Symposium Introduction

The notion that logic is a domain inhospitable to the prevarications of natural languages is an assumption that has long been held by philosophers and literary theorists alike. Of the logician, W.V. Quine once wrote that “he does not care how inadequate his logical notation is as a reflection of the vernacular, as long as it can be made to serve all the particular needs for which he, in his scientific program, would have otherwise to depend on that part of the vernacular.”1 Seeking, for his part, to disambiguate the roles of linguistic structure and the tendencies of speech in sense-making, Paul de Man noted that “rhetoric radically suspends logic and opens up vertiginous possibilities of referential aberration.” 2 Since Aristotle, it might be said, the means by which one determines the truth or falsity of a statement has necessarily excluded terms that sidestep, by virtue of their sheer utterability, the requirement to point decidedly to a referent. Yet what if, in the long history of this inheritance, such terms were never merely unruly elements that qualified for expulsion but, in fact, were inadvertently invited into the house of logic, registering their presence in language even as reason denied their admissibility into its lexicon of existence and possibility?

It is the great merit of Daniel Heller-Roazen’s extraordinarily rich and ambitious book, No One’s Ways, to have not only posed this very question but to have tracked its career through the centuries from the perspective of one such excluded element, the deceptively innocuous particle “non-.” From the earliest constructions of propositional logic, terms such as “non-man” have been consigned to the domain of the “indefinite”: not a name for something, yet not quite the same as a negation either, such non-words have had to inhabit a region beyond the purview of the syllogistic proof, since they invoke neither something nor its logical negation, but rather everything besides the definite thing. Eventually, these indefinite terms acquired a new identity—the “infinite name”—under which the indefinite expanse of possible non-significations is simultaneously refused and affirmed, denied and contained, for fear that the “anything but” would undermine the “everything” that a “thing” can be said to be. Going far beyond a simple reconstruction of the history of logic and its companion disciplines, No One’s Ways proffers a wholly new look at the logics of exclusion from the standpoint of an equivocation that underlies the very construal of that logic. This equivocation, it argues, is traceable from the very first treatises devised for the purpose of securing logic’s borders against the possibility that infinitization would fracture the unicity of the predicative statement. Moreover, inasmuch as we, as speaking beings, are endowed with the sheer grammatical capacity to enunciate the name of anything that is not some definite thing, this threat cannot not persist. For Heller-Roazen, much of the history of philosophy, ranging from Aristotle’s medieval interpreters to Leibniz and Kant and through the German Idealists and their epigones, can thus be recast as a preoccupation with controlling the extrinsic and intrinsic partitioning that the infinite name promises to visit upon what is.

Indeed, Heller-Roazen writes, “whenever thinking encounters ‘non-being’ or ‘non-art,’ the ‘non-event’ or a ‘non-person,’ the question of infinite naming arises anew” (250). The persistence of this question therefore also serves as an appeal to examine the singularity of no one’s forays into other domains, an appeal which is taken up by Julie Orlemanski in the first response of this forum. Reflecting on the possibilities of infinite self-naming that remain just beyond the purview of the philosophical investigations thereof, Orlemanski speculates on the applicability of Heller-Roazen’s study to one such domain: disability studies. Disability per se is not thematized in No One’s Ways, and as Orlemanski points out, Aristotle himself mobilizes the example of sightlessness as an illustration of mere privation in the Categories and the Metaphysics. But disability also falls within the remit of the third species of “non-” terms, specifically indefinite verbs, which do not so much exclude sight as demarcate a limit beyond which the affirmation of whatever is non-seeing or non-recovering is itself without limit. As Orlemanski asks, might disability studies therefore be another as yet unrecovered chapter in the story of infinite naming? Could disability prove to be another refuge of “non-man” where physical existence manages to evade judgment, whether medical, legal or political?

One of the major claims that is reasserted in various ways throughout Heller-Roazen’s book is that the very process by which an edifice has been constructed for logic and in the name of logic itself produces that which requires bracketing out in order for logic to maintain its definition. In De Interpretatione, for instance, Aristotle summons the indefinite name as something he would soon take leave of for the sake of his science; as Heller-Roazen writes, it is as though Aristotle intended for non-words to persist, inasmuch as they sound out the condition of being without perceptible boundary, in the very architecture of certainty, determinacy, well-ordered contrariety, and reference. In other words, indefiniteness serves the science of definition; the science of sense relies, in a sense, on its own admission of non-sense into the territory of sense-making. Moreover, at a crucial juncture in the reception of Aristotle, indeterminate terms come to signify not just a single uncertainty, but an infinite range of excess signification. This leaves its mark all the way through to the twentieth century. As Juliette Kennedy remarks in the second response of this forum, a straight line might indeed be drawn from Boethius, who was the first to bring the infinite into view in the designation of “indefinite,” to Russell, who similarly affirms a range of non-denotable non-designations by virtue of his pinning reference down to a knowledge of the particular. But, for Kennedy, Heller-Roazen’s account thereby also leads directly into the mathematical controversy that was generated from Cantor’s discovery of the transfinite numbers. This was, in her words, a rediscovery of the question of whether objects with infinite names were also legitimate objects of mathematical propositions. In mathematics, actual infinites might be said to behave like the infinite name inasmuch as they, too, make intuitable an infinite array of items that sit within a range of denotability. Unlike the grammarians and logicians, however, Cantor regarded the infinite not as an indefinite morass of ontologies or interpretations, but as complete and ramified into a single ontology. Thus, Kennedy asks, might it not be the case that the centuries-old quarrel between logic and grammar encounters a solution in mathematics—or, conversely, that one of modern mathematics’ most radical advances was made with the help of the very problem that inaugurates the story of logic and its uneasy, fricative relationship with natural language?

Indeed, one of the most intriguing proposals of Heller-Roazen’s study is that logic, language and mathematics are not just proximate domains but share permeable borders across which they can, given the opportunity, affect one another. At stake in this notion is not just that modern mathematics, in virtue of its new valuation of its symbols, begins to behave like a natural language, but that mathematics changes the very nature of symbolization, language’s position in relation to thought, and therefore naming itself. As Markus Hardtmann remarks in the third response of the forum, it is no accident that the variable gains in importance in the history of logic just as the infinite name recedes from view. Switching the focus from ω to x, Hardtmann argues that Frege’s introduction of the function-argument analysis of predication represents a turning point in the history of logic, not least because the statement “there is an x, such that not f(x)” is simply not mappable onto “x is a non-man”; in modern logic, notation takes leave of grammar altogether. The x in modern formal logic is no longer the x of classical logic, for the specific reason that it no longer needs to refer to the meaning of a specific term, a res, or thing; in other words, x is no longer a name but a variable. And in the event that x denotes “any” term without ranging across the border into the domain of “everything” a thing is—simply because such a border is no longer relevant in the context of a function—are we not, Hardtmann asks, in need of revisiting what it is we might still mean by name, or sign, or indeed signification?

The question of what is left of the name after nominalism is also the question that, for Heller-Roazen, inaugurates the defining event of contemporary philosophy, namely the “parting of the ways” of logical positivism and phenomenology circa 1929, which is the moment with which No One’s Ways culminates. At the same time as Carnap and others were occupied with emancipating logic from the constraints of natural languages, Heidegger sought a “prelogical foundation” (227) in the consciousness that makes predicative assertion possible in the first instance. And though Heidegger, like Carnap, remained unconcerned with the destiny of the grammatical particle “non-“, this very insensitivity on the part of philosophy demonstrates, for Heller-Roazen, that the afterlife of the particle would hereafter persist in language. In the fourth and final response of the forum, Eleanor Kaufman explores the ramifications of this insight for our understanding of how the particle ne operates in Lacanian ontology inasmuch as it appears in language in a negative yet non-negating function. This ne, Kaufman notes, is prelogical yet discursive; it operates in speech, not on the level of the unconscious. Indeed, inasmuch as it articulates negation it seems to undermine negation per se; the Lacanian ne has, in this sense, a unique approximation to the real. Might there be an affinity between Lacanian non-being and the non-words uncovered by Heller-Roazen in his history of the infinite name? And if so, might this imply that there is something like an “unconscious” to logic—that the “unconscious,” too, is an infinite name, something non-identical with itself that needs to be summoned in order for discourse to constitute itself?

Tracking down the multifarious ways in which non-being makes an appearance within the logical and discursive structures that have been erected on the promise of its disappearance might seem like an infinite task but is precisely the achievement of No One’s Ways. As the following exchanges attest, the history of this apparently unremarkable fact of natural grammar named “non-” has proven to be precisely the incitement for thinking in the name of which Heller-Roazen introduces his project. I hope you will agree.

  1. W.V. Quine, “Mr. Strawson on Logical Theory”, in The Ways of Paradox and Other Essays, Revised and enlarged edition. Cambridge, MA: Harvard UP, 1966, p. 150.

