A Closer Look at Material Composition and Modal Discernibility Arguments

(For David Brightly, whom I hope either to convince or argue to a standoff.)

Suppose God creates ex nihilo a bunch of TinkerToy pieces at time t suitable for assembly into various (toy) artifacts such as a house and a fort.  A unique classical mereological sum -- call it 'TTS' -- comes into existence 'automatically' at the instant of the creation ex nihilo of the TT pieces. (God doesn't have to do anything in addition to creating the TT pieces to bring TTS into existence.)   Suppose further that God at t  assembles the TT pieces (adding nothing and subtracting nothing) into a house.  Call this object 'TTH.'  So far we have: the pieces, their sum, and the house.  Now suppose that at t* (later than t) God annihilates all of the TT pieces.  This of course annihilates TTS and TTH.  During the interval from t to t* God maintains TTH in existence.

I set up the problem this way so as to exclude 'historical' and nonmodal considerations and thus to make the challenge tougher for my side.  Note that TTH and TTS are spatially coincident, temporally coincident, and such that every nonmodal property of the one is also a nonmodal property of the other.  Thus they have the same size, the same shape, the same weight, etc.  Surely the pressure is on to say that TTH = TTS?  Surely my opponents will come at me with their battle-cry, 'No difference without a difference-maker!'  There is no constituent of TTH that is not also a constituent of TTS.  So what could distinguish them?

Here is an argument that TTH and TTS are not identical:

1. NecId:  If x = y, then necessarily, x = y. 

2. If it is possible that ~(x = y), then ~(x = y). (From 1 by Contraposition)

3. If it is possible that TTS is not TTH, then TTS is not TTH. (From 2, by Universal Instantiation) 

4. It is possible that TTS is not TTH.  (God might have assembled the parts into a fort instead of a house or might have left them unassembled.)

5. TTS is not TTH. (From 3, 4 by Modus Ponens)

The gist of the argument is that if x = y, then they are identical in every possible world in which both of them exist.  But there are possible worlds in which TTS and TTH both exist but are not identical. (E.g., a world in which the pieces are assembled into a fort instead of a house.)  Therefore, TTS andf TTH are not identical.

If you are inclined to reject the argument, you must tell me which premise you reject.  Will it be (1)? Or will it be (4)? 

Your move, David.

 

Comments

[David Brightly]

Bill,

If we accept Unrestricted Composition as given in the Lewis post then there exists a sum of the TT pieces which appears to be discernible from each of the pieces. You are entitled to call this TTS and I think your argument does indeed show that TTS is discernible from TTH. However, if we don't accept UC but instead take the Varzi/Van Inwagen definition of 'sum of parts' then I think that the argument I gave under the Varzi post shows that the somewhat differently defined TTS is identical with TTH. The only modification to that argument I'd make is to say that we don't need to assume UC to show that the concept Sum(zs) is instantiated. It's sufficient that TTH exists.

So I think the difference between us rests on differing starting assumptions. If you say that UC is a non-negotiable component of classical mereology then I concede. But the AV/PVI view seems to hold up well and as far as I can see does not lead to the singularly strange results that flow from UC. And Varzi is the author of the SEP entry for Mereology.

[Bill Vallicella]

David,

Here is what you wrote in the Varzi thread:

>>Would it also help if I were to suggest that, on the evidence of your quote, Varzi is in the PVI camp with regard to interpretation of mereological talk. He is not saying that a mereological sum is a radically different kind of object that, for example, transforms from cube-shaped to house-shaped. I think he is just introducing a new bit of language or conceptual machinery. He is proposing that whenever some parts zs and an object x are in a certain relation then we should say that the x is a 'mereological sum of the zs'. That relation holds whenever the zs are all parts of x and every part of x has a part in common with at least one of the zs. Or, in my terminology, the zs comprise x. As Halmos says of general set theory, it's 'pretty trivial stuff really'. With this stipulation as to how to use the term 'sum of zs' I think it does follow straightaway that TTH is a sum of parts. Forgive me if I'm wrong, but are you not bringing the other, concrete, sense of sum to the Varzi paper, and inevitably finding a clash?<<

I'm afraid I don't understand the above. A sum of concreta is itself concrete, no matter how one thinks of a sum. 'Concrete' means 'causaaly active/passive.' 'Abstract' means 'causally inert.' So I don't know what the concrete sense of sum is supposed to be. As opposed to what?

