Between the Void and Emptiness: Ontological Paradox and Spectres of Nihilism in Alain Badiou ’ s Being and Event and Graham Priest ’ s One

: In this study, I reconstruct and compare Alain Badiou ’ s Being and Event ( 2005 ) and Graham Priest ’ s One ( 2014 ) , arguing that the ontologies pursued within the two texts are intriguingly analogous in a number of ways. Both Badiou and Priest are committed to thinking through classically ontological problems without denying the validity of the paradoxes they raise; both regard Plato ’ s Parmenides as an early and formative account of these paradoxes; both establish conclusions to the e ﬀ ect that unity – or “ oneness ” – is indeed a contradictory phenomenon; and both, as a corollary of this conclusion, develop frameworks which confer ontological prominence upon the void and emptiness respectively, thereby arriving at what appears to be a shared ontological nihilism. In this comparative study, I not only enquire whether the frameworks adopted by Badiou and Priest really do lead to nihilist conclusions, but also attempt to clearly outline their trajectories of thought for the bene ﬁ t of those who might be tempted to dismiss their seemingly outlandish theses out of hand.


Introduction
William Watkin describes Alain Badiou's Being and Event (1988;English translation 2005) as "a work of unashamed ontology that grappled with the topics of the One and many while straddling the intelligible languages of both continental and analytic thought, using a strict formalism."¹ This description could work, word for word, as a subtitle to Graham Priest's One (2014)the actual subtitle of which is the no less bombastic "Being an Investigation into the Unity of Reality and its Parts, including the Singular Object which is Nothingness." This convergence may come as a surprise, for Badiou and Priest seem, at first glance, to be very different philosophers. The work of the formeralthough increasingly popular in the Anglosphereis rarely studied outside of continental philosophy or comparative literature departments, whilst that of the latterdespite its recurrent engagement with the continental traditionhas been easily subsumed into (if not always willingly embraced by) mainstream analytic philosophy. Yet Badiou and Priest occupy, in their most overtly ontological texts, very similar conceptual territory. Both are committed to thinking through classically ontological problems without either denying the validity of the paradoxes they raise or casting them aside as archaic metaphysical sophistry. Both regard Plato's Parmenides as an early and formative account of these paradoxes, and both deploy in-depth analyses of this text to catalyze their own investigations. Both establish conclusions to the effect that unity, or "oneness," is indeed a contradictory phenomenon. And both, as a corollary of this conclusion, develop frameworks that confer ontological prominence upon the void and emptiness, respectively, thereby arriving at what appears to be a shared ontological nihilism. In this comparative study, I will reconstruct Badiou and Priest's arguments, allowing the parallels and divergences found within their works to alternately illuminate and problematize one another. In so doing, I wish to not only enquire whether their frameworks really do lead to nihilist conclusionsontological or otherwisebut clearly outline their trajectories of thought for the benefit of those who might be tempted to dismiss such seemingly outlandish theses out of hand.

The Platonic Inheritance
The Parmenides is perhaps the most notoriously perplexing text in the Platonic corpus, not only because the arguments pursued therein are complex, but also because many of the rebuttals to Platonism they contain seem inadequately answered or even wholly ignored by the textual Socrates. Indeed, many of the refutations of the Theory of Forms provided in the Parmenides would, famously, be reiterated by Aristotlewho clearly took them to be knock-down argumentsin his own attacks on Plato. Interpreters of the text are therefore tasked not only with decoding an immensely dense and bewildering series of deductions but also with explaining why on Earth Plato would have penned a dialogue in which his own mouthpiece appears to come off so poorly. Neither Badiou nor Priest, however, shy away from this obligation. The ontologies pursued in both Being and Event and One pivot around original readings of the Parmenides, with the authors arguing that the impasse reached by Socrates in the dialogue is not the result of resolvable issues with his own system, but rather symptomatic of ineluctable contradictions at the heart of all ontological enquiry. If we wish to do ontology at all, these readings suggest, we must attend to the Parmenides. Let us heed this advice, then, and briefly recap the text.
The Parmenides opens with Antiphon, an old associate of Pythodorus, recounting to the narrator, Cephalus of Clazomenae, a tale he heard from Pythodorus regarding a meeting between Zeno, Parmenides, and Socrates. Zeno, Parmenides' student, has just composed a treatise upholding the Parmenidean thesis that what exists cannot be "many"; the first part of the dialogue proper is, accordingly, a summary of Zeno's argument, followed by Socrates' rebuttal. As the cogency of Socrates' reply relies upon his Theory of Forms, it is this theory that Parmenides, entering the discussion, goes on to attack. The second part of the dialogue, which is perhaps the best known, contains a series of counter-Platonic arguments, including the so-called "third man regress" later adopted by Aristotle and the notorious objection that "trivial" substances like hair, mud, and dirt could not possibly have forms (which Socrates, bizarrely, concedes). It is the dialogue's third part, however, which is of most interest to our purposes. This part is divided into four "hypotheses on unity," which are themselves divided into several subsections comprising individual "deductions." In the first deduction of the first hypothesis, Parmenides establishes a number of claims concerning the nature of unityspecifically, unity as a distinct essence or form, rather than merely a property or relation. He argues that this unity must neither have parts (as it must be one) nor be a whole (as all wholes are partite) and that it must thus be unlimited, shapeless, unable to be situated in space, neither in motion nor at rest, neither different from nor the same as itself or another, neither like nor unalike itself or another, neither equal nor unequal to itself or another, and atemporal. Reading this chain of apophatic declarations, a reader may feel that the argument is approaching a hazardous vacuity, and indeed, the deduction ends with the conclusion that, given the claims established, unity must have "no share of being," such that it "in no way is."² However, this state of affairs poses a paradox, for it seems that we have been speaking and thinking of this apparently inexistent unity, which must thereforeas Parmenides himself recognizespartake of some kind of (at least conceptual) being. The second deduction, then, proceeds instead from the assumption that unity exists and that it thereby does possess "a share of being." From this assumption, Parmenides establishes that unity, being an existent and singular object, must be whole with parts and must therefore be both limited and unlimited, shaped, situated in space, in motion and at rest, both different and the same as others and itself, both like and unlike others and itself, both in contact and not in contact with others and itself, both equal and unequal to others and itself, and situated in time such that it is both older and younger than others and itself. As one might suppose, many of these arguments are questionable to say the least, making use of several tenuous assumptions (such as the notion that wholes must be partite). However, the important fact is that the two deductions unfurl according to a parallel logic, so as to reach the conclusion that the form of unity, if it exists, possesses contradictory properties. The third deduction of the first hypothesis confirms this conclusion, exploring complexities arising from the idea that unity might shift in time and space between contradictory states.