  2. Paul de Man, “Semiology and Rhetoric,” in Allegories of Reading. New Haven: Yale UP, 1982, p. 10.



No Body’s Ways

Daniel Heller-Roazen has written a mesmerically lucid book about a disturbance at the interface of logic and language. The tremor is perceptible across the surface of the Western philosophical tradition, from Aristotelian syllogistic theory to logic’s transcendental refashioning and onward to the present. This is the disturbance effected by the negative particle—in English, the non—and by the act of tracing a limit that the particle makes possible. Non-man touches everything excluded from the species. Non-seeing is the attribute of whatever, existent or inexistent, does not see. Non-mortal names a restricted yet inexhaustible domain of conceptual possibility. The indeterminate reference of such designations has garnered them the title “infinite” starting with Boethius, who called them nomina infinita. Infinite names might have remained a largely untested reflex of grammar, if not for the systematic bent of logical and metaphysical theory. After Aristotle’s introduction of them, philosophers have periodically returned to plumb the difference that the non makes, and Heller-Roazen tracks their efforts.

One of the pleasures of No One’s Ways is the contrast between the topic’s delicate austerity—a minor and glancingly understood philosophical theme—and the book’s virtuosic ranging through the plural traditions of Western logic and metaphysics. It is the work of a scholar singularly possessed of philological and conceptual brilliance, the clarity of whose prose becomes ever more spectacular as the complexity of the ideation mounts. This will not surprise readers of Heller-Roazen’s previous titles for Zone Books, each of which traces a particular topic across an array of discourses while turning exacting details toward the revision of our accounts of language, consciousness, the human, and form’s power. Similarly, No One’s Ways pursues its subtle quarry, the indeterminate name, through the various regions of philosophical understanding.

As a scholar of medieval thought and an enthusiast of philosophy more generally, I read No One’s Ways with uninterrupted delight. As a scholar of literature, I found the promise of the book’s first chapter tickling the edges of my concentration during the rest, the fourteen chapters wending through their eclectic way over the breadth of the Western philosophic tradition. But the first chapter is different. It retells Odysseus’s escape from the Cyclops Polyphemus, “a brute who killed his guests and ate them raw” (7). The wily hero famously tells his captor, “Outis is my name. My father and mother call me / Outis, as do all others who are my companions” (7). Once Odysseus and his men manage to gouge out the Cyclops’ eye and flee, Polyphemus calls for help. But at this point, the name’s “two syllables fall apart,” and “Oūtis, in the Cyclops’ mouth, becomes oú tis, ‘no one’” (8). The term cancels and nullifies the assertions that Polyphemus rages to make. Heller-Roazen unfolds several meanings for the hero’s self-canceling name to arrive at “the most literal, if barely grammatical, sense, as the refusal of the indefinite article ‘a’ or ‘one’: as ‘not-any,’ ‘not-one,’ or ‘non-one’” (9). Heller-Roazen continues:

Everything follows from this act of language, which constitutes the most minimal and extreme of affirmations, as well as the most far-reaching and excessive of negations. . . . This act is simple yet profoundly perplexing in sense and effects. In it, an entire company of linguistic shadows comes loose. They are the most indeterminate of spoken beings: meanings more tenuous and more fleeting even than that of being “a” or “one,” for that, precisely, they are not. (9)

Odysseus’s self-naming is the tactical, not systematic, activation of a possibility of language, deployed to survive, to keep his and men’s bodies from becoming meat. Or: the poet uses it to glorify speech over brute strength and to carry out the weaving of poetic narrative. Are the “company of linguistic shadows” this name unleashes the same shadows that Heller-Roazen tracks through the philosophic tradition?

Yes and no, the final chapter of No One’s Ways suggests. The ultimate chamber in the book’s architecture is stocked with the “grammatical” interest, or the fully connotative and cultural implications, of the first, in contrast to the philosophical restraint of the rest. In concluding, Heller-Roazen takes the step of formulating something not stated outright in philosophy’s corpus of indeterminate names, although, as he says, it “remains obstinately audible from text to text” (253). This is the idea that “the ‘infinity’ of non-’s sense is our own” (253). And so the first Aristotelian example of the indefinite name, “non-man,” is decisive. It marks the distinctly “human possibilities of non-being, inexistence, and privation” (254, my emphasis). It is here that No One’s Ways points toward the future for which it has been prolegomenon. What Heller-Roazen calls “our speaking nature” “has still not received the attention it demands” (255). The central matter, he urges, lies less in our faculty of understanding than our faculty of naming, availing ourselves of language, and doing so necessarily without having first traversed the sense and entailments of what we say. The possibilities of infinite self-naming “have hardly come into focus in the philosophical investigations of infinite naming,” and the philosophical tradition “fails to confront the crucial question that it raises,” especially through its examples and illustrations (257). “Today,” Heller-Roazen writes, the “non-subject” of Aristotle’s first foray into infinite naming “calls for our study” in other domains—in “literature and law, in anthropology and mythology, psychoanalysis and linguistics,” where “No One’s masks”—Odysseus’s ruse no doubt among them—“demand to be examined, one by one, in their disquieting force” (257–58).

Another such domain may be disability studies, or critical scholarship on how corporeal and cognitive variation become stigma and impairment. Having myself worked in the history of medicine, the “history of the body,” and historical disability studies, I wonder about nobody, non-body, and the corporeal possibilities for the logical language of absence and cancellation. “Disability,” after all, is a term that tarries with the negative. Its particle “dis-” refers to what is parted, sundered: “Latin dis- was related to bis, originally *dvis = Greek δίς twice, < duo, δύο two, the primary meaning being ‘two-ways, in twain.’”1 Etymologically, what is disabled is parted from capacity, from its suitability to purpose, from fitness, aptitude, proficiency. The current meaning of disability has arisen from a two-step historical process: first, the social-scientific invention of “normality” and “abnormality” starting in the late eighteenth century and, then, disabled persons’ activist reclamation of the category from the 1970s onward. Studying disability in earlier periods entails finding its synonyms, which together constitute a catalogue of negative particles: what is infirm, unnatural, misshapen, deformed, disfigured, invalid, diseased, abnormal, nonfunctional, deviant, disordered, inoperative.

In terms of Aristotelian logic, these terms are privative. As Heller-Roazen observes, Aristotle uses three kinds of logical terms for “three conditions of non-being” (34). The first is simple negation—“A is not B”—where the “not” modifies the copula “is.” The second condition is privation, and to explain this, Aristotle uses the example of sight: “Privation and possession are spoken of in connection with the same thing, for example sight and blindness in connection with the eye.” True privation concerns “whatever the possession naturally occurs in. . . . For it is not what has not teeth that we call toothless, or what has not sight blind, but what has not got them at the time when it is natural for it to have them.”2 The logical category depends on knowing what “naturally” ought to be present and the “time when it is natural” to possess whatever attribute.

Aristotle returns to the example in the Metaphysics where he differentiates the sightlessness of plants, “which would not naturally” have vision; the sightlessness of the mole, whose species was thought to lack eyes “in contrast with its genus”; and the sightlessness of the blind man, “in contrast with his own normal nature.”3 The unnaturalness of the privation, its character of deviation and violent loss, is meant to grow over the trio of sightless animals, and it is the blind man whose sightlessness is privative in the strict sense laid out in the Categories. To clarify the point, Aristotle’s early commentator Alexander of Aphrodisias introduces the rather strange example of a sightless wall. The negative predicate “does not see,” Alexander writes, “is true both of a blind man and of a wall, which is not at all receptive of sight; for the negation is true of an indefinite number of things.” However, “blindness is not this way: for it is only true in the case of things that have been deprived of sight.”4 Later, Alexander resumes: If “a wall and plants are said to be deprived of sight,” then “a privation of this sort would be equivalent to a negation.”5 As Aristotelian logic is taken up by Arabic philosophers, the elaboration of privation in corporeal terms continues; Averroes lists the attributes “bald,” “blind,” “naked,” “poor,” “squint-eyed,” and “cripple-bodied” as privative.6

The logical category of privation, then, demands adjudication between absence and loss, between the regular unfolding of natural form and its failures. A statement may be grammatically affirmative—“She is blind”—while still being privative in meaning. In the often exquisite abstraction that logic enjoys, it is startling to come across a basic classification that depends so completely on the contingent determination of normalcy. As Heller-Roazen notes, language “furnishes no unequivocal criterion for deciding whether a given word expresses a negative or a privative attribute” (35). The decision for privation comes not from the grammatical makeup of the sentence but from the social norms that undergird the semantics of language.

And yet there is a third condition of non-being, signaled by what Aristotle calls indefinite terms, and it holds out other possibilities. When Aristotle characterizes indefinite verbs in the Categories, he turns again to the body: “‘Non-recovers’ and ‘non-ails’ I do not call verbs. For though they additionally signify time and always hold of something, yet there is a difference—for which there is no name. Let us then call them indefinite verbs.”7 As Heller-Roazen writes, “Were one to hold to the philosopher’s example of ‘seeing,’ ‘not seeing,’ and ‘blindness’ to designate respectively a property, its negation, and its privation, a correspondingly indefinite expression would take the form of an attribute such as ‘non-seeing’ or ‘non-blind’” (35). To call someone non-seeing or non-recovering is to affirm a predicate itself constituted by drawing a limit, by excluding sight, excluding healing, and the domain of affirmation is itself unlimited. Heller-Roazen writes of the indefinite name, “Whatever the domain of the things that one refers it to, be it massive or minute, real or imaginary, the nomen infinitum in the Aristotelian-Boethian sense inscribes its cut in it” (72). Non-seeing “does more than evoke an indefinite expanse of signification. It also fractures it” (72–73).

