Dupréel (6.3.2) Essais pluralistes, ch.6: Théorie de la consolidation, sect 6.3.2, ‘Les consolidés de coexistence’, summary

 

by Corry Shores

 

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[The following is summary and not translation. Bracketed commentary is my own, as is any boldface. Proofreading is incomplete, so typos are present, including in the quotations. Please consult the original text to be sure about the contents. Also, I welcome corrections to my interpretations, because I am not especially good with French.]

 

[May I please thank the sources of the puddingstone images:

East Herts Geology Club:

http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/

]

 

 

Summary of

 

Eugène Dupréel

 

Essais pluralistes

 

Ch.6

Théorie de la consolidation.

Esquisse d’une théorie de la vie d’inspiration sociologique.

 

6.3

[Purpose/Finality in Sociology]

 

6.3.2

Les consolidés de coexistence

 

 

 

 

Brief summary:

(6.3.2.1) There are two stages in the manufacture of an object: firstly, the parts are manually given the arrangement they will finally hold on their own, and secondly, these structural relations between the parts are then fixed so that the object stands by itself, without the laborer’s interference. (6.3.2.2) We see these two phases of object construction in the molding process: {1} first the mold places the molding material’s parts together into a certain arrangement and holds them there. {2} Next, the material hardens into that form and keeps it all on its own. Note that two things are transferred from the mold to molded material: {1} the parts’ proper arrangement of mutual relations, and {2} the capacity to hold those relations intact over time, which is called solidity. Whenever there is such a transfer, we call it consolidation. (6.3.2.3) In manufactured things, the ordering of the parts is a spatial one. We call such things consolidations of coexistents (consolidés de coexistence). (6.3.2.4) This process of consolidation that we saw in human industry can also be found in natural processes, as for example in the formation of puddingstone. Here pieces of flint are fixed in place within binding materials by the soil and gravity. As the binding material solidifies, a solid rock is formed which no longer relies on the exterior supporting factors to maintain the compositional arrangement of the pebbles in the hardened binding cement. This is a natural example of consolidated coexistents. (6.3.2.5) Consolidations of coexistents are quite common in nature, as  all bodies with connected parts – be they solids or things with more loosely bound parts – are consolidations of coexistents. They are all formed by this two-step process where the exterior order gives arrangement and support to the parts until they solidify. (6.3.2.6) The world of our sensible perception is a totality of consolidations of coexistents.

 

 

 

 

 

 

 

Contents

 

6.3.2.1

[The Two Phases of Object Manufacture: The Arrangement of the Parts and the Fixing of the Arrangement]

 

6.3.2.2

[Molding as a Great Example of the Two-Phased Process. Solidity as Consistency. Consolidation as Exterior-to-Interior Structuration-Support Transfer]

 

6.3.2.3

[Consolidations of Coexistents]

 

6.3.2.4

[Natural Consolidated Coexistents: Puddingstone]

 

6.3.2.5

[The Prevalence of Consolidations of Coexistents in Nature]

 

6.3.2.6

[Our World of Sensible Perception as Being Composed of Consolidations of Coexistents]

 

 

 

 

 

 

 

Summary

 

 

6.3.2.1

[The Two Phases of Object Manufacture: The Arrangement of the Parts and the Fixing of the Arrangement]

 

(p.158: “Les Consolidés de coexistence. – Dans toute fabrication ...”)

 

[There are two stages in the manufacture of an object: firstly, the parts are manually given the arrangement they will finally hold on their own, and secondly, these structural relations between the parts are then fixed so that the object stands by itself, without the laborer’s interference.]

