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In the philosophical field of metaphysics, the problem of change is the question of how change can occur.
Let's begin with a definition of the word "change." First, notice that when an object changes, it always changes in some particular way. A baby grows up, and so changes in the respects of size and maturity; a snake sheds its skin, and so changes in the respect of the skin it has. So here's a definition of "change":
An object O changes with respect to property P iff O has P at one time and at a later time O does not have P.
That seems to be what it means for a thing to change: it has a property at one time and later it does not have that property. If a banana becomes brown we say: at one time the banana is yellow; several days later the banana is not yellow, but instead brown. This is all appears fairly straightforward at this point. No apparent problems yet.
But what about the sort of change after which a thing is destroyed? When a person dies, we don't say that the person's life has changed. We don't go around saying, "Harry just isn't the same sort of guy after he died." We say that Harry's life has ended. Or when a building is demolished, we don't say that the building changes; we say that it is destroyed. So what sort of events, on the one hand, result in a mere change, and what sort of events, on the other hand, result in a thing's destruction, in the end of its existence? Now that's one aspect of the problem considered here; we can call it the problem of change and identity.
There is one particular problem of identity that is generally explained with the story of the Ship of Theseus: One day in ancient times Theseus had a ship -- call the ship "the Theseus" for short -- and it was launched into the Aegean Sea. As the years wore on the Theseus started getting creaky and weak and otherwise less-than-perfect. So as boards started getting particularly creaky, they were removed and put into a warehouse, and replaced with new boards. Then masts started tottering, and so soon they were warehoused and replaced with new masts. And in this way, after fifty years, there is a ship moored in the harbor, but this ship now has all new boards, masts, everything. So is the ship in the harbor now -- call it S2 -- the same ship as the ship that was in the harbor fifty years ago -- call that original ship S1? In other words, is the ship we're looking at now really the Ship of Theseus?
There's one answer which is a little too easy and quick. You might say: "No, of course not. The Theseus has changed a lot, so it's not the same ship. At the end of your life you're not going to be the same person you were when you were a teenager. You're going to change a lot in the meantime." However, this is not quite answering the intended question. What is intended by the question is a sense of the word "same" in which an old woman is the same person at the end of her life as she is at the beginning of her life. Certainly, the word "same" has such a sense. After all, we implicitly depend on it when we say, for example, "She has changed a lot." In order for someone to change a lot, there has to be one person who underwent the change. (One could perhaps reject that sense, saying that objects do not change over time.)
Look at it this way. Go back to our definition of "change." I said an object changes with respect to a property if the object has that property at one time and at a later time the object does not have the property. What changes is the fact that an object has a particular property. The only way that that fact can change is if the object remains in existence. So you can think of a continuing object as the ground of change, or the arena where change occurs, as it were. So to get back to the Theseus, the question is: Has the Theseus merely changed a lot, or is the Theseus gone and a new ship now in its place?
Maybe you'll say, "Sure; it's just a refurbished Theseus, greatly changed to be sure, but still the Theseus." If you think that, then consider an addition to our story. Suppose someone buys all the planks and masts and whatnot that is stored in the warehouse, and out of all of those materials, and no other materials, he builds a ship according to the same plans that were used to build the ship christened the Theseus. And this ship, call it S3, is launched and sits on the other side of the harbor where we find S2. Might we not say that S3 is the same as S1 -- that this recently-constructed ship is the same ship as the ship originally called "the Theseus"? After all, S3 was built out of the same materials, and according to the same plans, as S1.
But then we have a problem. We can't say that both S2 and S3 are the same ship as S1, the original Theseus. Because if they were both the same as S1, then they would have to be the same as each other. There is a rule that states that if x=y, and x=z, then y=z. And S2 and S3 are clearly different ships: they are sitting on opposite sides of the harbor. So we have three choices: either, first, S2 is the same ship as S1; or, second, S3 is the same ship as S1; or, third, neither is the same ship as S1, and S1 has ceased to exist. How do we decide what is correct in this case?
It's hard to tell. Well, let's get some more theory on the table. Whenever we make an identity claim -- that is, a claim which states that two things are the same -- we almost always use two different descriptions. Not always -- sometimes we say "x=x," like "I am myself," but such claims are not particularly interesting or informative. The interesting identity claims are claims where we use two different descriptions for one and the same thing. For example, take these two descriptions: "the Morning Star" and "the Evening Star." Sometimes you can look in the sky just before dawn and see a very bright point of light in the sky -- that has been called "the Morning Star." And then also you can look in the sky just after sunset and see a very similar bright point of light -- that has been called "the Evening Star." In fact, the Morning Star is identical to the Evening Star -- both are the planet Venus. So they are "two" things only in description; in fact they are one and the same thing under two different descriptions.
Now it is a similar case with S1, S2, and S3: those are three different abbreviations, which stand for descriptions. "S1" means the ship which sat in the harbor fifty years ago, newly christened "the Theseus." "S2" means the ship which sits in the harbor now with the new planks. "S3" means the ship which sits in the harbor, recently constructed out of the old planks. So when we ask a question like "Is S2 the same as S1?" we can be understood to mean this: "Is the ship which sits in the harbor now, with the new planks, the same ship as the ship which sat in the harbor fifty years ago, newly christened 'the Theseus'?" Do those two descriptions pick out or refer to the same thing, or don't they?
Philosophers aren't interested in the Ship of Theseus problem simply because it is an interesting little puzzle. In fact, it's just an example of a more basic problem. The more basic problem is this: How do we decide that X is the same as Y, where "X" describes something at one time, and "Y" describes something at a later time? If you like we can call this the problem of identity over time, or alternatively, the problem of change.
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