I will express the 'Ship of Theseus' argument using an 'aporetic' set of propositions. An aporetic set of propositions is where each proposition seems to express an obvious truth, but all of them together are logically inconsistent. As follows.
A. An artifact remains numerically the same if one of its components is replaced, and the rest remain the same.
B. An artifact is numerically the same if all of its components are numerically the same.
C. Identity is transitive (if A=B and B=C, then A=C)
D. If artifact a and artifact b have numerically different components at the same time, a and b are numerically different.
E. Artifacts exist.
It is fairly easy to show they can't all be true*. Therefore one of them is false. In particular, if we accept A-D, we have to drop E. There can be no objects that conform to our intuitive idea of an artifact, which is the implied conclusion of the Ship of Theseus story.
In this post and three later posts, I will examine the four assumptions A-D to examine if the proposition-set is really aporetic (usually what appears to be an aporetic set dissolves upon enquiry, when we find that one or more of the propositions are not obvious or self-evident at all).
I will begin with the assumption that an artefact remains the same, when one of its components is replaced. My son recently wrecked the mudguard of my bike so badly that it fell off and is still lying in the front garden. I plan to take the bike to the repair shop to fit a new mudguard. Will the bike plus new mudguard be numerically identical with the bike as it was with the old mudguard (this is precisely what Peter van Inwagen appears to deny)? The question appears to involve deep metaphysical concepts like temporality and numerical identity, which makes it tempting to challenge it (on the grounds that deep metaphysical assumptions involve doubt and uncertainty from their very nature). But the question is actually quite mundane. Suppose, after getting the bike repaired, I say
(*) This bike had its mudguard replaced at the repair shop yesterday.
Is that true? Surely, on the assumption that I did take this bike to the repair shop, and that the crappy and bent mudguard was replaced with a shiny new one, what I say is unquestionably true. Yet it involves both 'metaphysical' assumptions. It involves identity through the very structure of the proposition. Identity: this bike = the bike that was taken to the repair shop. Temporality: the present reference to 'this bike', and the past 'had' of 'had its mudguard replaced'. Without the 'metaphysical' assumptions embedded in the proposition, we could not make the most simple, mundane and matter-of-fact statements about our ordinary everyday life. Such as replacing mudguards on bikes.
Note also (see my previous post) that we cannot re-tell this story by means of expressions like 'bikewise arrangement of parts'. For it is false to say that this bikewise arrangement of parts (which includes the new mudguard) is the arrangement of parts taken to the repair shop (which included the old mudguard).
I conclude that the principle of identity under replacement - which involves merely the assumption that we can refer to an artefact that exists now (this bike) by means of a past tense statement (the artefact that was taken to the repair shop) - is unquestionably true. Tomorrow (or whenever I can be bothered to write it): the transitivity of identity.
* Roughly as follows. If artifacts exist, suppose that some artifact X exists. Then suppose we replace one of its component to give artifact X1. Then (from A above), X=X1. Then replace one another component in X1 to give X2, as in the Ship of Theseus story. From A above, X1 = X2. And from C above, X = X2. Repeat this process for all the n original components to give Xn. Then X = Xn, by the previous argument.Now take the n replaced components, and assemble appopriately to make artifact Y. From B above, X = Y. From transitivity, X=Y and X = Xn so X = Y. Xn and Y are co-existing, i.e. exist at the same time. But Xn and Y have different components (since Y is made from the original components of X, and Xn is made from entirely new and different components). Thus, from D above, Xn /= Y. Since X = Xn, X /=Y. Contradiction: X=Y and X /= Y.