The Principle of the Identity of Indiscernibles

According to Leibniz’s Principle of the Identity of Indiscernibles (PII), it is logically impossible for two completely identical objects to exist.  If any two things share all their properties, they are but one thing.  (In other words, if two objects are qualitatively indiscernible, they must also be identical.)  Michael Loux expresses this principle more formally in Metaphysics: A Contemporary Introduction:

Necessarily, for any concrete objects, a and b, if for any attribute, φ, φ is an attribute of a if and only if φ is an attribute of b, then a is numerically identical with b. (112)

This is both a controversial claim and a complex one because each philosopher who evaluates this claim first has to give his definition of an attribute (or property).  Does it include spatiotemporal location?  Do relationships with other objects qualify as properties?  I believe that the truth of the Principle of the Identity of Indiscernibles depends on these questions, so I will provide a complex argument: (1) PII is true but trivial if spatiotemporal location is a property; (2) Leibniz’s theological justification for PII is wrong; (3) PII is true but trivial if relationships with other objects are properties; (4) PII is false otherwise; (4’) the trope theorist’s defense of PII is valid, but it is false because trope theory is false.

If we include spatiotemporal location as a property, we can safely accept the PII.  Only maniacs and quantum theorists would argue that two different objects can occupy the same space at the same time.  Such an occurrence would violate the very concept of space.  Thus, if two objects occupy the same space at the same time, and they share all their properties, they must be identical.  Under these circumstances, the Principle of the Identity of Indiscernibles is true, but it is also trivial and carries no interesting results for philosophy.

Leibniz, the progenitor of the PII, would have agreed.  He does not believe that an object’s locations in space and time qualify as properties.  These attributes place an object in a certain position in relation to all other objects, but they do not tell us anything about the object itself.  If I pick up my book, the book does not change.  The passage of time itself does not make milk spoil; rather, the milk changes because of the chemical reactions that happen during this time.

Space and time are purely relational properties to Leibniz.  Since the Earth is rotating and orbiting around the Sun, and our very solar system is in motion, it is ludicrous for us to consider anything in our world to be stationary.  Due to inertial effects, even if something in the universe was stationary, we would not notice it.  Furthermore, our universe is practically infinite.  Thus, it is ludicrous to say that an object A could have an absolute location; it would require boundaries which we don’t have.  We can only define an object’s location in terms of its proximity to other objects.  Time, like space, is apparently infinite; thus, we can only place objects in time by saying what came before and after them.  Our BC/AD system of counting years is as arbitrary as any other.

In his correspondence with Samuel Clarke, Leibniz proposes the Principle of the Identity of Indiscernibles as a necessary consequence of the Principle of Sufficient Reason (PSR) and his theological beliefs.  According to the PSR, there must be a sufficient reason for everything which happens.  According to Leibniz’s theology, God has an active role in the world.  He is all-powerful, so He can move the parts of the universe however He likes.  Since He is all-knowing and also all-good, all His actions are perfect.  Thus, He must have a logical, a sufficient, a good reason for everything He does.

Now, if God created two identical but separate and distinct objects, A and B, He would not know what to do with them.  Should he place A in Position X and B in position Y, or should he place B in X and A in Y?  Because A and B are identical, it does not matter where God puts them.  A and B will perform exactly the same function in X.  Thus, whatever decision God makes about the two will be entirely arbitrary.  Since God is always logical, though, He is incapable of arbitrary decisions.  Such a God would be stunned, trapped inside a paradox of His own making.

God, being perfect, knows that He cannot allow this to happen, and for this reason, He would never allow any two objects to be absolutely identical.  He would not create two such things, and He would not allow any two objects to become identical, either, because He might have to use them at any time.  A and B might be “practically” identical – in other words, alike in any way our senses can discern – but there will always be something, perhaps even something as small as a missing quark, which distinguishes the two.  This state of affairs which God has arranged is also known as the Principle of the Identity of Indiscernibles.

Leibniz’s argument is valid, but given its theological nature, we can easily call its soundness into question.  Atheists can reject this argument out of hand.  Other theists can argue that though God created the universe, He now allows everything inside it to function according to the laws of physics which He created.  Since he does not actively move objects like A and B, He does not need to regulate their identities.  (This is a rather Newtonian way of viewing God, so it’s easy to see why an anti-Newtonian like Leibniz didn’t consider it.)  Samuel Clarke argues that God would not need the PII at all because His Will is sufficient reason for anything He wishes to do.  If He wishes to make an arbitrary decision between A and B, so be it.

