Following some severe strictures by Brightly on my earlier posts I offer reformulations of the 'aporetic set' of propositions, as follows.
A. If an artefact X1 has a replacement component, and X contained the replaced component, and the other components of X1 have been components of X since the replacement, and if the arrangement of components is the same in X1 as in X, then X1 = X.
B. If the components of Y are now the components of X, similarly arranged, then X=Y
C. If Fx and x=y, then Fy
D. If it is not the case that components of X = the components of Y, then it is not the case that X = Y.
E. Artifacts exist.
The only notion that may need elucidation is 'plural identity'. I propose the following definition: the x's = the y's if every one of the x's = some y, and every one of the y's = some x, which uses singular identity. Note I have made a significant change to C. That is because substitutivity is more fundamental than transitivity, and because (I think) we can derive transitivity. For let x = y, and y = z. The identity x = y allows us to substitute 'x' for 'y' in 'y = z'. Then x = z, which is transitivity.
We can then prove inconsistency as follows.
1. Suppose there are an X1 and an X such that A is true. Then X1 = X.
2. Suppose next that there is an X2 such that A is true regarding X2 and X1. Then X2 = X1, and from C above, X2 = X.
3. Repeat this process, ensuring that each replaced component is always one of the original components (and not a replacement component). Then, repeating this n times for the n original components, we have Xn such that Xn = X. Note that while all of the components of Xn are components of X (because of the identity, and because of C above), it is nonetheless true that at one time none of the components of Xn was a component of X. Indeed, it is true (again because of C above) that none of the components of X was at one time a component of X.
4. Take all the original components of Xn/X and reassemble them so that they are arranged exactly as they were when they were part of Xn/X, to make Y. Then (from B above) X = Y.
5. But from D above, since the components of Xn/X are currently different from those of Y, it is not the case that X = Y.
6. Thus X = Y and not X=Y. Contradiction.
4 comments:
Since 'the components of X' is ill-defined if we don't specify a time (B) might be tightened up a bit by referring to 'the one-time components of Y' but, yes, I agree that these propositions are inconsistent.
The time specified is now. I am assuming that any predication in the present tense is predication as of now (ut nunc) instead of tenseless (simpliciter).
Thanks for the helpful comments. There is some way to go.
On second thoughts, although I still think it is not necessary, it would be best to say 'If the components that once belonged to Y are now the components of X, similarly arranged, then X=Y
You are right.
Glad we are in agreement. From yr comments at BV's I take it that you don't intend to end up with PVI? Will be interested to see where you go!
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