 

[(In rough form: The Consolidateds/Consolidations of Coexistence. Generally speaking, in any manufacturing process, we may distinguish two well-characterized successive states: In the first state, the parts of the object under construction are collected and arranged in the order that they should continue retaining. But at this phase of the labor, this order is only maintained by external and temporary means. It is only in a second and final state that, through an internal arrangement, the parts all on their own will hold together the parts’ positional relations in the completed object. For example, if it is a matter of making a crate, for a few moments, it is the hands of the worker that hold the boards upon each other, which she will fix together with nails. When they are hammered in, the crate “stands on its own”: it went from the first to the second of the two states whose succession we have just mentioned.) (Main ideas in refined form: In a manufacturing process, there are two stages of the object’s production. In the first stage, the object’s parts are arranged into the order they will finally hold. But at this point, the establishment and the maintenance of that order is made by the laborer. In the second stage of production, those arrangements are given a self-standing, fixed form, so that the object’s arrangement holds on its own. Consider for illustration the production of a wooden crate. In the first stage, the laborer’s hand sets up the proper arrangement of the crate’s parts. (Were the laborer to remove the influence of her hands, the crate would collapse). In the second stage, the crate’s boards are hammered together so that it can hold together even without the support of the laborer’s hands.)]

Les Consolidés de coexistence. – Dans toute fabrication, en général, on peut distinguer deux états successifs bien caractérisés : Dans un premier état, les parties de l’objet à construire sont rassemblées et mises dans l’ordre où elles devront demeurer. Mais à ce moment du travail cet ordre ne se maintient que par des moyens extérieurs et provisoires. Ce n’est qu’à un état second et définitif que, par un aménagement intérieur, les parties garderont d’elles-mêmes les rapports de position que comporte l’objet achevé. S’agit-il de faire une caisse, pendant quelques instants, ce sont les mains de l’ouvrier qui retiennent l’une contre l’autre les planches qu‘il va réunir par des clous. Ceux-ci étant enfoncés, la caisse « tient toute seule » : elle est passée du premier au second des deux états dont nous venons de rappeler la succession.

(158)

[contents]

 

 

 

 

 

 

6.3.2.2

[Molding as a Great Example of the Two-Phased Process. Solidity as Consistency. Consolidation as Exterior-to-Interior Structuration-Support Transfer]

 

(p.158-159: “Cela est encore plus apparent ...”)

 

[We see these two phases of object construction in the molding process: {1} first the mold places the molding material’s parts together into a certain arrangement and holds them there. {2} Next, the material hardens into that form and keeps it all on its own. Note that two things are transferred from the mold to molded material: {1} the parts’ proper arrangement of mutual relations, and {2} the capacity to hold those relations intact over time, which is called solidity. Whenever there is such a transfer, we call it consolidation.]

 

[(In rough form: This is even more apparent in the process of molding; the duality of the operation’s times is marked by that of the mold itself and of the molded object. Before “the setting” of the cement, the parts of the object are already placed in the proper order, but the force that maintains this order is external to them, and it is the solidity of the mold. The mold can only be removed when it is no longer needed to provide its sustaining role, as the parts in the molding now stand on their own. The order of the parts of the molded object was first supported or determined by the order of the parts of the mold, or its form; the accomplished operation consists in a consolidation of this order, which was initially precarious and inconsistent. Something has been transported from the mold to the molded object, namely, solidity, (which may be understood as a property shared by a number of terms (here, the parts of the object) by which they maintain themselves in a certain mutual relationship and keep this ordering). We will call consolidation any operation where we discern a transporting (of constituent structural relations) of this kind, that is to say, where an order, maintained initially by its dependence on an external order, comes to be supported by an internal capacity, such that the sustaining role of the external order, having become superfluous, can be eliminated). (Main ideas in refined form: This two-step process of first arranging the parts of an object under construction by external means and secondly for that arrangement to reify is seen quite clearly in the molding process. Here, the structural relations that the object’s parts will finally take-on are built into the mold’s form, and the mold itself, under this form, is the external factor that initially gives the molded material’s parts its proper arrangement and continued support in that formation. We then see the second stage when the material hardens, and the mold’s supporting ability becomes superfluous as the material now holds its internal structure without need of additional supporting aid. This process of the relations of the object’s parts becoming self-standing is called consolidation. This happens when solidity is transferred from the mold to the molded object (or more generally from the formational process or structure to the formed object.) Solidity is the property of the parts by which they can  maintain their mutual relations and conserve this ordering over time. Thus, consolidation is the operation by which there is a transfer both of constituent, structural relations and as well the capacity to maintain them. This process starts from an external influence that the constructed object initially depends upon; it then moves into the object itself, thereby endowing it with the internal capacity to maintain its constitution without the need for external support, on account of it obtaining its own solidity.) (Commentary: Here, solidity seems to be similar to what Dupréel calls “consistency” in “La consistance et la probabilité constructive” (see especially section 1.2). And “consolidation” seems to be similar to the amalgamation of similars and their formation of solids (see section 1.4).)]