The other component of Leibniz’s argument for the PII, the PSR, is not as controversial.  Other thinkers merely regard it in a non-theological manner; for instance, one of the basic axioms of science is that every physical phenomenon has a cause.  Even those who believe that human beings have free will believe that each person has a reason, no matter how legitimate, for his actions.

I do not know the thoughts of God, but I am inclined to reject Leibniz’s argument for the Principle of the Identity of Indiscernibles.  I am sympathetic toward the Newton idea of God: He created the world and His laws, and though He sometimes overrides physics in the form of miracles, He does not do it so much that He has to worry about identical As and Bs.  Even if He does, I do not think God is as beholden to the Principle of Sufficient Reason as Leibniz imagines.  There are plenty of theological points, from the Trinity to omniscience itself, which I don’t think I will ever be able to comprehend.  Surely, God has a way to solve this problem which we do not understand.  Perhaps, He would instantaneously put two identical objects in two different places, so he wouldn’t have to agonize about where he should move each one.  Leibniz’s argument is nothing but an unverifiable assertion, so we cannot contemplate it as a philosophical principle.

Future philosophers were not as theologically inclined as Leibniz was, so they did not argue about the PII on his terms again.  Instead, the PII resurfaced as a component of the argument between the bundle and substratum theories of objects.  Under this framework, we can consider a variety of incarnations of the PII.  Loux details the bundle-substratum debate very well in pages 111-117 of the grey book, much of which I will summarize here.

The bundle theory (BT) states that each object is nothing but a collection of its properties.  Put formally,

Necessarily, for any concrete entity, a, if for any entity, b, b is a constituent of a, then b is an attribute. (112)

In response, substratum theorists argue that BT requires the PII and that PII is false.  They note that the bundle theorist, like all other ontologists, is committed to the Principle of Constituent Identity (PCI), which states that any whole is the sum of its parts.  Put formally,

Necessarily, for any complex objects, a and b, if for any entity, c, c is a constituent of a if and only if c is a constituent of b, then a is numerically identical with b. (113)

According to BT, all objects are the sum of their properties.  According to PCI, if two objects share all their constituents, then they are the same object.  Therefore, if two objects share all their properties, they must be identical (the PII).  So, the Principle of the Identity of Indiscernibles is a necessary consequence of BT and PCI.

Substratum theorists deny the PII on the grounds that it is possible for two objects to share all their properties.  Because each object contains millions of atoms and countless electrons and quarks, it is extremely improbable (practically impossible) for two objects to be identical, but it is not logically impossible like the PII claims.  They then challenge bundle theorists to prove them wrong.

Max Black proves this with a thought experiment in his dialogue “The Identity of Indiscernibles” (Red Book 104-113).  He proposes a world in which only two objects exist: two spheres which are qualitatively identical and which stand a certain distance from each other.  Black then challenges his imaginary critic to prove that this world is impossible.

The critic first suggests that if he enters this world, he will be able to distinguish the spheres because one will be to his left and one to his right.  Black responds that he would indeed be able to do this, but it would entail introducing a third party to the world and ruining the example.

The critic also argues that the spheres are different because they have different identity properties.  In other words, A has the property of being A, but it does not have the property of being B.  B has the property of being B, but it does not have the property of being A.  Therefore, A and B are distinct.  In response, Black says that “A is A” is a tautology, not a property.

This impasse pushes Black and the critic into an argument about relationships as properties.  Black is quite hesitant to allow them; he accepts “being one meter from a sphere” as a property but rejects “being one meter from sphere A.”  His argument is epistemological: because we cannot distinguish between the two spheres, it is impossible to give them names.  I disagree with his reasoning.  We may not be able to remember which sphere is A and which is B after we arbitrarily name them, but according to Black, the two are separate and distinct, so we must assume we can name them this way.  In this case, “being one meter from sphere A” is a relational property on the same level as “being one meter from a sphere.”  Since Black tries to pick and choose which relationships he will allow as properties, he fails to disprove his critic.

