Maverick Philosopher

Footnotes to Plato from the foothills of the Superstition Mountains

Mereological Innocence and Composition as Identity

DavidLewis This is the third in a series.  Part I, Part II.  What follows is a 10th example of eliminativist/reductivist ambiguity.

One of the axioms of mereology is Unrestricted Composition.  Here is David Lewis' formulation (Parts of Classes, Basil Blackwell 1991, p. 74):

Unrestricted Composition: Whenever there are some things, then there exists a fusion of those things.

A fusion is a mereological sum, so I'll use 'sum.'  The axiom assures us that, for example, if there are some cats, then there exists a sum of those cats.  The cats are many but the sum is one.  So it is not unreasonable to think that if there are five cats that compose the sum, the sum is a sixth thing.  One could argue as follows:  (a) The sum is distinct from each of the cats.  (b)There are five cats, each of which exists, and by UC the sum also exists.  Therefore, (c) at least six things exist.

But consider this example, adapted from Donald Baxter.  You proceed with six bottles of beer to the supermarket 'six items or fewer' checkout line.  The attendant protests your use of the line on the ground that you have seven items: six bottles of beer plus one mereological sum.  This would be an outrage, of course.  The example suggests that the argument to (c) above has gone wrong.

Lewis avoids the mistake — assuming it is one — by pleading that "Mereology is ontologically innocent." (PC 81)  That means that a commitment to a cat-sum is not a further commitment over and above the commitment to the cats that compose the sum.  The cat-sum just is the cats, and they are it.  This is the thesis of Composition as Identity.  The xs compose the y by being identical to the y.  As Lewis says,

Take them together or take them separately, the cats are the same portion of Reality either way.  Commit yourself to their existence all together or one at a time, it's the same commitment either way.  If you draw up an inventory of Reality according to your scheme of things, it would be double counting to list the cats and also list their fusion.  In general, if you are already committed to some things, you incur no further commitment when you affirm the existence of their fusion. (PC 81-82)

I'm sorry, but this doesn't make much sense.  Glance back at Unrestricted Composition.  It is not a tautology.  It does not say that whenever there are some things, then there are some things.  It says that whenever there are some things, then there exists a fusion or sum of those things.  Now if the sum of the xs is just the xs, then UC is a tautology.  But  if UC is not a tautology, then Composition as Identity is false.  How can Unrestricted Composition and Composition as Identity both be true?

The problem is already present at the purely syntactic level.  'Y is identical to the xs' is unproblematic if the xs are identical to one another.  For then the open sentence collapses into 'y is identical to x.'  But if the xs are distinct from each other, then 'y is identical to the xs' is syntactically malformed.  How can one thing be identical to many things?  If one thing is identical to many things, then it is not one thing but many things.  A contradiction ensues: the one thing is one thing and not one thing because it is many things.  The gaps in the predicate '. . . is identical to ____' must either be both filled with singular terms or both filled with plural terms.

And now we come back to our main theme, eliminativist/reductivist ambiguity.  Lewis wants to say that there is the sum of the xs (by Unrestricted Composition) but that the the sum of the xs is identical to the xs.  So he seems to be making a reductionist claim: sums reduce to their members.  But I say the thesis is unstable and topples over into eliminativism:  there are no mereological sums.  For if the sum is just its members, then all that exists is the members so that the sum does not exist!

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14 responses to “Mereological Innocence and Composition as Identity”

  1. john Avatar
    john

    Hi Bill,
    As a preface, let me say that your arguments and those of most of comments are a bit over my head so I’ll try to keep it short. Nonetheless, I enjoy reading and following your arguments. It’s like sleeping in a philosophical Holiday Inn.
    My question is how the doctrine of Unrestricted Composition holds up with light. For example, we’re told that an infinite number of colors can come from pure white light. So does this mean that pure white light is really synonymous with an infinite number of colors? I can see how we can get the particular colors from an infinite array of possible colors, but the idea that pure white light is a composition of all those colors seems a little off. Wouldn’t we then be left with coming up with a sum by adding up an infinite number of particulars?
    In a similar vein, how would the idea of an infinite array of possible worlds, (analogous to an infinite array of possible colors), fit as ultimate reality or God? Could then our possible world be the filtering out of all the other possible worlds in a way that we filter out all colors other than blue to create the particular color blue out of an infinite array of possible colors?
    Best Regards,
    John

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  2. William Avatar
    William

