by Corry Shores
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[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
Part 1:
Unity
Ch.2
Identity and Gluons
2.4
Identity and Gluons
Brief summary:
(2.4.1) We define the identity statement in the following way:
a = b iff X(Xa ≡ Xb)
(2.4.2) A gluon is defined in the following way, keeping in mind that in our paraconsistent logic, identity is non-transitive: “Given a partite object, x, a gluon for x is an object which is identical to all and only the parts of x. By being identical to each of the parts and to only those, it unifies them into one whole” (20). Thus what we might call the “intimacy” of the paraconsistent identity binds the parts to the gluon and thereby together into one object, but the non-transitivity of the paraconsistent identity keeps the non-gluon parts distinct (being non-identical to one another). (2.4.3) Priest then illustrates with an example gluonic structure to show how a gluon may both have and not have a property if one part of the whole has it and another part does not have it.
[The Paraconsistent, Leibnizian Definition of Identity]
[Gluonic Unity Defined, with the Heterogeneity of the Parts Ensured]
[An Example Gluonic Structure]
Summary
[The Paraconsistent, Leibnizian Definition of Identity]
[We define the identity statement in the following way: a = b iff X(Xa ≡ Xb).]
[In the previous section 2.3, we discussed the material conditional in paraconsistent logic. We thought of truth-evaluation in terms of formulas being in either the true zone, the false zone, or in an overlap of both zones. A formula in the overlap is both true and false. If two formulas are in the same zone, then their material conditional is in the true zone. If they are in opposite zones, then their material conditional is in the false zone. But if one formula is in the overlap zone, and another is exclusively in the true or the false zone, then their material conditional will also be in overlap zone (see section 2.3.3). We learned also in section 2.3.4 that material equivalence in paraconsistent logic is reflexive and symmetric, but not transitive. Priest will now use the material conditional to define identity. He will use a second-order predicate logic. Recall some ideas from section 14.1 of Nolt’s Logics. (The following is taken from our brief summary).
In first-order logic, we can have quantifiers that quantify over variables that stand for individuals. In second-order logic, we can have quantifiers that quantify over predicates. In this way, we can express the following inference, for example: “Al is a frog. Beth is a frog. Therefore, Al and Beth have something in common.” We can write it as: ‘Fa, Fb ⊢ ∃X(Xa & Xb)’. Here we have the predicate variable ‘X’, which allows us to refer to some unspecified predicate as a variable.
[...]
We can use second-order logic to express a number of important logical ideas. One of them is identity. Leibniz’s law says that objects are identical if and only if they share exactly the same properties. It is written:
Leibniz’s Lawa = b ↔ ∀X(Xa ↔ Xb)
It is analyzable into two subsidiary principles.
The Identity of Indiscernibles∀X(Xa ↔ Xb) → a = b
This says that if two things are indiscernible, as they share exactly the same properties, then they are identical. The other is
The Indiscernibility of Identicalsa = b → ∀X(Xa ↔ Xb)
This says that if two things are identical, then they share exactly the same properties. [...]
(From the brief summary to Nolt’s Logics section 14.1)
Also recall some notation from section P7. is the universal quantifier, normally written ∀. And is the particular quantifier, normally written ∃. Priest will now define the identity statement in the following way:
a = b iff X(Xa ≡ Xb)
The X here is a variable for predicates. Normally in a classical logic, this is saying that a = b whenever a has exactly the same predicates as b has. (But things are more complicated, as we will see, now that we are using a paraconsistent logic. It will be possible for something to both have and not have some predicate. We return to this in a second.) Since we are using the material conditional here and using it to define =, that means = will also be reflexive, symmetric, but not transitive. Priest then shows this non-transitivity. We suppose that we have only one predicate, and object a has it, object b both has it and does not have it, and object c simply does not have it. Now, since a has it and b at least has it, then Pa ≡ Pb. And since c does not have it and b at least does not have it, then Pb ≡ Pc. But that does not mean that Pa ≡ Pc. (For, it is just true that a has it and just false that c has it.) With that being the case, we can see how this applies to the identity relations between a, b, and c: “Since P is the only property at issue, we have a = b and b = c, but not a = c.” (So as we mentioned above, matters are more complicated with a paraconsistent logic. We have some object b that both has and does not have property P. And we are assuming this is the only property, and a just has it and c just does not have it. So in a classical logic, we would say that b = c if b and c have exactly the same properties. But here, b has property P and c does not, yet b = c (this is because b also does not have property P. But it is odd, because we can no longer say that two things are identical if they have exactly the same properties). Perhaps we need to say now that they have “at least” the same properties, meaning that a first object that has a certain property can be identical to another object that lacks this same property, so long as the first one also at least lacks that property too. But I am not sure yet how to grasp this perfectly. But while all this is odd, we should keep in mind that the inconsistent objects here are the gluons, which are odd things already.)]
