Identity and Leibniz’s Law

Leibniz’s Law provides necessary and sufficient conditions for identity. I think Leibniz originally formulated the conditions in terms of resemblances or of being alike, but I will present them in terms of properties, as is typically done. In short, the identity of indiscernibles says it is necessary for identical things to share all their properties in common, while the indiscernibility of identity says that having all properties in common is sufficient for identity.

The indiscernibility of identicals is not controversial, although perhaps tricky to clearly state, at least informally. For a first attempt, It says that if two things are identical, then they have all the same properties. But this might be hard to follow on closer inspection, because aren’t identical things really one thing and not two things? So we might rephrase this as: if a thing is identical, then it has all its properties in common (with itself). And this is just to say that if a thing is one thing, and not two, it has all its properties in common with itself. This indeed should be all the claim is saying, and it looks entirely innocuous. Following quantified logic, we might also clarify the claim as saying that if the referent of two different names is the same, then the referent of the one name will have all the same properties as the referent of the other name.

This is all to say that it is necessary that identical things have all the same properties. This condition is not controversial because finding that two things that might have been considered identical differ in some property is a dead give away of their not being identical. In symbolic logic, the indiscernibility of identicals can be written as (where x and y are universal in scope):

x=y → ∀Φ(Φx ↔ Φy)

This says that if something x and something y are identical, then for any property Φ, then x has Φ if and only if y has Φ (i.e. then x and y have all properties in common).

The identity of indiscernibles is considered to be more controversial. It says that if two things share the same same properties, then they are identical. In other words, if something has a set of properties and something (maybe not something else) has that same set of properties, then they are identical. In yet other terms, if the referent of some name has all the same properties as the referent of another name, then the referent of those names is identical (note: I have to write ‘referent’ in its singular form because writing ‘referents’ is to commit to the view that there is more than one referent). All this is to say that sharing all the same properties is sufficient for identity. In symbolic terms, where x and y are universal in scope:

∀Φ(Φx ↔ Φy) → x=y.

This says that for any property Φ, if it’s the case that something x has Φ if and only if something y has Φ (i.e. if x and y have all properties in common), then x is identical with y. This condition is more controversial for at least a couple of reasons. One way to bring out potential problems involves invoking possible worlds that include two or more objects that intuitively exactly resemble one another, or that intuitively share all their properties in common, but which nonetheless are not identical. (This approach was devised by philosopher Max Blank.)

To illustrate, consider a world with two qualitatively identical spheres, placed a foot apart from one another. The thought goes that they resemble one another (or they completely share their properties with one another), and yet they are not identical. No doubt they are not identical, but do they really have all of their properties in common? I don’t think so.

To begin showing why not, any observer of the scene would see that the two spheres are in different locations. Yet what gives the observer the ability to observe this? Part of the answer must involve the spatial location of the respective objects, perhaps even in relation to the observer. But if the observer is equally between them, still they are distinguished (and if the “symmetry” of the universe is thought to be at issue, just place another observer on the opposite side of the original observer, and such that all the objects are placed as points of a square). So what provides the distinction? Plausibly, the distinction comes from the fact that the space that each object occupies is different. If so, the objects have different spatial properties. What’s more, these different spatial properties provide a qualitative dissimilarity between the objects, as evidenced by any observer immediately noting a difference between the objects. This is because, if not, then we must suppose that the fact of non-identity between the objects provides an observer with a ready distinction between those objects, but that it does so without involving a difference of properties that the objects have. Yet how is such a quick and clear distinction between the objects made without precisely a difference between them in some quality? And the qualitative difference that readily lends itself to being observed, comes from, I think is clear, a difference in the location that the objects occupy. In this world that we originally supposed to have no qualitative differences, it turns out that their spatial positions are readily capable of being distinguished, and that this is explainable as a difference in the spatial properties of the spheres.

However, we may tweak the example slightly to make a stronger case against the identity of indiscernibles. What if, instead of having two spheres be a foot apart, they also shared their spatial and temporal properties. In other words, what if both spheres precisely overlapped with one another? Here, it is very intuitive that such spheres share all their properties in common and yet are possibly non-identical.

Let’s pursue this thought. It seems possible that such non-identical objects might exist. At least it does not seem like it should be ruled out a priori. However, such a world seems to make non-identity into a feature of objects in a way that goes above their distinguishable properties (i.e. their properties not shared between them). This poses a dilemma:

  1. On the one hand, if non-identity is above or other than the distinguishable properties that objects have, then there is no basis for non-identity claims in the features of the objects themselves.  But if an object just is the totality of its features, then the basis cannot reside in the object at all. But where else could it reside?
  2. On the other hand, if we make non-identity itself to be a feature of the objects, then this feature is straightforwardly a property of the objects. But then how could that property be expressed? If A shares all of its properties with B, but A is not identical to B and B is not identical to A, then B’s being not identical to A, and A’s being not identical to B are themselves distinguishable properties, in which case Leibniz’s identity of indiscernibles stands after all.

Of course, 2) assumes that B’s being not identical to A, and A’s being not identical to B are themselves distinguishable properties from one another. It might be argued that both A and B have the same distinguishable property, namely that of being non-identical with one another. But I find this property difficult to make both clear and adequate to achieving that goal. We might think that A has the property of being not identical to B. Yet B clearly cannot have this property. And B may instead have the property of being not identical to A, which A in turn cannot have. So the property, while perhaps initially plausible, look entirely impossible to construct. Therefore, the dilemma stands, and Leibniz’ Law is safe. On the one hand, claiming non-identity becomes baseless, and on the other, non-identity confirms a difference of properties.

