qualitative) indiscernible, and so its viability depends on whether such a presupposition is true. Another option is to maintain that \(a = a\) is grounded in the fact that \(a\) bears only purely qualitative reflexive relations to \(a\). But this is vulnerable to the possibility of co-located purely qualitatively indiscernible objects, since some such objects will not stand in any purely qualitative irreflexive relations with respect to each other (bosons might be an example of such objects). Finally, Elgin has argued that what grounds the fact that \(a = a\) is a relation that obtains between the second-order universal quantifier and being a property that a has if and only if a has it. This avoids the circularity problem mentioned above, since a relation is not a fact and so the problem that the identity fact partially grounds itself does not arise (Elgin forthcoming). (Of course there are other accounts of the grounds of identity, but here the focus is on a very brief description of some accounts on which indiscernibility, or something very close to it, grounds identity. For more general discussion of the grounds of identity see Lo 2023, forthcoming and Shumener 2020).

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