So much for the background. Against this, we can define identity. The definition is the standard Leibnizian one. Two objects are the same if one object has a property just if the other does. In the language of second-order logic, a = b iff:
• X(Xa ≡ Xb)
The second-order quantifiers here are to be taken as ranging over all properties. Whatever these are exactly (and we will come back to the matter later) | the behaviour of identity is going to be inherited from the behaviour of ≡.4 In particular, it is going to be reflexive and symmetric, but, crucially, not transitive. Suppose, for the sake of illustration, that there is only one property in question, P, and that Pa, Pb and ¬Pb, and ¬Pc.5 Then Pa ≡ Pb, Pb ≡ Pc, but not Pa ≡ Pc. Since P is the only property at issue, we have a = b and b = c, but not a = c.6
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4. I note that the property of being identical with something is normally ruled out in a Leibnizian definition of identity on pain of triviality. For given that X(Xa ≡ Xb), it would then follow that a = b ≡ b = b, and so a = b. This is not the case in the present context, due to the non-detachability of ≡.
5. For ease of the informal exposition, I collapse the notational distinction between properties and predicates in a harmless fashion.
6. A consequence of this definition is that any object with contradictory properties is not self-identical. This consequence can be avoided by taking X(Xa ≡ Xb) to give the truth conditions for an identity statement, but giving different falsity conditions. One simple way to do this is to define a = b as: ⟨a, b⟩ satisfies ‘X(Xx ≡ Xy)’. Given the naive satisfaction scheme, this gives the appropriate truth conditions. But, arguably, negation does not commute with truth: T⟨¬A⟩ does not entail ¬T⟨A⟩. (See Priest (1987), 4.9.) Similarly, it does not commute with satisfaction. So the fact that ⟨a, b⟩ satisfies ‘¬X(Xx ≡ Xy)’ does not entail that ⟨a, b⟩ does not satisfy ‘X(Xx ≡ Xy)’; that is that ¬a = b.
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[Gluonic Unity Defined, with the Heterogeneity of the Parts Ensured]
[A gluon is defined in the following way, keeping in mind that in our paraconsistent logic, identity is non-transitive: “Given a partite object, x, a gluon for x is an object which is identical to all and only the parts of x. By being identical to each of the parts and to only those, it unifies them into one whole” (20). Thus what we might call the “intimacy” of the paraconsistent identity binds the parts to the gluon and thereby together into one object, but the non-transitivity of the paraconsistent identity keeps the non-gluon parts distinct (being non-identical to one another).]
[Priest next notes that when the middle, bridging object is consistent (not having contradictory properties), then identity can be transitive (see details in the quote below). Then Priest gives a more formal definition for a gluon. (Recall from section 2.1.1 that a gluon is the factor that binds parts into a unity, and it has the contradictory properties of both being and not being an object.) Here is the definition of a gluon now:
Given a partite object, x, a gluon for x is an object which is identical to all and only the parts of x.
(20).
So recall the diagram of a gluonic structure from section 2.2.3
x
xxxxb
xxxx||
ax=x中x=xc
xxxx||
xxxxd
xxxx
We have the parts a, b, c, and d. And the gluon 中 is identical to all the parts. But on account of the non-transitivity of identity, that does not make the parts be identical with one another. The next line is important but tricky.
By being identical to each of the parts and to only those, it unifies them into one whole.
(20)
Here, being identical is like a logical property of the factor that binds parts into whole. Being-identical is something like a full intimacy. But as a paraconsistent identity, it is not an exclusive, full intimacy. The gluon is identical to part a, but it is no less identical to part b, even though a is not identical to b. So in that sense of its identificatory immediacy, it binds a and b into one unity, but it does not by that intimate binding thereby reduce the distinctness of a or of b. In Dupréel’s La consistance et la probabilité constructive, section 1.4, he discusses something similar. He notes how things whose parts bind more strongly and strongly, thereby constituting a unified object whose wholeness and integrity likewise grows stronger, can take two paths of development. Either its parts fuse and homogenize, subtracting from their individuality as the whole increases its unity. Or instead, as in life forms, the parts continue to bond together and into a strengthening whole all while maintaining and increasing their individual diversity. In other words, what Dupréel calls consistance seems to share this paraconsistent logical property of Priestian gluonics, namely, a binding of the parts that constitutes a whole all without equalizing or homogenizing those parts. (And by extension, this would apply to Deleuze’s and Deleuze & Guattari’s similar theories of composition). Priest’s next observation is:
Note that a gluon is identical to itself; it follows that it is a part of x.
(20)
I think the idea might be the following here. Being a part of x means being paraconsistently identical to x’s gluon. Since the gluon of x is identical to x’s gluon (which is itself), then the gluon of x is also a part of x. Priest’s final point in this paragraph is:
Note also that the gluon of an object is unique. For suppose that g and g′ are gluons of an object, x, then, since g and g′ are parts of x, g = g′ (and g′ = g).
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(I think I do not follow this well, but maybe it is the following. We will conclude that the gluon of an object is unique, which I assume means there is only the one defining gluon. We show this by first proposing that there be two gluons for an object x, namely, g and g′. Next we recall that gluons are parts of their object. Every part of the object is identical to the object’s gluon. So g = g′, because gluon g as a part is identical with gluon g′, taken to be the binding factor; and g′ = g, because g′ as a part is identical to g, taken to be the binding factor. But that then means that g = g′, and thus the distinction between them was superfluous, and rather there is just one unique gluon. I may have that wrong, so please check it yourself.)]
