# The spirit (of our time) is and is not a bone.

[...] it belongs to the weaknesses of our time not to be able to bear the greatness, the immensity of the claims made by the human spirit, to feel crushed before them, and to flee from them faint-hearted (LHP II 10) (1)INTRODUCTION: SLAVOJ ZIZEK AND GRAHAM PRIEST IN AN UNEXPECTED ALLIANCE.

Slavoj Zizek in his second English book, For They Know Not What They Do, has noted the following:

What analytical [sic] philosophy does not know is that it already has what it is desperately looking for elsewhere: its own paradoxes (self-referential vicious circles, and so on) already produce the 'subject', the 'unspeakable' [...] (2)

This comment has been made in reference to, among others, Ludwig Wittgenstein's Tractatus, (3) however, Zizek appears that this point to be unaware that paraconsistent logic had been in development for more than twenty years. In fact logicians like Jaskowski had been developing the ideas of paraconsistent logic in Poland from the end of the nineteen-forties. They developed formal logics (inference systems, that is, theories that allows one to construct valid deductive reasonings) that could handle inconsistencies (4). However, paraconsistent logic on its own didn't lead to new ontological or epistemological status for contradictions, rather it was only with dialetheism that this was the case. Briefly stated, dialetheism suggests that contradictions can be true and as such it is the inheritor of an important aspect of Hegelianism.

In his book In Contradiction, a book that appear some four years before Zizek's For They Know Not What They Do, the dialetheist Graham Priest made the following the rather 'cool, responsible, and fairly technical' statement:

[...] according to Kant, the categories of thought provide a framework for categorising the objects of experience. Thought itself produces objects--objects of thought--but the categories cannot be legitimately applied to them. According to Hegel, it is not possible to stop the categories applying to the objects of thought, and that the attendant contradictions have to be accepted. The logical paradoxes can be seen to make the Hegelian point. (5)

So, what we finally find in Priest's In Contradiction, and he can be regarded as the most clear and far-reaching dialetheist in this respect, is a defender in analytic philosophy of the idea that the only proper way to deal with certain contradictions, as we will try to show, is simply characterized by the following slogan: 'This is it!'.

In this way Priest complies with Zizek's ideal for analytic philosophy. In agreement with this, we will argue that Zizek and Priest are, in an unexpected alliance, two Hegelians who make us 'able to bear the greatness, the immensity of the claims made by the human spirit' (LHP II 10) of our time. These claims can, from a proper Hegelian point of view (see section 9 of this paper), be described as differential and consistent, their greatness as monist and contradictory. Moreover, it is not simply a truth which relies on the authority of the original Hegelian frame of reference, often regarded as modernist and totalitarian; on the contrary, the Hegelian claims are shown to be a step past respectively postmodernist, differential and analytic thinking.

This requires first of all that we outline the Hegelianisms of Priest and Zizek. To do this we stress their formal Heglianism, while contrasting their thinking against two leading figures of what is often referred to as Anglo-Saxon and continental traditions: Bertrand Russell and Jacques Derrida. On the basis of these figures' theories we will show in which way their Hegelianism is, contrary to settled opinions, a step beyond the more recent philosophical theories.

THE SPECULATIVE PROPOSITION: HEGELIANISM AS A CHALLENGE TO POST-MODERNIST THINKING.

Firstly, we return to the above stated point that Priest corresponds to Zizek's analytic philosophical ideal. What must have struck the reader is that Priest's acceptance of contradictions just like that still doesn't show that he believes them to produce the subject or the unspeakable, as Zizek suggests. In fact, he doesn't, as is shown by his comment on Wittgenstein's famous confession in the Tractatus that his statements are senseless.

[...] some have tried to read profound or even mystical significance into the penultimate proposition of the Tractatus. I cannot agree with this line of thought. To admit that one's own view is nonsense is as damning a self-indictment as one can find--particularly when it is clear that the views, however wrong they may be, are intelligible, and so not nonsense. (6)

This reveals an important difference between Zizek and Priest, which comes down to their different uses of the Hegelian frame of reference. Zizek, unlike Priest, assumes what we could call a 'formal Hegelianism', which is best explained using the structure of 'speculative proposition'. (7) Zizek's frequently used example of this is Hegel's famous statement that 'The spirit is a bone', (8) that expresses the last speculative step in Hegel's dialectics of phrenology (PS [paragraph] 336-340). (9) Phrenology was of course a growing science in Hegel's time. But more importantly, it reveals the structure of a speculative proposition which is simply the equivalence of contradictory terms. Likewise, with regard to the above mentioned comment about analytic philosophy, we can read Zizek's speculative proposition concerning analytic philosophy as the following: the release of the transcendental (subject, unspeakable) is nothing but the deadlock of the paradoxes. The form of the speculative proposition is correct, though misleading as we will further see.

Translating the speculative proposition in its proper form will allow us to argue the analogy between Priest and Zizek. It will also allow us to explain the difference between Priest and Zizek, relating to the just mentioned 'speculative proposition', which expresses the Hegelian synthesis. We can already announce that the difference will essentially concern the synthesis. (10) The analogy will also help us to explain the way in which their Hegelian thinking is a step past the dominant forms of postmodernist and analytic thinking.

First, we have to describe what we will consider to be dominant postmodernist and analytic thinking, and why we see it as dominant. The main figures of postmodernism: Bataille, Lyotard, Foucault, Derrida and Deleuze have all taken position against Hegel. They have done that in the name of 'difference'. In general this can be taken to mean the irreducible openness, displacement or violence. Hegel is accused, then, of pacifying the difference, through an ultimate 'Aufhebung' in which the Spirit (the totality) mediates every particularity (every difference). Zizek has been arguing against this view from the beginning of his writing.

On the other side, we have analytic philosophy, which was from the beginning is concerned with mathematical logic. We have to note here that Bertrand Russell and George Moore rejected Hegelianism as a form of holism in the first place (11) but also as a form of contradictory thinking. In his Principles of Mathematics, Bertrand Russell privileges Hegelian philosophy with regard to contradictions when he implies that it is the only doctrine which doesn't need to solve contradictions. (12) We have to add here that the logical paradoxes, that fit the Hegelian dialectical frame, as will be demonstrated further, have lead to theories, constructed in order to avoid them. These we can characterize as 'multiple', such as Russell's theory of types and Zermelo Fraenkel set theory. We will show, by Priest's criticism of one of these theories, the way in which his Hegelian thinking is a step beyond it.