Cela est encore plus apparent dans l’opération du moulage ; la dualité des temps de l’opération y apparaît marquée par celle du moule et de l’objet moulé. Avant « la prise » du ciment, les parties de l’objet sont déjà placées dans l’ordre qui convient, mais la force qui maintient cet ordre leur est extérieure, c’est la solidité du moule. Celui-ci ne peut être ôté que lorsque son rôle sustentateur est devenu inutile, les parties du moulage se tenant désormais d’elles-mêmes. L’ordre des parties de l’objet moulé était d’abord soutenu ou déterminé par l’ordre des parties du moule, ou sa forme ; l’opération accomplie consiste dans une consolidation de cet ordre, d’abord précaire et inconsistant. Quelque chose s’est transporté du moule vers l’objet moulé, c’est la solidité, ou cette | propriété pour un certain nombre de termes (ici, les parties de l’objet), de se maintenir dans un certain rapport mutuel, de conserver leur ordre. Nous appellerons consolidation toute opération où l’on discerne un transport de cette sorte, où un ordre, maintenu d’abord par sa dépendance à l’égard d’un ordre extérieur, arrive à se soutenir par une capacité interne, de telle sorte que le rôle sustentateur de l’ordre extérieur, devenu superflu, peut s’abolir.

(158-159)

[contents]

 

 

 

 

 

 

6.3.2.3

[Consolidations of Coexistents]

 

(p.159: “Dans le cas du moulage ... ”)

 

[In manufactured things, the ordering of the parts is a spatial one. We call such things consolidations of coexistents (consolidés de coexistence).]

 

[(In rough form: In the case of molding, and in general in all manufactured materials, the consolidated order is a spatial order; it is the relation of two or more extended parts that exist simultaneously or that endure together. We call such objects consolidations of coexistents (consolidés de coexistence). All our manufactured materials are consolidations of coexistents, from a pin to a bible or even a railway network.) (Note: consolidations of coexistents is not a very literal translation, but I do not know how better to render a natural translation. I invite your suggestions.)]

Dans le cas du moulage, et en général dans tout fabricat matériel, l’ordre consolidé est un ordre spatial, c’est le rapport de position de deux ou de plusieurs parties étendues, existant simultanément ou durant ensemble : Nous dirons de tels objets que ce sont des consolidés de coexistence. Tous nos fabricats matériels sont des consolidés de coexistence, depuis une épingle jusqu’a une bible ou un réseau de chemins de fer.

(159)

 

[contents]

 

 

 

 

 

 

6.3.2.4

[Natural Consolidated Coexistents: Puddingstone]

 

(p.159: “Nous venons de dégager cette notion ...”)

 

[This process of consolidation that we saw in human industry can also be found in natural processes, as for example in the formation of puddingstone. Here pieces of flint are fixed in place within binding materials by the soil and gravity. As the binding material solidifies, a solid rock is formed which no longer relies on the exterior supporting factors to maintain the compositional arrangement of the pebbles in the hardened binding cement. This is a natural example of consolidated coexistents.]

