Priest (P6) One, ‘P.6 Dialetheism and the Inclosure Schema’, summary

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Corry Shores
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[The following is summary. All boldface, underlying and bracketed commentary are my own.]

 

Summary of

Graham Priest

One:
Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness

Preface

P.6 Dialetheism and the Inclosure Schema

Brief Summary:

There are paradoxes of self reference. One example is the liar paradox (‘This sentence is false’. Is it true or false?), and another is Russell’s paradox (The set of all non-self-inclusive sets: is it included in itself or not?). Graham Priest uses his ‘inclosure schema’ to describe them, and he calls them inclosure paradoxes. This inclosure schema allows us to depict and understand how these paradoxes fit within a dialetheic logic. The paradoxical cases are not ‘enclosed’ within themselves or within an exterior set. They are ‘inclosed’, being on the boundary shared by both the interior and exterior of the sets, ‘in’ a special place where both sets overlap.

Summary

There is a kind of paradox called the inclosure paradox. The basic idea is that by modifying groups in a certain way, you will get a group that both belongs within itself and outside itself. Russell’s paradox and all other paradoxes of self-reference are types of inclosure paradoxes (Priest xx-xxi). Let us start with a famous and simple example: a barber who only shaves men who do not shave themselves. The question is, does he shave himself? If he does shave himself, then he is not the barber who only shaves non-self-shaving men. So he does not, meaning that he is not a self-shaving man. But by that criteria, he should then shave himself.

But Russell’s paradox is about groups. We will work our way gradually to this broader articulation of the inclosure paradox. So think of the barber as a group of one member. The question is, does the barber as a member of his grouping belong to that grouping or not? We think of him first as just a barber whose job it is to shave men. Then we apply a modification to him. We may think of it as enforcing a law on him that was not previously in place. He may now only shave men who do not shave themselves. So now we have both a category (‘barbers who only shave non-self-shaving men’) and a member (that barber himself), and the two will have to be identical; for, there is only one barber in this town. (If there were two barbers, then each could shave the other and never his own self). So this single member group, the barber himself, now both fits within the group of non-self-shaving men and does not fit within it.

This applies as well to larger groupings where self-reference can create paradoxes. Russell’s paradox is more generally the task of classifying the set of all non-self-inclusive sets: does that set belong within itself or outside itself? Like the barber example, if it does, then it does not, and if it does not, then it does.

Priest’s explanation of inclosure paradoxes is a bit technical, and for me personally it is tricky to grasp. Nonetheless, it is formulated with elegant simplicity. Let us take a look at it. I will take it apart piece by piece, and I invite corrections to my imperfect analysis.

We will work through it with the specific example of Russell’s paradox, then we will see how it can be generalized to other paradoxes of self reference.

The diagram and formulation is called the “inclosure schema”. There are enclosures which include items, and those outside the enclosure are excluded from the grouping. (‘Inclosure’ as we will see is being ‘in’ the boundary where the two groupings meet.) We will call the set in its entirety Ω (omega). In this Russell’s paradox example, it is the set of all sets.

Within it are items, other sets. These sets themselves contain members. These members of course are other sets, meaning that they themselves contain members. One such subset of Ω is x.

Now at this point we mention just one condition for inclusion in Ω and x, which is that they be self identical. We call this condition ψ (psi). So in our notation, ψ(Ω) means Ω is identical to itself (Ω = Ω), and ψ(x) means x is identical to itself (x = x).

[I suggest we here wonder about a possible formulation for this self-identity. We can think of x in terms of its totality, or we can think of it in terms of its plurality. It is both, of course. The totality is no more or less than the sum of the plurality, and the plurality of parts together define the totality. So both senses of x are identical. However, they can also be treated separately. Some sets can include themselves (the totality itself can be included among the plurality of its members), for example, “things”, “categories”, “nameables”, “ideas”, “abstractions”, “demonstrables (things you can point-out)”, “distinguishables (things you can distinguish from other things)”, and so on. The set of all things is itself a thing. If we call the set X, and if in it are such things as a, b, c, …, we would add X to that list. So,

a, b, c, … ∈ X,

but as well,

a, b, c, …, X ∈ X.

There is no paradox here. By the same token, we can consider a set which does not include itself, for example the set of all fruits (here the totality itself is not among the plurality of its members). That set is not itself a fruit. So for fruits a, b, c, …:

a, b, c, …, ∈ Y

but

Y ∉ Y

Here again there is no contradiction.] So we have described one condition or property, being self-identical, ψ. But we note another one, φ. In our Russell’s paradox example here, in order for something to belong to Ω, it must be a set that includes other sets, and we call this defining property of ‘includes other sets’ φ. (More generally speaking, φ is the defining property of Ω, which will be different for each application of the schema.)

This is important, because we will take a subset of x, which we will call y, but we want to be clear first that a certain operation works on the level of these subsets. (In the Russell’s paradox example) when that operation, called δ, is applied to a set, it converts that set in such a way that a subset of it is now a set of sets that do not include themselves. [I think the reason we complicate things now with a subset of x is because we need to distinguish the set of x’s members that includes itself and the set of x’s members that does not include itself. So y seems to be its members excluding x, which then allows us to ask whether or not x itself as a totality (and not the bare plurality of its members) belongs within or outside its own boundaries.]