The following hypothesis pertains to a set of mysteriously named "others," the referent of which is unclear. Badiou, like many commentators, simply takes "others" to mean all phenomena that might participate in unity, whilst Priest regards the term as pertaining specifically to the other forms.³ The deductions under this hypothesis begin from the assumption that these "others" must possess some form of unity, as they are necessarily either parts possessing unity or united assemblages of parts. However, the section closes with Parmenides suggesting that these others cannot have a share of unity, for then unity itself would be split; therefore, "if unity is, unity is both all things and not even one."⁴ So, things are looking bad again for the coherency of Plato's understanding of unity. Quite reasonably, then, the third hypothesis takes up the possibility that unity simply is not. If unity is inexistent, then we know for sure that it neither pertains to nor joins the ranks of the "others." This means that unity is something that we can have knowledge of; that it possesses difference, unlikeness, and inequality with regard to the others; that it must be like and equal to itself; that it possesses largeness and smallness; and that it "must also have a share of being in a way," such that "unity is, if it is not" (this is possibly the least convincing transition of the whole dialogue).⁵ Finally, the fourth hypothesis explores what would happen to the "others" if unity is inexistent. These others, Parmenides argues, cannot be onefor obvious reasonsbut they also cannot be many, for then each of them would possess the property of unity. Thus, "there will be many masses, each appearing one but not being so."⁶ We have reached some tangled depths here, and it may be tempting toas many commentators havewrite off the Parmenides as sophistry or satire. Even if we endeavour to take the dialogue seriously as a philosophical work, it seems immediately clear that the arguments for the ontological paradox provided by Parmenides rely on several idiosyncratic ideas surrounding the theory of forms itselfperhaps most importantly, the notions that existence is a form to be participated in and that participation in and by other forms or objects renders a form or object partite. These assumptions are what lead to the conclusions that, if unity indeed has being, it cannot truly be "one" (as it would then participate in something other than itselfthe form of existence), and that the form of unity's "oneness" would be corrupted if any other object participated in it. One might thus be tempted to suggest that we can do away with this paradox by doing away with the theory of forms or even simply with this particular model of "participation." And, unsurprisingly, many commentators have suggested that this is the dialogue's purpose: to iron out features of the theory of forms that Plato had come to see as problematic.⁷ However, this is not the route that Priest or Badiou will take. Whilst neither philosopher suggests that all of the arguments presented within the Parmenides are legitimate, they both claim that Plato's paradox of the "one and the many" is one with which all subsequent ontologies must grapple.
We will explore the precise ways in which each philosopher pursues these claims below, but it is worth underlining that a fundamental assumption of the Parmenidesand one that will be affirmed by both Badiou and Priestis the notion that unity or "oneness" is coextensive with being itself. We see this in the dialogue's bombastic concluding interchange, in which Parmenides straightforwardly attests that "if unity is not, [then] nothing is."⁸ This idea, as Priest points out, makes intuitive senseto be a thing is surely to in some sense be one thing (One 50)and follows a venerable non-Platonic tradition, with Aristotle writing in the Metaphysics, for instance, that "being and unity are the same and are one."⁹ For both Badiou and Priest, then, the Parmenides is not simply an exercise in thinking the form of unity, but in thinking being-quabeing, or Being as subtracted from beings (the Heideggerian formulation of the problematic is referenced explicitly in both One and Being and Event). The two thinkers are distinctive in the modern philosophical landscape precisely because they regard the paradoxes produced by way of the problem of the one and the many as comprising the heart of all ontological enquiry, instead ofas in much of the early analytic traditionwriting them off as linguistic confusions oras in much post-Heideggerian continental thought attributing them to the human limitation in the face of mystical unintelligibility.

Badiou's Axiomatic Ontology
To reconstruct Badiou's exploration into the Parmenides, we can begin by questioning whether the equation outlined abovethat of unity and beingis indeed correct. Could it not be the case that, as many proponents of the various (Russellian-Wittgensteinian) theories of semantic vagueness might suggest, unity is simply a function of our language or conceptual scheme, such that the property of being a thing is little more than a contingency imposed on the world by the mind? Well, if we wish to make this claim, we shall need to provide an explanation as to what there really is in place of objects and their parts. Given that we are denying objecthood as such, we cannot simply suggest that some illusory objects (e.g. chairs) can be decomposed into more fundamental objects (e.g. subatomic particles), and we cannot claim that there are indeed objects, but that they are perpetually in the process of changing into other objects. Rather, in maintaining that objecthood itself is illusory, at any (to draw on Hilary Putnam's terminology) "level" of our conceptual scheme, we will need to claim that the world in-itself subsists in some kind of Heraclitan perpetual flux.¹⁰ There are two problems with this conclusion. The first is that it is unclear whether the concept of "being-qua-being" is the correct one with which to capture this flux. For it seems likely that we are here, like Plato's famous rendition of Heraclitus, explicitly denying the notion of being-qua-being in favour of a true reality of, for instance, "becoming" (or even "becoming-qua-becoming"); if this is the case, the equivalence of unity and being-qua-being would be maintained.¹¹ The second problem concerns the validity of the denial itself. For if we wish to suggest that all unity is simply an illusion, then we are required either to deny that we do indeed experience unity or to, so to speak, "step out" of the linguistic, cognitive, or conceptual faculties which impose this illusion, both of which seem like difficult or even untenable gestures (certainly Wittgenstein thought that the latter was only possible at the high price of sense itself).