 

[(In rough form: We have just formulated this notion by considering human industry, which is an eminently finalistic activity. Now, let us note that consolidated coexistents are as much a fact of nature as they are a fact of humans. Consider a piece of puddingstone that  contains two flint pebbles embedded in ferruginous cement. (Here are some images of puddingstone, from the East Herts Geology Club website:

(These beautiful images come from the East Herts Geology Club: http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/)

There is no doubt about the way this composite and solid body was formed. The two pebbles, rolled by the water, stopped, and then landed flat in each other’s neighborhood. Sand deposited around them and filled the gap between them; then, with the help of moisture, the sand hardened, making it all one same solid. Before this agglomeration, the order constituting the two stones considered as terms was maintained by the underlying soil combined with the attraction of the Earth (gravity) (fig. 1).

This supported ordering was therefore exterior to what would constitute our conglomerate. The change in the consistency of the sand made this support – which was once exterior – an internal support, if not to the two pebbles, at least to the object they constitute with the cement that binds them. The soil and gravity are displaced from their sustaining role; I was able to take away this piece of stone, and I can turn it in a hundred ways without altering it; it is a consolidation of coexistents (fig. 2).

)]

Nous venons de dégager cette notion en considérant l’industrie humaine, c’est-à-dire une activité éminemment finaliste. Constatons maintenant que des consolidés de coexistence sont aussi bien le fait de la seule nature que le fait des hommes. Voici un morceau de poudingue. Il contient deux cailloux de silex enrobés dans un ciment ferrugineux. La manière dont ce corps composite et solide s’est formé ne fait pas de doute. Les deux cailloux, roulés par les eaux, se sont arrêtés, posés à plat, dans le voisinage l’un de l’autre. Du sable s’est déposé autour d’eux et a comblé l’intervalle qui les séparait ; puis, l’humidité aidant, ce sable s’est durci, faisant du tout un même solide. Avant cette agglomération, l’ordre que constituaient les deux cailloux considérés comme termes, était maintenu par le sol sous-jacent combiné avec l’attraction de la Terre (fig. 1).

Cet ordre de sustentation était donc à l’extérieur de ce qui allait constituer notre conglomérat. Le changement de consistance du sable a fait que cette sustentation, d’extérieure qu‘elle était, est devenue intérieure, sinon aux deux cailloux, du moins à l’objet qu’ils constituent avec le ciment qui les lie. Le sol et la pesanteur sont évincés de leur rôle sustentateur ; j’ai pu enlever ce morceau de pierre et je peux le tourner de cent façons sans l’altérer ; c’est un consolidé de coexistence (fig. 2).

(159)

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6.3.2.5

[The Prevalence of Consolidations of Coexistents in Nature]

 

(p.160: “Loin que ce processus soit rare ou exceptionnel ...”)

 

[Consolidations of coexistents are quite common in nature, as all bodies with connected parts – be they solids or things with more loosely bound parts – are consolidations of coexistents. They are all formed by this two-step process where the exterior order gives arrangement and support to the parts until they solidify.]

 

[(In rough form: Far from this process being rare or exceptional, there is nothing more common in nature than such consolidations of coexistents: they are everywhere. All solid bodies, all consistent beings that are formed from parts that are fused together or that are simply attached to one another, have come into existence by means of the operation we have been discussing. They all share the same history/story. A time always passes, with a duration that is sometimes quite short but often very long, during which the parts are only maintained in the relations that constitute the whole only by an exterior order of support which is followed by it obtaining its own consistency when the thing is supported by itself and from within. There are cases where the order of primitive and external support does not disappear even though it became superfluous. This would have been the case for our puddingstone had it remained, with the thousands of similar fragments, “in situ”; the sand would have maintained the two pebbles in the same position where gravity sufficed to hold them. Sometimes, on the contrary, the object at this point has become so detached from the circumstances of its formation that it is no longer possible to reconstitute it. Nevertheless, in this case, like in any other, something subsists in the object which we cannot explain by the nature of its parts, something irreducible to their properties, something which is compatible, on the contrary, with other elements, and which comes from this exterior order that is currently being eliminated.) (Main ideas in refined form: The consolidations of coexistents is quite common in nature, as it is what forms all groupings where parts come together, like in solid bodies where parts are fused or like in more loosely constituted bodies. In all cases, they follow the same course of formation: first their parts and their arrangements are supported by an external order, and following that they obtain their own consistency when they find structural support from within. Sometimes the external support remains even if it becomes superfluous to the structural supporting of the parts. For example, suppose the puddingstone remained in the place of its formation. The gravity would still be acting on the pebbles in the same supporting way, even though the hardened binding is now sufficient to keep the pebbles in their places. (I cannot grasp the following ideas well, so please see the quotation below.) Sometimes, however, the object becomes so detached from the circumstances of its formation that it can no longer be reconstituted. (I am guessing that for instances the binding of the puddingstone does not harden. But I have no clue what the idea is here.) But even in this case, there is something about the parts which is not intrinsic to them but rather is the result of external relations coming from the exterior supporting order. (So perhaps, to continue guessing, even if the binding material of the puddingstone does not harden, the arrangement of the pebbles has its ordering on account of the exterior factors like the gravity and soil structure below it.)]

