### Identity in replacement

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.

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.

Labels: identity

## 14 Comments:

I think you're being too quick. To say that "this bike had its mudguard replaced [...] yesterday" is to refer not to the bike now, but the one before.

It seems like you've just declared that this is the same bike. In a literal sense however I don't think it is. It is a different bike, unless you are positing that the mudguard is a part that is unessential to the identity of the bike.

The talk of bikewise arrangements amplifies this point nicely, but introduces other problems for the reason BV introduced over at MP.

That we can establish an equivalence relation does not, I think, imply that there is (numerical) identity.

>>"this bike had its mudguard replaced [...] yesterday" is to refer not to the bike now, but the one before.

Surely when I use the demonstrative 'this book' - pointing to the bike here and now, I am referring to the bike that exists here and now?

What if I say 'the bike that is in front of us right now was taken by me to the repair shop yesterday'?

Also what exactly do you mean by 'the bike now'? Do you not mean *this* bike?

I am suggesting, on the interpretation that to refer to an object--in this case artefacts--is to refer to either the concept of the things or the constitutive stuff of the thing, that on the distinction between this bike and the one that was repaired is to refer to two different objects, but to formally establish an equivalence--not identity--relation between them.

>>I am suggesting, on the interpretation that to refer to an object--in this case artefacts--is to refer to either the concept of the things or the constitutive stuff of the thing

No. There is an old objection to this. The expression 'The concept of my house' refers to the concept of my house, not to my house. And the expression 'the constitutive stuff of my house' refers to the constitutive stuff of my house.

>>that on the distinction between this bike and the one that was repaired is to refer to two different objects, but to formally establish an equivalence--not identity--relation between them.

The question is, whether it is true to say "this bike, which is now repaired, was taken to the repair shop yesterday". Surely it is. The same sentence predicates 'is now repaired' and 'was repaired yesterday' of 'this bike'. That's all. Very simple!

>>No. There is an old objection to this. The expression 'The concept of my house' refers to the concept of my house, not to my house. And the expression 'the constitutive stuff of my house' refers to the constitutive stuff of my house.<<

I am sure that I am being imprecise, or else being otherwise unclear in conveying my meaning. I am suggesting that to refer to an object using an expression 'this bike' or 'this house' is to pick out a particular things as it is at that spatio-temporal moment--I don't want here to get into an argument about temporality or any related physical arguments, I am merely trying to capture my premise in this argument. On this interpretation, how we refer to things, is necessarily imprecise, a defect of our use of language and the conceptual vagueness in how we refer to objects--which in ordinary circumstance, as you indicate in your second paragraph is perfectly intelligible. I am assuming though that the point of the SoT problem is to actually get at, if it exists at all, a fact of the matter.

>>The question is, whether it is true to say "this bike, which is now repaired, was taken to the repair shop yesterday". Surely it is. The same sentence predicates 'is now repaired' and 'was repaired yesterday' of 'this bike'. That's all. Very simple!<<

I don't disagree when you imply a commonplace intellibility, but again I am suggesting that we are, on the interpretation I have been advancing, really being sloppy. The sentence "this bike, which is now repaired, was taken to the repair shop yesterday"is fine. This is because it really is, on my interpretation, the same object. Whether, however, it is the same object--artefact--as the one

taken tothe repair shop is, I think, a different question.There is another interpretation that says that the object just is its constitutive stuff, and on this interpretation, the bike and houses are different. This interpretation suggests that objects are identical only under conditions of compositional invariance, and suggests that our commonplace use of identity is a linguistic foil for talking about a stream of objects that are equivalently related but not strictly the same object.

There is an unclarity here which I have been trying to bring out for analysis by introducing this latest interpretation. Clearly, I am advancing conditions on the way identity relates objects with respect to an artefact, especially when we can speak/point to compositional change. In this sense, I holding that (B)-(D) are the fact of the matter.

>>Whether, however, it is the same object--artefact--as the one taken to the repair shop is, I think, a different question.

Well it's not the same in the sense that it has changed. Its mudguard has been replaced. But the sense in which it is different is clearly not the sense in which it is the same.

Note also that a bike cannot have a 'new' component without the numerical identity. If the bike was created de novo as a result of the 'new' mudguard being fitted, then it has only ever had one such mudguard - the 'old' mudguard' belonged to a different bike.

Similarly for the idea of having a component 'replaced', or of being 'repaired'. If the bike is created de novo as a result of the 'repair', then it has been created, not repaired. And no component was ever replaced. The 'replacement' mudguard is the first such mudguard it ever had.

>> I am suggesting that to refer to an object using an expression 'this bike' or 'this house' is to pick out a particular things as it is at that spatio-temporal moment

I will address this in a separate post.

O, I think your argument is confusing me because I can't tell if your axioms are defining what it is to be an artifact or defining what it is for an artifact-already-understood to be numerically the same (as another?) I'd prefer to treat this problem by startiing with simple parts which we accept endure over time and which can be re-identified over time and see if we can construct something that looks artifactual from them, ie, something for which a well-behaved definition of 'numerical sameness' can be given. So,

Define a complex (at t) to be a maximal set of connected parts (at t)

Define an artifact1 to be a sequence of complexes s.t. Cn and C(n+1) differ by at most one part. This is roughly your (A). Things endure by changing slowly.

