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Monday, September 06, 2010

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>>Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:
>>4. B is a proper part of (B + C) & C is a proper part of (B + C).

No. If we are interpreting our ordinary-language in terms of modern predicate logic, then 'B + C' is a referring term, referring to a singular object distinct from B and C. It is not the case that C = B+C, and not the case that A=B+C, and not the case that B=C. So by definition there are three entities. Van I's argument is valid so long as the interpretation of our ordinary way of speaking is correct. But his interpretation is not correct. By equal reasoning, it is clearly wrong to interpret the 'one' in 'one hundred' as denoting 'one' entity distinct from the 100 entities.

In summary, Van Inwagen's argument is perfectly correct from the part 'Suppose that we have a bunch of sentences containing quantifiers, and …'. The error consists in interpreting the referring expression 'my big parcel of land' as 'a possible value of our variables'. It would be the same error as taking 'one hundred-things' be a possible value of the singular variable x, then a hundred other variables besides, none of which is identical with x. Perhaps we could call this the Fallacy of Grammatical Singularity?

Please note I am not necessarily denying that the composite parcel of land is a single entity in its own right. It has a certain legal status, perhaps (the fact that you have title to all the component parts, or perhaps the fact that you have a legal right to dismantle the fencing and make them into a single plot for redevelopment, all that sort of stuff). What I am denying is the argument that gets us there, attempting to translate a grammatically singular term ('the composite parcel of land') into a logically singular term (an instantiation of a variable or a logically referring term).

That's an admirably clear dismantling of PvI's argument!

I am with you in opposing Unrestricted Composition, but "adjoining" seems neither necessary nor sufficient for constituting a whole out of parts, so I am curious how you will develop the idea. Not necessary, because whole objects are not constituted by adjoining objects at the microphysical level; nevertheless, various arrangements of microparticles do constitute various midsized goods. Not sufficient, because putting two books right next to each other does not create a new whole object out of them. Nor does gluing them together. (van Inwagen has a paper where he goes through a whole list of such possibilities I think, sorry I can't remember the name of it.)

I do think B and C have to be connected in some suitable sense in order to compose into A, but doubt that adjoining can do the job. Perhaps only in special cases. For material objects, I propose that L-composition instead of unrestricted composition is required: two material object-types B and C L-compose into a third material object-type A iff (i) there is a law of composition L applying to B and C such that (ii) they are combined into a new object A with novel properties whose existence cannot be inferred from the properties of B and C alone. Condition (ii) allows for the possibility that A can be reduced to or eliminated in favor of B and C.

Boram,

Thanks. As for 'adjoining' I am not making the universal claim that for any xs, there exists a y such that the xs compose y iff the xs adjoin or are in contact. (I am indeed aware of the chapter on Contact in PvI's Material Beings.) I am making a claim just about the two subparcels B and C. Intuitively, they are disjoint proper parts of a concrete whole, the parcel A.

Now ask yourself: Is B-adjoining-C identical to the sum (B + C)? I say no: the sum can exist whether or not B-adjoining-C exists. Suppose an earthquake occurs which causes B and C to split apart with a huge chasm between them. In that scenario, the sum continues to exist while B-adjoining-C ceases to exist. Therefore, B-adjoining-C is not identical to the sum (B + C).

And yet the whole parcel (B-adjoining-C) and the sum have exactly the same parts. What then must be added to the parts to get the whole? Mereology alone will not give us the extra ontological ingredient. In this case it is the adjacency of the two disjoint subparcels that makes them into a whole.

Have I persuaded you that in this case at least the whole is not identical to the sum of the parts?

You seem to take me to be denying Unrestricted Composition, but I don't see that I am (unless I am confused, which is possible!). Suppose there are all the sums that UC allows. Thus there is a sum of the rocks in a certain rock pile in my backyard. My claim is that what makes that pile one pile is not the summation of its parts. For the sum exists at times and in possible worlds at which the pile does not. If you scatter the rocks, that is without prejudice to the existence of the sum; but it will destroy the pile. A pile of rocks is not a sum of rocks.

Thus I suppose I am committed to a distinction between summation and composition, or perhaps between mereological composition and 'real' composition. Why can't one accept UC, and consistently with that, deny that for every sum there is a 'real' or 'natural' whole?

Have you read PvI's 2006 JP article "Can Mereological Sums Change Their Parts"? I am puzzling over that one now. Maybe you can help me understand it.

