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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.