Thus, it is not only to impugn the negative particle’s semantics of “deflation and deprecation” that disability studies might turn to it, not simply to parry the specter of the disabled non-man, a “ruined man, or the ruining of ‘man’ as such . . . an individual perilously close to man: a failed or ruined man, who by misfortune or fault, has somehow proved himself to be unworthy of his own name” (256). There are also poetic possibilities in the philosophical exercises Heller-Roazen gathers, something for the poiesis of disabled self-naming and world-building, in terms of an infinitized disability. The mole, the plant, the wall, the man: all fall within a sphere of non-seeing. The infinite name provides a rhetorical alternative to the currently favored figure of posthumanism, the list or “Latour litany” as Ian Bogost calls it: “floodlight, screen print, Mastercard, rubber, asphalt, taco, Karmann Ghia, waste bin, oil stain. The Latour litany gathers disparate things together like a strong gravitational field.”8 Such a list “reveals a few unfamiliar corners of being’s infinity through naming.”9 But infinite naming does something different, operating not through metonyms of “being’s infinity,” but via a predicate that traverses being by making its cut. If at the end of No One’s Ways Heller-Roazen wishes to seek out further archives of “non-man,” one might also pursue novel occasions and ways of saying “non-seeing,” “non-recovering,” “non-able,” “no-body,” that is, of taking an infinitized disability into our speaking nature. What is fractured in infinite names is not natural form, or the arc of regular biological development, but the unsorted domain of entities, real and merely possible. The bracing abstraction of logic may have something to contribute to the radical poetics of corporeal variation.

In closing, I note another moment in No One’s Ways when the body and its determinations come to the fore. Perhaps the most “disability studies” moment in the corpus of G. W. F. Hegel occurs near the end of the section on “Observing Reason” in the Phenomenology of Spirit, when reason is at “its worst,” Hegel says, as it endeavors to apprehend the human via psychology, physiognomy, and phrenology.10 This nadir is also where Hegel’s protagonist, Spirit, confronts the “infinite judgement,” or the category of “sentence that ‘collapses into itself,’ expressing the ‘complete incommensurability of subject and predicate,’” as he explains in the Encyclopedia Logic of 1830.11 The infinite judgment is where Aristotle’s indefinite names have arrived by the nineteenth century, on the twisting course that Heller-Roazen tracks. Judgment meets its collapse in the opposition between simple physiological determinism and the self’s mobile responsiveness. This takes the form of antagonism between expert and embodied subject, doctor and patient; as Hegel ventriloquizes the phrenologist: “I regard a bone as your reality.”12 But it also takes place as an unsustainable contradiction in thought itself. The collapsing proposition is exemplified by the claim that the “being of spirit is a bone.”13 To think, or to undergo, such an infinite judgment, on Hegel’s account, entails not only a rejection of diagnostic authority, but a rejection of the very form of judgment, which “seeks to define the thinking self by means of the duality of subject and predicate,” as Heller-Roazen explains (188). For Hegel, Spirit’s fundamental power lies in its imaginative criticism of the present as part of the emergence of the future—which is to say, in negation. In the end, “Observing Reason” does not by any means reject embodiment, but it dismisses the idea of the body as an inert and unyielding fact. Hegel insists that far from being defined by static materiality, this self delivers “a box on the ear” to would-be experts and recalls in the very failure of the infinite judgment its power to “produce itself by its own activity.”14 Disability studies’ rejection of the “medical model” of the embodied self is here the figure and means to a new form of speculative thought.

The non, the dis, the infinite judgment, the indefinite name: these may be means for speaking new corporeal personae into existence. No One’s Ways captures the strange speculative power of such terms. These may turn out to be good for eluding danger, evading contemporary vignettes of shipwrecked precarity and physical fear, where the cleverness of knowing one’s way around no body’s ways might produce surprising escape.

  1. OED online, s.v. “dis-, prefix” (June 2017).

  2. Categories 12a. Aristotle’s Categories and De Interpretatione, trans. J. L. Ackrill (Oxford: Clarendon, 1963), 33.

  3. Metaphysics 1022b. The Complete Works of Aristotle, ed. Jonathan Barnes (Princeton: Princeton University Press, 1984), 2:1615.

  4. Alexander of Aphrodisias on Aristotle’s Metaphysics 4, trans. Arthur Madigan (Ithaca, NY: Cornell University Press, 1993), 122.

  5. Alexander of Aphrodisias on Aristotle’s Metaphysics 5, trans. William E. Dooley (Ithaca: Cornell University Press, 1994), 104.

  6. Epitome of the Metaphysics I, 47; cited from H. A. Wolfson, “Infinite and Privative Judgments in Aristotle, Averroes, and Kant,” Philosophy and Phenomenological Research 8 (1947) 177.

  7. 16b; translation cited from No One’s Ways, 19.

  8. Ian Bogost, Alien Phenomenology; or, What It’s Like to Be a Thing (Minneapolis: University of Minnesota Press, 2012), 49.

  9. Bogost, Alien Phenomenology, 55.

  10. Hegel, Phenomenology of Spirit, trans. A. V. Miller (Oxford: Clarendon, 1977), 206.

  11. Cited from No One’s Ways, 178–79.

  12. Hegel, Phenomenology of Spirit, 205, emphasis original.

  13. Hegel, Phenomenology of Spirit, 208, emphasis original.

  14. Hegel, Phenomenology of Spirit, 209.

  • Daniel Heller-Roazen

    Daniel Heller-Roazen


    Response to Julie Orlemanski

    Learning that one’s book has met with a sympathetic reader is a rare and felicitous event, yet discovering that it has met with a reader so generously discerning as to put it to a new and compelling use is an occurrence still rarer and more felicitous. For me, Julie Orlemanski’s response is an example of just such a happening. Orlemanski has not merely been willing to follow through the steps in the argument and demonstration of No One’s Ways with patience and exactitude; she has also shown how they open to an important domain of study to which they hardly allude. Recalling the terms of the book’s unresolved conclusion, Orlemanski points out that the “‘non-subject’ of Aristotle’s first foray into infinite naming” demands scrutiny not only in literature and law, anthropology and mythology, psychoanalysis and linguistics, as the book’s last chapter announces. “Non-man” also calls out for exploration in disability studies, and more particularly, “critical scholarship on how corporeal and cognitive variation become stigma and impairment.” Orlemanski thus sets the book’s project in a new perspective, illuminating threads that run through the chapters while raising a far-reaching question that the book itself does not broach.

    “‘Disability,’” Orlemanski writes, “is a term that tarries with the negative. Its particle ‘dis-’ refers to what is parted, sundered. . . . Etymologically, what is disabled is what is parted from capacity, from its suitability to purpose, from fitness, aptitude, proficiency.” To what degree are the “negative” senses of the dis-, she asks, those of the non-? That non- constitutes a persistently equivocal word-part or particle is a guiding motif in No One’s Ways, and one of the book’s principal theses is that this minimal operator of linguistic “in-definition” poses a challenge to the apparent unity of the concept of negation. Throughout the book, I seek to show some ways in which non-being is, as Aristotle might have put it, “said in many ways,” including in the very expression “non-being,” which points to such modes of difference as opposition, contradiction, falsity, and impossibility. As Orlemanski observes, the sense of non- that comes closest to that of dis- in disability is the one that Aristotle identified as signifying lack or, in his word, “privation.” It is irreducible to negation. From treatise to treatise, commentary to commentary, Aristotle and his commentators restate this fundamental distinction: while a negation simply denies that some predicate belongs to a subject, a statement of privation does more, indicating that what has been denied of a subject might also have been affirmed of it, that the predicate, in other words, is “missing,” being somehow present even in absence.

