Defining the nature of identity can be a perplexing task.  For every definition or principle of identity that is put forward, a host of problematic thought experiments will inevitably arise to challenge it.  In this post we will be applying Leibniz’s Law to the problem of the statue, the vase, and the hunk of metal.

One of the most well-known principles of identity that has ever been offered was given to us by the seventeenth century philosopher, Gottfried Wilhelm Leibniz.  His principle, which has come to be known as Leibniz’s Law, asserts that necessarily, for any X and Y, if for any property, P, X has P iff Y has P.  This principle actually contains two separate principles which can be listed as follows:

1. The Indiscernibility of Identicals: Necessarily, for any X and Y, if X is identical with Y, then for any property, P, X has P iff Y has P.

2. The Identity of Indiscernibles:  Necessarily, for any X and Y, if for any property, P, X has P iff Y has P, then X and Y are identical.

The second of these two principles is considered by some philosophers to be controversial.  This paper, however, will only be considering the first principle.  The Indiscernibility of Identicals is generally not thought to be a controversial principle, and it can be an immensely helpful tool in solving puzzles about identity, as we shall see below.

            Imagine the following scenario that involves a hunk of metal:

            1. The hunk of metal was a statue on Monday.

            2. The hunk of metal was a vase on Tuesday.

            3. On Wednesday, the hunk of metal was neither a statue nor a vase.

It appears therefore that:

            1. The hunk of metal = a statue on Monday.

            2. The hunk of metal = a vase on Tuesday

            3. The hunk of metal = neither a statue nor a vase on Wednesday.

This line of reasoning becomes very problematic, however, when we introduce the principle of the Transitivity of Identity.  This principle states that if a=b and b=c then a=c; or to apply it to the previous scenario.

            1. If the hunk of metal = the statue

            2. And the hunk of metal = the vase

This leads to the absurd conclusion that

            3. The statue = the vase.

Some would argue that the conclusion can be avoided by restating Leibniz’s Law so that it is time contextual.  Thus, necessarily, for any X and Y, and any time, t, if X is identical to Y then at any given time, t, X and Y must have the same properties.  Yet this does not avert the conclusion at all.  Indeed, it leads to the same inconsistency.  Using the above restatement of Leibniz’s Law we come to the following conclusion:  if the hunk of metal is identical to the statue on Monday, then on Tuesday, the hunk of metal must still be identical to the statue.  This is obviously not the case, however, since the hunk of metal is identical to the vase on Tuesday and is identical with neither the statue nor the vase on Wednesday.

Another proposed solution to this problem is to assert that the only thing that actually exists over the three days is the hunk of metal itself.  Thus, we would say that on Monday the hunk of metal is statue-shaped, while on Tuesday the hunk of metal is vase-shaped.  This theory – a form of Mereological Nihilism – would deny the existence of things like statues and vases and would propose that the only things that truly exist are basic substances (like hunks of metal, or rather, the mereological atoms that compose the hunks of metal) that are arranged in various ways.  While this is a possible solution to the problem, it is hard to reconcile this theory with our basic intuitions that clearly acknowledge the existence of things like statues and vases.

There is yet a third solution to this problem, however, that is much more in agreement with our intuition.  This solution asserts that the hunk of metal is not identical to either the statue or the vase.  According to the principle of the Indiscernibility of Identicals, two entities are identical iff they share the same properties.  Therefore, to demonstrate that the hunk of metal is not identical to either the statue or the vase, we must point out some property that they do not share in common.  Yet, this may be more difficult than it sounds.

Since the hunk of metal and the objects that it constitutes occupy the same place and have the same mass and extension, it appears that the hunk of metal is indeed identical with the statue and the vase.  In other words, if on Monday, the statue has the property of being two feet tall, then so does the hunk of metal; and if on Tuesday, the vase has the property of being cylindrical, then the hunk of metal possesses the same property.