Badiou will respond to this problem as follows: by explicitly equating unityor "oneness," in his terminologywith being-qua-being; by suggesting that "oneness" is indeed presented to consciousness; and then by straightforwardly affirming that the one "is not." Well, it seems that he has run head-first into  8 Plato, Plato's Parmenides, 64. 9 Aristotle, Metaphysics, Book IV, Part 2. 10 On Putnam's understanding of different "levels" of conceptual schema, see De Caro, "Hilary Putnam on Perspectivism and Naturalism" in Knowledge from a Human Point of View, 57-71. 11 Excepting Heraclitus, this philosophical manoeuvre is most notably associated with Nietzsche and, following him, Deleuze. the problem we just located! What legitimizes Badiou to make this claim pertaining to oneness, given that both unity and being are, as he freely attests, "presented" to ordinary consciousness? The answerand one that might seem highly unsettling to those of us with solidly post-Kantian philosophical temperamentsis nothing. "My entire discourse," proclaims Badiou in the second meditation of Being and Event, "originates in an axiomatic decision: that of the non-being of the one" (BE 31). In his study Being and Event and the Mathematics of Set Theory, Burhanuddin Baki associates this axiomatic meta-ontological methodology with the structure of mathematical consistency proofs. In a consistency proof, writes Baki: The proposition is only decided to be true, and that is already enough. … Instead of deriving the proposition from the premise, one derives the statement that the conclusion is consistent with respect to the premises, if the premises are themselves consistent. If the premises are true, then the end result need not be true, unless it can also be proven that its negation is inconsistent with respect to the premises. But the point is that it can be true (emphasis mine).¹² Still, an axiomatic decision being unlicensed does not mean that it is necessarily unwarranted. Why is it that Badiou founds his metaontology with this particular claim rather than any other? Crucially, the thesis that "the one is not" is put forth as an "interruption" of the "impasse" of the Parmenidesan impasse that, Badiou believes, has haunted Western philosophy since its inception. His axiomatic intervention can thus be regarded as motivated by a kind of epistemic pragmatism, which allows him to push stridently past the endlessly cyclical paradoxes of the Parmenidean quarrel by simply deciding on a single line of argument. Now, Badiou's attestation of both the prominence and relevance of the conclusions reached in the Parmenides may seem surprising in light of our above reconstruction of the text. For, as we noted, it seems that many of Plato's arguments rely on his own particular metaphysic and follow rather dubious trajectories even if this metaphysic is granted. So let us set out more clearly Badiou's understanding of the Parmenidean paradox. First, Badiou states that, in our apprehensions of the world, we are clearly presented with oneness (a chair, a cup, a book) but that this oneness is only ever presented in the form of collections of ones, or "multiples," and never manifested in-itself. We are then faced with the choice of whether to affirm or deny oneness. If we wish to affirm oneness, then we are committed to the notion that there is some kind of unity in itself, or being-qua-beingfor, as Badiou writes in the very first sentence of Being and Event, "what presents itself is essentially multiple; what presents itself is essentially one" (BE 23). Yet this is "unacceptable to thought," because we only ever access multiples of ones and never pure onenesseven to say "there is oneness" is a "localization" and thus corruption of the one (BE 24). This, Badiou suggests, is what Plato is getting at in the first hypothesis on unity. However, if we instead assert that there is no such thing as unity, then we are immediately faced with the issue raised by Parmenides in the fourth hypothesisthat the multiples we are presented with can no longer be collections of ones, but will instead be "many masses, each appearing one but not being so." Badiou's insight here is that these "many masses," in order to escape contradiction, must not be multiples of ones, but rather multiples of multiples. He deems this latter kind of multiplicity "inconsistent," as opposed to the "consistent" multiplicity of the former (BE 25). If we were to suggest that there is no such thing as onenessthat there is no being-qua-beingthen we would be committed to the view that there is only pure, inconsistent multiplicity. But how could this be the case? Badiou has already affirmed that all we ever experience is presented consistent multiplicity or multiples of ones (which he will term "situations"). His solution is to propose that "the one, which is not, solely exists as an operation" (BE 24). Oneness is always the effect of an operational "count," which converts inconsistent multiplicity into consistent multiplicity. "Structurization" is an "effect" of the one, but being-qua-being is in itself inconsistent. And yet, as Badiou clarifies in an interview collected in Infinite Thought: The operation is the situation itself. The operation is not distinct from the multiplicity in itself. There is no presentation of multiplicity and the operation. The operation is the same thing as the presentation.¹³ It is for this reason that William Watkin writes that it is useful to think of Badiou's proclamation "the one is not" as translating to "the one is-not": the one, a contradictory phenomenon, both is and is not.¹⁴ Thus, "the one is-not" is not a denial of the concept of being-qua-being, only an assertion that this concept must be an inconsistent one. According to Badiou, Plato was unable to conceptualize inconsistent multiplicity and thus stumbled on the aporia of the fourth deduction, because there was "no form of object for thought" available to him which would be "capable of gathering together the pure multiple, the multiple-withoutone, and making it consist" (BE 34). But how, one might well ask, do we access a "form of object for thought" in which inconsistent multiplicity might remain inconsistent? Even if we accept that this is what being-qua-being comprises, how do we avoid what Badiou terms the "Great temptation" (BE 26), which is to deny that this multiplicity can be signified to thought and thus invoke what amounts to a (negative) theology?
Here is when Badiou makes his famous statement that ontology is mathematics and that it is within mathematics, specifically ZFC set theory, that inconsistent multiplicity can be grasped.¹⁵ This proclamation of Badiou's is renowned and often misunderstood, or at least written off as obviously false. What often goes unappreciated is the genealogy of the claim itself: if we accept that the paradox outlined in the Parmenides is a genuine aporia, and we utilize the axiom that the one is-not as a means of pragmatically progressing from this static point; then, we must find a system capable of thinking inconsistent multiplicity, and one such system is ZFC set theory. Of course, one may challenge Badiou's reasoning at any of these steps, but the point is that there is a cumulative rational structure here; the proclamation "ontology is mathematics" does not come out of nowhere. But the obvious question at this point is clearly: how does ZFC allow one to think of inconsistent multiplicity? Badiou's reasoning proceeds from the fact that ZFC is an axiom system. For, he asks: What is a law whose objects are implicit? A prescription which does not namein its very operationthat alone to which it tolerates application? It is evidently a system of axioms (BE 29).
