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by Corry Shores
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[Logic and Semantics, entry directory]
[Graham Priest, entry directory]
[Priest, One, entry directory]
[The following is summary. You will find typos and other distracting mistakes, because I have not finished proofreading. Bracketed commentary is my own. Please consult the original text, as my summaries could be wrong.]
Summary of
Graham Priest
One:
Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness
Ch.2
Identity and Gluons
2.2
Breaking the Regress
Brief summary:
(2.2.1) To explain how gluons bind parts into a unified whole, we need to break the Bradley regress, which prevents gluons from simply being object-parts. (2.2.2) We might name the parts of a unified object with letters, as for example, a, b, c, and d. The gluon, symbolized 中, is what binds all the other parts into the unified whole. If the gluon were distinct from the other parts (in the sense of not being identical to them), there would always be room for another gluon to intervene between the first gluon and the given parts, which leads to the Bradley regress. To avoid it, we say that the gluon is identical to each of the parts, thereby closing those “gaps”. (2.2.3) The gluon is non-transitively identical with each and every part. That means that although each part is identical to the gluon, they are not thereby identical to one another. And, parts can themselves be composed of parts by means of another internal gluon.
x
xxxxb
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ax=x中x=xc
xxxx||
xxxxd
x
(2.2.4) Gluonic unity involves non-transitive identity, meaning that a = 中 and 中 = c, but not thereby a = c.
[Gluons as Needing to Break the Bradley Regress]
[The Gluon as Identical to the Parts]
[The Non-Transitive Identity of Parts to the Gluon, Part 1]
[The Non-Transitive Identity of Parts to the Gluon, Part 2]
Summary
[Gluons as Needing to Break the Bradley Regress]
[To explain how gluons bind parts into a unified whole, we need to break the Bradley regress, which prevents gluons from simply being object-parts.]
[Recall from section 1.4.2 that we cannot think of the gluon as an object-part, because as such, it would require yet another gluon to explain its binding among the parts it binds, and that newer gluon will need its own to bind it into the unity, and there can be no end to this regress. Priest reminds us that to explain how gluons bind parts into the unity, we need to break this regress.]
The problem of unity is to explain how it is that gluons glue. What stands in the way of an explanation is the Bradley regress. As we saw in Section 1.4, this is vicious, and so it must be broken. But how?
(16)
[The Gluon as Identical to the Parts]
[We might name the parts of a unified object with letters, as for example, a, b, c, and d. The gluon, symbolized 中, is what binds all the other parts into the unified whole. If the gluon were distinct from the other parts (in the sense of not being identical to them), there would always be room for another gluon to intervene between the first gluon and the given parts, which leads to the Bradley regress. To avoid it, we say that the gluon is identical to each of the parts, thereby closing those “gaps”.]
[Priest will symbolize the gluon with the Chinese (and maybe also Japanese) character 中. We think of a thing having a number of parts, which we can give letter names to, like a, b, c, and d. The gluon 中, then, is what binds these parts into the whole. Now, the gluon cannot simply be another part, for then it will cause our account to fall victim to the Bradley regress. Priest notes that the regress happens when we come to think of the gluon as being another object for which yet another gluon could intervene between the first and the other parts. Priest then says that if we make the gluon identical to the parts, that closes the metaphysical “gap” that otherwise would stand between the gluon and the parts. So we will need to make the gluon identical to the parts to avoid the Bradley regress.]
Suppose that an object has parts a, b, c, and d, and that these are held together by a gluon 中.1 The Bradley regress is generated by the thought that 中 is distinct | from each of the other parts. If this is the case, then there is room, as it were, for something to be inserted between 中 and a, and so on. Or to use another metaphor, there is a metaphysical space between 中 and a, and one requires something in the space to make the join. Thus, the regress will be broken if 中 is identical to a. There will then be no space, or need, for anything to be inserted.
(16-17, boldface mine)
1. The character 中 (Chinese: zhong; Japanese: chu) means centre, which seems like a pretty good symbol for a gluon. (By coincidence, it is also sometimes used as part of one of the Chinese names for Madhyamaka Buddhism: zhong dao zong.) As the amount of logic increases, it also seems a good time for Western logicians to move to some less familiar languages in search of symbols. Unfortunately, | I will use the character in this section only, due to the current difficulty of typesetting Chinese characters in heavily symbolic contexts.
(16-17)
[The Non-Transitive Identity of Parts to the Gluon, Part 1]
[The gluon is non-transitively identical with each and every part. That means that although each part is identical to the gluon, they are not thereby identical to one another. And, parts can themselves be composed of parts by means of another internal gluon.]