However, when we begin to look at things from a temporal perspective, the indiscernability of these objects begins to break down.  Take, for example, the hunk of metal and the statue which it constitutes.  If these two substances are not identical, then on Monday they each occupy the same place at the same time.  Yet at the end of the day, when the metal is melted down, the statue is destroyed while the hunk of metal continues to exist.  Thus we discover that the hunk of metal possesses the property of existing past Monday while the statue lacks this property.  Now, we can inverse the principle of the Indiscernibility of Identicals to read as such:

Necessarily, for any X and Y, and for any property, P, if X possesses P and Y does not posses P, then X and Y are not identical.

Therefore, according to the principle listed above, the hunk of metal and the statue cannot be identical due to the fact that the hunk of metal possesses a temporal property which the statue lacks.

We are left, then, with two possible solutions to the problem of the hunk of metal, the statue, and the vase.  We can either espouse a form of Mereological Nihilism or – by introducing Leibniz’s Law and properly defining the properties of each object- we can assert that the hunk of metal, the statue, and the vase are all separate entities.  Considering the highly unintuitive conclusions of the first solution, the latter solution seems to be the most favorable to answer this problem.

Two Spheres

For any ontological category, it is preferable to have a principle of individuation that will make it possible to distinguish entities within it.  However, discovering exactly what it is that makes one substance diverse from another can be a difficult task; this is particularly true when the substances in question are exactly alike (share the same internal properties).

Imagine a universe in which the only things that exist are two spheres.  These spheres are exactly the same in their internal properties.  The question that must be answered is this: “What is it that makes these spheres different?”  Or, in other words, what is the principle of individuation for these two objects?  I will examine the merits of the following four theories.

  1. They are individuated by space and time.
  2. They are individuated by the matter of which they are composed.
  3. They are individuated by their properties.
  4. Individuation is a brute fact.

1. Individuation by Space and Time
One possible theory is that the two spheres could be individuated by space and time.  Thus, Sphere A is distinct from Sphere B because A and B occupy different places at the same time.  This theory may seem very plausible at first glance; it is quite obvious that the two spheres are not in the same place at the same time, so it seems that this must indeed be what dinstinguishes them.  However, after closer examination, we find that this theory brings some heavy baggage along with it: namely, this theory presupposes the Absolute Theory of space. 

The Asbsolute Theory of Space, championed by Sir Isaac Newton, asserts that space is a container that holds the objects that exist within it.  Therefore, space is not dependent upon objects and would exist regardless of whether anything else did.  Modern physics, however, has powerfully discredited this theory of space and it is no longer espoused by many scientists or philosophers.  Furthermore, we are sitll left with the question, “What would the principle of individuation for each place be?”

One may ask, then, why it is not possible to use the competing theory of space (the Relational Theory) instead of the Absolute Theory in formulating a principle of individuation for the spheres?  The Relational Theory, famously held by Gottfried Wilhelm Leibniz, asserts that space is dependent upon objects.  In other words, what we refer to as space is merely the realtion between objects and their observers within the universe.  The Relational Theory of space will not accomodate a place-centered principle of individuation, however, due to the fact any appeal to “place” within the relational framework will lead to a circle of interedependence of places and objects since space exists in relation to objects.  Thus a principle of individuation based upon space and time does not appear to be plausible.

2. Individuation by Matter
Another principle that could be used two differentiate between the two spheres is that of matter: so that A is distinct from B because A and B are composed of different matter.  This theory assumes the priority of understanding substance as “stuff” (formless matter) over the priority of understanding substance in a counting sense (individual substances).

Aristotle (and consequently, Aquinas) held that matter is prior to individual substances and that one could individuate substances by the matter of which they were composed.  In other words, an individual substance is matter that has form imposed upon it.  Thus it seems that Aristotle’s principle of individuation relied heavily upon the forma indentity of a substance.

Leibniz, on the other hand, asserted that individual substances are prior to stuff.  Indeed, he believed “stuff” to be composed of individual substances which he referred to as ‘monads’.  Although Leibniz’s monads lead to an idealistic understanding of reality, the basic principle (if there are complexes, then there must be simples) holds true.  In the view of modern science, these simple substances would not be monads, but the basic particles which exist on the atomic level.  Thus, individual objects seem to have more priority than formless matter.