    Orlemanski has rightly drawn attention to a central feature in the ancient theory of privation: the logical and metaphysical accounts of lack find their canonical illustrations in human disability. Alexander of Aphrodisias thus explains that a negation has the logical form of a proposition such as “The wall is not seeing,” which denies to the wall the property of vision. A privative statement, by contrast, possesses the logical form of a proposition as “The person is not seeing”—in other words, “He or she is blind.” “As Aristotelian logic is taken up by Arabic philosophers,” Orlemanski reminds us, “the elaboration of privation in corporeal terms continues; Averroes lists the attributes ‘bald,’ ‘blind,’ ‘naked,’ ‘poor,’ ‘squint-eyed,’ and ‘cripple-bodied’ as privative.” To this degree, the twelfth-century Arabic philosopher remains faithful to his source. In both Categories X and in the Metaphysics V, Aristotle exemplifies his theory of privation by appeals to variations in the human body, variations which he judges to be declinations with respect to a single state of perfection. What is “invisible” is not so much without color, he explains in the Metaphysics, as what is missing the proper color, having a “poor” one; in the same way, what is “footless” either “has no feet at all or . . . has imperfect feet”; and people called “sightless” are not so much “the one-eyed” as the ones who, being “sightless in both eyes,” are called “blind.”1

    “Privation” names a condition of metaphysical restriction, in which a lack appears as such. For this reason, I would wager that privation also constitutes a category of what one might call meta-metaphysical restriction: in it, non-being appears to assume a single shape, being tethered to the idea of a plenitude. For Aristotle, the plenitude is often that of the human body. Aristotle repeatedly takes the most common bodily forms of a given species to instantiate a rule, even as he represents the spectrum of differentiation as a set of variegated “deprivations.” Although he himself does not linger on the exact status of the examples that he draws from the field of anatomy, his presentation may well commit him to an implicit doctrine of disability. Each of its elements is well worth questioning. What is the status of the link Aristotle establishes between the metaphysics and logic of privation, on the one hand, and, on the other, variations in human bodily conditions? What, more generally, is the place of human anatomy in the theory of the relative non-being that is privation? If one moves from the theory of privation per se to the analysis of bodily forms, one may also ask: On what grounds does Aristotle conceive of bodily variation according to a single form of privation? To what degree can Aristotle’s account of life also admit the existence of differentiation without “deprivation”?

    Aristotle’s remarkable attention to the Greek lexicon of privation indicates that he was well aware of a major difficulty, which is worth recalling. If he must pause to explain the proper meaning of the word “invisible,” it is, he suggests, because an ambiguity threatens the clarity of the term. Implicitly, Aristotle admits that such a privative expression might adhere to the pattern illustrated by such words as “footless” and “eyeless,” each of which admits of several interpretations: being without one of two parts or organs, for example, or being without them both, or being with them, albeit in some fashion that alters their customary form and function… Simply put, there are many ways to be lacking. Even if one restricts the meaning of non-being to “being without,” or “missing what ought to be present,” the senses of absence can proliferate.

    If, following Orlemanski’s suggestion, we reread the classical example of “non-man” from the perspective of disability studies, there is, therefore, every reason to ask: To what degree is the disabled body a subject of the doctrine of infinite naming and judging? Yet I think one may also take a further step. “Non-man” may be read in such a way that the “non-” appears as the mark of disability, without, however, bearing exclusively on the human body. There are also non-corporeal privations, which diminish the status of a human life without necessarily implying a representation of a regular or irregular corporeal form. I am alluding to the set of instituted reductions of the person that have been called “civil disabilities.” In the book on which I have been working since finishing No One’s Ways, which is in some sense the second part of this project, civil disabilities and the stigmas that they induce play a central role. They bring into existence a social and legal person to which the logic of infinite naming seems to me uniquely apt: someone of whom one may say neither, “It is a person,” nor “It is not a person.” The civilly disabled constitute a major and persistent example of the ouk-anthrōpos, non-homo and “non-person.”

    The Roman jurists, the authors of the most systematic and influential of ancient legal systems, conceived of a practice by which “non-persons” in such a sense were to be brought into existence. They gave it the name of capitis deminutio. A literal translation would be “decrease of the head.” Yet caput, “head,” stands less for the anatomical part, for which the Romans possessed the word testa, than for the “living human body,” and, by extension, the individual being: in legal parlance, the capacity to become the subject of civil rights and prerogatives. That a “head” could be “decreased” meant that civic status could be formally diminished: a persona could be disabled, according to several degrees. The compendium of ancient Roman law known as the Digest sets forth the theory of such reductions. When the head is decreased such that status within the family is lessened, while rights pertaining to liberty and citizenship remain intact, the deminutio is least. When citizenship is suppressed, the decrease is moderate. When liberty is withdrawn from a man, finally, the reduction is most extreme: “maximal.”2 Then a free Roman citizen would become a slave.

    The institution of slavery is perhaps the clearest and the most extreme example of civil “disabling.” It plays a crucial role in classical Roman law. In the Latin code, slavery appears at the foundation of the law of all persons, which opens with the difference between free persons and slaves. Civil law begins, in other words, with the positing of persons defined by privation. Incapable of entering into contracts, possessing nothing of their own, being without any official name other than their masters’, Roman slaves are less “persons” in the usual legal sense of the term than “things”: more exactly, chattel. They have “no head,” as the Institutes establish.3 Yet there is still more, so to speak, to the deprivation of this legal condition. Slavery, for the Romans, is “without duty,” in the sense that citizens can owe slaves nothing. “There is no obligation towards the person of the slave,” as a classical adage has it.4 When a judgment is declared against a servus, it is therefore judged void of effects. Some Roman jurists thus allege that “the slave is akin to the dead.”5 Others maintained that a slave is precisely pro nullo: “as no one.”6 He or she is a human being whose condition draws close to that of a non-human. Yet one might also reason that this state is precisely that of a human being—and nothing more. Homo, in classical Latin, means not only “human being” or “man,” but also “slave.”

    Even where medieval legal theory and practice attenuate or ignore the institution of slavery, the possibility of civil disability remains present. As examples, it suffices to evoke the status of those persons variously exiled, deported, or banished from the societies of which they were members. Such individuals, in the Middle Ages, are often could to be treated by the law as dead, even when, in fact, they remained alive. Excommunication provides another case, as fascinating as it is troubling: What becomes of the person sundered from Christianity by such a ban? With the emergence of mendicant orders, further cases arise. Francis bids his brethren to abandon all property, including that held in common; through adherence to his rule of life, they are thus to dwell in “highest poverty.” That resolution raises new problems concerning the legal status of such monks. In the fourteenth century, Bartolus of Saxoferrato proposes treating friars, legally speaking, as “disabled persons” (incapaces), even as he admits that, upon joining the order, they retain the ability to bequeath their possessions to those still living. Baldus of Ubaldis, his contemporary, offers a different account of this state of monastic deprivation: he reasons that, in joining the brotherhood, the monk renounces not only worldly goods but also the “person” that lays claim to rights. He induces his own legal diminishment.

    In the European and American legislations of the modern age, the forms of civil disabling are legion. The most extreme case is perhaps that introduced by the canon lawyers, “civil death.” By this institution someone is considered to be “dead to the law,” even while alive. This legal state was famously defined under the Ancien Régime, but it outlasted the French monarchy. Under the Terror, Anne Simonin has written, “a civilly dead man’s goods are confiscated; his marriage is dissolved; his legitimate children are treated as bastards; he loses his aristocratic titles; he can have no heir, nor receive any inheritance. Infamous, he is deprived not only of the exercise but even the enjoyment of the set of his civic and civil rights.”7 Such provisions were once known to many European civil codes. Even in the United States, where civil death was never formally prohibited, many states passed special civil death statutes. As late as 1939, eighteen states contained such provisions.8 The laws of civil death in the United States justified neither the seizing of property of the condemned nor the blocking of the transmission of his possessions; yet they allowed a living convict’s properties to be passed on to his heirs, as if he had died, when he was in fact alive.

    By the mid-twentieth century, many of these provisions were repealed or voided. Yet a number of the consequences of such civil diminishments persist in the United States. Today, conviction for a crime or misdemeanour still results in the entry of a person into a state of civil “privation.” In many jurisdictions, when someone is sent to prison for the commission of a crime, his person is irreparably tainted. As Itzkowitz and Oldak note, “the ex-convict who has completed his prison term is subject to a variety of civil disabilities that remain in force for the rest of his life.”9 The most striking among such “civil disabilities” concerns what is often viewed as the democratic right par excellence:, namely, suffrage. It was estimated in 2004 that as many as four million adult citizens of the United States were indefinitely barred from voting because of a criminal conviction, regardless of whether they had been or were incarcerated.10 United States law also lessens its subjects’ civil abilities in further ways. Following conviction, a non-citizen may be deported; criminal registration and community notification may be obligatory; an individual may be judged ineligible to work, reside, or be present in a certain location; and such a person may be subject to occupational debarment or deemed ineligible to initiate or hold family relations.11 Such “collateral consequences” of conviction form a contemporary complex of civil disability.

    Each of these examples merits a study in itself. I enumerate them to indicate some of the ways in which one “non-person” of infinite naming and judging remains a persistent and troubling reality. In the new book project I am now concluding, I have sought to treat some of these varieties of “non-personhood,” attending to the their structural features. What joins No One’s Ways to my current project is the thesis that the “non-” of “non-person” remains irreducible to any single sense of “negation,” whether it be that of opposition, contradiction, impossibility, or privation. In the reading that seems to me most provocative – and also closest to ordinary speech – a “non-person” is a person in exactly the sense in which a “non-event” is an event, or a “non-entity” an entity: it is, in other words, a person deemed to fall below or beneath the threshold necessary for fulfilling the category it would instantiate. Of such a “non-person,” one may state, for this reason, neither that “it is a person,” nor that “it is not a person.” It is, each time, someone in whom the possibility of membership in the community—be it legal, social, or political—becomes an urgent and unavoidable question. Orlemanski has drawn attention to some of the ways in which the reality of physical disability weighs upon the classical theory of the non- of logical and metaphysical privation. Her insight deserves to be unfolded in all its implications, and with attention to the entire range of human forms of non-being that we hastily, and all too indefinitely, call “privation.”