[The gluon is non-transitively identical with all of the parts, including itself. We can depict it by having all of the parts equal the gluon, but none of the parts equaling one another.
x
xxxxb
xxxx||
ax=x中x=xc
xxxx||
xxxxd
x
Priest then gives an analogy to explain why the identity here is not transitive. We think of how the mortar between bricks binds the bricks without making them one solid brick. Likewise, the gluon binds the parts by being identical to them, without making those parts be identical with each other. The footnote here is important, but I did not quite grasp it all. The basic idea seems to be that each part can be consider as being made of parts, bound with their own subgluon of sorts (not his term). These subparts still form parts of the whole, but how all this works I did not quite follow. Let me just go slowly through that footnote, line by line:
Of course, the parts of an object can themselves have parts.
This we noted.
Thus, it could be the case that, for example, c has parts m and n.
That simply sets up a naming convention.
These will be joined by a gluon, 中′.
That would seem to be what I called the subgluon, namely, the gluon that unifies the subpartitions of a thing’s main parts.
So we will have m = 中′ = n.
This simply says that the subgluon bears the same non-transitive identity relation with respect to the subpartitions it unifies.
If one takes the parthood relation to be transitive, m, 中′, and n, are also parts of the original object. So we will have 中 = m, 中 = 中′, and so on.
Here is where I get lost. As far as I can tell, such main parts as a, b, c, d, 中 are bound by non-transitive identity. And I would assume also that for m and n we would also not want them to be identical. So when Priest says, “If one takes the parthood relation to be transitive,” I am guessing by “parthood relation” he refers to the subpartition’s relation to the main part. But how does that work? Let us try to make a figure for it:
x
xxxxb
xxxx||
ax=x中x=xc {m = 中′ = n}
xxxx||
xxxxd
xSomething we have not established here, which would show the
In order to get enough levels of parthood for a transitive relation, let us call the whole object Ω:
x
Ω {xxxxb
xxxxxx||
xxax=x中x=xc {{m = 中′ = n}}
xxxxxx||
xxxxxxdxx}
x
My best guess at the moment is that when Priest says, “If one takes the parthood relation to be transitive” he does not mean that the parthood relation is one of identity. I am guessing he simply means that if m is a part of c, and if c is a part of Ω, then m is a part of Ω. But then, I do not understand the line using equations: “So we will have 中 = m, 中 = 中′, and so on.” Because if 中 = m and 中 = n, and this is a transitive identity relation, then we have n = m. In the two given formulas, 中 = m, 中 = 中′, there is in both cases an equation of a higher scale part with a lower scale part. So the only way I can think of this all working is if we think instead of the parthood relation as a non-transitive identity relation. And maybe specifically it is the non-transitive identity relation between the subpartitions specifically with the main gluon. This would bind each subpartition into the whole without equating them. I apologize; please read the text below to see what is really meant here.]
Of course, 中 must be identical with b, c, d, for exactly the same reason. Thus, 中 is able to combine the parts into a unity by being identical with each one (including itself). The situation may be depicted thus:
x
xxxxb
xxxx||
ax=x中x=xc
xxxx||
xxxxd
x
The explanation of how it is that the gluon manages to unite the disparate bunch is, then, that it is identical with each of them.2 Consider, if it helps, an analogy. Suppose that one wants to join two physical bricks together with physical glue. The glue is inserted between the bricks. It bonds to each one, and so joins them. It does not make the two bricks one, but the molecules of the glue and each brick become physically indissoluble. In the metaphysical case, the parts of an object do not become identical either, but the gluon bonds with each part in the most intimate way, by being identical with it.
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2. Of course, the parts of an object can themselves have parts. Thus, it could be the case that, for example, c has parts m and n. These will be joined by a gluon, 中′. So we will have m = 中′ = n. If one takes the parthood relation to be transitive, m, 中′, and n, are also parts of the original object. So we will have 中 = m, 中 = 中′, and so on.
(17)
[The Non-Transitive Identity of Parts to the Gluon, Part 2]
[Gluonic unity involves non-transitive identity, meaning that a = 中 and 中 = c, but not thereby a = c.]
[Priest now explains how the transitivity of identity fails under this conception of gluonic unity. For, “We have a = 中 and 中 = c, but we will not have a = c.” Priest will now provide a more precise theory of non-transitive identity to show how it is a coherent notion.]
It should be immediately obvious that the relation of identity invoked here will not behave in the way that identity is often supposed to behave. In particular, the transitivity of identity will fail. We have a = 中 and 中 = c, but we will not have a = c. Two bricks of a house are not identical. It might be doubted that there is any such coherent notion, or that, if there is, it is really one of identity. These concerns cannot be set aside lightly, and the only way to assuage them is to provide a precise theory of identity which delivers what is required. Let us turn to this.
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Priest, Graham. 2014. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University.
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