Although Badiou's assertion that such a conclusion is "evident" may seem unconvincing, his reasoning on this point can be easily explicated. An axiom system "alone avoids having to make a one out of the multiple" (BE 30). Due to its entirely self-legislating nature, it does not have to presume the existenceor, more properly, the beingof any given phenomenon; all it has to do is lay out operative rules to which speculative phenomena must be beholden. In ZFC, this capacity of axiom systems in general is particularly evident: neither sets nor their elements are defined by way of the axioms themselves; the axioms do not legislate the existence of any particular set (with one important exception); the lexicon contains a single, primitive relationnamely, "belonging." Therefore, as Badiou writes, an "explicit definition of what an axiom system counts as one, or counts as its object-ones, is never encountered" (BE 30). For set theory, moreover, the fact that axiom systems can operate in this manner is particularly integral, for it was the intuitive conceptualization of sets as collections of atomic objects adopted by early (or "naïve") set theory that gave rise to the paradoxes of self-belonging famously identified in the early twentieth century. The Axiom of Separation, adopted as one means of circumventing these paradoxes, ensures that a set cannot be constructed by way of a predicate alone, but only determined as a subset of an existing set. This means that one cannot refer to the set of everything possessing a certain property, but can only specify elements in an already existing set that possess a particular property. It is this feature of ZFC to which Badiou is referring when he claims that "every multiple is composed of multiples" (BE 29).
Yet how can it be that all sets must be subsets of "pre-existent" sets, if it is also the case that there are no sets whose existence is legislated by the axioms of ZFC? Or, to put the question in Badiouian terms: "What  14 Watkin, Badiou and Indifferent Being, 8. 15 Badiou has suggested that other mathematical systems would be just as capable as expressing an ontology of the pure multiple as ZFC set theory and goes some way to exploring this claim in his work subsequent to Being and Event. However, this claim is made tenuous, as we shall see, due to certain specificities of ZFC that would need to be shared by alternative systems in order to perform the ontological work Badiou demands. existent is seized upon by the Ideas of the multiple whose axioms institute the legislating action upon the multiple qua multiple?" (BE 57). Well, ZFC does proclaim the existence of one set: the void set, Ø, the set with no elements. With only this set, along with the other axioms, one can construct an entire "universe" of sets: this is the "Von Neumann universe," or hierarchy of well-founded sets, all of which are constructed from the empty set alone. The Von Neumann universe can produce representations of the ordinals, as the void set, Ø, can be taken to represent 0, whilst the set whose only member is the void set, {Ø}, can be taken to represent 1, as so on. To reiterate, however, the existence of anything within such a system is yet to be presumed (including even numerals): the only thing that possesses being in the Von Neumann universe is nothing, which has no being at all. As "the sole term from which ontology's compositions without concept weave themselves is necessarily the void" (BE 57), "the unpresentable is presented, as a subtractive term of the presentation of presentation" (BE 67). This, then, is how ZFC allows us to "think" inconsistent multiplicity, which is impossible to think without imposing oneness: by presenting multiples of multiples, but withholding all being from them. "The one of the count," writes Badiou, "being merely a result, implies as its ghostly remainder the fact that multiplicity does not originally have the form of the one; thus, although "pure or inconsistent multiplicity is … excluded from the whole, and hence excluded from presentation as such," it is also "included as what presentation itself or in itself 'would be', were that which the law forbids as inconceivable to be conceivable: that the one is not, and that the being of consistency is inconsistency" (BE 53). Now, even if one grants that ZFC provides a way of accessing the kind of multiplicity that Plato would have needed to conceptualize the contradictory nature of unity (and one may well not; Priest, as we see, would dispute this claim), it might nonetheless be queried whether its introduction into a contemporary ontology is beneficial. In The Philosophy of Set Theory, Mary Tiles observes: The universe of sets given by the ZF axioms is peculiarly a mathematician's universe. There are no individual objects, such as people, stones or trees, or collections of them in this universe. It is a wholly abstract universe generated, as it were, out of nothing.¹⁶ Is it not the case, however, that a functional ontology should refer at least on some level to a non-mathematical universeto people, to stones, to trees, even? This suggestion seems to be denied in the early chapters of Being and Event, in which it is precisely ZFC's indifference to the external universe, indeed to any stable presence at all, that allows it to function as a direct discourse on being-qua-being. However, throughout his own work Badiou himself appears to forget this point, and very often uses his ontological terminology to discuss political, sociological, or psychological phenomena. Numerous commentators have attempted to clarify this discrepancy, suggesting that Badiou has (for instance) discovered a "structural homology" between the ontological, the mathematical, and the political, but such equivocations are dubiousas we have seen, the referents of ZFC need to be existentially indeterminate in order to inscribe inconsistent multiplicity and thus illustrate the claim that the one is-not.¹⁷ There seems no way to justify Badiou's application of set-theoretical terminology to non-mathematical contexts aside from claiming that its usage is purely metaphorical; equivocation on this point serves only to weaken the metaontological component of his project (I will leave to other commentators the task of interrogating whether Badiou manages to adequately conjoin this indifferent ontology with the phenomenology set out subsequently in Logics of Worlds). And, of course, the objection still remains that Badiou's strict allegiance to the Heideggerian ontological difference renders his ontology little more than a scholastic exercise in circumventing the sophistic confusions of the Parmenides. For if ontology's status as the "science of being-quabeing" (BE 252) can be maintained only at the price of rendering it incapable of discoursing on beings at all, then perhaps it is this status, or indeed the discipline itself, that should be jettisoned. Perhaps, Badiou's real wager is not that the one is-not, but that we can produce a science of being-qua-being and that the labyrinth of paradox into which such enquiries lead us is not a nonsensical or avoidable one. However, if  this is indeed the case, then this wager is not one taken by Badiou alone. Let us now turn to Priest's One and observe an alternative trajectory that may be opened up by such an engagement.