Define an artifact2 to be a sequence of sets s.t. Sn is a fixed complex C or Sn is {}. This is (B). Something can be disassembled, left in pieces, and reassembled later.

Define an artifact to be either an artifact1 or an artifact2.

Define artifacts (Cn) and (Dn) to be disjoint iff Cn and Dn are disjoint for all n.

Then, Theorem: there are pairs of non-disjoint artifacts. Proof: the Ship of Theseus construction. But is this so terrible? After all, there are pairs of non-disjoint living things too. Perhaps Bill will explain how PVI thinks living things differ.

>>O, I think your argument is confusing me because I can't tell if your axioms are defining what it is to be an artifact or defining what it is for an artifact-already-understood to be numerically the same (as another?)

I'm sorry if it was confusing. Assumptions (A) and (B) are simply assumptions meant to correspond to certain intuitions we have about artefacts. The first assumption (A) is that if we replace any small component (a brick, a plank, even an atom) then the artifact remains 'numerically' the same. As you put it, things endure when they change slowly.

And A-E are not an argument. They are simply a set of assumptions which are logically inconsistent. Roughly, A and C entail the possibility of all the components of an object changing completely, so that no original component remains. B captures the obvious truth that if all the components are identical, the artefact is identical. It corresponds roughly to the axiom of extensionality: if the elements of X and Y are the same, so are X and Y. Assumption D is an 'add on' which we need to counter the objection that perhaps the ship made from the new planks, and the ship reconstructed from the old planks are one and the same. Clearly they can't, but we need the assumption nonetheless. Assumption E is that things with the properties defined in A-D actually exist. So they are not an argument, but we can turn them into one by assuming any 4 of them, and concluding that the remaining one is false.

The question is: if we accept the inconsistency of the five assumptions together, which one is false? I am going to examine each one in turn. We are still stuck on (A).

>>Theorem: there are pairs of non-disjoint artifacts. Proof: the Ship of Theseus construction. But is this so terrible?

I don't follow this. What is the meaning of disjoint? What is terrible about A-E is the result of the transitivity of identity. It entails that (if artefacts exist) there are artefacts with different components which are both identical (because of A and C and B) and not identical (because of D). Contradiction.

1. We assume that an artefact X exists.

2. A+C entails the existence of Xn (ship made of new planks) such that Xn=X

3. B gives us a Y (the ship reconstructed from the old planks) such that X=Y

4. D gives us Y /= Xn. The ship reconstructed from old planks, and the visibly next door ship constructed from new planks, are not the numerically the same

5. hence (given Xn=X and X=Y as above) X /= X. Contradiction. The two ships, which are not identical, are in fact identical.

Re 'disjoint'. Well, I define disjoint for artifacts in terms of its usual meaning for sets: A and B are disjoint iff A meet B = {}.

I don't understand how you deduce X=Y from (B) when (B) mentions only one artifact. Do you mean

(B*) artifacts X and Y are ni iff every part of X is ni to some part of Y and vice versa.

I'm happy that the Xi are just aliases for the ship X that gradually has its planks renewed. What I don't grant is line (3) in yr last comment because X is (present tense) made of completely different planks from Y. There was never a time when both X and Y existed and were made of the same planks.

>>What I don't grant is line (3) in yr last comment because X is (present tense) made of completely different planks from Y. There was never a time when both X and Y existed and were made of the same planks.

Yes, X is made from completely different planks from Y. Therefore (from D) X and Y are not identical. But assumption B is that "An artifact is numerically the same if all of its components are numerically the same." If the components of the reconstructed ship Y, the original planks, are identical with the components of the original ship - and surely they are, for the original ship was made up of the original planks - then X = Y, present tense.

It seems to be assumption (B) you disagree with. But I am going to defend that in a future post. If not, can you clarify?

I'm still not clear at all about (B)---I think it should talk about two ships.

>>>(B) says that if the components of the reconstructed ship (i.e. the original planks) are identical with the components of the original ship - and surely they are

No, I disagree. I think you are falling into the trap you warn us against in your 'Bikes now and then' post, viz, attaching a temporal modifier, 'original', to a noun 'ship'. The 'original ship' is just X and by the time Y is created X's planks are disjoint from those it *had* at the begining of the process and which now make up Y.

We do need to get (B) formulated more clearly, I think.

>>I think you are falling into the trap you warn us against in your 'Bikes now and then' post, viz, attaching a temporal modifier, 'original', to a noun 'ship'. The 'original ship' is just X and by the time Y is created X's planks are disjoint from those it *had* at the begining of the process and which now make up Y.

You are correct again (just proves my point about how tricky this is). Let's change (B) to

(B') if (for any time t1 t2) the components of x at t1 = the components of y at t2, then x = y.

While 'x at t1' is suspect for the reasons we agree on, 'the components of x at t' looks perfectly OK. It simply means: the components that belonged to x at t1. Formulated like this, (B') looks less intuitively obvious. But there are strong arguments for it. Next post.

OK, I'm happy with B'. Given the startling consequences of B' I'll be very interested to see where you take this. Looking forward to the next post!

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