Thanks for the comments and best wishes.

Bill, can you clarify what you mean by 'B+C'? If you mean a summation, like arithmetical summation, then my counter-argument stands. If you mean something like the 'and' of 'Peter and Paul preached in Jerusalem', then

A = B+C

is ill-formed.

Thanks for the reply. In the special case, yes, I agree that the adjoined whole is not identical to the mereological sum of its parts. More generally, connected wholes are not identical to the mereological sums of their parts, because they seem to have different persistence conditions, and "real" or "natural" wholes seem more like connected wholes than mereological sums. I think Achille Varzi takes this line in his book Parts and Places and other essays, where (if I correctly recall) he takes the parthood relation and the connectedness relation as two independent primitives formalized respectively by mereology and topology, and "connectedness" as a broader concept than spatial connectedness, including for instance, nomological connectedness as well.

I was too quick in assuming that you were denying Unrestricted Composition when imposing certain restrictions on certain wholes. Of course you are right that you weren't. I was thinking of the restricted domain of composite material objects, and two rival hypotheses regarding objects in that domain. Namely, either (i) composite material objects are just mereological sums of their material constituents, or (ii) some further restrictive condition must apply to material constituents in order for them to compose into material objects.

Sorry I haven't read that article; any of PvI's articles deserves close scrutiny that I can't give now that I'm trying to finish my dissertation in ethics. I've just scanned it quickly and it seems to me that one of two key moves might be how he fixes the referent of the term "mereological sum". He does not fix the referent in the usual way, via definite description involving the proper parts unrestrictedly composing it. Instead, he claims that, given mereology and the definition of "mereological sum", it follows that any ordinary object is a mereological sum. (By mereology, any object is a part of itself, and so, by definition of "mereological sum", any object is a mereological sum of at least itself.) Then he argues that, since any ordinary object can undergo change in parts, and ordinary objects are mereological sums, mereological sums can undergo change in parts.

The second key move I think is PvI's denial of Uniqueness of Composition (a.k.a. Extensionality). This permits him to say that the same object can undergo change in parts without becoming a different object. (Varzi has a nice discussion of arguments for and against Uniqueness of Composition in his "Mereological Commitments".)

I'll stop here. Call me lazy, but I'd rather await your own more careful analysis of his argument. ;)

Please ignore my guesses above as to what van Inwagen's argument might be. I hope they don't mislead anyone; it will require a great deal of investment on my part to understand the paper.

Boram,

You are certainly not lazy. What you gleaned from a quick scan of PvI's article seems accurate -- but I haven't fully digested it. Thanks for the reference to Casati and Varzi, Parts and Places, which I discovered I have in my library.

William,

You appear to be conflating values and substituends of variables. In the open sentence 'x is a man,' 'x' is a variable, 'Socrates' is a substituend and Socrates is a value.

Hi Bill,

One question I have pertains to your response to Boram. You say "Mereology alone will not give us the extra ontological ingredient. In this case it is the adjacency of the two disjoint subparcels that makes them into a whole." Now, as your ensuing dialogue with Boram suggests, you are not denying Unrestricted Composition. You just want to say that, granting the existence of arbitrary mereological sums, it is not the case that all these sums are on a par. Some are wholes, and some are not, and in your view mereology cannot tell the difference. As Boram notes, this is basically Casati & Varzi's view as well.

First question: do you take this line out of a preference for it over an attempt to deny Unrestricted Composition? Or, remaining agnostic about UC, or at least not averse to it, you just want to accommodate our general intuition that arbitrary mereological sums differ in important ways from "ordinary objects"? The reason I ask is that someone might choose, in an effort to avoid all this talk of connectedness, to attempt to restrict composition, and to do so in wholly mereological terms. So I'm curious as to your thoughts on that.

Second question: supposing that UC is true, and that we want to draw some ontological distinction between arbitrary mereological sums and other objects, should we have a general rule for doing so? I ask this because you say, in the quoted passage, that "In this case" it is the adjacency of the parts that provides the other ontological ingredient. But is this going to be the case in general? Some of Boram's points suggest not. So do you think there needs to be some general way of distinguishing arbitrary mereological sums from other objects, or that whatever succeeds in an individual case will suffice?

I've been inclined to think for some time that there is a distinction between summation and composition. PVI himself discusses it in "Material Beings". If I recall, it has to do with the fact that composition, but not summation, has a requirement that the parts of the object not overlap. I haven't given this account of the distinction much thought, but perhaps it works.

