Sunday, August 29, 2010

On artefactual identity

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.

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Blogger David Brightly said...

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.

2:13 p.m.  
Blogger Ocham said...

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.

3:45 p.m.  
Blogger Ocham said...

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.

3:51 p.m.  
Blogger David Brightly said...

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!

5:04 p.m.  

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