Priest's Paradoxes of Form and Emptiness
Priest is most renowned for his work on paraconsistent logicsthat is, logics in which contradiction does not entail explosionand dialetheismthe epistemological theory that contradictions can be true. It is perhaps unsurprising, then, that when he undergoes an in-depth reading of the Parmenides in the second part of One, he makes the case that the dialogue acts as a means by which Plato could outline the contradictory nature of the forms. Plato, this novel interpretation argues, did not adhere to the law of noncontradictionthe earliest and perhaps most famous formulation of which appears in Aristotle's Metaphysicsand penned the Parmenides to suggest that the forms were indeed contradictory objects, a possibility that has been overlooked as a result of later commentators' latent Aristotelianism. However, Priest intends this interpretation not merely to decode what Plato intended the dialogue to convey, but to speak to and in a sense answer a general problem for ontology. Although he is no Platonist, Priest agrees that in order to explain unity we must have some notion of "form," where "form" is not a transcendental universal but simply "something that binds … parts into a whole," whatever it is that we assume such "parts" comprise (One, p. 9). This allows Priest to (for the time being, at least) remain agnostic on the question of whether unity is a conceptual imposition or real metaphysical phenomenon: even if "form" is but a product of our language or conceptual processing, it must nevertheless be accounted for. However, Priest goes on to claim that the very notion of form, whether metaphysical or conceptual, provides us with a contradiction. He states: [The form] is, after all, something, an object. (I have just spoken about it). On the other hand, it cannot be an object. If it were, the collection of parts plus the form constitute a plurality, just as much as the original. So the problem of binding would not be solved (Priest, One, 9). The paradox outlined above does not dissolve when we abandon the Platonic theory of forms and its assumptions regarding participation. The minute we consider unity as a form in the most abstract sense as a thing, of whatever kindwe run into this issue. Notice that this is a very similar problem to that identified by Badiou, in that it is a contradiction which arises when we try to consider unity-in-itself and find the notion immediately encroached upon by the multiple. Indeed, one might think of Priest as here attempting to affirm oneness and then stumbling across the first of the aporias outlined by Badiouthat we can only ever access multiples of ones and never pure oneness. However, Priest observes that the attempt to conceptualize unity in this context leads not merely to a reassertion of plurality, but to an infinite and vicious regress -Bradley's regress, which asserts that if two objects A and B are of one kind (in this case, objects), and their relation C is of the same kind, then one is forced to posit a new relation between C and AB to explain the original link.
Using his dialethic grounding, however, Priest reaches an ingenious solution to this dilemma: he affirms that the "forms" to which he refers above do exist and that they are indeed contradictory phenomena, being both whole objects and mere multiples of parts. He terms the forms which account for unity "gluons"named, with a nod to particle physics, for their capacity to "glue" parts into unitiesand states that they operate by being identical with each of the parts of an object, including themselves. In order to theorize such phenomena, Priest must adopt a definition of identity that is reflexive (a = a) and symmetrical (if a = b, then b = a), but not transitive (if a = b and b = c, then a = c). This is because the fact that a gluon is identical to all parts of an object does not mean that these parts are themselves identical; if an object has parts a, b, and gluon g, then g = a and g = b, but a/= b. The fact that the gluon possesses contradictory properties means that it terminates Bradley's regress after one iteration: it both is and is not an object, so it does not constitute a second-order plurality. Lest we feel squeamish at the idea of permitting such odd and inaccessible phenomena into our ontological landscape, Priest reassures us that gluons may well be "mental entities" (One, p. 14). This allows him to eschew a potentially problematic attempt to attain a gods-eye perspective external to our unifying perceptions, whilst nevertheless satisfying a potential Heraclitan objector (who could absorb the theory into their own by concluding that gluons are mere structures of thought, perhaps in the vein of Kantian categories).
Gluons provide a means for Priest to develop a theory of identity in terms of relationwhich is to say, what makes a thing a thing. His ontology will be fleshed out, however, with regard to an extended exploration of identity in terms of a "functional expression," or what he, borrowing the traditional medieval terminology, terms quiddity: what makes a thing the thing it is. Priest's theory, however, is that quiddity itself is merely relational: "any object is what it is in virtue of the properties it bears," meaning that "the quiddity of an object is constituted by its locus in a network of relations." (One 172). This suggestion provides a means of theorizing trans-temporal identity: for something to remain the same object over time is simply for the relational network determining it to remain the same (One 177). Of course, the notion that some things have a merely relational quiddity is, Priest concedes, a relatively common one in the history of philosophy. Take a structuralist understanding of mathematics, which many people seem to find quite intuitive: the abstract object "3" is what it is due to "its place in (relation to) the number sequence, being the successor of 2, the predecessor of 4, and so on," such that "any object which related to those things in those ways would ipso facto be the number 3" (One 172). Other theories which deploy the notion of relational quiddity include structuralist linguistics and Leibnizian physics. However, Priest claims that the notion that all things have a merely relational quiddity is "largely unknown" in the history of Western philosophy, despite being common to many Eastern philosophical traditions (One 174).
Whether Priest is correct on this point is a question that I leave the historians of philosophy to answer in full, but some readers might be surprised at the suggestion, given the apparent proximity of the theory at hand to another popular contemporary epistemology. Is Priest's view merely a structural realism, with his agnosticism with regard to realism and idealism corresponding to an equivocation between the ontological and epistemic versions of this theory? Priest claims not. Structural realists, he suggests, "take structure itself to be self-standing" (One 175), whilst for Priest the relations involved in the determination of quiddity are "just as empty as anything else." So: how can structure be empty? Well, we might, Priest proposes, begin by thinking of any given structure as a kind of locus, composed of points (objects) and their relations. Yet if our aim is to demonstrate the emptiness of structure, we should wish to avoid thinking of this locus as determined by the free-standing (e.g. non-empty) objects that compose it (One 189). So instead, we can think of a locus as a set of relation-instances. However, once this manoeuvre is performed, it appears that we are back to square one, given that the relation instances would themselves be free-standing objects, with "a bunch of relations between them" (One 189). So, we must conceive of this new structure in terms of relation instances and repeat the analysis ω times. "In the limit," Priest writes, "everything will have been analysed" (One 191). Lest we worry that this state of affairs appears to preclude the existence of anything at all, Priest reassures us that at each stage of the iteration, what is obtained is not nothingness but simply "a richer structure," such that the result at the limit is Xω, not the empty set. Moreover, if we subject "not the sets, but the relation-instances themselves to the same kind of recursive analysis" (One 191), and adopt a paraconsistent set theory, we may arrive at a model of structure in terms of "purely non-well-founded sets" (One 192). We need not even reiterate this analysis infinitely, since we could take the initial set subjected to analysis to be the universal set of everything, V.