    1. Metaphysics Delta, § 22, 1022b33–1023a8.

    2. Paul, Sabinus, book 2, in Digest, IV 5.11.

    3. Institutiones, 1.16.4.

    4. Ulpian, Sabinus 28, in Digestum 50.17.22 pr.

    5. Ulpian, Lex Julia et Papia 4, in Digestum 50, 17, 209; cf. Novellae 22.9; Gai Institutiones 3.101.

    6. Ulpian, Edict 60, in Digestum 28.8.1. pr.

    7. See Anne Simonin, Le Désohonneur dans la République. Une histoire de l’indignité (Paris: Grasset, 2008), 319.

    8. See “Civil Death Statutes—Medieval Fiction in a Modern World,” Harvard Law Review 50.6 (1937) 968–77.

    9. Howard Itzkowitz and Lauren Oldak, “Restoring the Ex-Offender’s Right to Vote: Background and Developments,” American Criminal Law Review 11 (1973) 721–70, here at 721.

    10. Alec C. Ewald, “‘Civil Death’: The Ideological Paradox of Civil Disenfranchisement Law in the United States,” Wisconsin Law Review (2002) 1045–132, here at 1046.

    11. Gabriel J. Chin, “The New Civil Death: Rethinking Punishment in the Age of Mass Conviction,” University of Pennsylvania Law Review (2012) 1789–833, here at 1790.



From “Non” to “Trans”

Noun terms in language refer—and yet sometimes it seems they do not. It is difficult to explain how reference happens; how a speaker’s utterance, coming from “inside,” manages to pin down an object “outside,” even in the very simple positive cases, such as “The apple on the table is wholly red.” The problem of how to account for reference when negated terms are involved, for example when the term “non-thing” is substituted for the variable X in the proposition “Any X is not a man,”1 and beyond that the problem of how to treat judgments involving such terms, has always been, and remains, highly contested.

In No One’s Ways, Daniel Heller-Roazen tracks the philosophical career of the particle “non-” from its beginnings in the logical literature of the classical period to its treatment by the language philosophers of the twentieth century, when a certain essential, language-sensitive, philosophical discussion around the metaphysics of indefinite terms2 went, in Heller-Roazen’s view, to ground.

It is a history of an idea, the idea of “non-man”; but at the same time, it is a chronicle of a deep, long-lasting and largely unresolved puzzlement, if not outright confusion, over how to treat negated or indefinite terms in language, terms that cause language to rupture and collapse. And there is no therapy in view. In that sense what No One’s Ways draws out for us is how the workings of a single particle, “non-,” throws the speaking self into view, as well as into its own view.

Speakers following the rules of grammar wander easily into these unworkable linguistic domains. “This sentence is false” is syntactically correct and yet it has no truth value, being neither true nor false. The sentence “All unicorns are white” has the same form as the sentence “Every non-thing is not a man,” except that now the filament-like term “non-thing” is slotted into the subject position (where one expects to see an existent) and the predicate appears in negated form. Should a general theory of propositions treat these sentences in the same way?

What is the negation of the sentence “Ten men rowed across the lake”?3 Nine men rowed? Nobody rowed? It is difficult to frame a theory of the proposition when what we can say so very often outstrips our capacity to make sense of what was said. “Philosophical analysis must go further than grammar,” Heller-Roazen observes (38). And indeed, if nothing else the language of negation is a tremendous source of philosophical opportunities, perhaps one of the greatest that has ever presented itself to the philosopher.

No One’s Ways is an astoundingly rich and multifaceted work, but I will confine myself here to Heller-Roazen’s treatment of the discovery of infinite names on the part of Aristotle’s fifth-century commentators, namely their realization that the infinite is buried in ordinary language in the form of terms such as “non-man.” Specifically, they noticed that the range of the term “non-man” is infinite, including “absolutely all things,” with the exception of “man.” The discovery appears at the end of the shift in focus from Aristotle’s analysis of indefinite terms towards a more expanded framework designed to handle all propositions involving such terms. In particular, an indefinite term such as “non-man” is now thought of as a name:

Aristotle himself had not explicitly treated the question of whether a term such as “non-man” signifies “the particular existence of any thing” or a “non-existence.” . . . Recalling the distinction between the implications of negation and privation,4Ammonius resolves an old problem by a single gesture: he sets infinite names among negations. He then introduces a new specification. Whereas an ordinary negative term merely denies a certain quality, a “transposed” designation . . . has a double function. By its “non-,” it first “destroys one thing.” Then, from that initial elimination, it “introduces” a panoply of non-existences: “everything beside the definite thing” that has been evoked. For Ammonius, an indefinite name thus un-names and names at once. . . . “Non-man” . . . can then designate “absolutely all things,” with the exception of “man.” (39–40)

Heller-Roazen points out that (contrary to appearances) Ammonius viewed the range of the name “non-man” as restricted to the genus of which man is a species: thus, including “horse, dog, goat-stag and centaur,” but not pear, spoon, or piety. It is precisely here that the problem with “non-man” emerges, the problem of whether the term should indicate the full “contrary” (as I will discuss below), or whether “non-man” should indicate a class of things nearby, so to speak, while still not falling under the predicate “man.” Boethius went further in attaching a genuinely infinite object to the name “non-man”:

With a terminological decision whose consequences were to resonate in Europe for centuries, Boethius forges a single Latin expression for Aristotle’s “indefinite” words. . . . Boethius translates the Greek aoriston, “indefinite,” by the Latin infinitum, adding, by way of explanation: “for it signifies many things, indeed infinite things.” (41)

It was an innocent move, made in passing, but it had monumental consequences, to wit: the infinite comes into view, simply through an idle running of language, the language of indefinite terms. “To utter an infinite name” such as “non-man,” Heller-Roazen writes—a “double act of meaning” that both annihilates and resurrects in a single act of thought—“is . . . both to refuse a finitude and to arm the infinitude that exceeds it” (43).

Boethius makes an interesting distinction at this point. The term “non-man” can be admitted into grammar; but because it fails to “signify definitely,” because the term fails to possess a “well-delimited meaning” (est circumscriptivae significationis), “definitely” signifying whatever it is that it designates” (41), “non-man” fails to be a name for the philosopher.

This view keeps together what would nowadays be separated into two separate theories of reference, the descriptivist (possessing a “well-delimited meaning”) and the causal (“signifying definitely”). According to Boethius the utterer of “non-man” is in no position to have knowledge of the particular of which “non-man” is a name—if “non-man” denotes a particular at all. In that sense he did not stray very far from the more or less empiricist view of these things laid down by Aristotle, and which remains influential:

To understand a name for a particular the only thing necessary is to be acquainted with that particular.5

It is interesting to track the troubled emergence of the mathematical infinite against the narrative of No One’s Ways. Of course, mathematicians have always been deeply engaged with the infinite. But it was only in the 1870s that infinite totalities were introduced as numerical entities, as numbers in the usual sense. This step was taken by Georg Cantor and it was met with a virulent response at the time, and indeed the status of the mathematical infinite remains controversial among mathematicians.

No One’s Ways leads straight into the mathematical controversy. Following Boethius centuries later, Cantor introduced infinite names into mathematics, reviving the fifth-century debate over whether or not objects with infinite names are legitimate objects of the, now, mathematical proposition.

Even across many centuries their arguments turn on the same sort of issue: determinateness, acquaintance and the question of what constitutes an object of thought. Albert the Great had asked whether infinite names signify “any particular substance with quality” (55), that is, whether such names ought to be admitted into logic; and the thirteenth-century philosopher Simon of Faversham argued against admitting infinite names on the basis of their indeterminateness:

Note that the infinite name and the infinite verb are excluded from the consideration of the logician because the name and verb which the logician considers should be parts of the proposition. But infinite names are not parts of the proposition, because everything that can be a part of the proposition must signify some concept of the mind, for the proposition is principally for the sake of truth. But we cannot have truth except through that which expresses a definite concept. For [the infinite name and verb] are said indifferently of being a non-being; they are therefore neither verbs nor names for the logician, and thus they are not his concern. They are not, however, excluded from the grammarian’s consideration. (Simon of Faversham, cited in No One’s Ways, 55)

Meanwhile, Cantor argues for admitting infinite names into mathematics—or something very much like them—on the basis that they do determine clear and definite constituents of our thought:

We may regard the whole numbers [e.g., ω, JK] as “actual” in so far as they, on the ground of definitions, take a perfectly determined place in our understanding, are clearly distinguished from all other constituents of our thought, stand in definite relations to them, and thus modify, in a definite way, the substance of our mind. . . . In the introduction of new numbers, it is only obligatory to give such definitions of them as will afford such a definiteness. . . . As soon as a number satisfies all these conditions, it can and must be considered as existent and real in mathematics.6

To put it concretely (in the case of numerical infinities), the question was whether an infinite set such as {0, 1, 2, 3 . . .}, to which Cantor gave the name ω, ought to be interpreted as a “completed” as opposed to a “potential” infinite. The distinction turns on whether an object like ω is transparently visible to thought or intuition qua object, as Cantor thought, or whether the set {0, 1, 2, 3 . . .} is simply (or only) indefinitely extendible, so that for any number (belonging to the set), one can name a larger number. The Aristotelian view that ω ought to be regarded as only potentially infinite was dominant among the mathematicians of the nineteenth century and persists (somewhat) to this day. The term “ω,” in this view, would then be not so much the name of an object as shorthand for a kind of call and response.