We emphasize the analogy between analytic and postmodernist thinking here. It shows that two things are closely connected: consistency and difference. Analytic philosophy, through its attempt to retain consistency, has resulted in multiple theories while postmodernist philosophy, in its attempt to think the differential, has lead to explicitly consistent systems. (13) These corresponding theories of both 'movements' are what we name 'the claims made by the spirit of our time'. We will now investigate their 'greatness'.

ZIZEK'S FORMAL HEGELIANISM: FROM SPECULATIVE PROPOSITION TO CHIASM.

Let's start with Zizek. When in his For They Know Not What They Do he outlines what is indicated as 'the formal aspect' (of Hegelianism), he makes use of the Kantian concepts of Understanding and Reason:

Reason marks the moment of reduction of all content of thought to the immanence of Understanding. The categories of Understanding 'become fluid', put in motion by dialectics, when one renounces the conception of them as fixed moments, 'objectivizations' [sic], of a dynamic process that surpasses them--that is to say, when one locates the impetus of their movement in the immanence of their own contradiction (FT 160).

When later in the text Zizek explains the movement of Understanding to Reason, in terms of appearance and essence, we read the following:

The crucial point here is how the 'significance' of the essence consists in the movement performed by the subject, in the procedure by the way of which he posits an entity as the appearance of some essence.

To make this clear, we have to reconstruct the dialectics preceding this positing of 'the appearance of some essence'. First of all, we will name the elements in this part of a sentence. The appearance must be regarded in Aristotelian-logical terms as the predicate. The essence as the subject. The predicate indicates a determination in relation to the subject, which is clear by the use of the copula 'of '. Other ways of uttering this relationship is by means of the copula 'is' or 'as' in which cases the position of subject and predicate is exchanged. For example 'the appearance of some essence' can also be expressed as 'essence is appearance' or 'essence as appearance'.

Now we can reconstruct the dialectics: first we have 'appearance': appearance is experienced unmediated here. We are still in Eden's garden, where appearance does not have to be supplemented by something else, it is still undifferentiated. Then we have 'essence of appearance'. There is now some essence, which formally indicates the negativity of the appearance or 'that which is not appearance'. So the second moment is the moment in which a certain negativity of appearance is formulated, outside appearance. The third moment is when we have 'appearance of essence (of appearance)' or, more clearly, 'essence (of appearance) is appearance'. So, appearance reveals itself to be the subject of the predicate that is its own negativity (essence of appearance). Because of the many clauses in this paragraph, we will reconstruct the preceding in a schema. First, we note what some may have noticed, that the reconstruction makes use of the style figure of 'chiasm'. (14) The chiasm is a grammatical operation in which the subject and predicate exchange positions crosswise, we have indicated this with two arrows in the schema:

[ILLUSTRATION OMITTED]

We will now connect this to the movement from Understanding to reason by means of the speculative proposition. Zizek notes about the speculative proposition that it results from the failure of a first, immediate reading (SO 208). The failure is precisely the second sentence. Essence appears only as the negativity of appearance, independent from it. In dialectical terms, this sentence is the antithesis. It shows the predicate as the negativity of the subject. The last sentence indicates that just this negativity of appearance is appearance. (15) And it is precisely this structure that is expressed by the speculative proposition: 'The spirit is a bone'. which in that interpretation comes down to '(the spirit is) a bone is the spirit' or 'the bone (of the spirit) is the spirit. This is the reason why we claimed before that the form of the speculative proposition is misleading. Because it does not show how the predicate is identical to the subject only as a negativity of that subject. And, again in formal terms, this last sentence is naturally the synthesis. It sublates the negativity of the antithesis in the subject. (16)

We are led in the direction of the kantian opposition between Reason and Understanding by the following remark of Zizek: '[The] logic of an object which, by its very inadequacy, "gives body" to the absolute negativity of the Idea, is articulated in Hegel in the form of the so-called "infinite judgment"' (17) (FT 118). The Ideas are in Kant's philosophy the units of Reason. Zizek's remark about Understanding and Reason at the beginning of this section can be stated in the form of a speculative proposition. 'The Reason (of Understanding) is Understanding'. So the Ideas don't put in motion the categories of Understanding from the outside, on the contrary, the categories are put in motion by their own immanent contradiction. This contradiction of Understanding is Reason.

THE PHALLUS OF DECONSTRUCTIONISM: THE TWO HEGELIANS ABOUT DERRIDA

Let's return to Zizek. His Hegelianism is tightly related to the Lacanian psychoanalytic conceptual frame. It may be no surprise then, that central to his conception of formal Hegelianism is the 'phallus'. Zizek refers, in his first English book, to Derrida's critique of Lacan offered in his 'Le Facteur de la Verite'. (18) The critique, in Zizek's words, is as follows:

Derrida repeatedly reproaches Lacan for the paradoxical gesture of reducing lack through its affirmation of itself. Lack is localized in a point of exception which guarantees the consistency of all the other elements, by the mere fact that it is determined as 'symbolic castration', by the mere fact that the phallus is defined as its signifier (SO 154).

Why this is a problem for Derrida is developed a little further, with an immediate reaction from a Lacanian frame of reference:

For Derrida the localization of the lack is supposed to tame the 'dissemination' of the process of writing, while for Lacan only the presence of such a paradoxical 'at least one' sustains the radical dimension of the gap. The Lacanian name of this paradoxical element is, of course [sic], the phallus as signifier, a kind of negative version of 'truth as the index of itself'. The phallic signifier is, so to speak, an index of its own impossibility. In its very positivity it is the signifier of 'castration'--that is, of its own lack. The so-called pre-phallic objects (breasts, excrement) are lost objects, while the phallus is not simply lost but is an object which gives body to a certain fundamental loss in its very presence. In the phallus, loss as such attains a positive existence (SO 157).

As such, the concept of 'phallus' is analogous to the concept of 'Understanding'. Our above treatment of the speculative propostion allows us to easily demonstrate why. In the penultimate quote, phallus is said to be determined as castration, as we saw above (section 3) 'phallus as castration' can be transformed into 'castration of phallus', and this is naturally the antithesis which is sublated in 'castration (of phallus) is phallus'. The analogy with Understanding is obvious now (see section 3).