    Here is the ‘hinge’ of your argument, I think.
    “The cats are many but the sum is one. So it is not unreasonable to think that if there are five cats that compose the sum, the sum is a sixth thing. ”
    Let me replace this with an argument of a similar if not identical logical form.
    (*) The cats are one hundred, but the hundred is one. So it is not unreasonable to think that if there are a hundred cats that compose the hundred, the hundred is a hundred and oneth thing. ”
    Do you agree:
    (1) That the modified argument is obviously and patently invalid.
    (2) That the unmodified argument is invalid for a similar reason
    (3) If you don’t agree with (2) above, at least there is cause to question the unmodified argument.
    Turning to your point about the E/R ambiguity. The reductionist says that “There are F’s, but F’s are only G’s’. We found a problem with that, in that in many cases the correct definition of ‘F’ includes features that are non-G, meaning that the eliminativist is right, because he has the correct definition.
    But what if the eliminativist does not have the correct definition? In this example, the eliminativist would have to be saying that there is no ‘one’ hundred, because the grammatical ‘one’ signifies one thing different from any of the hundred. But he is clearly wrong.
    >>But I say the thesis is unstable and topples over into eliminativism: there are no mereological sums. For if the sum is just its members, then all that exists is the members so that the sum does not exist!
    You might just as well say that if ‘one’ hundred things is just a hundred things, then all that exist are the things, so the ‘one’ hundred does not exist. That is clearly wrong.

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  3. William Avatar
    William

    If I may (since you don’t have comments box enabled on your Ground Zero post), Kripke (in an early chapter of Naming and Necessity) was actually quoting Voltaire: “Ce corps qui s’appelait et qui s’appelle encore le saint empire romain n’était en aucune manière ni saint, ni romain, ni empire”.

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  4. Bill Vallicella Avatar
    Bill Vallicella

    Yes, he was quoting Voltaire. I did not suggest otherwise.
    I see where you are going with your parody argument, but there is a difference, namely, that I am deducing a conclusion from Unrestricted Composition. As I argued, UC is not a tautology: it says that whenever there are some things, THERE EXISTS (in reality!) a fusion (mereological sum) of those things. So how can the sum be identical to its members taken collectively?

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  5. Stuart Avatar
    Stuart

    The composition of more that one entity may constitute a different entity, in that the new constituted entity has different properties from its parts. For example, I am a human being with a mind, but I am also composed of atoms. I am not just a collection of atoms, though. You can count all of my atoms as entites, but there is a further thing-my functioning mind and body-that is an entity in itself in addition to the atoms that compose me.

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  6. Bill Vallicella Avatar
    Bill Vallicella

    As for your three points, I reject all of them.
    Are you prepared to assert the following?
    It is never the case that whenever there are some things, there is a whole with those things as parts. Equivalently:
    For any xs, if the xs are two or more, there is no y such that the xs compose y.

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  7. Bill Vallicella Avatar
    Bill Vallicella

    In the thread just before this one we were achieving a marvelous agreement. Now we seem to be disagreeing again. I knew it couldn’t last.
    We need to sort out the meanings of ‘one.’
    Frege (Phil. of Arithmetic) gives the example of a copse composed of five trees. Do you agree with Frege that ‘ . . . is five in number’ is a second-level predicate, a predicate of such first-level predicates as ‘ . . . is a tree’? If yes, then presumably the same goes for ‘ . . . is one.’
    Do you think that ‘one’ can be predicated only of concepts/properties/predicates?

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  8. Bill Vallicella Avatar
    Bill Vallicella

    John,
    You are using ‘composition’ in a sense different from the one relevant here.

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  9. William Avatar
    William

    You raise so many points here. To some of them.
    1. If I say there are no more than *one* dozen things, then I say that there are no more than 12 things. Do you agree? That seems fundamental.
    2. The ‘one’ in ‘one dozen’ does not qualify any of the 12 things. E.g. if there are 12 apostles, we have one person who is Peter, another one who is Paul and so on. The use of the 12 ‘ones’ to qualify each ‘one’ person. Yes?
    3. But if ‘one’ qualifies ‘dozen’ in the way that it qualifies ‘person’, i.e. if a dozen is a kind of Aristotelian substance the way a man is, it would follow there are more than 12 things. But we agreed there aren’t.
    4. So ‘one’ qualifies ‘dozen’ in a different way, and ‘dozen’ is not a kind of thing. I would liken the ‘one’ here to the ‘1’ in ‘100’. If I count ‘100, 200, 300 …’ I am not counting *things*.
    5. Turning from ‘dozen’ to ‘fusion’. If ‘fusion’ is a term for a kind of thing, an Aristotelian substance, then it is clearly false to assert that it has no existential commitment beyond the things it is a fusion of.
    6. If – on the other hand – ‘fusion’ is like ‘dozen’, then it clearly has no commitment.
    Your other points.
    >>Do you agree with Frege that ‘ . . . is five in number’ is a second-level predicate, a predicate of such first-level predicates as ‘ . . . is a tree’? If yes, then presumably the same goes for ‘ . . . is one.’
    I don’t agree. In ‘The Apostles are 12’, we have a non-distributive predicate. ‘- is an Apostle and – is another Apostle and – is another Apostle …. and – is another Apostle and there are no other Apostles’. There is no singular predicate F and no object a such that Fa expresses the proposition that there are n things. Frege is fundamentally mistaken here.
    >>For any xs, if the xs are two or more, there is no y such that the xs compose y.
    Agreed, if you are using xs as a plural quantifier, and by implication y as a singular quantifier. This leaves us with the problem of how to quantify over dozens or hundreds. I don’t know of any logical system that has addressed this.
    >>In the thread just before this one we were achieving a marvelous agreement. Now we seem to be disagreeing again. I knew it couldn’t last.
    Agreement is bad / disagreement is good. Disagreement means unclarity and confusion which there is further work to dispel. This is work and enjoyment for philosophers. If there were no possible disagreement, we would be out of a job. Aristotle’s god exercises himself purely by thinking and philosophising, which is the greatest and most pleasurable activity.

