The logical law of identity is epistemologically self-evident because it would still by necessity be metaphysically true even if it was false, making the latter thus impossible; a thing can only be what it is. If it was not, it would be something else, but that would still be what it is. Reason is reason, an apple is an apple, a specific apple is that apple and not another, and so on. Anything contrary to this is logically impossible because it would still require the law of identity's veracity. Unlike logical axioms, it is not self-evident as to whether something like a ship is still the same vessel if a certain amount of its individual components are replaced over time (this depends on axioms and cannot be true or knowable only in light of itself). If one plank of wood was removed and a new piece used, the ship would scarcely have changed.
The other parts would be both intact and the originals. Still, it would not have the same components, and every single one of them could be switched for a new part given enough time. This is what the concept of the Ship of Theseus is about: if a ship, such as the one used by Greek mythological figure Theseus, the founder of Athens, has its pieces exchanged, is it still the same ship? At what point is it not? Any truth about the matter would of course have to be the case with any other ship, whether made of ancient or modern materials, and of any other physical structure or item that can be repaired or altered. As long as not every part is replaced simultaneously, it is clearly the same object being modified to a limited extent in its immediate physical composition. The rest of the whole does not change because of any individual part alone.
A house that no longer has any of its original parts, moreover, is still the same house in a sense if it was not remade all at once. If our bodies are constituted of trillions of cells that die and are replaced over the course of our lives, then the same heart of matters like this and the Ship of Theseus is relevant to our very physical being. Our bodies usually retain their appearance, shape, and so on across time, but individual cells regularly die (apoptosis) in a healthy, biologically "normal" manner before cells divide (mitosis) in two to replace the former ones. The body can still maintain much of its ordinary nature despite how abnormalities like excessive mitosis can lead to new states of the body (in this case, cancerous growths). If our bodies, though, entail an even more complex process of physical transformation than a ship with its materials, what of our minds?
Of course, the self is the conscious mind, so the persistent replacement of one cell out of many for another over the course of biological life would not mean that the mind also necessarily undergoes this process of substitution. In fact, since the contents of one's consciousness cannot be misinterpreted if one makes no assumptions, for one's mind is directly experienced, unlike something such as external objects, a persistent, smooth stream of experience from one moment to the next refutes this already. Yes, a thought can rise and then dissipate, and many thoughts can voluntarily or involuntarily manifest and depart in even a single day even as at least a million cells reportedly undergo apoptosis each moment.
Still, thoughts occur inside of a mind, so they are not the same as the mind that perceives the thoughts. The appearance and disappearance of a particular thought only changes the immediate content of a consciousness rather than the enduring nature of the consciousness itself. Consciousness is also immaterial, even if it turns out that it is spawned and sustained by physical matter and its processes, so parts of the body being replaced like the pieces of Theseus's ship would not automatically entail that there is not a unified, ongoing consciousness presiding in one's body. Someone's thoughts can surface and flee, and their personality can shift, but it is the same consciousness that experiences all of this as the exact contents change.
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