However, even if Leibniz’ Law is vindicated on logical grounds, it is perhaps unsettling that two objects might have all of the same properties except properties of being not identical with one another.

We need to get into what identity is supposed to mean. I take it identity means being the same. If properties are features of objects and objects are just a collection of features, then it just logically follows that the same objects will have the same collection of features and different objects will differ in their collection of features. Of course, making being the same into one of the features that an object can have is possible, but ultimately it is empty unless such a feature references all of the object’s other features. The feature of sameness relies on all of the other features and whether they are shared with some other object. It is not itself a feature that is absent a reference to these other features. Someone who believes that the identity of indiscernibles is false (or even not necessary) must either have a different view about what identity means, or have a view of objects without features. I don’t have much to say about such views. A view of objects without features looks highly self-defeating, for if I press for a name or a description of some supposed non-feature of an object, the very naming or describing of it, insofar as it refers to anything at all, suffices to name and describe a feature of that object.

All the same, upholding Leibniz’s Law is paradoxical if there is something we could have in mind when considering objects that perfectly overlap in their spatio-temporal qualities and that share all their qualitative or causal properties besides. If this thought is possible, there must be something we have in mind that would confer non-identity upon such objects. But what could that be? Based on the above remarks, it must be we have in mind, if anything at all, a distinction between them that would readily amount to a difference in properties. But, all the same, consideration of two-qualitatively identical objects that are nonetheless non-identical seems possible. Let us dig into the thought a little more.

Consider a world of spheres and cubes. Here, spheres are frictionless in regard to one another. That is, they go right through one another when their boundaries enter the same place, and they transfers no energy between one another. Cubes, on the other hand, do bump against spheres, and cause them to change directions. Besides being spatially separated, we will say that cubes otherwise share all the same properties with other cubes and that spheres otherwise share all the same properties with other spheres.

Let’s say some cubes bumped into some spheres in such a way that the spheres ended up occupying the exact same space with one another (e.g. maybe two cubes sandwiched two spheres together from opposite sides). In such a case, the two spheres that have been sandwiched together must be overlapping objects. There would be no way to pry such overlapping objects apart, because any causal influence from a cube upon the thing in that exact spot would have to apply to both spheres, since they are both in that spot. If a cube hits these spheres, they would have to both be causally affected in the same way. To consider that they do not have to be is to consider that the cube can somehow discriminate its causal powers between them. But I don’t know a causal notion that would make this power to do so clear. (Maybe a cube alternates between being able and being unable to hit a sphere each time it enters the space of a sphere. Yet, no matter which cycle it is on, when it enters and overlapping sphere it enters the space of two spheres. So which sphere does it jettison? We might think we could build on top of the causal model the one it jettisons, but this looks impossible, for how do we distinguish between the overlapping spheres? We cannot. So the overlapping spheres are forever wedded.)

When a cube next bumps into the overlapping spheres, the cube would transfer some of its kinetic energy to the spheres. Yet, the energy would have to go to both spheres. So half the work would result from the same cause. That is, an overlapping sphere would be affected as if it were a singular sphere that got half the cube’s inertial energy.

The point of this illustration is to highlight a ready alternative way to view the spheres in light of the change in causal capacity that occurs once sandwiched together. That is to view the spheres as themselves powers that exist separately but that, when combined together into an exact space, become an entirely new object that is singular. This remark is probably uncontroversial. But, furthermore, the original powers are nowhere in the object itself, but, if they exist at all they exist as one singular object that just is the overlapping sphere. Observe all you want, and no original spheres can be discovered. Any causal interaction with the sphere (including observation) would confirm that the object of that space is a singular and entirely new object, with no spheres to be found as elements.

An objection may be that because these elements cannot be discovered, this does not necessarily mean that they no longer are. Well, answering this objection would likely involve getting into what objects are. I mainly view objects as a set of causal capacities in particular places and times. (That probably sounds reductive to many at first glance, but causes themselves are qualitative and vary widely in my view.) At least, it should not be controversial that objects have qualities and causal relations, as well as being in a place at a time. But if these notions provide a good framework for providing a theory of objects, then it would prove difficult to argue that the original spheres are elements of the overlapping sphere. For instance, note that the causal capacities of the spheres are altered, and, more importantly in my view, that the two sets of causal capacities are now just one set.

This view of objects should not be viewed as competing with Leibniz’s Law for an understanding of identity. It rather works alongside it. Causal capacities are properties that objects have, and this convergence of causal properties in the case of the two spheres coming together, as well as the loss of those previous causal properties that the two spheres had, makes the case that the two objects are no more, and that there is nonetheless a new object. The picture is akin to two powers coming together to form a new power that does not have the original powers as parts. Is this necessarily what is occurring? Maybe not, but examining this possibility would require articulating a different view of objects. Based on the notion of objects in play, however, Leibniz’s Law holds. At the start of this illustration, we assumed that two non-identical objects had all the same properties. Yet the logical result from a fairly straightforward view of objects is that the original objects are replaced by one with new causal capacities, and the original two sets of causal capacities are no more. This leads to the conclusion that the two spheres, once together, are no longer two, but are a singular object.

We had to blast off on a metaphysical journey to get this far, but I think it has been worth it. I really see no straightforward objections to Leibniz’s Law emerging, save some competing views of objects and identity that do not seem to me to be forthcoming.

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