That said, perhaps you'd be interested in some of Jonathan Schaffer's work on this issue. He is not interested in the question of whether UC is true, because his view is that existence questions (and thus, questions about the existence of arbitrary mereological sums) are not the primary subject matter of metaphysics. He thinks that metaphysics is about what grounds what. Accordingly, he draws a distinction between arbitrary sums and genuine wholes in the following way (Schaffer 2009):

x is an integrated whole =df x grounds each of its proper parts
x is a mere aggregate =df each of x's proper parts grounds x

Most of his paper is concerned with explicating, at least to an extent, the relevant notion of grounding. He says it is a transitive, reflexive, and (I think) anti-symmetric relation (he might've said asymmetric; I don't recall). But that's about as much of an explanation as he gives. Anyway, with that in hand, I wonder what your thoughts are on this.

At any rate, it appears that I have several questions, not one. Oh well.

>>You appear to be conflating values and substituends of variables. In the open sentence 'x is a man,' 'x' is a variable, 'Socrates' is a substituend and Socrates is a value.

No. Your argument above is completely and unbelievably wrong. You argue

>>Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus: 4. B is a proper part of (B + C) & C is a proper part of (B + C).

There are three entities, not two, namely B, C and B + C. The expression 'B+C' is a substituend, yes. The object referred to by that expression is a value. And that value is numerically different from B and C. Ergo three entities, not two.

You are biting on bedrock, as you say.

Regarding my last post: I apologize. Schaffer (2009) argues that grounding is irreflexive, transitive, and asymmetric. My mistake. It was late :)

William:

You state "There are three entities, not two, namely, B, C and B+C. The expression 'B+C' is a substituend, yes. The object referred to by that expression is a value. And that value is numerically different from B and C. Ergo three entities, not two."

This ignores the distinction between identity taken distributively and identity taken collectively. Nobody is claiming that B+C is identical to either B or C. That goes without saying. The claim is not this distributive identity claim, but rather the COLLECTIVE identity claim that B+C is identical to B and C taken collectively.

Hence expressions like "the whole is nothing more than the sum of its parts". The whole is not an additional object, above and beyond its parts. It IS its parts, and they ARE it. To claim, therefore, that B+C is a third entity is already to deny composition as identity. And this, of course, was Bill's point when he said that if PVI required there to be 3 entities in the domain of quantification to make (4) come out true, then PVI was begging the question against Lewis.

@John
>>x is an integrated whole =df x grounds each of its proper parts
>>x is a mere aggregate =df each of x's proper parts grounds x

In both cases the translation of the ordinary language variable 'integrated whole' or 'mere aggregate' into the 'x' of the predicate calculus guarantees an 'ontological distinction' of the kind you may want to avoid, so I don't see the point of the distinction.

>>This ignores the distinction between identity taken distributively and identity taken collectively. Nobody is claiming that B+C is identical to either B or C. That goes without saying. The claim is not this distributive identity claim, but rather the COLLECTIVE identity claim that B+C is identical to B and C taken collectively.

Then we are back to the point I mentioned further up, namely that if 'B+C' is to be read like the 'Peter and Paul' of 'Peter and Paul preached in Jerusalem', then the statement A = B+C is ill-formed, from the standpoint of predicate calculus. Simple first order predicate calculus does not understand 'collective identity'.

Some writers have developed plural or collective versions of predicate calculus, to capture statements of the form

These people = Peter and Paul

Or

Some X's = A & B

But then the Van Inwagen argument fails, for the following statements are all consistent:

These people = Peter and Paul
Not (These people = Peter)
Not (These people = Paul)
There are only 2 people

It is only within simple predicate calculus that the inference from 2 things to 3 things is valid. But then the interpretation of the ordinary language statement 'this parcel of land consists of B and C' is not necessarily the right interpretation.

>>And this, of course, was Bill's point when he said that if PVI required there to be 3 entities in the domain of quantification to make (4) come out true, then PVI was begging the question against Lewis.

But this was *not* Bill's point. I am specifically objecting to the section here.

-------------------------------------------------------------------
3. There is an x, y and z such that x is a proper part of z & y is a proper part of z.
(3) contains three distinct variables, but it does not follow that the domain of quantification must contain three distinct entities for (3) to be true. Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:
4. B is a proper part of (B + C) & C is a proper part of (B + C).
--------------------------------------------------------------------

He claims 'it will then be possible to existentially instantiate (3) using only two entities'. But 3 is in the language of the predicate calculus. It is absolutely not possible to instantiate (3) in the way he suggests.