Loin que ce processus soit rare ou exceptionnel, il n’est rien de plus commun dans la nature que de tels consolidés de coexistence : ils sont partout. Tous les corps solides, tous les êtres consistants formés de parties soudées ou seulement rattachées les unes aux autres, sont venus à l’existence en passant par notre opération. Ils ont tous la même histoire. Un temps, parfois très court, souvent très long, s’est toujours passé pendant lequel les parties n’étaient maintenues dans le rapport qui constitue le tout, que par un ordre de sustentation extérieur, puis est venue la consistance propre, la chose s’est soutenue d’elle-même et par le dedans. Il y a des cas où l’ordre de sustentation primitif et extérieur ne disparaît pas, quoique devenu superflu. Tel aurait été le cas pour notre morceau de poudingue s’il était demeuré, avec des milliers de fragments analogues, « in situ » ; le sable aurait maintenu les deux cailloux dans la même position où la pesanteur suffisait à les conserver. Parfois, au contraire, l’objet s’est à ce point détaché des circonstances de son élaboration qu’il n’est plus possible de les reconstituer. Il n’empêche que, dans ce cas comme dans tout autre, quelque chose subsiste, dans l’objet, qu’on ne saurait expliquer par la nature de ses parties, quelque chose d’irréductible aux propriétés de celles-ci, de compatible, au contraire, avec d’autres éléments, et qui vient de cet ordre extérieur, actuellement aboli.

(160)

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6.3.2.6

[Our World of Sensible Perception as Being Composed of Consolidations of Coexistents]

 

(p.160: “Le monde de notre perception sensible …”)

 

[The world of our sensible perception is a totality of consolidations of coexistents.]

 

[(ditto)]

Le monde de notre perception sensible est un ensemble de consolidés de coexistence.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Dupréel, Eugène. (1949). Essais pluralistes. Paris: Presses universitaires de France.

 

 

Image Source:

East Herts Geology Club. “Puddingstone Use”.

http://ehgc.org.uk/hertfordshire-puddingstone/puddingstone-use/

Thank you very much for the beautiful pictures.

.

.

Priest (2.4) One, ‘Identity and Gluons,’ summary

 

by Corry Shores

 

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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.

 

 

 

 

 

 

Contents

 

2.4.1

[The Paraconsistent, Leibnizian Definition of Identity]

 

2.4.2

[Gluonic Unity Defined, with the Heterogeneity of the Parts Ensured]

 

2.4.3

[An Example Gluonic Structure]

 

 

 

 

 

 

Summary

 

2.4.1

[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 Law

a = 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 Identicals

a = 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

(20)

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.

(20)

[contents]

 

 

 

 

 

 

2.4.2

[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).

(20)

(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).

(20)

7. To keep the account as general as possible, I leave it open here whether ‘part’ includes the improper part which is the whole.

(20)

[contents]

 

 

 

 

 

 

2.4.3

[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.]