If you tell me that a book on the table is nothing other than a mereological sum of its parts, then I don't understand how that could be true. For if I pull all the pages way from the binding, then the sum continues to exist but the book doesn't.

[David Brightly]

Bill,

Please don't take my 'concrete' and 'abstract' too seriously. I needed a pair of contrasting labels for the two notions of mereological sum I thought I had detected. The notion underlying Unrestricted Composition seemed, in the end, a referencing device for arbitrary parcels of matter such as Brick Sum. Hence 'concrete'. In contrast, the idea in Varzi and Van Inwagen seemed to behave more like terms such as 'mammal'. Hence, perhaps misleadingly, 'abstract'. Having thought about it a bit more, a better comparison might be the matter/form distinction. The UC derived notion seems to track the matter, whereas the V/VI notion seems to follow the form.

Regarding the mutilated book: Suppose we say that a book is a species of document. If the book is the only document in a room and you pull the book apart then there is no longer any book or document in the room. A (V/VI) sum of parts behaves in the same way. Destroy the book and you destroy the (V/VI) sum. Recall the V/VI definition

x is a sum of the zs =df The zs are all parts of x and every part of x has a part in common with at least one ofthe zs. 

If there is no x that the zs comprise then there can be no sum of the zs.

Am I getting any nearer convincing you that there are two ideas in play here?

[Bill Vallicella]

David,

You seem to be saying that the definition of 'sum' -- which is standard -- supports your understanding of sums over mine. But why?

All the definition does is tell us what is meant by 'sum.' To spell it out: it tells us that, necessarily, a sum is a sum of some things, at least two. That is indicated by the plural variable 'zs.' There are no mereological singletons as there are set-theoretic singletons. The first clause of the definiens tells us that all of the zs are parts of x, while the second tells us that there is no part of x that does not overlap one of the zs. In other words, the first clause says that the sum is 'big enough' to comprise all of the zs, while the second says that it is not so big that it comprises items other than the zs and their parts.

Are we agreed that that is what the definition says?

If yes, then we can ask about the existence of sums. Obviously the definition is neutral on the question whether there exist any sums. Right? Unrestr Composition (which is perhaps better called Unrestricted Summation) gets us to existence: as Lewis puts it, "Whenever there are some things, then there exists a fusion [sum] of those things."

What that says is that, given any items at all, of whatever nature or whatever category, there exists a sum of just those items. A sum, not THE sum. To get uniqueness, you need a further axiom, Uniqueness of Composition (or rather Summation).

Now why do you assume that a book, which is undoubtedly a whole of parts, is a mereological sum?? There is nothing in the df of 'sum' that require me to say that an ordinary artifact like a book is a sum.

Seems to me you are reading out of the definition something that it does not contain.

[David Brightly]

Bill,

There is nothing in the df of 'sum' that require me to say that an ordinary artifact like a book is a sum.

I think this highlights a basic disagreement between us. My standpoint is this: We start with some entities before us over which the part-of relation is defined. I read the definition from right to left as follows: if we can find an entity x and some entities zs such that a certain condition specifiable in terms of the part-of relation holds between them, then we shall say that the entity x is a sum of the entities zs. Reading from left to right: a claim about some entity x and entities zs that the x is a sum of the zs can be translated into a claim that a certain condition involving the part-of relation holds between the x and the zs.

So, if we have a book before us and we identify its parts---pages, signatures, casing, whatever---and that certain condition between the book and its parts does indeed hold, then I do think we are required to say that the book is a sum of the said parts.

[Bill Vallicella]

David,

You are right, and what I said in the quoted sentence is wrong. If we go just by the definition of 'sum,' then the book is a sum, i.e., it falls under the concept *sum.*

But I stipulated earlier that we are talking only about the sums of classical mereology, and that the full meaning of 'sum' in this context is determined by all the definitions and all the axioms of classical mereology including Unrestr Composition and Uniqueness of Comp.

If you take 'sum' in this full sense, then I insist that TTH is not a sum.

Your response to me could be: well, we need to modify the classical axioms, and then perhaps we will be able to say that books and such are sums while avoiding the duifficulties you raise.

I should in a separate post list the definitions and axioms of classical mereology so that we know exactly what we are talking about.

[David Brightly]

Hello Bill,

A list of definitions and axioms would be very helpful. I've read through Varzi's SEP article several times. He offers a smorgasbord of axioms but it's not clear what combination counts as a widely-accepted core mereology.