Indeed, if we accept relationships as properties, we must accept the Principle of the Identity of Indiscernibles.  Each spatial location carries a unique set of relationships with all the objects placed in spatial locations around it: 5 miles west of X, perhaps, or 2 meters under Y or 5 inches from Z.  For A and B to be separate and distinct, they occupy different locations.  If they occupy different locations, they have different sets of relationships, and thus they have different relational properties, and so they are not identical, and it is impossible for them to be so.  Even in the world of the two spheres, the relationship which A has with B is different from the relationship B has with B, and so the objects are different.

Thus, we must conclude that if relationships are properties, the Principle of the Identity of Indiscernibles is true.  Since this case is so similar to the first case we discussed (two objects share the same location), though, it also seems trivial.  It is interesting to consider that our relationships to our fellow objects are always changing, but that is the only new information this Principle can bring us.

In response to this PII, some ontologists, including James Van Cleve, draw the distinction between “impure” spatial, temporal, and relational properties and “pure” properties (all the rest).  (Leibniz himself seemed to endorse this view, as I noted earlier.)  According to these philosophers, “impure” properties assume the existence of concrete particulars.  For A to have the property of being identical to A, A must already exist.  For Sphere A to have the property of being one meter away from Sphere B, we must assume A’s and B’s existences.  Bundle theorists cannot accept dependent properties, however, because they have already said that objects are dependent on properties for their existence.  They cannot use properties which depend on objects to prove that objects depend on properties.  Instead, they must work with the “pure” properties alone.  We can then amend BT to BT*, which reads

Necessarily, for any concrete entity, a, if for any entity, b, b is a constituent of a, then b is a pure property/attribute. (116)

Our new Principle of the Identity of Indiscernibles, PII*, says

Necessarily, for any concrete objects, a and b, if for any pure property/attribute, φ, φ is an attribute of a if and only if φ is an attribute of b, then a is numerically identical with b. (116)

I think that this PII, free of location and relationships, is easily proved false.  Black’s example of the two spheres does that very thing.  Though, like I said earlier, it would be highly improbable (practically impossible) for two objects, even atoms, to change to the point that they share all their properties (same mass, hue, shape, etc.), it would not be logically impossible.  All objects can change their physical characteristics, after all, so they could simultaneously evolve into qualitatively identical beings.  We can thus deny the PII* or Principle of the Identity of Indiscernibles which considers only “pure” properties.

Loux does provide one way for bundle theorists to save the PII from the razor: they must embrace the trope theory.  Trope theorists are nominalists; in other words, they deny that any properties can be shared.  Each property that an object has is unique to that object.  The blue on my shirt is unique to my shirt; it could not exist anywhere else.  The same goes for the blue on my pants.  Since my shirt and pants are both blue, they seem to share blueness, but this is merely a mistake of the language.  The two colors are similar but intrinsically different.

Under this framework, the Principle of the Identity of Indiscernibles is necessarily true.  Each object contains a set of properties which is specific to that object.  Therefore, no objects which are separate and distinct could have anything in common.  If two objects to share the same properties, they must be one and the same.

This is a valid argument, but I reject it because I believe the trope theory is false.  How are the blue in my shirt and the blue in my pants similar?  Is it because they each share the color blue?  In that case, isn’t blue a universal after all?  Either there are universals, or all properties are unanalyzable.  The theory fails Occam’s Razor.

The trope theory seems to be to be a clumsy expression of the principle of imminent universals.  According to this principle, universals do not exist independently.  Rather, they are contingent upon the existence of the objects which carry them.  Each time an object exhibits a universal property, we call it a particular instantiation of that property.  Trope theory also says that properties are imminent and that each instantiation of a property is particular.  It errs because it denies or does not explain that objects can have common properties, after all; the principle of imminent universals addresses this problem.

Leibniz formulated the Principle of the Identity of Indiscernibles in order to justify his theology.  The PII was a disappointment then, and it continues to be one today.  When PII is true, it is trivial: everyone already knows that two objects cannot occupy the same position or stand the same distance from every other object.  When PII could have interesting implications – what if all objects were required to be physically different from each other in some way? – it is false.  The best thing we can say about it is that it helps us to discard bundle theory, paving the way for the Aristotelian Categories I will describe in my next paper.

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