δ(x) = {y ∈ x : y ∉ y}

So, the reason we speak of a subset of x is because as we will see, x itself is not placed within itself. [To understand why this might be, let us first conduct an exercise that we will then later conduct with Ω. Assume first on the one hand that x is included within itself. That means it is self-inclusive. But x only includes non-self-inclusive sets and thus cannot be included within itself, ultimately making it directed outside itself. If on the other hand it is not included in itself, then it lies outside itself, belonging to another set that includes certain non-self-inclusive sets. There is no contradiction in this case. It seems we conclude that it lies outside itself, because this situation is both a possibility and is also the least problematic option.]

[Let us reflect more on how we can interpret the inclosure schema diagram. We return to our examples above, X is the set of things, and is itself a thing. When we apply an operation making its members things which are not themselves things, X itself now does not lie within itself but with another set which can include non-self-including things. What seems to be important in the diagram above is that the arrow begins from x’s enclosing boundary itself, not from within it. This means that x can be seen as having a bounded limit which can be understood either as containing all its plurality of members including the whole totality itself or otherwise as excluding that totality itself. In this case, the arrow tells us that the boundary itself is directed outside itself, meaning that the set is not included in itself. The exterior encroaches entirely atop the boundary but not over it, and the interior comes up to but not atop of that boundary. If x would be included in itself, the interior would encroach atop but not over the boundary, while the exterior would come right up to it but not encroach atop it. The third possibility as we will see is that the exterior and the interior both encroach atop but not across and over the boundary, meaning that the set itself both belongs and does not belong within itself.]

[[But bare with me please once again. Before we continue, let us think of this diagram in terms of the boundary encroachment idea, as it might give a different visualization that can further help us grasp Priest’s insights. So we begin with Ω and its subset x.

We start first by making the set x be a set of non-self-inclusive sets (but not the one and only such set, just an instance of them, with there being other unnamed instances as well. So outside x can still exist non-self-inclusive sets, with x itself belonging to one of these exterior sets.)

Outside x is the rest of set Ω.

Let us think of these thick borders as representing the set itself in its entirety. The question is, is x included in itself? In that case, we would color the border the same as the set, to indicate that the set itself is enclosed within that set.

Or perhaps x is not included within itself. In that case, we would color the border blue, meaning that x as a totality that is distinguishable from the raw conglomerate of its parts lies outside itself and is included in the rest of Ω.

As we concluded above, it would lie outside itself, because that way it can itself be a set of non-self-inclusive sets without being included within itself.

Now we turn to Ω. It is the set of all sets. Outside it are no sets into which it may be included. We will designate that null space with yellow.

However, it can still be that the null outside encroaches over the border, meaning that Ω would not be enclosed in itself. Next we apply that operation δ to it, and we say that it is now the set of all non-self-inclusive sets. The question is, is Ω included within itself? In that case the border would be colored blue to show that the set itself is within that set.

Or is it not included within itself? In that case the border would be colored yellow to indicate that all within the set are included in the set, but the set itself is not.

This situation is similar to the barber paradox we mentioned above, and below we will look more at this paradoxical structure. But note for now how we saw that in a dialetheic logic, we could say that there is no logical problem here. Perhaps the set of all non-self-inclusive sets both does and does not include itself. We might depict that by shading the border  with green, the mixture of blue and yellow, to show that Ω itself both is found within itself, giving it blue tone, and not found within itself, giving it yellow tone.

Now let us return again to Priest’s diagram, with this possible visualization in mind.]

Not all of x remains in x after operation δ is applied, only subset y remains. Within x are y, which are non-self-including sets. But x itself cannot belong in itself, yet it does not have to, because x does not contain all possible sets, only some of them. So x is placed outside itself [its exterior encroaches atop its boundary but does not cross it]. What is important about y being φ (being a set that includes sets) is that we see a clear case where the operation δ works fine on items which have the defining traits of Ω. The question is, how does it work when we consider larger and larger sets x until they grow to exactly the size of Ω itself (the set of all sets)? [Since the operation works fine in all cases of subsets up to Ω, we would expect it to work on the limiting case of Ω itself, especially since all cases up to the limit share the same defining properties as the limit case itself.]

We apply the operation δ to Ω, and now all sets within Ω are non-self-inclusive sets. What do we say of set Ω itself? Can it be found within itself? If so, then it is a not a self-inclusive set, and thus its boundary is directed outward.

But if it cannot be found within itself, then it is a non-self-including set, qualifying it for inclusion within itself, and thus its boundary would be pointing inward.

[Unlike the case of x, there is no exterior set to which it may belong. So there is no unproblematic situation. It can only be that it both belongs to itself and does not belong to itself.] Priest depicts this double status of Ω by placing a cross on the border, meaning that it falls squarely on the boundary where the two contrary statuses meet.

