Note: New material (in red) added on March 11, 2022.
Please make sure that you pay at least some attention to the idea that physical continuity (AKA "Spatio-Temporal Continuity") is actually the best solution to this problem. (I'm getting tired of reading papers that ignore physical continuity, or which assert that it isn't the basis of object-identity while actually relying on it to make their favorite theory seem to work.)
First, let me discuss the issue of identity. Suppose your grandpa owns an antique pocket watch that he has willed to you. (Call this "watch 1".) Suppose he keeps it in a safety deposit box in a bank. Suppose he also periodically takes you in to the bank to look at the watch, so you get to know the appearance very well. Now finally suppose that your grandpa, alas, dies and two untoward things happen in the bank. First, a clerk accidentally drops Grandpa's watch and it receives a long scratch across the back (Call the watch as it exists at this time "watch 2".) and second, when you come in to pick up your bequest, the bank manager makes a mistake and hands you a different watch, (Call this "watch B".) a watch belonging to a man called Smith that just happens to exactly resemble Grandpa's watch in every respect, and so you take it to be Grandpa's watch, and for the rest of your life you carry this watch ("watch B".) around in the happy belief that this is the watch your grandpa owned, even though it actually is not the same watch. Meanwhile, Smith collects the other watch, ("watch 2".) and, not minding the scratch, takes away Grandpa's watch in the belief it is his own.
So there's two watches. There's the watch that Smith now holds, which is actually the same watch your grandpa used to own, even though it is different, owing to the scratch. And there's the watch you now hold, which is not the same watch your grandpa used to own, even though it is physically the same in every respect.
Does this story make sense to you? Or do you
find yourself saying things like, "oh no, watch B is actually
Grandpa's watch", or finding some other problem with the story.
When the Ship of Theseus problem asks, if some ship is the same ship as the ship Theseus used to sail, it is asking if it is that ship in the sense that the watch held by Smith, with the scratch, is the same watch that your grandpa used to own. It is not asking if it looks the same, feels the same, sounds the same or smells the same, it is asking if it is the same ship.
Before you start, you should get clear on the concept of "Physical continuity". This is the property of being the same coherent discrete object over time.
In the story of the two watches, watch 2 is physically continuous with watch 1, while watch B is not physically continuous with watch 1. In fact, physical continuity is the only thing that connects watch 2 with watch 1, so if you think watch 2 is watch 1, physical continuity is your only reason for thinking that they're the same watch.
Human bodies are physically continuous because although they grow and change over time, this is always because of small quantities of material, contained in food, drink, and air being added to the body, and other bits of material leaving the body as exhalations, discarded skin and hair, and other ways. Thus the body you have now is physically continuous with the body you had when you were born, and indeed, with the fertilized ovum you were just after you were conceived, even though the vast majority of atoms now in your body were not there at that time.
Another way to illustrate physical continuity is to consider two houses sitting side by side in a subdivision in 1950. Being recently made by very precise and conscientious builders, they are exactly similar to each other. They are so similar at this point that the only way to tell which one is which is by seeing (or remembering) where the other one is. While the houses are still indistinguishable from each other, they are purchased by Connor Tinuity and Samantha Form. Sam Form thinks that her new house is perfect. To her, everything about her new house, the facade, the floor plan, the colors, the doors, the windows, everything, is just exactly as she wants it. Con Tinuity looks at his new house and thinks, "it's a good start." Over the following years, a little bit at a time, Con makes many alterations to the house, moving windows, tearing out whole rooms, adding extra rooms, making a courtyard in the middle of the hourse, adding porches, adding a whole extra story, and a tower, and an observatory. This all takes a very long time, since Con only makes one change every year or so, but eventually, Con's house looks nothing like it's original form. In contrast, Sam works hard to keep her house exactly the same. Putting preservatives on its surfaces, fixing small problems immediately they arise, and so on. finally, in the year 2000, Con's house is a wild extravaganza that looks nothing like the original structure, and Sam's looks exactly the same as when she first bought it. Now, at this point, Con's house attracts tourist who come to gape at the strange house, and Sam haaaaaates the tourists, and haaaaates even looking at Con's house, and so she gladly agrees to sell her house to Con when he offers to buy it so he can put in a giant Tesla coil. Sam is especially happy to sell because she has found out that the builder who made her house also built several more exactly like it on the other side of town. Sam buys one of these, and moves all her stuff other there, setting up her new house exactly the way she had the old one. Meanwhile, Con has completely demolished Sam's old house, burning all the wooden bits and grinding everything else up for landfill. Now, the point of this story is to point out that Con's present house has physical continuity with his original house, even though it now looks nothing like his original house, and Sam's present house does not have physical continuity with her original house, even though it looks exactly.the same. This is what the term "physical continuity" means! If you have a house and you alter it in some way, say by moving a bit, adding a bit, or taking a bit away, the house after the alteration has physical continuity with the house before the alteration, even though the house is no longer the same.
Now watch this Video read one or more of the following articles by people who may or may not know what they are talking about.
The Ship of Theseus
The Ship of Theseus and Personal Identity
Identity, Persistence, and the Ship of Theseus
The Ship of Theseus and the Question of IdentityWhen you think you have a good understanding of the issues involved in the Ship of Theseus problem, take some more "think" time do your best to come up with the most logically justifiable answer you can think of. When you have thought about the issue, write a paper explaining the problem, explaining what you think about the problem, and saying exactly why you think what you think.
Practical solutions are preferred as opposed to arbitrary abstract, or speculative solutions that would never be used for ordinary objects in real life.
Alternate assignment: Remember Grandpa's watch? Turns
out that Smith, the owner of the other watch (the one you have) has found
out about the switch, which is verified by security camera footage, and
wants his original watch back. You don't want to swap, and don't want to
believe that the watch you have isn't Grandpa's, so he takes you to court.
You go into court with Smith, each believing that the watch they
hold is your grandpa's watch. Now, what rule
should the judge use to decide which watch is yours, and which is Smith's?
Should he decide that the watch Smith has, the one that has physical
continuity with the watch your grandfather owned is your
grandfather's watch, or should he decide that the watch
Smith has, the one that you believe to be Grandpa's watch, but
Smith believes to be not Grandpa's watch is Grandpa's watch? After you've thought
this through from all angles, write a paper explaining which rule
you favor (continuity, belief, or whatever) and why you reject the