Priest, recall, suggests that the notions of relational quiddity and the groundlessness of relations are unknown to Western philosophy but familiar in Eastern Traditions, and it is at this point that his ontology's proximity to Buddhist metaphysics becomes explicit. The connection is initially established by way of the notion of "interpenetration." Two objects "interpenetrate" if their quiddity is dependent on one another, such as is the case withto use an intuitive examplethe north and south poles. Moreover, every object, in order to have being, interpenetrates with nothing, as "its not being nothing makes it possible to be" (One 180). Priest regards nothingness as a contradictory phenomenon, due to its being both an object (since we can speak of and conceptualize it) and the absence of all objects; if this definition seems close to that of a gluon, then this is because nothing is a gluonthe gluon of itself. Priest therefore agrees with Heidegger's proclamation that, as nothing "is neither an object nor any being at all," it is the case that "for human existence the nothing makes possible the openness of beings as such."¹⁸ Moreover, as interpenetration is a symmetrical, transitive relation, all things interpenetrate with all other things (as if a interpenetrates nothing and nothing interpenetrates b, then a interpenetrates b). This state of affairs is captured in the Buddhist metaphor of the "Net of Indra," a lattice of jewels which is "infinite in all dimensions," and in which each jewel reflects all other jewels, so that the "process of reflection" is itself infinite.¹⁹ One might wonder, at this point, precisely how Priest's affirmation of the emptiness of all objects relates to his theory of gluons. Well, Priest elaborates, if to be is to be one, then "what it is that makes something onenamely, its gluonis what it is that makes it empty, that is, is relatedness to all things" (One 202). The gluon is the essence or "self-nature" of the object, which renders it both one and empty. But, hang onis it not contradictory to suggest that an object's self-nature is what renders it empty, given that self-nature is precisely what the object of empty of? Perhaps unsurprisingly, Priest asserts that this is indeed contradictory. Indeed, it is a contradiction long since recognized within Buddhist metaphysics, with Nagarjuna referring in the first century CE to the "non nature [of objects] that is their nature."²⁰ Just as a gluon both is and is not an object, it both possesses and does not possess essence.

Ontological Nihilism?
We have seen, then, that in attempting to tackle the ancient ontological problematic of being-qua-being, or oneness, both Badiou and Priest have reached a similar, and startling, conclusion: that the essence or condition of being is, for the former, "the void," and for the latter, "emptiness." Does either position amount to an ontological nihilism? Well, let us be more precise about what these characterizations of being-quabeing entail. For one thing, they are not Heideggerian statements to the effect that being and nothingness, in their dependence upon one another, are in some sense equivalent. Nor are they assertions that what there "really is" is nothingness, a claim that, due to the equivalence of being and oneness, would be subject to every problem engendered by the Parmenidean denial of unity. Importantly, Badiou's "void" and Priest's "emptiness" are not equivalent to nothingnessor, indeed, to one another. For Badiou, the void is a strictly mathematical formalization; ordinary philosophical notions of "nothingness," like (in his view) nonmathematical conceptions of infinity, are mere (mis)approximations of what can be nominated only within the strictures of the axiom system. For Priest, nothingness is a separate (if interdependent) phenomenon to being, which interpenetrates with beings as one component of the relational matrix which constitutes their quiddity and therefore renders them empty of essence. For both philosophers, then, naïve ontological nihilism is simply not an option: we are presented with being; we are presented with oneness; there is no escaping this fact (and indeed, those few theorists who have attempted to defend an explicit ontological nihilism tend to take a strictly non-phenomenological approach).²¹ And yet the ostensible alternative optionto affirm the reality of being and of onenessis also impossible to sustain in light of the paradoxes first identified in the Parmenides and relevant, in altered forms, to this day. The only solution is to locate technics of thought that allow one toin an ontological context, at leastembrace these particular 21 See Westerhoff, "An Argument for Ontological Nihilism." It might be argued that ontologies of becoming, which do tend to take a phenomenological approach, must be nihilistic in the sense referred to above. In these cases, the argument is generally that we are not in fact presented with being or oneness, but rather impose these notions for historically contingent reasons. paradoxes: ZFC set theory for Badiou; paraconsistent logics and the dialethic epistemologies they enable for Priest.
Of course, these approaches are not necessarily compatible. Recall that Priest considers and then rejects a model of the structure in terms of well-founded setsthat is, sets composed from the empty seton the grounds that this would imply a nihilism. A Badiouian would object here that the empty set is not equivalent to nothingness and perhaps additionally suggest that modelling structuresomething which is, in the Badiouian scheme, always a product of well-founded, consistent presentationin terms of paradoxical, groundless multiples is the real route to nihilistic thought. Nevertheless, the philosophers' strategies remain paralleled in their shared attempts to eschew pure nihilism whilst nevertheless placing a paradoxical absence of being at the heart of ontology, even if they would disagree on what is required to perform this eschewal.
Interestingly, then, the responses to a canonical, discipline-defining text of Western philosophy found in Being and Event and One end up defending a viewpoint that is articulated in a canonical, disciplinedefining text of one strand of Eastern philosophy: Nagarjuna's Mulamadhyamakakarika. In this dense and complex verse treatise, Nagarjuna writes "those who see essence and essential difference,/And entities and nonentities" do not see the truth taught by the Buddha, who, "through knowledge of reality and unreality … refuted both 'it is' and 'it is not.'" He continues: To say "it is" is to grasp for permanence. To say "it is not" is to adopt the view of nihilism. Therefore a wise person Does not say "exists" or "does not exist." "Whatever exists through its essence Cannot be nonexistent" is eternalism. "It existed before but doesn't now" Entails the error of nihilism.²² Both Badiou and Priestthe former implicitly, the latter explicitlyform conclusions that mirror those of Nagarjuna, but both suggest that the developments in philosophy, philosophical logic, and mathematics that have occurred since the time of the Mulamadhyamakakarika's composition allow for a fuller realization of his ideas.²³ ZFC, for Badiou, provides a discourse of inconsistent multiplicity and thus a means of thinking the statement "the one is-not"; paraconsistent set theories and logics, for Priest, respectively, explain how the structure itself can be empty of essence and demonstrate that the law of non-contradiction is not a requisite for logical analysis. In light of these developments, we do not have to remain agnostic or quietist on the question of being-qua-being. With regard to oneness, we do not have to refrain from saying "exists" or "does not exist": we can say "does-not exist" (or perhaps something less clunky, but with similar implications). And, indeed, it is arguable that an experiential acknowledgement of such contradictions (which is of course very different from an abstracted, theoretical understanding) has long since been regarded by Buddhists as a component or device for the production of enlightenment. Priest goes so far as to claim that the ancient adherents to the Madhyamaka and Huayan doctrines of ontological emptiness would have been "absolutely delighted" to learn of the mathematical legitimacy of non-well-founded structures (One 193).