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  10. David Brightly Avatar
    David Brightly

    (1) Can we say that Unrestricted Composition is tautologous by virtue of being a *definition*? If I say ‘Whenever you have an unmarried man then you have a batchelor’, am I not giving a definition of ‘batchelor’?
    (2) Arguments based on counting ‘things’ do seem unreliable. A pack of cards is one deck, four suits, and 52 cards. I can’t say how many things that adds up to. Fusions don’t add up like ‘things’ either. If you give me a cat I have one cat and one cat-fusion. If you give me another cat (one more cat-fusion?) I have two cats and three cat-fusions. Give me a third cat and I have seven cat-fusions. And so on.
    (3) I don’t see how we can classify Lewis as either eliminativist or reductivist. He’s not saying that we once thought there were things we called ‘fusions’ of cats but now we see there are just pluralities of cats. Nor does he say, Yes there are fusions of cats but they are only pluralities of cats. On the contrary, he seems to be saying, You didn’t realise it but there are things we might call ‘fusions’ of cats; here are their identity conditions and examples of ways we can talk about them. We might call this definitionist or extensionist. It appears to make the world bigger rather than smaller. Perhaps what Lewis is doing is more akin to mathematics than philosophy. Like the square root of minus one, or, for an earlier time, zero and negative numbers, if the concept ‘fusion’ doesn’t lead to outright contradiction and proves itself useful to us in dealing with the world, then it comes to have an existence for us.
    (4) Lewis has to tell us what it means for two fusions to be the same fusion, speaking loosely. What I think he is doing is *extending* the predicate_is-identical-to_ to a new predicate, _is-identical*-to_, so that it becomes syntactically correct to fill either placeholder with either a singular term or a plural term. This is somewhat analogous to the way the equality relation over the integers is ‘contained in’ the equality relation over the rationals, say.

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  11. Bill Vallicella Avatar
    Bill Vallicella

    Hi David,
    Ad (1). UC is not a definition because it makes an existence claim. Surely one cannot define anything into existence. The negation of UC is not a contradiction. I can’t see the ‘bachelor’ comparision as apt.
    Ad (3). As a mathematician you seem to want to skirt the question of the real existence of what you are talking about, whether imaginary numbers or negative numbers or mereological sums. As long as no contradiction is encountered, no problem. I can’t fault this attitude, but then you are not engaging the philosophical questions.
    Ad (4). You are quite right. Lewis ends up broadening the ordinary one-to-one relation of identity that each thing bears to itself and to nothing itself. Thus he takes the many-one relation of composition as identity in a broadened sense and ordinary one-to-one identity as a special limiting case of it.

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  12. Bill Vallicella Avatar
    Bill Vallicella

    Read ‘else’ for ‘itself’ in the 2nd sentence of ad (4).
    William,
    I will respond to you in separate posts.
    How would you respond to Brightly’s comments?

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  13. William Avatar
    William

    >>How would you respond to Brightly’s comments?
    David said:
    >>Arguments based on counting ‘things’ do seem unreliable. A pack of cards is one deck, four suits, and 52 cards. I can’t say how many things that adds up to.
    Counting things is reliable. But (as stated above) counting ‘three hundred’ is not counting three things. It’s counting three hundred things. 12 things is 6 2’s, 3 4’s, 4 3’s and so on. How many things does that add up to? 12.
    On David’s point that Lewis is making the world bigger – this is not consistent with Lewis saying that fusions have no ‘commitment’.

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  14. David Brightly Avatar
    David Brightly

    (1) UC as definition. Let me try another example. It has been known for millennia that a square object can be picked up, rotated, and set down again in the same place in four ways, but it wasn’t until the mid 19C that these operations came to be seen as forming an instance of what became known as a mathematical group. I take W’s point that this invention hardly makes the world bigger. Nevertheless, it expands our language and the concepts by which we can think about the world.
    (2) Perhaps I am skirting the question of the real existence of what I’m talking about. I’m not sure what resources we have for answering this. PVI’s argument suggests that the ideas comprising our notion of composite object are inconsistent, yet most readers will find themselves resisting this conclusion. We are quite unclear as to how this conviction arises. The argument assumes the existence of re-identifiable simples, but physics tells us that the best candidates we have for simple entities are not re-identifiable. I’ve no idea where this leaves our ordinary idea of physical object—it seems to have evaporated altogether!
    (3) Counting things. Even after we have cleared up the issues surrounding dozens and hundreds we will be left with the problem that, faced with a row of brick houses and asked to count the things in the vicinity, we don’t know whether to count houses or bricks or both. Things are too general to be counted.

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