John,

Thanks for your helpful comments. Can you tell me the title of Jonathan Schaffer's book or article? You write:

>>One question I have pertains to your response to Boram. You say "Mereology alone will not give us the extra ontological ingredient. In this case it is the adjacency of the two disjoint subparcels that makes them into a whole." Now, as your ensuing dialogue with Boram suggests, you are not denying Unrestricted Composition. You just want to say that, granting the existence of arbitrary mereological sums, it is not the case that all these sums are on a par. Some are wholes, and some are not, and in your view mereology cannot tell the difference.<<

First of all, I suspect you and Boram know more about this subject than I do. I am not an expert in it it but I am studying it with some intensity at the moment because of its relevance to certain other projects of mine. So I could easily fall into some serious errors.

A lot depends on what we take mereology to be. I have been assuming that it is the general formal theory of parts and wholes developed primarily by Lezsniewski. ('meros' Gr. means 'part.') I think this is called classical mereology. I am aware that there are variant mereologies. Among the axioms of classical mereology are Unrestr Comp, Unique Comp, and Transitivity of Parthood.

Let's use 'sum' as short for 'classical mereological sum.' I don't believe I said that some sums are wholes and other are not. It seems obvious that every sum is a whole of parts. How could there be a sum that is not a whole, and how could there be a whole that is not a whole of parts (or of one part, in the case of a simple that is a part of itself and thus a whole of itself alone)?

My point was rather that class. mereology does not suffice for the understanding of a house of bricks, say. The sum of the bricks is a whole of parts, the parts being the bricks. But the house is not identical to that sum.

Now you could say that the house is a whole of parts, but then you are using 'whole' in a way that diverges from the use of 'whole' when 'whole' means 'sum.' I hope that's clear!

So I see the task along the following lines. Don't tamper with Unrestricted Composition. But try to figure out which mereological wholes have corresponding to them 'real' or 'natural' wholes. Clearly there is no such whole corresponding to the sum of the number 9 -- an abstract object -- and my left foot. But that is a perfectly good sum 'let in' by the axioms of class. mereology.

What we need are nonformal principles to move us from the wholes of mereology to 'natural' or 'real' wholes.

To answer your question, I suspect that there are a number of different principles, different for different subject-matters.

John,

I take it you agree with Boram that my argument in the main post against PvI is a good one.

Thanks for defending me against Willim. I'm afraid I don't see what his point is.

William,

The expression is "biting on granite." I borrow it from chess grandmaster Aron Nimzovich. Think of a pawn being attacked by a bishop when the pawn is supported by two pawns.

>> I'm afraid I don't see what [William's] point is.

It's a very simple point, namely that your argument below is wrong :)

-------------------------------------------------------------------
3. There is an x, y and z such that x is a proper part of z & y is a proper part of z.
(3) contains three distinct variables, but it does not follow that the domain of quantification must contain three distinct entities for (3) to be true. Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:
4. B is a proper part of (B + C) & C is a proper part of (B + C).
--------------------------------------------------------------------

I've explained very clearly several times above why it is wrong. I'm not saying that anything else is wrong. Only that piece. And yes, you are biting on granite, big time. (Sorry)

Bill,

I'm sure you are quite right that an existentially quantified statement involving n bound variables can be true in a domain of fewer than n entities. To satisfy (3) I'd suggest x-->{1}, y-->{1}, z-->{1,2} and interpret 'proper part of' as 'proper subset of'.

But I have to agree with W that something is very wrong with (4). The presence of the term 'B+C' suggests that this is a sentence in the mereological language we are trying to interpret, not a statement about values in some interpretation. A interpretation of (3) has to have a valuation function that maps x, y, and z to values in some domain and maps the _is a proper part of_ predicate to some binary relation over that domain. The only way to do this with fewer than three distinct values is to map both x and y to one value and z to a second, distinct value, as in my example above. The values must be distinct to preserve the antireflexive character of _is a proper part of_.

Bill:

Schaffer's 2009 article is called "On What Grounds What". I think a penultimate draft is available on his website. It was published in a volume titled "Metametaphysics: New Essays on the Foundations of Ontology" (ed. Wasserman, Manley, & Chalmers). The volume is a very good one, overall.