Furthermore, the principle of individuation given above involves circular reasoning.  If we individuate the two spheres, A and B, by the matter that they contain, what then is the principle of individuation for the matter itself?  One may reply that we can individuate the matter by the form imposed upon it, but this leads us back, once again, to the two spheres.  How do we individuate the forms?  By the matter which composes them.  On and on we go.

3. Individuation by Properties
A third (and quite promising) possible principle of individuation for the two spheres is individuation by the properties which the spheres possess.  The reason that this theory is more feasible than the previous two is that it appeals to an ontological category outside of the concrete category.  Whenever a theory attempts to individuate an entity on the basis of something within the same category as that entity, it leads to serious problems (usually a vicious circle).  This theory of individuation is based upon an inverse understanding of the Indiscernability of Identicals known as the Diversity of the Dissimilar:

Necessarily, for any A and B, and for any property, P, if A possesses P and B does not possess P (or vice versa), then A and B are not identical.

All that is necessary, then, to individuate the two spheres is to demonstrate that they do not actually share all of their properties.

Leibniz actually proposed that it was impossible for any two substances to have the exact same properties.  He believed that any similar objects would differ at some point even if that point were so miniscule as to escape observation.  However, Leibniz’s reasoning behind this was pure theological speculation.  He asserted that God would never create any two substances with the exact same properties because there would be no reason to do this and that doing so would contradict God’s good and reasonable nature.  Again, pure speculation.

Duns Scotus disagreed with Leibniz.  Scotus proposed that substances could be identical in every general property, but that each individual substance possessed a nongeneral property that was unique to that substance.  He called this special property a haecceity or “thisness” (from the Latin pronoun for “this”).  Thus Sphere A is distinct from Sphere B, because A possesses the property of ‘A-ness’ and B does not (and vice versa).  While the idea of haecceities may seem almost too convenient for the problem at hand, there does seem to be strong evidence for their existence when one considers the concept of self-recognition.  How is it that we are able to simply know that we are indeed ourselves?  What is it that we appeal to when we do such?  It appears that I am appealing to the property of “Josh-ness” when I consciously recognize myself.  Thus the principle of individuation based upon abstract properties seems much more viable than the theories we have looked at thus far.

4. Individuation as a Brute Fact
A fourth, and final, response that can ve offered to the problem of explaining the diversity of the two spheres is that individuation, itself, is a brute fact that has no explanation.  It seems that the nominalist would be forced to accept this fourth proposal since any appeal to individuation that he would make would be within the same ontological category (concrete) and would lead to circular reasoning.  Brute facts, however, are not welcome within the theories of most philosophers, since a good system of metaphysics will seek to explain as much as possible.  Therefore, after reviewing the four theories given above, it is the property-based principle of individuation that seems to be the most promising explanation to the problem of the two spheres.


Three Cats

          In the debate between realists and nominalists, one of the most noteworthy arguments that realists have employed is known as the Formal Cause Argument.  This argument takes its name from one of the four causes that Aristotle listed as accounting for the reasons why a certain thing exists.  Among these four causes, the formal cause can be thought of as the exemplar for something else.  Take, for example, a statue of Socrates.  The formal cause for this statue would be Socrates, himself, since the sculptor is seeking to copy the form of Socrates in stone.

            The Formal Cause Argument can be given like so:

  • 1. Felix is a cat, Garfield is a cat, and Morris is a cat.
  • 2. There must be something about Felix, Garfield, and Morris that accounts for all of them being cats.
  • 3. This “something” can only be their felinity or “catness”.
  • 4. Therefore, the universal, “catness”, exists.

            The above argument is valid; therefore if the nominalist desires to counter it, he must attack one of the argument’s premises.  Since the nominalist would have to agree with the first premise, the only premises that remain are premises two and three.

            In rejection of premise two, some nominalists assert that there is no explanation for why Felix, Garfield, and Morris are cats: this is simply a brute fact.  While this is certainly a position that an individual can take, it tends to be quite an undesirable one because of the need to appeal to a brute fact.  Most philosophers desire to be able to explain as much as possible about the universe, but brute facts deny the possibility of explanation.  Therefore, whenever possible, it is better to give a reasonable explanation for something than no explanation at all.