The crucial point here is that instead of simply taking the term “infinite” to mean “non-finite,” as was the convention among the mathematicians of his time, Cantor saw that the term “non-finite” effaced a vast, ramified hierarchy of infinite names. Thus, in a single act, the various ontologies that philosophers had associated with indefinite terms, the oversupply of interpretive options urged on them by the natural language cases, were fused by Cantor into a single, massive ontology, that of the transfinite.

The great theme of No One’s Ways, the philosophical problem that naming is a form of reification, was thus not only not solved by Cantor (or for that matter by any mathematician who followed him), but also advanced—albeit with a change of particle from “non-” to “trans-.”

A positive definition of the term “infinite” was given by Richard Dedekind in his 1874 Was sind and was sollen die Zahlen?: a set is said to be infinite if it contains a mirror image of itself (in one of its proper parts).7 But “non-man is too uncanny to be avoided,” Heller-Roazen remarks (256), and indeed Dedekind allowed himself to be lured into attempting a proof of positive existence, of even a single infinite set, which led to a disaster, mathematically speaking. The “theorem” appears in the now notorious paragraph 66 of Was sind and was sollen die Zahlen? and is stated as follows: “My own realm of thoughts, i.e., the totality S of all things, which can be objects of my thought, is infinite.”8

Dedekind’s proof goes like this: He posits the thought of his own ego; let us call this thought E. He then considers the thought of E, and then the thought of the thought of E, and then the thought of the thought of the thought of E, and so on. David Joyce speaks for most mathematicians when he says of the proof:

Dedekind’s so-called proof of this statement leaves much to be desired. It’s worthwhile quoting, but as it has no mathematical merit whatsoever, it’s not worth discussing. The statement should be an axiom, not a theorem.9

Cantor was wise to sidestep any proof of existence—though of course he argued for it—and this is the way the existence question is handled by mathematicians nowadays: they adopt the existence of infinite sets as an axiom, as a principle beyond proof. Just as the early descriptivists once excluded infinite names from logic while allowing them into grammar, so mathematicians deny the existence of infinite objects the status of a theorem while allowing them into their own grammar with the phrase façon de parler—a term of art in mathematics that is often used to refer to the infinite.

As for the transfinite hierarchy itself, which Hilbert once defended with the statement “No one will expel us from the paradise that Cantor has created for us,”10 it has been vastly elaborated by set theorists, even in the face of the somewhat ubiquitous use of the phrase façon de parler. In fact, for the set theorist, “non-finite” turned out to be the key to the kingdom.

The magnificent problem posed by No One’s Ways is that “man” and “non-man” cannot coexist, in the sense that the act of prefixing a term with the particle “non-” eliminates all trace of the negated object. One might call this the Outis Problem, after the passage in the Odyssey with which Heller-Roazen opens his work. There is a sense in which the Outis Problem was overcome by set theory: the finite is not only not obliterated in the passage from finite to transfinite, it is always there—it is the way in. Other instances of this retention of the negated object in contemporary logic might begin with Boethius’ observation that “non-man” has an infinite range, an idea that can now be expressed in infinitary logic.11 This logic was introduced by D. Scott and Tarski, and its idea was to eliminate the vagueness associated with the existential quantifier by replacing existential statements of the form “There is an x with property P” with an infinite disjunction, over all names.12 For example, the sentence “There is a number which is not even” then becomes “Either 1 isn’t even or 2 isn’t even or 3 isn’t even or 4 isn’t even . . .” Note that all the numbers are mentioned, including all the even numbers. Infinitary logic is the focus of very active logical study nowadays.

A final logical point: Heller-Roazen remarks at the end of his book that philosophers have ignored the rich possibilities “non-man” offers, and this is astutely observed:

Although multiple, conflicting and arresting, such possibilities have hardly come into focus in the philosophical investigations into infinite naming. Only rarely, and as if unaware of themselves, have philosophers drawn close to a troubling possibility of logical determination, which none, however, has dared to state. It is the possibility that “non-man” names not the “contrary” or “complimentary” class of man, as Boole and others naïvely assume, but a shadow that belongs to man and to him alone. Were this to be the case, the indefinite state of being “non-man” would pertain solely to “man.” The obscure power of the “non-” would inhere to the very essence to which it is tied. Some non-being would “be” enclosed in it. (257)

Boole was working at a time when logicians did not really understand the concept of definability in logic, the idea that an object can be uniquely described in a logical language. Logicians understand definability better now, and this may bring us closer to what Heller-Roazen is asking for here. For suppose a proposition φ defines a unique object m. Then ¬φ names the contrary of m, while still indicating m in the form of φ. Moreover, we can recover the object m from ¬φ by adding a second negation, because ¬¬φ and φ are equivalent in classical logic. Heller-Roazen is correct, though, that logicians have not yet devised systems in which terms are negated directly, as in ¬m.13

The call for a new analysis of the Outis Problem is very interesting. It would be truly sublime to develop a discourse of “non-man,” but with “man” inside. I do see logic giving way—but perhaps there are new ways of thinking about definability, and this will save us from the wreckage.

As for pure philosophy, the distinction philosophers once drew between infinite names that may be considered as legitimate objects of grammar and those that are legitimate objects also of logic went underground in the analytic philosophy of these matters for a while, as Heller-Roazen notes of the broader discussion. The doctrine of ontological commitment due to W. V. O. Quine, for example, which has influenced generations of philosophers since its introduction in the 1950s, conflated the scientist’s theoretical and natural language discourses, such that all noun terms are now to carry ontological commitments tout court.

The so-called “linguistic turn” in philosophy inaugurated by Russell, Frege, and Moore in the beginning of the last century channelled philosophy towards the development of formal languages and the project of unearthing the formal, logical skeleton of all that we do in language. But Heller-Roazen’s remark, at the end of No One’s Ways, that “the doctrine of infinite naming silently spirals back, time and again” (255), applies as much to the discourse of indefinite terms as to philosophers themselves. Philosophers have returned to natural language in large numbers.14 They see now that what happens in between the logic of our discourse matters, too; that becoming attuned to the odd contingencies of language, to its linguistic residue, is as revealing of what we do even in our scientific languages as is their putative structure. Heller-Roazen seems to call for precisely this at the end of his book, and just as he sets this down it has indeed materialized.

  1. This is a useful example from Heller-Roazen’s book.

  2. I.e., terms of the form “non-X.”

  3. Thanks are due to Jouko Väänänen for this example.

  4. In the sense of Aristotle’s On Interpretation.

  5. Bertrand Russell, “The Philosophy of Logical Atomism (1918),” in Logic and Knowledge: Essays 1901–1950 (Spokesman, 2007), 202.

  6. Georg Cantor, Contributions to the Theory of Transfinite Numbers, translated and provided with an introduction and notes by Philip E. B. Jourdain (New York: Dover, 1952), 67.

  7. For example, the set of natural numbers {0, 1, 2, 3 . . .} is infinite because it can be mapped into a proper subset of itself, namely the set of squares of natural numbers {02, 12, 22, 32 . . .}. This fact was noticed already by Galileo, who referred to it as a paradox. See Richard Dedekind, Was sind und sollen die Zahlen? (Braunschweig: Friedrich Vieweg, 1961). Republished as What Are Numbers and What Should They Be?, translated by H. Pogorzelski, W. Ryan, and W. Snyder (Orono, ME: Research Institute for Mathematics, 1995).

  8. Dedekind, What Are numbers and What Should They Be?, 64.

  9. David Joyce, “Notes on Richard Dedekind’s Was sind und was sollen die Zahlen?,” 13,

  10. “Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können,” in Georg Cantor, “Über das Unendliche,” Mathematische Annalen 95 (1926) 161–90; 70.

  11. Infinitary logic allows for infinitely long propositions.

  12. D. Scott and A. Tarski, “The Sentential Calculus with Infinitely Long Expressions,” Colloquium Mathematicum 16 (1958) 166–70.