I think we're having a bit of a verbal dispute over the word "whole". I've heard some people speak in a fashion closer to mine (where they draw a distinction between sums and wholes), and others speak in a fashion closer to yours (where all sums are wholes, but only some wholes are 'real' wholes). These people are all kindred spirits. They don't want to tamper with UC, they just think that mereology is too weak to draw any kind of ontological distinction between, as you say, a house and the mereological sum of, say, your left hand and the Eiffel Tower.

And I would agree that when we've been speaking about 'mereology', we've been speaking about classical extensional mereology, whose three axioms are the Transitivity of Parthood, Unrestricted Composition, and Uniqueness of Composition. As you say, other mereologies have been developed. Casati & Varzi have something of a nice discussion of this in their "Parts & Places" (MIT: 1999). Also, though I've not read it, I understand that Peter Simons' "Parts: A Study in Ontology" is something of the Bible of mereology. It's supposed to have the clearest development of classical extensional mereology around, followed by a rejection of it (I think Simons rejects both Unrestricted Composition, and the Uniqueness of Composition, but it might be that he rejects only one of these).

William: I'm afraid I don't follow your remarks regarding Schaffer's distinction between integrated wholes and mere aggregates. I don't want to avoid any ontological distinctions; I want to draw them, by employing Schaffer's terminology. So I'm not sure I follow. Maybe you can help me out?

Meanwhile, eturning to what you said about plural quantification, it seems to me that you've said the following. In a first-order language without plural quantification, composition as identity fails because such a language makes no sense of the idea of collective identity. In a first-order language supplemented with plural quantification, then PVI's argument against composition as identity fails. Is this right? I'm just not sure I've followed everything.

David & Bill: Here's a worry I've only just now picked up on. PVI's original example, as quoted by Bill, speaks in terms of parts; that is, when posing his question about quantification, PVI uses as an example: "y is a part of x & z is a part of x & y is not the same size as z". Bill, when using (4) as an example, speaks in terms of PROPER parts: "x is a proper part of z, y is a proper part of z". Parthood and proper parthood have different features. Whereas proper parthood, as David notes, is irreflexive, parthood is reflexive. Different mereological systems opt for different primitives, depending on when and how they want to introduce identity. Assuming that we have a first-order language with identity, we can start with proper parthood as our primitive, and define parthood in the following way:

x is a part of y =df x is a proper part of y or x=y.

Other systems, assuming a first-order language without identity, take parthood as primitive and define proper parthood and identity in terms of it:

x is a proper part of y =df x is a part of y and it is not the case that y is a part of x.
x is identical to y =df x is a part of y and y is a part of x.

It seems to me that this is relevant to evaluating PVI's argument. Bill uses an example employing proper parthood, and argues that it can be instantiated with only two entities. David has objected to this. But what about PVI's example employing parthood? Consider:

(5) x is a part of z & y is a part of z & x is not identical to y.

The last clause of (5) requires there to be at least two entities in our domain of quantification. Do we need three? Maybe if we have plural quantification, we don't? Consider this alternative:

(6) x is a part of the zzs & y is part of the zzs & x is not identical to y.

Here, perhaps we could say that the zzs are identical to x and y (taken collectively). Then, we could have the following interpretation:

Domain: {a, b}
x:a
y:b
zzs:a+b

Does this work? I'm genuinely asking, because I'm genuinely confused. It seems plausible to me that the difference between PVI's original example and Bill's example (where PVI speaks in terms of parthood and BIll speaks in terms of proper parthood) should be relevant to our evaluation of both. But perhaps it makes no difference?

Gentlemen, may I try to offer an irenic solution to the impasse between Bill and William? I believe both are right. What we need, perhaps, is the idea of different counts and of a special sort of identity, i.e., cross-count identity.

So, take the restricted domain D containing just a six-pack of beer. According to one count, it contains 1 thing, a six-pack. According to another count, it contains 6 things, six bottles of your favorite beer. Each count is of exhaustive-and-non-overlapping divisions of D. On no single count would it have 7 things such that 1 is the six-pack itself and 6 are the individual bottles; that will be a form of double counting.

When a friend of composition as identity claims that 1 six-pack = 6 bottles, the identity claim in question here is not the usual kind that can be made in first-order logic. It is an assertion that the two counts are of different ways of dividing the same portion of reality, or as Baxter might put it a cross-count identity claim. One important way in which cross-count identity is different from ordinary identity is that indiscernibility of identity does not apply to the former.