            Thus the nominalist is left to counter the third premise of the Formal Cause Argument.  One such attempt is the Verbal Definition theory of universals.  This view was popularized by the medieval monk, Roscelin, who asserted that universals are nothing more than flatus vocis or “puffs of air”.  In essence, Roscelin was asserting that what are often mistaken for universal properties are nothing more than verbal definitions.  For example, the property of “catness” can be reduced to the technical definition of “cat”.  Therefore, Felix, Garfield, and Morris are all cats in virtue of the fact that they all conform to the definition of “cat”.

            The problem with this theory is that it is not so easy to define things as one may think.  While it may be a somewhat simple matter to define a basic element like gold or iron; just how do we define the word “cat”?  Perhaps we may say that a cat is a four-legged mammal with whiskers and a tail; however this very same definition applies to thousands of other species of animals (and perhaps some species that have not been discovered as of yet).  And what are we to do with a cat that is born with only three legs?  Is this animal no longer a cat?  Our base intuition would say that it is, but according to the above definition it could not be.  In short, the problem with the Verbal Definition theory of universals is that it requires something that we cannot give; in this case, a perfect and exhaustive definition for “cat”.

            Another objection to the third premise of the Formal Cause argument is the Causal Theory.  This theory seeks to dismantle the Formal Cause argument by appealing to another one of Aristotle’s four causes: the efficient cause.  The Causal theory would assert that Felix, Garfield, and Morris are all cats based upon the fact that each one of them had another cat as its efficient cause.  In other words, we know that a creature is a cat because it came from a cat. 

            The problem with this theory, however, is that it is based upon the assumption that “like begats like”.  This means that each cat would have to be produced by another cat ad infinitum.  There could never come a point in which a cat was formed by something that was not a cat (such as Darwinian evolution or Divine creation).  The problem here is that this theory requires that every kind of concrete particular be eternal.  In other words, cats would have always existed and will always exist.  There are not too many individuals who would be willing to accept this as a fact.

            A final and more promising alternate theory offered by nominalists is called Trope Nominalism.  In this theory, nominalists introduce the concept of tropes.  Tropes are asserted to be the particular, non-sharable, concrete aspects of a certain concrete particular.  For example, if two squares are the same shade of blue, the realist would say that both of these squares exemplify the universal “blueness”.  However, the trope nominalist, while stating that the two squares resemble one another, would assert that neither one of these squares share a universal property but that each one of them exhibits a particular shade of blue that is unique to that square (a trope). 

            Indeed the very idea of “resemblance” is crucial to this form of nominalism (trope nominalism is sometimes referred to as resemblance nominalism).  A trope nominalist explains their position as such:

X, Y, and Z belong to a kind F iff X, Y, and Z each have a particular F-ness and these F-nesses form a circle of similarity (or resemblance).

            Applying the above definition to Felix, Garfield, and Morris would yield the following result:

Felix, Garfield, and Morris are all cats iff all three of them have a particular “catness” and these “catnesses” form a circle of similarity.

            Some realists would argue that this brings the trope nominalist perilously close to realism due to the fact that the relation of resemblance between each concrete particular appears to be a universal itself.  However, the nominalist reply is that each resemblance relation is a particular resemblance relation (just as each trope is a particular aspect) based upon its resemblance to other resemblance relations.  While some realists would (correctly) argue that this leads to an infinite regress in resemblance relations, the nominalist can easily defend himself by pointing out that realism, itself, is dependant upon an infinite regress of shared qualities.

            Therefore, it appears that the concept of tropes provides the nominalist with an effective counterexample to premise three of the Formal Cause Argument.  One can accept either the existence of universals or the existence of tropes to give an account of the “something” that causes Felix, Garfield, and Morris to all be cats.  And in the case of this particular argument, it seems that the nominalist (with Ockham’s razor in his corner) would be in a better position with his theory of concrete tropes then the realist with his theory of universals.