Zizek adds to this defence immediately a criticism of deconstructionism: 'the poststructuralist poetic style itself--the style of continuous ironic self-commentary and self-distance, the way of constantly subverting what one was supposed to say literally--exists only to embellish some basic theoretical propositions' (SO 155). We will show that these basic theoretical propositions can be termed as analogous to the concept of phallus as we explained it in the preceding paragraph. What is the concept of deconstruction that functions as a phallus, as a central index of its own impossibility? The following criticism sends us in the right direction:

[...] the position from which the deconstructionist can always make sure of the fact that 'there is no metalanguage'; that no utterance can say precisely what it intended to say; that the process of enuncation [sic] always subverts the utterance; is the position of metalanguage in its purest, most radical form (SO 155).

Zizek notes that this 'Nietzschean reference' (SO 154) is totally absent with Lacan. (19) And we add to this that there is a Hegelian reference instead of a Nietzchean one. So what we see here, is something similar to what Stephen Houlgate has argued regarding the relation between Nietzsche and Hegel. (20) In Nietzsche's works we find the criticisms against traditional metaphysics, that we are dominated by the will to power that falsifies the diversity of life and deceives us through our language and that the conditions of life might include error etcetera. Houlgate shows that Nietzsche, in this way, remains trapped in the oppositions he attacks. His views are still presented as the true ones, independent from language and the will to power. What escapes Nietzsche is the self-reference of his own views. His own utterance is arbitrarily excluded from the extension of his criticism. The Hegelian move is of course to make this his own utterance 'for oneself '. This results in the contradiction that the stating of the will to power is the pinnacle of the will to power, that the stating of the diversity of life is the ultimate reduction of the diversity of life and so on. So it is only at the cost of supposing its non-existence that we can state the will to power, the diversity of life, the deceit of language, but this contradicts the statement itself.

We can translate the structure of this move by making use of the chiasm. The move of Nietzsche to Hegel is exactly this, from 'the will to power is stated' to 'the will to power is stated is the will to power'. This is the position of true infinity, as opposed to bad infinity, to which we will return later below. Zizek refers to this notion in his criticism of deconstructionism: 'post-structuralist commentaries often produce an effect of 'bad infinity' in the Hegelian sense: an endless quasi-poetical variation on the same theoretical assumption, a variation which does not produce anything new' (SO 155). What is being repeated again and again is precisely the non-existence of metalanguage. The step from bad to true infinity is exactly the step from antithesis to synthesis, as explained above through the chiasm. So, similar to the move from Nietzsche to Hegel, we can state the move from Derrida to Zizek as follows: from 'the non-existence of metalanguage' to 'the non-existence of metalanguage is metalanguage'. As such, this theoretical presupposition is analogous to the concept of phallus.

But there is a more fundamental theoretical poposition that is related to the method of deconstruction itself. consequently, it can be named 'the phallus of deconstructionism'. In the same way as Nietzsche's criticism applied self-referentially to itself, deconstruction applies to itself. In this way we can understand Zizek's often repeated remark about differance, that it is, as the condition of possibility, (21) the condition of impossibility (22) of an entity. This analysis is to be complemented by Graham Priest's excellent analysis of Derrida. (23) In the article, suitably entitled 'Derrida and Self-Reference', Graham Priest offers a complementary reading of Derrida. At the end of the article, we find a reference to Derrida's own characterisation of difference: 'Differance produces what it forbids, makes possible the very thing it makes impossible'. (24) There is a literal correspondence with Zizek' description of phallus mentioned above: phallus is, likewise, an index of its own impossibility. It is precisely as such that differance is the phallus of deconstructionism.

How can we understand this? To get to this, we have to explain what deconstruction is. Simply stated, (25) deconstruction is the lecture of a text which proceeds in different steps. In the first instance, a central opposition in the text is being located, for example speech and writing. One of the two concepts is always dominant. Deconstruction reverses the order and shows the other, opposite term to be more fundamental to the text than the central term. In the second instance (displacement), a new term, that does not belong to neither of the two opposed terms, is being produced, for example a 'supplement', (26) and consequentially can't be expressed in terms of it. In this way, the opposition is removed. Now, differance is the third term that results from the deconstruction of the whole of language, which is constituted by the opposition between presence and absence. (27) By the definition of displacement though, differance cannot be expressed in language. But it is clearly being expressed. Read in isolation from his other works, this can give the impression that Priest is just stating the usual criticism of logicians on relativism, that it is self-refuting so incoherent. But with Priest, obviously for a dialetheist, inconsistency is detached from incoherence. (28) So the fact that Derrida's statements entail this contradiction puts him in line with, among others, Wittgenstein as we already saw in this article (section 2) and reveals that 'there is something about the very possibility of language that is like this' (BTL 245). As we noted before: 'This (contradiction) is it!'

Below we will work out our engagement on the part of analytic philosophy, to analyse Russell's famous solution for the paradox he found in Fregean set theory, the Russell paradox.

THE STRUCTURE OF THE LOGICAL PARADOXES: THE INCLOSURE SCHEMA.

As we don't expect readers interested in Hegel to know a lot about the history of formal logic, we will start by explaining the Russell paradox. Russell found it, inspired by 'Cantor's proof that there can be no greatest cardinal number'. (29) cantor's construction, called 'diagonalization', plays a very central role in Priest's philosophy, similar to that of self-reference for Zizek. An explanation of this concept in the original context would require too much space, so we will restrict ourselves to an intuitive definition for the moment. Further on, we will put a stress on its function in the Russell paradox, which will result in a metaphorical, though sufficient understanding of it.

To be perfectly clear, we will start from Priest's inclosure schema (BTL 147), that he puts forward as a schema that expresses the structure of all logical paradoxes. It goes as follows:

(1) [OMEGA] = {x; [phi](y)} exists, and [psi]([OMEGA])

(2) if x is a subset of [OMEGA] such that [psi](y): (a) [delta](y) [??] y

(b) [delta](y) [member of] [OMEGA]

If y = U then

(a) [delta]([OMEGA]) [??] [OMEGA]

(b) [delta]([OMEGA]) [member of] [OMEGA]

x and y are sets, [OMEGA] the universal set. [phi] and [psi] are properties. [delta] is a diagonalizing function. Now we can define the diagonalizing function: it is a function which, when applied to a set, gives a value that is not identical to any member of that set (BTL 143). So what is stated in the schema is the following:

(1) The universal set is defined as the set containing any member that has the (1) property [phi], and it exists. The universal set has the property [psi].