(21)

[contents]

 

 

 

 

 

 

 

 

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.

 

 

 

 

Priest (2.2) One, “Breaking the Regress”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[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

xxxx||

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.

 

 

 

Contents:

 

2.2.1

[Gluons as Needing to Break the Bradley Regress]

 

2.2.2

[The Gluon as Identical to the Parts]

 

2.2.3

[The Non-Transitive Identity of Parts to the Gluon, Part 1]

 

2.2.4

[The Non-Transitive Identity of Parts to the Gluon, Part 2]

 

Bibliography

 

 

 

 

 

Summary

 

2.2.1

[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)

[contents]

 

 

 

 

2.2.2

[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)

[contents]

 

 

 

 

2.2.3

[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.

(17)

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)

[contents]

 

 

 

 

2.2.4

[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.

(17)

[contents]

 

 

 

 

 

Bibliography:

 

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.

 

 

 

.

Priest (2.1) One, “How Gluons Glue”, summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[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

 

2.1

How Gluons Glue

 

 

 

Brief summary:

(2.1.1) The gluon is the factor that binds parts into a unity. It has the contradictory properties of both being and not being an object. We now will see how gluons bind parts into unities, which involves breaking the Bradley regress. (2.1.2) The binding action of gluons involves non-transitive identity.

 

 

 

 

Contents:

 

2.1.1

[How Gluons Will Glue]

 

2.1.2

[Gluons’ Non-Transitive Identity]

 

Bibliography

 

 

 

 

 

Summary

 

2.1.1

[How Gluons Will Glue]

 

[The gluon is the factor that binds parts into a unity. It has the contradictory properties of both being and not being an object. We now will see how gluons bind parts into unities, which involves breaking the Bradley regress.]

 

[In section 1.3.1 we noted how things have parts that together compose the unity they belong to by means of a factor that binds the parts together in a unifying way. This unificatory binding factor is called the gluon. We discussed the problems with seeing it just as an object-part and with seeing it just as not being an object-part (as for example being instead a relation) (see section 1.4, section 1.5, and section 1.6). As a result, in section 1.6.6, Priest proposed that it is best to hold the dialetheic position that gluons both are and are not objects. In sections 1.3.4 and 1.6.6, Priest noted that the gluon is a contradictory entity, having the contradictory properties both of being an object and of not being an object (and see section 1.3.5). But we still need to explain how the gluon binds the parts into a unity. (On this importance of the how question when accounting for something philosophically, see section 1.5.5.) This is what we now endeavor, which involves us seeing how we will break the Bradley regress (see section 1.4 especially).]

In the previous chapter we saw that there must be something which accounts for a unity composed of parts where one exists – a gluon; and we saw that a gluon may be expected to have contradictory properties. But we have not yet faced the question of how the gluon does its job: how does it bind the parts (including itself) into a whole? Its having contradictory properties does not immediately address this question (though, one might suspect, it is going to play an important role). In this chapter, we look at the answer. The key is breaking the Bradley regress. We will start by seeing how.

(16)

[contents]

 

 

 

 

2.1.2

[Gluons’ Non-Transitive Identity]

 

[The binding action of gluons involves non-transitive identity.]

 

[In order to explain how gluons glue, we will also need to change the logical properties that we normally ascribe to identity, namely, we need to conceive it as being non-transitive. First Priest will give an informal account of gluonic non-transitive identity, and at the end, he provides a more technical account.]

This will immediately launch us into a discussion of identity. Identity cannot work in the way that orthodoxy takes it to if gluons are to do their job. In particular, it must be non-transitive. How so? The rest of the chapter explains, and articulates the nature of gluons more precisely in this theoretical context. The ideas are spelled out informally. Full technical details can be found in the technical appendix to the chapter, Section 2.10, which can be skipped without loss of continuity by those with no taste for such things.

(16)

[contents]

 

 

 

 

 

Bibliography:

 

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.

 

 

 

.