Yet even this highly caveated entertainment of the notion of ontological nihilism may appear cause for wariness. Nihilism is an extremely disquieting spectre, especially given that its presence in one area of a philosophical system is liable to breed reappearances in others. The retort that references to "the void" and "emptiness" are not means of smuggling in nothingness but rather conclusions emergent from attempts to  22 Garfield, The Fundamental Wisdom of the Middle Way, 40. 23 Badiou himself has not acknowledged the similarities between his ontology and Buddhist metaphysics, presumably due in part to the radical left's distaste for the ease with which Buddhist values have been subsumed into the ideological terrain of neoliberalism, but other commentators have recognised these overlaps: see Boon, "To Live in a Glass House is a Revolutionary Virtue Par Excellence." sustain the co-reality of oneness and multiplicity may not be particularly reassuring to the sceptic, replacing as it does the threat of ontological nihilism with the threat of epistemic nihilism. For if being itself is contradictory, then what are the implications for those areas of thought that we would very much wish to keep free from contradiction: moral values, for instance, or scientific truth?
In Badiou's case, this worry is easily dismissed. His ontology, as we have noted, remains firmly on one side of the ontological difference, so any unexamined extrapolations from the ontological discourse of set theory to epistemic, moral, or scientific discourses would be out of place. Moreover, the decision that the one is-not is the only point in Being and Event at which Badiou explicitly affirms a contradiction: his ontological proclivities in general are not openly dialethic. Indeed, the Event, which is conceptualized as a self-belonging multiple and thus a phenomenon productive of paradox, is "prohibited" by ontology, cancelled out by the numerous ZFC axioms devised to impede its possibility (184)(185)(186)(187)(188)(189)(190). That said, Badiou does wish Events to occur on the other side of the ontological impasse, for beings in the worldor, at least, real decisions to be made which retroactively legislate the taking-place of an Event. It is unclear whether these Events are simply phenomena which, like paradoxical multiples, are impossible and yet (in Badiou's account) productive of Truths, or if they somehow actually are paradoxical multiples; as ever, the slippage between metaphor and application on this point does him no favours. As Badiou's political work is better known than the details of his ontology, I will not rehearse it in detail here, but suffice to say that his Maoist principles are explicated and even justified by way of recourse to set-theoretical terminology, with the proletariat as a multiple that belongs but is not included, the revolution as the paradoxical multiple, etc. Again, there is no problem with the use of this terminology in a metaphorical context, but Badiou often seems to write as if his ontology implies his political commitments, which it could only be argued to do at the price of its ability to confront the Parmenidean paradox. For even if Badiou were to adopt as his ontological discourse of choice a paraconsistent set theory that permitted paradoxical multiples, the Evental multiple could not be a concrete phenomenon in the world, for this would be to inscribe it with being, and thus defeat the original point of the adoption of ZFC.
In Priest's case, things are a little more complicated. Priest's ontology relies, not on an axiomatic assertion, but on a fully fledged epistemologythat is to say, dialetheism. Priest has retorted at length to the claim that dialetheism induces an epistemic relativismas one might imagine, given that the contention seems the most obvious rebuttal of a dialethic epistemology. His arguments against the claim are complex and various, so I will not attempt to capture them here, but the basic contention is that the slide from "some contradictions are true" to "all contradictions are true" makes about as much sense as the slide from "some things are true" to "all things are true."²⁴ Perhaps more intriguingly for our purposes, however, is the fact that Priest not only denies that his paradoxical equation of essence with emptiness induces a nihilism but claims, like Badiou, that his ontology implies ethics and thereby politics. As all things interpenetrate with one another, "there can … be no radical disjuncture of being between myself and others; as such, "interbeing grounds an important solidarity between people" (One 226). Priest adheres to the Buddhist tenet that existence is suffering, along with the notion that the opposite state to suffering is not happinesswhich always requires worldly attachmentbut "peace of mind." In order to cultivate one's own peace of mind, one must attempt to cultivate the peace of mind of others; this is why "emptiness is the ground of compassion" (One 222). There are, however, obvious problems with this ethical schema as a guide for concrete moral action. In its lack of didacticism, it leaves open justificatory space for a wide range of ethical manoeuvres; the notion that a moral act must precipitate a general flourishing of "inner peace" is a vague guide when weighing up various options in real-world moral dilemmas. Moreover, Priest has just affirmed that "all things interpenetrate with all things," raising the possibility of a disquieting flattening of value-priority: do we have as much of an ethical duty to a rock or a table as to a mammal or a human, due to our shared participation in the Net of Indra? Finally, Priest's transition from this ethical program to Marxism is unlikely to convince those anti-Marxist who believe that their preferred ideologies would be more effective at lessening suffering and promoting inner peace (even if they are mistaken).

Competing Fidelities: Badiou or Priest?
There is, we have seen, an affinity not only between Priest and Badiou's interpretations of the Parmenides, but also their answers to the problems the text raises. Where Badiou claims that the one is-not, Priest claims that gluons both are and are not (as to be is to be a thing, such that to be an object is to have being and vice versa). One might think of Priest's gluon as taking on the ontological baggage that Priest ascribes to oneness in general, thereby enabling the presentation of unity that both philosophers agree constitutes our phenomenological condition. It may seem that Priest would reject the notion that the only way to model being devoid of oneness is by way of inconsistent multiplicity, given that the multiplicity of a gluon (as a collection of parts) is that which Badiou would term "consistent." Yet of course the fact that a gluon is simultaneously a whole object renders it inconsistent in a different but resonant manner. Moreover, whilst Priest may not agree that some form of inconsistent multiplicity is necessary to conceptualize the contradictory nature of unity, he certainly thinks it necessary to conceptualize structure devoid of essence. Indeed, it could be argued that Priest's adoption of non-well-founded sets allows him to draw upon a multiplicity even more inconsistent than Badiou's, which builds itself not from the void, but from infinite clusters of speculative sets.