(2) If x is a subset of the universal set such that x has the property [psi], then: (2)

(a) the diagonalization of x is not a member of x (b) the diagonalization of x is a member of the universal set So far for the first part of the schema. At this stage there is no problem. There is a value that results from the application of diagonalization to the set x, this member is just another element, so belongs to the universal set. Now we go to the second part of the schema.

If the set we apply the diagonalizer to is the universal set, then:

(a) the diagonalization of U is not a member of U (b) the diagonalization of U is a member of U What has happened in this second part, is simply that we have replaced x, as an argument of the diagonalizing function, by the universal set. Now we clearly have a contradiction because the diagonalized value is by definition not identical to any member of the set it is applied to, but the value that results from it is another set, so it also is a member of the universal set.

In the schema, we see some important elements of Hegelian thinking. Further on we will show the analogy between this diagonalizer and 'negativity'. considered on its own, the schema might look like witchcraft to some or simply like rubbish to others. To prove this impression wrong we are going to show how the schema expresses the structure of the Russell paradox. (30)

THE RUSSELL PARADOX AS AN INSTANCE OF THE INCLOSURE SCHEMA.

We will restrict ourselves to an outline of the proof. Because a technical outline of the proof would require more attention to syntactical requirements than useful and also because the reasoning, leading to the Russell paradox, is generally recognized as sound (within first order logic + the axiom of comprehension).

We start by defining a set Px as the set which is not a member of itself. (31) There is no problem at this stage. Px is not a member of itself, though it is a member of another set, among which the universal set. Note that we have in the very definition of the set the diagonalising function incorporated: the set is defined as a variable which is not a member of itself, Px:

[there exists]x[for all]y: y[member of]x y[??]y

We want to stress that the contradiction is normally derived from the hypothesis that Px is a member of itself, and by showing that it is not (by its very definition). But this does not deliver us a contradiction. It simply shows the hypothesis to be wrong (through the logical rule 'reductio ad absurdum'). To derive a contradiction, we should be able to derive the negation of the negation (32) of the hypothesis (which is the original hypothesis, by the rule 'double negation') from the negation of the hypothesis. But that is not possible, since the set of any member of x which is not a member of itself can still be a member of another set (for example, the set of any member of z which is not a member of itself). So we only have:

Px[??]Px

We only get the contradiction when we have to make a small, though philosophically speaking major modification. We have to define the set of all sets which are not a member of themselves, Pv:

[there exists]V[for all]y: y[member of]V[and]y[??]y

This is the 'totalization' of the diagionalizing function. We enlarge, figuratively speaking, the diagonalizing set to its limit. Likewise, we can derive the negation of the hypothesis. But we can also derive the negation of the negation from the negation, because every set which is not a member of itself is a member of the set of all sets which are not a member of themselves. In this way, we get the conjunction of the two derived sentences.

(Pv[??]Pv) [and] (Pv[member of]Pv)

The last sentence is what it appears to be, the Russell paradox. Resuming, we see the following: we first derive the fact that this set is not a member of itself, making use of the diagonalizing property of the set itself. But then, we derive the negation of the latter statement, making use of the fact that this set comprises all sets which have this diagonalizing property.

We can identify the elements of the Russell paradox now in the inclosure schema.

First, the sets x and y fulfil the same role. Property [phi] is the condition y [member of] x. The property [psi] is the property P. [delta] is the condition y [??] y. [OMEGA] is the universal set V. The first part of the schema just states the definition of Px, namely: (33)

(a) Pxx

(b) Px[member of]V

If we put V in the place of y, we get the Russell paradox:

(a) Pv[??]V

(b) Pv[member of]V

We will briefly show now how the paradox fits the Hegelian dialectical frame before showing how Russell thought to be able to avoid his paradox and how it doesn't work.

THE PARADOXES AS SPECIFICATIONS OF THE HEGELIAN TRUE INFINITE.

It is by means of the Hegelian distinction between the already mentioned bad infinity and true infinity (SL [section]270-304 in BTL 116-120) that Priest explains the analogy with the paradoxes. To explain this, we have to explain the 'limit of thought' or the 5th antinomy, analogous to Kant's, that Priest constructed. It is surprisingly simple and contrary to the original antinomies (BTL 97-106) a decent argument.

To start with, we define the elements of the contradiction as a kantian antinomy. The diagonalizer (34) has generating properties, what means that, when applied to an object, it may produce a new value. In the case of the limit of thought, this diagonalizer is 'thought of x'. By iterating the application of the diagonalizer as often as possible, we become the limit of iteration, in this case 'the totality of all thoughts' (T). As is well known, the dialectical arguments of the antinomies are always double: one for the thesis and one for the antithesis. The thesis states that the limit can be transcended, using the diagonalizer: there still is a thought of the totality of all thoughts. The antithesis states that this transcendence is, in its turn, immanent to the limit: this thought of the totality (of all thoughts) is still a thought, so part of the totality (of all thoughts).

Now we can identify the various elements of the inclosure schema (BTL 143). The sets x and y remain the same. [phi](y) is the condition y[member of]T. The property [psi] is T[member of]T. [delta](x) is the condition t(x). [OMEGA] is the universal set T. So we get:

(a) t(x)x

(b) t(x) [member of] T

If we put T in the place of x, we get the Russell paradox:

(c) t(T) [??] T

(d) t(T) [member of] T

The bad infinite now is a concept of the infinite which inevitably leads to a thought of the finite (BTL 117). In the schema, this is basically the third statement (c). But the structure of the bad infinite also covers the other direction, a thought of the infinite leads to a thought of the finite (BTL 117). In the schema, this is the fourth statement (d). What we eventually get here is what Priest calls a 'flip-flop' which amounts to the following: 'another thought of the finite, if x is a thought of the infinite; another thought of the infinite, if x is a thought of the finite' (BTL 118). (35) This is analogous to the 'badly infinite' position Zizek ascribes to Derrida (cf. supra), but gets a more precise meaning than the one he ascribes to it himself: 'another metalinguistic utterance, if x is an utterance of the nonexistence of metalanguage, another utterance of the non-existence of metalanguage if x is a metalinguistic utterance'.