[[As we noted in the previous section, this is a problem in classical logic, which does not allow for contradictions; but in dialetheic logic there are true contradictions. Russell’s paradox and other paradoxes of self-reference are good candidates for such dialetheias. Here we have situations where all the parts of the paradox make sense on their own, and their combination is made according to the normal rules of language, grammar, and logic. Some philosophers, dialetheists, think that instead of these paradoxes being problematic, they rather are evidence for the claim that there are true contradictions. For, how can the paradox be otherwise than true? Its meaning and formation are clear and proper. Just its logic is in question. Russell’s solution is to legislate that one level of a hierarchy refers to its lower level and not to itself. But language does not work by such rules. Language allows for many grammatically and semantically sensible combinations, and the trouble is found only on the level of logic. Maybe it is wiser to make our logic conform to the real way that language works, rather than insist on making language work in ways that it fundamentally does not, namely to say it cannot make logically self-contradictory combinations, when all its basic principles and mechanisms in fact allow for those combinations. Furthermore, philosophically speaking, these paradoxes, rather than being outlaws to the system, could lie at the very heart of its structure, so rather than exclude them, perhaps they should be treated as our best insights into the deeper structures of significance and meaning.]]

[Jc Beall looks at Graham Priest’s rendition of this enclosure schema as it is in Priest’s Beyond the Limits of Thought. He writes (citing Priest’s aforementioned text) (my comments are in double curly brackets):

The inclosure scheme involves a set Ω, two unary predicates φ and  ψ, and a function δ, where the following conditions are satisfied (see Priest, 2002, §9.4ff):
1. Ω = {y : φ(y)} and ψ(y)

{{The whole is made of subsets that have properties/predicates φ and ψ.}}

2. For any X ⊆ Ω such that ψ is true of X,

{{For any subset X of Ω where X has the predicate ψ, …}}

(a) Closure: δ(X) ∈ Ω

{{… the function applied to X leaves X belonging to Ω, …}}

(b) Transcendence: δ(X) ∉ X.

{{… but the function applied to X also makes it not included within itself}}
In the limiting case, where X = Ω, we have a contradiction: δ(Ω) ∈ Ω and δ(Ω) ∉ Ω.

{{In the largest set X of Ω, X is the same as Ω. In this case, the operation makes the set both included in itself and not included in itself.}}

Example: let Ω be the ‘collection’ of all truths (so that φ is truth). {{Ω is the set of all true matters. The defining property of that set, being true, is the predicate φ.}} Let ψ be the property of being well-defined {{I think this means in this case ‘unambiguous’, so before the function is applied, it is unambiguous, but in some cases that creates an ambiguity}}.  In turn, δ can be thought of as taking subsets of Ω to sentences, namely, the sentence ‘I am not in X’ {{Perhaps the “I” is like a variable into which the members are placed, and when applied to the set X itself, it just means that the set is not included in itself.}} By standard liar-like reasoning, we get a plausible argument for Closure and Transcendence: namely, that δ(Ω), which is the sentence ‘I am not true’ or | ‘I am not in the collection of truths’, is both in Ω and not in Ω. {{Perhaps Beall is saying that the liar sentence ‘I am not true’ can be understood with the “I” meaning not simply that very statement itself but the collection of all true statements. This equivalence between the singular statement itself and the collection of all true statements comes about through a limiting process. The difference would be between saying “I am not in a collection of truths” and “I am not in the one and only collection of all truths.” When we take the partial collections to the limit of the whole, its meaning is identical to the particular self-reference of “I am not true” or “I am not in the collection of truths”, because the “I” can refer to nothing other than that collection of truths.}} And precisely the same sort of situation exists in the case of the Russell-Zermelo-Cantor paradox (concerning all non-self-membered sets/collections, etc).
(Beall 11-12)

]

Priest continues by explaining how the inclosure paradox applies in other cases, namely, König’s paradox

Some of the paradoxes in question are paradoxes of definability. A paradigm of these is König’s paradox. Something is definable if there is a (non-indexical) noun phrase that refers to it. If a is a definable set of definable ordinals, then (since this is countable), there is a least ordinal greater than all the members of a. It is obviously not a member of a, but it is definable by the italicized phrase. Since the set of all definable ordinals is itself definable, we may apply this operator to it to obtain a set that cannot be referred to (defined), but which yet can. In this case, Ω is the set of all definable ordinals; ψ(x) is ‘x is definable’; and  φ(x) is the least ordinal greater than all the members of x.
(Priest xxi)

Priest, Graham. One: Being an Investigation into the Unity of Reality and of its Parts, including the Singular Object which is Nothingness. Oxford: Oxford University, 2014.

Beall, Jc. “End of Inclosure.” Mind (2014) 123 (491): 829-849.
http://mind.oxfordjournals.org/content/123/491/829.abstract
Quotation and page citation taken from the copy provided by Beall at:
http://homepages.uconn.edu/~jcb02005/papers/inclosure-Mind-wrphc.pdf