  13. At least this logician has never heard of such a logic.

  14. See, for example, the work of Frederike Moltmann.

  • Daniel Heller-Roazen

    Daniel Heller-Roazen


    Response to Juliette Kennedy

    While at work on No One’s Ways, I was all too keenly aware that, when completed, the book would run a double risk. On the one hand, the study seemed to bear on something minute, even infinitesimal, in magnitude: a word or word-part, non-, as it had attracted the attention of philosophers in their accounts of the meaning of terms, sentences, and judgments. Attending to linguistic and logical phenomena that often remain beneath the threshold of critical perception, the book could well pass by unobserved. On the other hand, the project on which I had embarked seemed to demand that some traditional disciplinary divisions be ignored. Following the ways in which naming and judging by means of non- became the occasion for analysis and invention meant moving from logic to metaphysics, without ignoring grammar, passing from the Greek doctrine of the parts and relations of statements to the Arabic theorists of “thing-ness,” from Scholastic and post-Scholastic accounts of transcendentals and “super-transcendentals” to Kant’s typology of judgments, Cohen, Carnap, and Heidegger on the types of “nothing,” and the fracture that arose in the twentieth century between phenomenology and various attempts to found philosophy in the light of Frege’s logic. No One’s Ways therefore ran the risk of appearing trivial or overweening, if not both. In any case, it seemed perfectly poised to miss its potential readers. And even if I set aside the trajectory of the examination I hoped to conduct and the various materials it would involve, I could hardly forget that any contemporary inquiry into what one might, for lack of a better term, call “the theory of language” runs up against a partitioned terrain: readers are often divided between opposing traditions, discourses, and research programs, and the shared appeal to rigor rarely includes a call to cultivate interest and knowledge in methodologies and references of other disciplines.

    Juliette Kennedy’s response suggests that at times, the least likely and the luckiest possibility can come to pass, dispelling doubts of such a kind. The generosity with which she has read No One’s Ways, discussed its theses, and developed its questions in new terms indicates that the features of natural languages that attracted the attention of ancient, medieval, and modern thinkers are still of acute interest to scholars of the most diverse of disciplines—literature and mathematics being, I think it is fair to say, disciplines as distant from each other as can be. Kennedy’s response, however, offers more than the happy evidence that an author’s apprehensions can miss the mark, real readers outstripping his imagination. It also proves more than that those who study that equivocally named phenomenon, “language,” do, despite some appearances, share an interest in a single subject. Kennedy has brought into sharp focus crucial tensions in the elements of the theory that I tried to develop. She has sketched some of the responses that mathematical logic has proposed to the difficulties that philosophy encounters. Drawing attention, in particular, to the ways in which the modern theory of the transfinite has reconfigured earlier representations of the infinite, she has, finally, suggested some new terms and forms in which the obscure subject of No One’s Ways—“non-man”—may yet be productively redefined. Each of these contributions merits commentary.

    Kennedy opens her response by alluding to the diverse types of analysis to which natural language has been submitted. A grammatical study may break down the flow of speech into such units as phrases, words, word-parts, or morphemes, and phonemes. A logical analysis will reach different elements, such as subject and predicate. But it is not only the “parts of speech,” so to speak, that vary among disciplines; the rules defining their possible combinations also depend on the type of analysis that is to be conducted. To recall Kennedy’s example, the sentence “This sentence is false” is a perfectly acceptable and even paradigmatic example of a correct utterance, at least according to the rules of English grammar. In logic, by contrast, that sentence defies analysis, for it has no truth value. As Kennedy notes, one of the commitments that unites the thinkers discussed in No One’s Ways is that of going beyond the grammatical analysis of speech. From Aristotle to Kant, from Maimon and Hegel to Frege and Heidegger, philosophers propose their own accounts of the parts of discourse and the rules of their combination.

    Yet here again, the paths of analysis diverge. At least since Aristotle, philosophy proposes a division of the statement into subject and predicate. Twentieth-century philosophers object, for various reasons. In different ways, Husserl and Heidegger take the distinction between subject and predicate to be superficial because it is derivative; the task they set themselves, in their distinct phenomenologies, is to reach a dimension of experience that is “pre-predicative.” In his Begriffsschrift, Frege issues a different challenge to the classical logical analysis. He takes subject and predicate to be unnecessary because they have no place in the “concept script” that he devises for thinking, in “imitation of arithmetic.” Frege’s invention was to be massively influential in early twentieth-century analytic philosophy, which sought to submit ordinary speech to logical study, unearthing, as Kennedy so well puts it, “the formal, logical skeleton of all that we do in language.”

    Yet this project supposes that the “forms” of a formal language and natural languages are homogenous, or that they may be superimposed—a supposition that, with time, has only become increasingly contestable. Thinkers as diverse as Heidegger and Adorno objected to it in the earlier parts of the twentieth century. But the development of logic itself casts doubt on “logicist” impulses in philosophy. In 1930–31, Carnap evoked “The Old and the New Logic,” celebrating the “creation of a new and efficient instrument in the place of the old and useless one.”1 Today, Carnap’s “new and efficient instrument” is known to logicians as “classical logic.” There are now many logics, which are at least as removed from Aristotle’s and Kant’s as was the “concept notation” developed by Frege and Russell: intuitionism, conditional logics, paraconsistent logics, free logics, fuzzy logics, and many more.2 If one aims to bring “logic” to bear on the structure of natural language, one thus runs up against a question: Which logic is to be evoked? A further question is never far away: According to what criteria does one decide on the pertinence of one logic or another for the analysis of ordinary speech?

    No One’s Ways orients itself with respect to one feature of some natural languages: the possibility of affixing a particle such as “non-” to a word or term. Even in its final chapters, which seek to measure the novelty and consequences of Frege’s “concept script” for the fate of the theory of infinite naming, the book concerns logic solely insofar as it has borne on this aspect of ordinary speech. The formal treatment of terms and propositions therefore plays an ancillary role, and the book is silent on the fascinating question that Kennedy has raised: the relation of “indefinite” or “infinite” expressions, such as “non-man,” to the mathematical infinite of set theory.

    Kennedy has recalled that the Greek term aoriston, which Boethius translates as infinitum, is equivocal. While both Ammonius and Boethius take it to designate particular meaning of a term bearing the prefix non-, they disagree as to the range of its “infinitude.” Ammonius tacitly treats “non-man” as meaning anything, be it real or imaginary, that is not man and that belongs to the genus of which man is a species: “horse,” “centaur,” “goat-stag” are all, then, “non-men.” Boethius goes further. For him, “non-man” may signify absolutely anything but man; he suggests that the stone and lifeless human body, therefore, are “non-men.” Subsequent philosophers, particularly in the medieval period, are divided on the question of whether or not such “infinite names” can be of any use in logic. That they are grammatically admissible seems to have been a matter of agreement. But what, they asked, does one know in knowing anything to be “non-something”?

    There are many modern echoes to that question, only some of which are explored in No One’s Ways. The so-called “paradox of confirmation” attributed to C. G. Hempel furnishes an illuminating example. The proposition, “All ravens are black,” is logically equivalent to the proposition, “No black thing is a raven.” And just as one may confirm the statement that all ravens are black by indicating a raven that is indeed black, so one may also confirm that thesis by pointing to a “non-black thing” that is not a raven. In I. J. Good’s terms: “The hypothesis, H, that all crows are black is the same as that all non-black things are not crows, and this is supported by the observation of a white shoe.”3 One of the questions that any treatment of the “paradox of confirmation” must confront is what it means to know that anything is “non-something,” or “non-x,” where “something” or “x” is a referring name of some kind. In other words: What is the value of an infinite name?

    Kennedy suggests that the controversies concerning the admissibility of infinite names in traditional logic have notable parallels in the modern debates about the status of infinite sets, which she likens to “infinite names in mathematics.” “Infinite” in the modern mathematical sense is not equivalent to “indefinite.” While the “indefinite” is lacking in definition, the mathematical “infinite” possesses a major property, which Kennedy, following Dedekind, defines as follows: “A set is said to be infinite if it contains a mirror image of itself (in one of its proper parts).” For this reason, as Russell explains in his Introduction to Mathematical Philosophy, an infinite number in the modern sense is “unchanged by adding 1 or subtracting 1 or doubling or halving or any of a number of other operations which we think of as necessarily making a number larger or smaller.”4

    The novelty of such a concept of infinite magnitude cannot be doubted. Yet it is curious that, as I indicate in passing (119), Kant seems to have tacitly envisaged a quantity of this kind. He explains that when I say, “‘The soul is non-mortal,’ I have certainly made an actual affirmation, for I have placed the soul within the unlimited domain of non-mortal things. . . . What is mortal contains one part of the whole domain of possible beings; that which is non-mortal contains the other. Nothing is said by my proposition but that the soul is one of the infinite multitude of things that remain if I take away everything that is not mortal.” It appears that, for Kant, the “whole domain of possible beings” is unchanged in size when one subtracts one determination (that is, being mortal) from it. “The infinite sphere of the possible is thereby limited only to the extent that that which is mortal is separated from it, and the soul is played in the remaining space of its domain. But even with this exception, this space still remains infinite, and more parts could be taken away from it without the concept of the soul growing in the least and being affirmatively determined.” It is perhaps only by reference to the figure of an “exception” that Kant can allow such an infraction of the rules valid for traditional magnitudes.