What is the true infinite then? It is simply an object whose finitude is its infinitude. This object is precisely the totality of all thoughts. Once we consider it as a finite totality, we can apply the diagonalizer to it and get something beyond that finitude t(T)T. But this infinitude is again part of that totality, being of the same kind: t(T) [member of] T. Likewise, in the case of the Russell paradox, the true infinite is the universal set of sets that are not a member of itself. Once we have it, we can say that it is not a member of itself but, as it is defined as the set of all sets that are not members of themselves, it is a member of itself. In the case of Derrida we can say that the true infinite is just the conjunction of the fact that the utterance of the non-existence of metalanguage is metalanguage and not metalanguage. But we will come back to this later on (section 9)

Now we will see how Russell's solution to the paradoxes doesn't work, more specifically, how it only moves the contradiction (instead of removing it).

8. THE THEORY OF TYPES AS THE RELOCATION OF A CONTRADICTION.

Russell's famous solution is known as the theory of orders (or types). It is to deny the existence of the totality, or the universal set [OMEGA]. Russell does this by ascribing to every statement an order which is indicated by an index, for example set1. Variables range only over one order. A variable of the first order could be something like a set1 that. A variable of the second order could be something like a [set.sub.2] that is not a member of [itself.sub.1] and a variable of the third order the [set.sub.3] of all [sets.sub.2] that are not a member of [themselves.sub.1], and so on. The order of a function is one more than the order of the highest order variable that it contains (BTL 150). The value that results from it is of the same order as the function, so one higher than the highest order variable in the range of the function from which it results. This is just a general expression of a diagonalizer ([[delta].sub.2]([y.sub.1]) [y.sub.1]). So the order of the function in the first example we just gave, 'that is not a member of itself', is two, the order of the function in the third statement 'that are not a member of themselves' is three. Now it is not possible for a variable to range over all variables. So there is no totality. This blocks the second part of the schema. What we get after all is the following:

(a) [[delta].sub.2] ([[OMEGA].sub.1]) [[OMEGA].sub.1]

(c) [[delta].sub.2] ([[OMEGA].sub.1]) [member of][[OMEGA].sub.2]

This might look like a nice solution to the paradoxes that are captured by the schema (among which the Russell paradox). It makes the second part of the schema analogous to the first part because the diagonalized value can always be an element of a 'bigger' totality. But, as we noted before, it merely moves the contradiction. As Graham Priest expresses it sharply: 'By his own theory, Russell's theory cannot be expressed [...]; but he does express it [...].' (BTL 154). What he means by this is simply the following:

[...] the theory of orders cannot be explained without quantifying over all functions, and hence violating it. For to explain it, one has to express the fact that every propositional function has a determinate order (BTL 152).

Indeed, Russell has to speak about the totality of propositional functions to state that they all have an order. We get a similar situation as in the case of Derrida, trying to express that there is no metalanguage, though being able to do this just by making use of metalanguage. The bad infinite can be stated in this case as 'another universal function, if x is the statement that there are no universal functions, another statement that there are no universal functions, if x is a universal function'.

It is important to stress an interesting property of contradictions, that is their capacity of being moved. In fact this is a motif in Priest's writings about logical paradoxes: the proposed solutions to the logical paradoxes haven't removed the contradiction, they have just moved them.

Now, any practicing logician will note that the statements underlying the construction of Russell's theory of types are not to be put in the structure that is constituted by it (36). But we have to be very clear about what is being attacked by this argument. Russell's theory of types is an attempt to make a distinction between reference and (linguistic) referent. The distinction between the statements underlying the construction and the construction is a variant of the former. Priest has shown that establishing the distinction between reference and referent is done by a statement violating this distinction. In this way it is abolished. In arguing against this, the analogous distinction between the statements underlying the construction and the construction is indeed of no use.

In relation to this and the already mentioned proposition of the Tractatus, in which Wittgenstein states the senselessness of the preceding propositions, the following is particularly interesting: in his introduction to the Tractatus, Russell already made an observation similar to Priest's (cf. section 2.):

What causes hesitation is the fact that, after all, Mr Wittgenstein manages to say a good deal about what cannot be said. (37)

So Wittgenstein says that his statements are senseless, but they clearly aren't. Now, Russell proposes his theory of types.:

[...] every language has, as Mr Wittgenstein says, a structure concerning which in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit. (38)

Russell is as bright to note that Wittgenstein would object that his propositons apply to the totality of these hierarchies, but Russell says that this does not apply when this totality is assumed to be non-existent. No additional arguments are added, so we can refer to the above argument against Russell's proposal.

9. THE DIAGONALIZING FUNCTION AS NEGATIVITY: PRIEST AND ZIZEK REVISITED

The remaining task consists in explaining how Slavoj Zizek's formal theory relates to that of Graham Priest. More specifically, what is the relationship between Zizek's speculative proposition and Priest's inclosure schema? Therefore we have to return to the speculative proposition. We stated before that it expresses the identity between subject and predicate as the negativity of subject. Here we can rely on the simple analogy of a function to a predicate. (39) The function that delivers the negativity of the argument over which it ranges, sis simply the diagonalizing function ([delta]). The antithesis is expressed then by '[delta]([OMEGA])', this is analogous to 'the bone of the spirit'. The synthesis is expressed by the last sentence of the schema '[delta]([OMEGA]) [member of] [OMEGA]', this is analogous to 'the bone of the spirit is the spirit.

After this transformation, we see a difference between an account of dialectics on the basis of Zizek and an account on the basis of Priest. In Priest's case, the synthesis is clearly the conjunction of [delta]([OMEGA])[OMEGA] and [delta]([OMEGA]) [member of] [OMEGA]. One of the two conjuncts alone will just deliver us the other one over and over again which bring us back to the badly infinite position. With Zizek, only the conjunct a(O) [member of] U is stated and this brings us to the simple objection against Zizek's speculative proposition that the spirit is (also) not its bone, that the essence is also not its appearance, that phallus is also not its castration etcetera or, formal logically that also [delta]([OMEGA])[??][OMEGA].

Strangely enough, our objection against Zizek's speculative proposition sounds very much like the postmodernist criticism against Hegelianism that we mentioned above: there is always a remainder that escapes the sublation. Zizek's own solution brings us at once to an important objection against our own account of his formal theory (see also section 3, footnote 8): the relationship between the two incompatible terms in the speculative proposition is not at all one of predicate to subject. The one term is not subordinate to the other, which is expressed by the relation 'is nothing but'. (40) And as such it is much stronger than the relation of membership (41) ([member of]) that Priest uses. So both the account (of Zizek) and the analogy (with Priest) fail.