Nevertheless, there are also integral differences between the two philosophers' approachesdifferences which make it difficult to neatly reconcile the ontologies they propose. The most important of these is simply that regarding the applicability, or lack thereof, of the ontology at hand. For Priest, "emptiness" is a quality that pertains to the real objects in the world, or indeed any possible worldprecisely those stones, trees, and people that Mary Tiles affirms are absent from the set-theoretical universe (along with their potentially non-existent other-worldly counterparts, as Priest outlines when discussing his "noneist" theory of modality (Priest, One, xxi-xxii)). These are the objects which possess a purely relational quiddity; whose essence is empty; and which are ensured a status as singular entities by way of gluons. For Badiou, by contrast, ZFC set theory can be a discourse of pure multiple and voida discourse that articulates beingqua-being as suchprecisely because it has no referents, and therefore pertains to but one side of the ontological difference. No ontology expressible in ordinary language, even a paraconsistent one, could operate as this kind of discourse.
We might additionally turn to the related question of the metaphysical commitments of these ontologies. One might start by noting that the mere proposition that philosophy may embark from an ontological, rather than phenomenological or epistemological, basis seems to be a solidly pre-criticalwhich is to say, non-idealistsuggestion. That said, both Badiou and Priest are centrally concerned with explaining how and why unity is presented to thought, a task that most realists would not find compelling or even necessary. Badiou, for his part, calls himself a materialist and has long since positioned himself at odds with the idealists of postmodern continental philosophy, but the metaontology set forth in Being and Event has been alternately praised and lambasted for idealist and realist implications alike.²⁵ There seems no reason, however, why he should be committed to either idealism or realism purely as a result of these ontological obligations. The wager that the one is-not appears at first glance to be a realist one, as it implies that there is something (not-oneness, or inconsistent multiplicity) that escapes from or resists our conceptual resources; however, given that unity's absenceinconsistent multiplicitycan, it transpires, indeed be conceptually accessed by way of mathematics, the statement could well be posed from within an idealist framework. The notion that a ZFC-centred ontology must be committed to materialism due to, as Watkin suggests, the Axiom of Separation's guarantee that sets are sets of pre-existent multiples which are "by definition real" is, moreover, misguided; although the Axiom of Separation posits that all sets are necessarily subsets of sets, none of these sets are inscribed with being and are thus certainly not "real" in the most obvious sense of the term.²⁶ Priest, for his part, attempts in the final sections of One to overturn his previous metaphysical agnosticism and posit a stance that bridges both idealist and realist perspectives, writing: The view that everything is emptyobjects in the world and conceptsis neither … a realism nor an idealism. It goes between the horns of the two positions. Realists were right in seeing that concepts depend for their nature on the world. Idealists were right in seeing that objects in the world depend for their nature on concepts. Both were wrong in seeing the dependence as one-way. (Priest,One,198) Priest wishes to suggest here that the essential emptiness of objects relies on their position in a network of relations that are conceptual as well as objective, but he also wishes to suggest that objects are not merely empty for us, but in themselves. Whether this stance is a dialectical overcoming of only ostensibly incompatible opposites or a sidestepping avoidance of the matter at hand is a question that is up for debate. Either way, it seems that, as per Badiou, there is nothing in Priest's ontology as outlined up to this point that would commit him to this position. An idealist may well contend that our understanding of the relations between objects is always conceptually mediated, meaning that any definition of the essence in terms of such relations must too be conceptual, whilst a realist could maintain that objects can only be empty in themselves because the relations that define them are similarly non-conceptual; both stances seem permissible within the framework hitherto provided.
If it is possible to adapt different metaphysical commitments to both the ontologies we have outlined, then the question remains as to how we are supposed to choose between them, if at all. It is all very well tracing the similarities between Badiou and Priest's stances, but whoultimatelyis right? Well, at risk of equivocation, it seems that any answer to this question will depend on prior commitments not metaphysical, but epistemic. One's preference for one of these ontologies over the other here will rely on one's perspective on what manoeuvres are possible within philosophy. Badiou's theory requires us to axiomatically state an injunctionthat the one is-notand subsequently argue that there are phenomena graspable by way of mathematics that cannot be otherwise thought. Moreover, given that Badiou's perspective is that, even in set theory, we are only privy to the spectre or negative imprint of inconsistent multiplicity, one may well argue that this wager requires us to posit the existence of something that remains despite Badiou's insistence that he is resisting the "Great Temptation"effectively noumenal. The precarity of Badiou's project lies in the fact that one can always refuse to accept his wager that the one is-not; there would be no, so to speak, epistemological repercussions if this were to occur. A Badiouian might suggest that we must exercise fidelity to the Event of the Cantorian revolution, which provided a means of thinking inconsistent multiplicity and thus facilitating the claim in question, but of course, there is nothing compelling this fidelity: that is a crucial component of the theory of the Event. Besides, the idea that the ontology required to accept the concept of Evental fidelity can only be accepted by way of fidelity to an Event possesses a circularity that is uncomfortable, if not necessarily vicious.
By contrast, Priest's gluon theory, if taken in a dogmatic-metaphysical spirit as a claim about the real nature of the world, implies the existence of contradictory objects, a claim which is surely at least as radical as anything in Being and Event. Even if taken as an idealist-friendly statement about the nature of conceptual unification, it implies that thought is productive of ineluctable contradictions, a claim that many would resist. I see Priest's theory as an attempt to resolve the ontological paradoxes addressed by Badiou without accepting the claim that the contradictory nature of oneness can only be conceptualized by way of an axiom system (which, recall, is not itself an axiomatic statement made by Badiou, but an argument from the specificities of ZFC). In a sense, then, Priest does take on Badiou's wagerhe too agrees that the one isnot. Yet this is not, in his scheme, a wager, but an argument, built upon the assumption that contradictions can be true and that the contradictions that arise when theorizing unity possess the same structure as others whose truth dialetheism admits. Perhaps one way of working towards some kind of Badiou-Priest philosophical fusion, then, would be to suggest that Priest, too, is exercising fidelity to an Event: the Event of dialetheism. If we wish to take the Parmenidean paradoxes seriously, our responses to them will be conditioned by whether we, too, pay heed to this opening of epistemic possibility.

Conflict of interest:
Author states no conflict of interest.