It should be noted that though we do not have transitivity of identity in general, we do have it when the “middle” object is consistent, that is, has no contradictory properties. For suppose that a = b = c, and that b is consistent. Consider any property, P. Then Pa ≡ Pb and Pb ≡ Pc. Hence, (Pa ≡ Pc) ∨ (Pb ∧ ¬Pb). Given that the second disjunct can be ruled out, we have Pa ≡ Pc. So a = c. There is much more to be said about identity, but we may leave the matter for the moment. Given this understanding of identity, we may now define formally what a gluon is. Given a partite object, x, a gluon for x is an object which is identical to all and only the parts of x.7 By being identical to each of the parts and to only those, it unifies them into one whole. Note that a gluon is identical to itself; it follows that it is a part of x. Note also that the gluon of an object is unique. For suppose that g and g′ are gluons of an object, x, then, since g and g′ are parts of x, g = g′ (and g′ = g).
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7. To keep the account as general as possible, I leave it open here whether ‘part’ includes the improper part which is the whole.
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[An Example Gluonic Structure]
[Priest then illustrates with an example gluonic structure to show how a gluon may both have and not have a property if one part of the whole has it and another part does not have it.]
[Priest will now show how this works with an example gluonic structure. We have four objects, g, i, j, and k. We put aside k for the moment, because it is like a distinct entity, but g, i, and j are parts of one entity x, with the g as its gluon. (Now, as the gluon, that means it has the paraconsistent material conditional relation with each of the parts, meaning that if it is true one of the parts has some property, then the gluon has that property too, and if it if false that the other part has that property, then it is false that the gluon has it. But under our paraconsistent logic, gluon g can both have a property (if one other part has it) and not have that property (if yet another part does not have it.) So look at the distribution of property possessions for the various parts of x, including the gluon g (and forget k for the moment. Just look at the first three, i, g, and j).
| P1 | P2 | P3 | |
| i | + | − | + |
| g | ± | ± | + |
| j | − | + | + |
| k | + | + | − |
As we can see, since i has the first property but j does not, then gluon g both has and does not have that property (since it is identical to both), and since both i and j simply just have the third property, that means gluon g just simply has that property too. So given the sharings and lackings of properties, we have: i = g (because i has the first and third properties, but lacks the second; and g at least does too), g = g (of course), and g = j (because j lacks the first property but has the second and third; and g at least does too). Priest next looks at object k. Look at the third property for all of the parts. We see that all parts of object x have property 3, but k does not. That means no part of x is identical to k. Priest’s final point seems to be the following. We will conclude that g ≠ g. (We see that g is ± for the first property. That means P1g is at least false and P1g is at least true, meaning that P1g ≡ P1g is at least false (it is also true) and thus that ¬(P1g ≡ P1g) is then at least true (it is also false). Now, since the first property both holds and does not hold for g, that means there is a property for which it both holds of g and does not, or: X(Xg ∧ ¬Xg), which furthermore means that it is not the case that for all properties that if they hold for g then then it cannot be that they do not also hold for g, or: ¬X(Xg ≡ Xg). Now, since a = b iff X(Xa ≡ Xb), and since ¬X(Xg ≡ Xg), that means g ≠ g. Please see the quotation to be sure.)]
Let me illustrate a gluon structure with a simple example. Suppose that we have four objects, g, i, j, and k. g, i, and j are the parts of some object, x, and g is its gluon. Suppose that there are just three properties, P1, P2, and P3, possessed as follows. ‘+’ indicates that the object is in (just) the extension; ‘−’ indicates that it is in (just) the anti-extension; and ‘±’ indicates both.8 |
P1 P2 P3 i + − + g ± ± + j − + + k + + −
It is easy to check that for each of the three properties, P, we have Pi ≡ Pg , and so X(Xi ≡ Xg), and similarly for g and j (and of course for g and g). Hence i = g, g = g, and g = j. However, we have none of the following: P3i ≡ P3k, P3g ≡ P3k, P3j ≡ P3k. Hence, none of i = k, g = k, and j = k holds. g is identical to all and only the parts of x.9 Note that ¬(P1g ≡ P1g), so X(Xg ∧ ¬Xg), that is ¬X(Xg ≡ Xg); that is, g ≠ g.10
(20-21)
8. Recall, from Section P.5, that we need to specify both the places where P holds—the extension of P—and the places where ¬P holds—the anti-extension of P—since, unlike the classical case, neither determines the other. (20)
9. Suppose that the object of our diagram had another part, l, which was in the anti-extension of P1, P2, and P3. Then g would be in the anti-extension of P3 too. Hence, k would be part of the object as well. This bespeaks a certain failure of atomism, but hardly a surprising one. If you build a room between a house and an out-house, and join them internally, the out-house becomes part of the house.
10. [Not included in this quotation. See p.21.]
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From:
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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