The stronger (equivalence) relationship that Zizek uses has the advantage of making him able to counter the standard postmodernist criticism against Hegelianism, because the terms can be turned around that way: The spirit is a bone, but a bone is also the spirit. More generally, the totality incorporates its ultimate negativity, but the negativity also incorporates the totality. In his own words, it allows him to defend:

how meaningless is the usual reproach according to which Hegelian dialectics 'sublates' all the inert objective leftover, including it in the circle of the dialectical mediation: the very movement of dialectics implies, on the contrary, that there is always a certain remnant, a certain leftover escaping the circle of [...] appropriation-mediation (SO 209).

Stated otherwise, his stronger thesis seems to allow him to maintain a certain irreducible difference, likely to that of other postmodernists: every leftover is identical to the statement which it escapes but every statement is also equivalent to its leftover. But drawback of this stronger relationship is that the incompatibility between the two terms completely disappears, so this solution has the problem of logically deducing the equivalence relation from the antithesis.

That's why propose to replace the equivalence relationship by the subordinate relationship between subject and predicate and the analogous relationship of membership. This obliges us to make a modification to the chiasmic schema we presented to explain Zizek formal theory (section 3.):

1. appearance

2. essence of appearance is not appearance

3. essence of appearance is appearance

The difference with the earlier proposed schema lies in the second statement to which 'is not appearance' is added. This makes the last two sentences analogous to the last two of Priest's schema. We remark that this version is also not subject to the postmodernist criticism. As we mentioned before, the essence (of appearance) is also not essence. This brings us to the proposal for a modification of Zizek's famous speculative proposition 'The spirit is a bone': 'The spirit is and is not a bone'.

10. CONCLUSION: THE TWO HEGELIANS AS NEGATIVITY OF THEIR MOVEMENTS.

As we noted, Zizek's Hegelianism is closely related to his psychoanalytic background. Several critics though have blamed him for not being very rigid in his references to Lacan. Perhaps he could be blamed the same in reference to Hegel. We haven't taken up the task to investigate this, for either Zizek nor for Priest. But our design is limited and our central claim to Hegel is minimal: first, we consider Hegelianism as monist, so involving totality (section 1)--this is a point universally agreed upon--secondly, we consider Hegelianism as involving contradictions in the formal logical sense (section 1)--this is a point not universally agreed upon but defended by Priest in his 'Dialectic and Dialetheic'. (42) We refer sceptical Hegelians to that source.

The advantage though of both theoreticians' approach exceeds any hermeneutical essentialism justly considered. It consists in the fact that (what we call) their Hegelianism is uniquely a critical position because they relate, as we have attempted to show in this article, closely to present day frames of reference or, in terms of Hegel himself, to 'the claims made by the human spirit' of our time (sections 4 and 8). In other words, they don't criticize present day thinking from an outside point of view. And this makes their criticism pertinent and a profound challenge to the foundations of postmodernist and formal logical thinking.

Both these foundations are shaken by self-reference and the two figures make diligently use of it in their arguments, as we have shown. Zizek does this on the basis of Hegel's treatment of Kant. As we have illustrated with the infinite judgement, the categories are being applied to themselves (section 3, footnote 9). Priest moves in a different conceptual environment and does this on the basis of logic and set theory. In his case, we have shown by the Russell paradox how the concept of set, established by the axiom of comprehension, is made self-referential (Annex 2, footnote 43). The same thing is necessary to think the greatness of the spirit of our time. It is necessary to think it precisely as that which escapes it. We have to think it as that which, by reference to it (and no alternative to this reference is disposable, since the Spirit in Hegel's philosophy is the totality, there is no outside point of reference) is defined as that which does not belong to it. So the greatness of the spirit of our time is, figuratively speaking, a bone and since it does not belong to it, it is also not a bone. As such this bone is what we hold to be, again in Hegel's words, 'the greatness, the immensity of the claims made by the human spirit' (section 2).

In fact, this is the way the two Hegelians' positions should be considered in the first place: not as Hegelians who each supplement their movements from the outside, from a third, Hegelian position. Their position can on the other hand best be described in terms of the formal frame presented in this article itself, as the negativity of their movements. They have arisen from their movements, and there certainly are pioneers before them (as we mentioned, in section 1), but their theories crystallize the defence of that which was historically radically excluded from their movements (as shown at the beginning of this article, section 2). In other words, they represent Hegelianism within two historically anti-Hegelian movements. If we consider their movements as one totality, by means of the joint characteristics consistency and difference (section 2) and their positions as analogous (section 9), their joint position is clearly that of the bone in relation to the spirit of their time (section 9).

Annex: The Different Elements of the Various Paradoxes in the Article in Terms of the Inclosure Schema. Paradox [delta](y) [psi](x) Paradox [phi](y) (([delta]([omega]) [not member of][omega]) & ([delta]([omega]) [member of][omega])) Derrida A supplement Differance is is not inexpressible and speech nor it is expressible. writing Russell y[??]y ((Pv[not member of]V) & (Pv[member of]V)) 5th antinomy t(x) (t(T) [not member of]T) & (t(T) [member of]T)

Annex 2: The Proof of Russell's Paradox in First Order Logic

We start by defining a set Px as the set which is not a member of itself (43). There is no problem at this stage. Px is not a member of itself, though it is a member of another set, among which the universal set. Note that we have in the very definition of the set the diagonalising function incorporated: the set is defined as a value which is not a member of itself, Px:

[there exists]x[for all]y: y[member of]x y[??]y

Now, we start by assuming the hypothesis that this set is a member of itself:

Px[member of]Px

Out of the definition, it follows that Px[??]Px, so we can state the following:

(Px [member of] Px) [right arrow] (Px[??]Px) (1)

What we have now, is that out of a certain proposition its negation follows. The valid logical rule reductio ad absurdum (RAA) states that if we can derive from a proposition its negation, that proposition is false, so its negation is true:

[logical not](Px[member of]Px) or, what is the same (Px[??]Px)

In this case, we can still hold that Px is not a member of x (Px[??]x), as the set which is not a member of itself can still be a member of another set than itself. The only requirement is that it is a member of (a set which is a member of) the universal set (V)

The contradiction only arises when we take the set of all sets who are not a member of themselves: Pv:

[there exists]V[for all]y: y[member of]V y[??]y

Now, if we consider likewise that the sentence that states that Pv is a member of itself, is this true of false?