I refer you to what Baxter has fondly called the "bicycle page" here:
http://www.people.vcu.edu/~csutton2/composition.html
I really ought to know more about all this, since Professor Baxter is on my dissertation committee, but in his capacity as a Hume expert, and not as a proponent of carefully defended maverick claims on metaphysics.

p.s. I haven't read John's latest message which may be relevant, and posted while I was composing this comment.

David,

Thanks for your comment. Here is what I wrote:

3. There is an x, y and z such that x is a proper part of z & y is a proper part of z.

(3) contains three distinct variables, but it does not follow that the domain of quantification must contain three distinct entities for (3) to be true. Suppose that Lewis is right, and that A = (B + C). It will then be possible to existentially instantiate (3) using only two entities, thus:

4. B is a proper part of (B + C) & C is a proper part of (B + C).

'(B + C)' denotes the mereological sum of the subparcels B, C. This is an individual just as B, C are individuals. But by Composition as Identity (B + C) is identical to B, C taken collectively. So in the domain of quantification there are only two individuals B, C.

You point out that proper parthood is anti-reflexive which I take it means irreflexive: nothing is a proper part of itself. So perhaps your point against me is that if there are only two disjoint individuals in the domain, B, C, then there is nothing in the domain to stand in the proper parthood relation. Is that your point?

John: >>Meanwhile, [r]eturning to what you said about plural quantification, it seems to me that you've said the following. In a first-order language without plural quantification, composition as identity fails because such a language makes no sense of the idea of collective identity.

Yes.

>>In a first-order language supplemented with plural quantification, then PVI's argument against composition as identity fails. Is this right? I'm just not sure I've followed everything.

Yes.

>>When a friend of composition as identity claims that 1 six-pack = 6 bottles, the identity claim in question here is not the usual kind that can be made in first-order logic.

Well it can. We can construct a set {a,b,c,d,e,f} for example. But because simple FOL does not allow for plural referring terms, we have to construct an expression for the set itself, using the curly brackets, and so we have a seventh entity. This is not the case with plural identity statements in ordinary language. The statement 'The Apostles were Peter, Paul ...' does not introduce a singular entity like a set, because the plural referring term 'The Apostles' does not refer to a single thing, the way an expression for a set does.

Bill: >>So in the domain of quantification there are only two individuals B, C.

Therefore the domain of quantification does not include B+C, and

Ez, z = B+C

is false, and so "A = B+C" is false.

David: >> To satisfy (3) I'd suggest x-->{1}, y-->{1}, z-->{1,2} and interpret 'proper part of' as 'proper subset of'.

But then sets would have to be within the domain, and there would have to be assumptions like 'for any x, y, for some z z= {x,y}' wouldn't there?

Hello Bill,
*Suppose Lewis is right*. Then it would appear that we can construct a domain consisting of distinct individuals B and C and their mereological sum B+C with the valuation v(x)=B, v(y)=C, v(z)=B+C. We have ~B=C, and also ~B=(B+C) and ~C=(B+C), the latter being by the irreflexivity condition. My question then is Can we consistently claim that the domain contains but two individuals? My feeling is No. In the penultimate para of your last comment to me you say 'This [B+C] is an individual just as B, C are individuals' and 'in the domain of quantification there are only two individuals B, C'. I can't see this as non-contradictory, I'm afraid.

William, here has a way through, I think.

W,
Apologies, but I don't follow. My understanding is that I'm constructing a model for Bill's formula (3) and I think I can do that with a domain consisting of just the sets {1} and {1,2}. If not, my mathematical logic is seriously corroded!

>>W, Apologies, but I don't follow. My understanding is that I'm constructing a model for Bill's formula (3) and I think I can do that with a domain consisting of just the sets {1} and {1,2}. If not, my mathematical logic is seriously corroded!

Sorry I misread your formula. But there x is identical to y, no? And I am assuming Bill meant that B must be non-identical with C. If not I owe him an apology. But then if he did mean that, the other parts of his argument make no sense. And in that case the argument does not refute Van Inwagen, who is clearly assuming that each of the proper parts is non-identical with any of the other proper parts.

Tom Mckay has commented at my blog http://ocham.blogspot.com/2010/09/concerning-plurals.html on a parallel discussion going on there.

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