I am finding this debate with Joseph Jedwab fascinating, but I can easily appreciate how others would find it hopelessly boring and pointless. Ontology is not everyone’s cup of tea. Is it conducive unto salvation? The very question smacks of an anti-intellectualism that needs to be addressed in a separate post. But now, back to work.
We are discussing how a whole is related to its parts. Our examples are of the simplest sort since if we cannot get clear about these simple cases, then it is highly unlikely that we will be able to get clear about more complex ones. Joseph Jedwab (JJ) maintains that in a case in which a whole W is composed of exactly two nonoverlapping proper parts P1 and P2, there are exactly three entities: P1, P2, and W. I find this unacceptable. (See here, here, here, and here.) Now back to the thread.
JJ: Let's consider your
3. ExEyEz[x is a proper part of z & y is a proper part of z].
This could indeed be true in a domain of two entities for it could well be that [a is a proper part of c & b is a proper part of c] but a=b. But this is not to the point.
BV: Right, that would not be to the point. But your example implies that you do concede my point that if a quantified formula contains n variables, it does not follow that there must be n entities in the domain of quantification. Therefore, if someone infers that P1, P2, and W make three entities because of the truth of (3), then that person commits a non sequitur.
JJ: If some things, the xs, compose something y, then the xs do not overlap each other, each of the xs is a proper part of y and every proper part of y overlaps at least one of the xs.
JJ: So a better example is this. If two simples compose a whole, the simples are distinct from each other and each is distinct from the whole and so there are at least three things in the domain.
BV: You are arguing as follows:
a. Two distinct simples compose a whole
b. Each simple is distinct from the whole
c. There are at least three things.
Trouble is, (c) does not follow from (a) and (b). To arrive at (c), you need some such supplementary premise as
d. If n distinct entities, each distinct from a whole, compose that whole, then there are n + 1 entities.
But (d) will be rejected by someone who holds that composition is identity. Therefore, you are begging the question against the composition-as-identity theorist. Why should anyone accept (d)?
You are quite sure that the whole W is a third thing, a thing in addition to its sole proper parts P1 and P2. But if W is composed of these parts but not exhausted by them, then what is the further ontological ingredient that distinguishes W from the two parts? Please tell me what that is. Please tell me what must be added to P1 and P2 to get W. And please don’t tell me that W must be added to P1 and P2 to get W. That would make no sense: W is not ‘over and above’ P1, P2. After all, W is composed of P1 and P2: P1 and P2 are the ‘building blocks’ out of which W is ‘built.’ If there is nothing which, when added to P1 and P2, yields W, then what makes W numerically distinct from (P1, P2)? Will you say that nothing makes them distinct, that their distinctness is a brute fact?
NOTE: if you simply help yourself to W as a third thing, if you simply assume that it is a third thing, then I say you have begged the question against the composition-as-identity theorist.
Here is another way to look at it. (P1, P2) is a complex. W is a complex. How can two complexes differ without differing in a constituent? Two sets differ numerically iff they differ in an element, i.e., iff one has a member (element) the other doesn’t have. That is a consequence of the Zermelo-Fraenkel Axiom of Extensionality. Now (P1, P2) is not a set, but it is some kind of complex. W is a still different kind of complex. There is more to W than (P1 P2). What is that more?
JJ: The simples are distinct from each other for they do not overlap each other. And each simple is distinct from the whole for each simple is a proper part of the whole.
BV: Yes to both of these sentences. But it does not follow that the whole is a third thing in addition to the two proper parts. Again, if the whole is composed of the two parts (and of nothing besides) and the whole is a third thing distinct from the two parts, then please tell me what distinguishes the whole from the two parts.
What justifies your positing of a tertium quid? How can you convince me that you are not illictly inferring three values from three variables? It seems to me that you are not entitled to assume that the whole is a third thing. That must be proven. You may assume that there is a whole, but not that it is a third thing.
To help convey my point, allow me to introduce a distinction between an entity and an object. An entity (thing) is something that exists in reality apart from our consideration. An object may or may not exist in reality apart from our consideration. A mere object exists only as an accusative of our consideration. Given that W is composed of P1 and P2, there must be some sense in which W exists. Well, suppose that W exists merely as an object of our consideration. Then it will be the case that there are two entities but three objects. In this way I avoid your conclusion that there are three entities, while accommodating your intuition that W is a third item.