Pv[member of]Pv

Just like before, it follows:

(Pv[member of]Pv) [right arrow] (Pv[??]Pv)

Just like before, we have by RAA

[logical not](Pv [member of] Pv) or, what is the same (Pv[??]Pv) (2)

We have derived the latter proposition. Nothing seems wrong. If we reason further from this situation though, by the definition of Pv, as Pv[??]Pv, it follows that Pv [member of] V and so Pv [member of] Pv. By RAA, we can derive Pv [member of] Pv (3)

To resume, we have derived that (PvPv) (2) and that (Pv [member of] Pv) (3), so we can rightly state that:

(Pv[??]Pv) & (PvPv)

The last sentence is the Russell paradox. For people not experienced with logic, the latter reasoning will look odd. For them, we will explain the structure of it a little more. We start by a hypothesis, we derive its negation. Through RAA, we derive the negation of the hypothesis. So, we can say this hypothesis to be false, so its negation to be true. Than we start from this true hypothesis, and we derive its negation, so by RAA we can state this hypothesis to be false, so its negation to be true. As the negation of the first hypothesis is the negation of the negation of the second hypothesis, we have a contradiction.

Finally, we note that by the definition of Pv, we can derive that Pv[??]Pv entails Pv[member of]V and Pv[member of]Pv entails Pv[??]V. Note that we couldn't do this in the case of Px, as there could be others sets than x that are members of themselves. In the case of the set of all sets that are not members of themselves, this is not the case.

REFERENCES

Badiou, Alain, Derrida ou la localisation de l'inexistance, Paris, Ecole Normale Superieure 22 octobre 2005.

Derrida, Jacques, Spurs, trans. Stefano Agosti, chicago, chicago University Press, 1979.

Derrida, Jacques, 'Le Facteur de la Verite', in La carte postale, Paris, Flammarion, 1989, pp. 439-524.

Fraenkel, Abraham e.a., Foundations of Set Theory, Amsterdam, North-Holland Publishing company, 1973.

Hegel, G. W. F., Science of Logic, trans. A.V. Miller, New Jersey, 1969.

Hegel, G. W. F., The Phenomenology of Spirit, trans. A.V. Miller, New York, Oxford, 1977.

Hegel., G. W. F., Lectures on the History of Philosophy, trans. E. S. Haldane and Frances H. Simson, vol. II Plato and the Platonists, 3 vols., Lincoln, University of Nebraska Press, 1995.

Houlgate, Stephen, Hegel, Nietzsche and the Criticism of Metaphysics, Cambridge, Cambridge University Press, 1986.

Moore, G.E., 'The Refutation Of Idealism', Mind, no. 12, 1903, pp. 433-453.

Priest, Graham, 'Derrida and Self-reference', Australasian Journal of Philosophy, vol. 72, no. 1, 1994, pp. 103-111.

Priest, Graham, 'Dialectic and Dialetheic', Science and Society, no. 53, 1989, p. 388-415.

Priest, Graham In Contradiction, Dordrecht, Martinus Nijhof Publishers, 1989.

Priest, Graham Beyond the limits of Thought, Cambridge, Cambridge University Press, 1995.

Russell, Bertrand The Principles of Mathematics, Cambridge: Cambridge University Press, 1903.

Russell, Bertrand 'Introduction' in: Ludwig Wittgenstein, Tractatus Logico-philosophicus, London, Routledge and Kegan Paul, 1922.

Zizek, Slavoj The Sublime Object of Ideology, London, Verso, 1989, 1989.

Zizek, Slavoj For they know not what they do, London, Verso, 1991.

Zizek, Slavoj Tarrying with the Negative: Kant, Hegel, and the Critique of Ideology, Durham, Duke University Press, 1993.

Zizek, Slavoj The Parallax View, Cambridge, MIT Press, 2006.

(1.) Hegel, G. W. F., Lectures on the History of Philosophy, trans. E. S. Haldane and Frances H. Simson, vol. II Plato and the Platonists, 3 vols., Lincoln, University of Nebraska Press, 1995, p. 10 (henceforth LHP).

(2.) Slavoj Zizek, For They Know Not What They Do, London, Verso, 1991, p. 173, n. 27 (henceforth FT).

(3.) It is to Wittgenstein that he explicitly ascribes the themes of the unspeakable and paradoxes a little bit further on in the same book (FT 223)

(4.) But they were modal logics which, like relevant logics, appeared to be mere extensions of classical Russell-Whitehead logic. It was only when the logical rule 'ex falso (sequitur) quodlibet' was dropped from the inference systems that the systems became paraconsistent. Ex falso quodlibet comes down to: a contradiction entails everything, or, inconsistency entails triviality.

(5.) Graham Priest, In Contradiction, Dordrecht, Martinus Nijhof Publishers, 1989, p. 47

(6.) Priest, Graham, 'Derrida and Self-reference', Australasian Journal of Philosophy, vol. 72, no. 1, 1994, p. 110.

(7.) Which is equivalent to the infinite judgment in Zizek's theory. (FT 139)

(8.) Slavoj Zizek, The Sublime Object of Ideology, London, Verso, 1989, 1989, pp. 207-209 (henceforth SO).

(9.) Hegel, G. W. F., The Phenomenology of Spirit, trans. A.V. Miller, New York, Oxford, 1977, [section]336-340.

(10.) Although in relation to Priest, it is improper to speak in dialectical terms of his formal account. He doesn't present it in dialectical terms in the concerned book, but he has done this elsewhere. See Graham Priest, 'Dialectic and Dialetheic', Science and Society, no. 53, 1989, p. 402.

(11.) Moore, G.E., 'The Refutation Of Idealism', Mind, no. 12, 1903, pp. 433-453.

(12.) Bertrand Russell, The Principles of Mathematics, cambridge: cambridge University Press, 1903, p. 105.

(13.) The classic example here is of course Alain Badiou's l'Etre et l'Evenement in which Zermelo Fraenkel set theory is used to establish a differential ontology.

(14.) Zizek, following Andrzey Warminski, mentions this as 'an (chiasmic) exchange of properties" (FT 40).

