My writing in the last couple of days has been exploring the ancient philosophical problem known as The Ship of Theseus, which you might know as grandfather's axe - when granddad says it's his favourite axe; and that he has replaced the head 3 times and the handle twice. The question philosophers usually as is, "Is it really the same axe?"
Unpacking this problem and establishing useful ways of thinking about it has been very enjoyable.
My way into the problem was to notice that no matter whether we think it is the same axe or a different axe, we never doubt that it is an axe. Because the parts are generic we can replace them at will without changing the intrinsic properties of the object. Any correctly assembled combination of axe-head and axe-handle makes up an axe. Change of a part does not affect the identity of the complex object as a whole.
So, at least at this level, the object has identity and continuity as an axe. It is an axe and we know it is an axe. These are objective facts. The first is an objective fact about what is (ontically objective), the second is an objective fact about what we know (epistemically objective). The object either has the relevant properties or it does not. The fact that it is an axe is dependent on the observer knowing what an axe is. But any observer who knows what an axe is (no matter what they call it) will correctly identify it as an axe.
But is this grandfather's axe? Ownership depends entirely on the minds of grandfather and his community. He asserts "this is my axe" and the community either ascent or they don't. So ownership is some kind of subjective fact. In which case, there is no one right answer. Some might feel that property is theft, in which case grandfather's assertion carries no weight. Or grandfather might have become confused with another similar axe.
Maybe it's not so much a matter of ownership, but of close association. In which case this is also a subjective fact. Recognition is a matter of seeing the object and having a feeling about it. In the bizarre neurological disorder, Capgras Syndrome, people visually recognise their loved ones (usually, but it might also include pets or familiar objects like one's home), but the identification does not set off an emotional reaction. The spouse looks exactly right but feels wrong. The person with Capgras is usually at a loss to explain this. And the explanation that they have suffered brain damage doesn't help much. They often confabulate stories - the spouse has been replaced by a duplicate or doppelganger for nefarious purposes. Again there is no right answer. If grandfather feels that this is his axe, then that is what he feels. That we do not feel it only tells us that we are not grandfather (which we already knew).
Objective facts are independent of observers. Metal is hard, it can be shaped into a cutting edge. Wood is firm but flexible and can be shaped into a handle. None of these statements depends on an observer or what they believe. Subjective facts are not always shared. They do depend on the observer. Money, for example, is based on us all agreeing that bits of paper or plastic represent units of wealth. A £5 note is intrinsically almost worthless. But £5 of wealth is enough to redeem for a cup of coffee and a slice of cake (outside of London). If we stop agreeing to those special bits of paper or plastic are valid tokens, then the system breaks down. This is what happens when there is hyper-inflation for example.
The Athenians maintained a boat that at one point in its history carried Theseus and his companions to Minos, where he overthrew the Minotaur, and then it ferried him back to Athens. Theseus went on to become a great general/admiral. So for Athenians the boat is a symbol of a national hero; of someone they feel epitomises their national character. For the Athenians it is definitely Theseus's boat. If we don't know who Theseus was, or his story, or anything much about ancient Athens, then we may not feel any connection with the symbol. We may conclude that it is not Theseus's boat. But even if we had lived at the time, what we believed would probably not have changed the minds of the Athenians.
If they had been celebrating a goat as the boat of Theseus, then we could have made an objective argument that a goat and a boat are not the same. A goat might be Theseus's goat, but it cannot be Theseus's boat. But because it was a boat, and remained a boat despite repairs, we can only make subjective arguments. And, frankly, why should the Athenians care what we think about their hero and his boat?
And of course it gets much more interesting when we get to the fact that the boat or the axe is a metaphor for ourselves.