JJ: Of course, we could easily define a function 'x+y=z', which takes as argument a pair of parts that compose something and gives as value the whole they compose. But this does not show that the parts are identical to the whole.
BV: Right. But the onus probandi is on you to show that the whole is distinct from the parts, and to explain exactly what this means given that the whole is composed of the parts.
JJ: I don't understand how your
4. B is a proper part of (B + C) & C is a proper part of (B + C)
is true in a domain of only two entities. 'x is a proper part of y' is a two-place predicate that expresses an irreflexive and asymmetric relation. Your point works if 'B+C' is not a singular term that refers to one entity in the domain, but is a plural term that refers to two entities in the domain. In that case, you do not use the two-place predicate 'x is a proper part of y' but the multigrade predicate 'x is a proper part of the ys' which holds if and only if x is one of the ys. But this can hold even if x is not a proper part of any of the ys or the composite of the ys. And it also looks like what is really going on is that mereological talk is being construed as plural quantification talk.
BV: You say in effect that ‘x is a proper part of the ys’ can hold even if x is not a proper part of the composite of the ys. Can you give an example of this? If x is a proper part, then x is a proper part of some whole. (Isn’t that analytic?) But every whole of parts is a composite. So if x is a proper part of the ys, then there is a whole of ys, a composite of ys, such that x is a proper part of it. For example, if Mungojerrie is a proper part of the cats in the neighborhood, then there is a whole composed of these cats and Mungojerrie is a proper part of this whole.
Perhaps you are building into the notion of ‘composite’ something I am not building into it.
Could you please explain the exact difference, as you understand it, between plural quantification talk and mereological talk? ‘The’ connotes uniqueness in this context; so ‘the ys’ is not purely plural. Thus, ‘the cats’ refers to a grouping together of cats. There is a difference between ‘the cats’ and ‘cats.’
JJ: Does this beg the question against Baxter? My argument that if two simples compose a composite, there are at least three entities at no point uses the premise that composition as identity is false. That, rather, follows from the conclusion. Look again. Suppose B and C compose A. By the definition of 'composition', B and C do not overlap each other and each of them is a proper part of A. By the definition of 'overlap', B and C are distinct from each other. By standard predicate logic, they are two. By the definition of 'proper part', B and A are distinct from each other and C and A are distinct from each other. Again, by standard predicate logic, they are three. No worries!
BV. You are a marvelously clear-headed fellow, Joseph, but I persist in my claim that you are begging the question. You do so when, from "B and A are distinct from each other and C and A are distinct from each other" you infer that "they are three." Three what? Three objects of consideration, say I, but not three entities.
JJ: Why isn't the connectedness detectable? Can't one tell the difference between some things that are properly connected and some things that are not properly connected? Why doesn't that count as detecting connectedness?
BV: Yes I can tell the difference between my pipe taken apart for cleaning and the same pipe assembled and ready for use. In the first case, the parts are not spatially contiguous; in the second case they are. But that is not the question. The question concerns the pipe with pipe stem properly inserted into pipe bowl. I can see (visually) that the two parts are in a certain spatial relationship. But that does not count as seeing the connectedness. Compare Hume on causation. One can see that event e1 is spatiotemporally contiguous with event e2, and that e1 precedes e2. But that does not amount to seeing e1's causing of e2. Causing is empirically undetectable.
JJ: What Bradleyan regress results? You ask what makes A distinct from B and C? Let me ask in turn: what makes anything distinct from anything?
BV: It is perhaps acceptable to say of two simples that they are just numerically distinct, that their being numerically distinct is a brute fact. But as explained above, A is a complex and (B, C) is a complex, and it is difficult to understand how two complexes could differ without differing in a constituent.
The Bradleyan regress comes into the picture if an entity is introduced to connect B and C. What connects this connector to B and C? Etc.
JJ: Finally, what did you think of my gloss of your claim that A is not a third thing in addition to B and C properly connected as what makes it true that B and C are properly connected makes it true that A exists and so that A exists is not a further fact in addition to the fact that B and C are properly connected?
BV: I’ll leave this for later.