(15.) This is what is called in 'hegelese' a reflection in itself, stating that essence as the negativity of appearance is appearance. The opposition between appearance and its counterpart is figuratively reflected into appearance itself.

(16.) Important with Zizek is that the emphasis in sublation is not simply on the negativity which is incorporated in the subject, but also in the other direction: the subject is incorporated in its negativity, for example 'the subject is nothing but the impossibility of its own signifying representation' or, similarly, 'the spirit is nothing but its bone'. We will come back to this later.

(17.) We want to stress that the infinite judgment is an illustration of how, as Priest claims (see section 1.), the categories are applied to objects of thought with Hegel. The categories are applied to themselves namely. The infinite judgment (Hegel, G. W. F., Science of Logic, trans. A.V. Miller, New Jersey, 1969.) is a part of the Hegel's theory of judgment whose sub-categories, as Priest observes, correspond to those of Kant (Graham Priest, Beyond the limits of Thought, Cambridge, Cambridge University Press, 1995, p. 115, (henceforth BLT)). It corresponds to Kant's subcategory of 'limitation', the negative judgment preceding it corresponds to the subcategory of 'negation'. The infinite judgment, again as Zizek notes, 'redoubles the negation already at work in the negative judgement, or rather brings it to its self-reference' (FT 118).

(18.) Jacques Derrida, 'Le Facteur de la Verite', in La carte postale, Paris, Flammarion, 1989, pp. 439-524.

(19.) This Nietzschean reference is, of course, present with Derrida. The most clear example of it is his 'Spurs', in which he also criticises Lacan, more specifically his phallocentrism and the related castration (Jacques Derrida, Spurs, trans. Stefano Agosti, chicago, chicago University Press, 1979, p. 61).

(20.) Stephen Houlgate, Hegel, Nietzsche and the Criticism of Metaphysics, cambridge, cambridge University Press, 1986, pp. 38-44.

(21.) This is naturally a reference to Kantian transcendentalism, to which the deduction of the conditions of possibility of knowledge is central (FT 70).

(22.) Among others: Slavoj Zizek, Tarrying with the Negative: Kant, Hegel, and the Critique of Ideology, Durham, Duke University Press, 1993, p. 2 and Slavoj Zizek, The Parallax View, cambridge, MIT Press, 2006, p. 266.

(23.) Priest, Graham, 'Derrida and Self-reference', pp. 103-111.

(24.) Jacques Derrida, Of Grammatology, trans. G.C. Spivak, Baltimore, John Hopkins University Press, 1974, p. 143 in: Priest, 'Derrida and Self-reference', p. 109.

(25.) The deconstructionists, rearing up on their hind legs when reading the following combination of deconstruction and simplicity, we can refer to 'Derrida ou la localisation de l'inexistance', Alain Badiou's fantastic lecture in which he illustrates that underneath the volatile fleeting ('fuidite volatile') of Derrida's writing, there is an authentic simplicity (See Alain Badiou, Derrida ou la localisation de l'inexistance, Paris, Ecole Normale Superieure 22 Octobre 2005).

(26.) This term is often used as a general name of the third term which escapes the opposition of a text.

(27.) Jacques Derrida, 'Structure, Sign and Play in the Discourse of the Human Sciences' in Writing and Difference, London, Routledge, 1978, pp. 280-281 and Jacques Derrida, Positions, trans. A. Bass, Chicago, University of Chicago Press, 1981, p. 27 in: Priest, 'Derrida and Self-reference', 107-108.

(28.) Priest, In Contradiction, pp. 5-6.

(29.) Russell, The Principles of Mathematics, p. 101.

(30.) It is important to add here that, as Priest notes, the paradox was at the root of Russell's own version of the schema (BTL 142).

(31.) This is closely related to a basic axiom of set theory which is the axiom of comprehension, sometimes referred to as the abstraction principle or, anachronistically, as the axiom of separation. It says that every element which has a certain property (in this case, P) belongs to a set (in this case, x).

(32.) This negation of the negation has to be clearly distinguished from the Hegelian negation of the negation. There is no semantic correspondence.

(33.) The attentive reader will remark that the step from PxPx to Pxx and, respectively, PvPv and Pv[member of]Pv to PvV and Pv[member of]V is not being explained. We can refer them to annex 2 for that.

(34.) There is a subtle difference between the major ingredient in the Kantian antinomies, a generator, and that in the paradoxes of self-reference, a diagonalizer, but it is not necessary to explain this in the scope of this article. (see BTL 143, 255)

(35.) In the case of the Russell paradox we get then 'another set that is not a member of itself if the universal set of all sets that are not a member of itself is a member of itself and another set that is a member of itself if the universal set of all sets that are not a member of itself is not a member of itself'. The formulation is too complicated though to be comprehensible in English, that's why it is better expressed in formal logic.

(36.) In fact, this objection has been made to me by my former teacher in logic.

(37.) Bertrand Russell, 'Introduction' in: Ludwig Wittgenstein, Tractatus Logico-philosophicus, London, Routledge and Kegan Paul, 1922.

(38.) Russell, 'Introduction'.

(39.) We note that the copula 'is' in our text is used for a functional relation as well as for a membership relation (see further). An example of the former is ' appearance is essence', which we see an analogous to 'appearance as essence' and 'the essence of appearance' (section 3.), which is analogous to the diagonal functional relation between [delta] and [OMEGA] (section 9). To avoid confusion we have tried to use consequently 'of ' for a functional relation and 'is' for a membership relation, like in 'the bone of the spirit is a bone'

(40.) For example, 'the subject is nothing but the impossibility of its own signifying representation' (cf. supra)

(41.) Fraenkel et. al. remark in relation to this that 'the copula "is" of Aristotelian fame' is also used in the sense of the relationship of membership ([member of]), next to the sense of the relationship of inclusion ([subset or equal to]) (Fraenkel e.a., Foundations of Set Theory, Amsterdam, North-Holland Publishing company, 1973, p. 26).

(42.) Graham Priest, 'Dialectic and Dialetheic' pp. 388-415.

(43.) This is closely related to a basic axiom of set theory which is the axiom of comprehension, sometimes referred to as the abstraction principle or, anachronistically, as the axiom of separation. It says that every element which has a certain property (in this case, P) belongs to a set (in this case, x).

Johan Vandycke

Universiteit Gent

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Publication: | Cosmos and History: The Journal of Natural and Social Philosophy |

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Date: | Jul 1, 2007 |

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