    In 1896, Jonas Cohn published his extraordinary History of the Problem of the Infinite in Western Thinking till Kant.5 It would be fascinating to follow that “history” into the period of the mathematization of the infinite, to which Kennedy has rightly drawn attention. I lack the means to do so. Yet I am struck by the terminological detail that affords Kennedy her title: “From Non to Trans.” “Cantor saw that the term ‘non-finite’ effaced a vast, ramified hierarchy of infinite names. Thus, in a single act, the various ontologies that philosophers had associated with indefinite terms, the oversupply of interpretive options urged on them by the natural language cases, were fused by Cantor into a single, massive ontology, that of the transfinite.” Cantor thus demonstrated that being non- is but one variety of being in-. There is also being trans-.

    Yet the “transfinite” never fully replaced the infinite, at least in theory. Cantor argues trenchantly against the so-called “infinite” of the Scholastics, which he casts as a merely “syncategorematic” quantity, produced by addition or diminution: “a variable magnitude, either growing beyond all limits or diminishing to an arbitrary smallness, always, however, remaining finite.”6 For this reason, he calls it “the improper-infinite” (uneigentlich-Unendliche), distinguishing it from transfinite, which is an actual “proper-infinite.” Yet Cantor also opposes the transfinite to the “Absolute”: “The transfinite, with its plenitude of formations and forms, necessarily indicates an Absolute, a ‘true infinite’ whose magnitude is capable of no increase or diminution, and is therefore to be looked upon quantitatively as an absolute maximum.”7 With Cantor, mathematics thus masters the traditional concept of the “non-finite”; non- is redefined as trans-, as a notion “indefinite” in philosophy receives a rigorous mathematical definition. Yet with the appearance of this “un-increasable” Absolute, does the infinite simpliciter not elude mathematics, too, withdrawing into the domain of philosophy—if not into theology? “The true infinite or absolute, which is in God,” Cantor writes, “admits of no determination.”8 “Consequently, I distinguish between infinitum aeternum increatum sive Absolutum, which refers to God and his attributes, and an ‘infinitum creatum sive Transfinitum.’”9

    With such an “absolute infinity,” we seem far removed from the modestly infinite name from which No One’s Ways takes its point of departure, and therefore also far removed from what Kennedy has eloquently dubbed the “Outis Problem”: “‘man’ and ‘non-man’ cannot coexist, in the sense that the act of prefixing a term with the particle ‘non-’ eliminates all trace of the negated object.” Yet with the tools of logic and set theory, Kennedy has proposed several solutions. One involves Peirce’s beautiful formulation of the meaning of the existential quantifier. Another emerges from a revised account of definability: “Suppose a proposition φ defines a unique object m. Then ¬φ names the contrary of m, while still indicating m in the form of φ. Moreover we can recover the object m from ¬φ by adding a second negation, because ¬¬φ and φ are equivalent in classical logic.”

    These proposals, as far as I understand them, seem to me most intriguing. Yet I still wonder: What if non-man were something other than the contrary of man? What if its logic were more disorderly than such accounts of definability allow? I propose distinguishing four meanings of “non-man.” In a first sense, “non-man” may be grasped as shorthand for the negation of “man”: the denial that “man” applies. Such a reading is difficult to refute; but it can also hardly be developed. A second possibility is to take “non-man” as the contrary of “man.” Does this thesis not assume, however, that whatever is signified by “man” admits of a contrary, and that it admits of one contrary alone? Can this be presupposed? A third possibility is to return the first sense, rendering it affirmative. “Non-man” can be the positive sign of anything that “man” is not. Then, however, one needs to concede, for the sake of a coherent usage, that such things that are not man are graspable as some unity, defining a concept. That much is also contestable.

    The fourth sense is the one that seems to me both most troubling and closest to ordinary speech. In the project on which I am now at work, which is the development of No One’s Ways, my hypothesis is this fourth sense leads furthest. “Non-man” is a “man” in the same way in which a “non-starter” is a “starter,” a “non-entity” an “entity.” In other words, it is not external to the category of “man”; it is internal to it. Here the rules naïve and classical logic, as I understand them, no longer hold: being “non-man” belongs to the determinations of “man” and to them alone. Only “man” can be or can become “non-man.” Of such a “non-man” it is, then, impossible to state either of these two contradictory propositions: “It is a man,” or “It is not a man.” Between being and non-being, a subject remains obstinately, uncannily, suspended. Is there a formal system that would be adequate to such a person or non-person? Does logic give way? Although I am far from an answer, I am certain that Juliette Kennedy’s rich and stimulating response helps to clarify the field of possibilities.

    1. Rudolf Carnap, “Die alte Logik und die neue Logik,” Erkenntnis 1 (1930–1931) 12–26, quoted here at 13; English in “The Old Logic and the New Logic,” trans. Isaac Levi, in Logical Positivism, ed. A. J. Ayer (New York: Free Press, 1953), 133–46, here at 134.

    2. See, e.g., Graham Priest, An Introduction to Non-Classical Logic: From If to Is (Oxford: Oxford University Press, 2001; 2nd ed. 2008).

    3. I. G. Good, “The Paradox of Confirmation,” British Journal for the Philosophy of Science 11.42 (1960) 145–49, cited here at 146.

    4. Bertrand Russell, Introduction to Mathematical Philosophy (London: Allen & Unwin, 1919; 2nd ed., 1920), 79.

    5. Jonas Cohn, Geschichte des Unendlichkeitsproblems im abendländischen Denkn bis Kant (Leipzig: Engelmann, 1896).

    6. Georg Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Eine mathematisch-philosophischer Versuch in der Lehre des Unendlichens (Leipzig, 1883), in Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, mit Erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor-Dedekind, ed. Ernst Zermelo (Berlin: Springer, 1932), 165.

    7. Cantor, “Mitteilungen zur Lehre von Transfiniten,” in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, 378–439, cited here at 405. See Michael Hallett, Cantorian Set Theory and the Limitation of Size (Oxford: Clarendon, 1984); Ignacio Jané, “The Role of the Absolute in Cantor’s Conception of Set,” Erkenntnis 42.3 (1995) 375–402.

    8. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, 175.

    9. Cantor, Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, 399.



Ways of Indeterminacy


Daniel Heller-Roazen’s book is an extremely rich study of what may look, at first sight, like a minor undulation in the sea of logic and grammar. As No One’s Ways shows, however, the grammatical possibility of attaching the particle “not-“ or “non-“ to a word ends up sending ripples through logic, metaphysics, and epistemology. If the negative particle is affixed to a noun such as “man,” “thing,” or “being,” then how is one to determine the range of beings—or non-beings—that are designated by the resulting ex­pressions? From Aristotle through Kant to Cohen, Heller-Roazen’s riveting book charts the deep conceptual disturbances caused by these “indefinite names.” Given how tightly the author weaves his narrative, it may be best, for the purposes of my response, to take an oblique approach and begin with a text that he does not mention: the Excursus on the Odyssey in Dialectic of Enlightenment. In this digression, Adorno and Horkheimer proffer, among other things, an interpretation of the Cyclops episode—the very same passage with which Heller-Roazen opens No One’s Ways.

For Adorno and Horkheimer, the cunning escape of the Homeric hero from Polyphemus marks a fundamental shift in “the historical position [Standort] of language.”1 Whereas mythical fate does not know the distinction between word and object, Odysseus discovers the difference and exploits it accordingly: the wily hero identifies himself as Oūtis, and just as he had hoped, this false name becomes, upon repetition, oútis, or “no one.” By means of this linguistic trick, the man known to be “turning many ways” (polútropos) is able to extricate himself from the mythical sphere in which word and object are one. In a surprising twist of their own, Adorno and Horkheimer link Odysseus’ ruse to modern mathematics and symbolic logic. If one describes, with Carnap, the variable as a sign that designates “something indeterminate” (etwas Unbestimmtes), it becomes easier to see, however, why Adorno and Horkheimer would want to associate Odysseus’ ploy with what they characterize as the “formalism” of these disciplines: not unlike the variable in modern mathematics and symbolic logic, the pseudo-name Oūtis refers “to all sorts of contents, both to nobody and to Odysseus himself.”2

Heller-Roazen assigns hardly less significance to Odysseus’ ruse than do Adorno and Horkheimer. In his reading, however, the wily hero saves himself “by a single act of speech: adding the particle ‘no,’ ‘not-,’ or ‘non-’ (ou) to the indefinite pronoun ‘a,’ ‘one,’ or ‘someone’ (tis)” (9). Everything follows from this grammatical analysis of oútis. Where Adorno and Horkheimer compare Oūtis to the functioning of a variable, Heller-Roazen transitions from oútis to a discussion of indefinite names. And where the former diagnose a radical continuity from Homer to Carnap, the latter detects a no less radical discontinuity. As Heller-Roazen argues, it is, precisely, the formalism of logic in Carnap and others that brings about a fundamental change in “the historical position of language.” To be sure, Heller-Roazen does find instances of a philosophically relevant use of the negative particle in later thought; among other things, he highlights the importance of the non-identical in Adorno’s Negative Dialectics (241). After the arrival of modern formal logic, however, “a detail of grammar such as the possibility of affixing non- to a word” (237) no longer becomes an object of sustained philosophical reflection. In what follows, I will not so much challenge Heller-Roazen’s claim as highlight the significance of another story, one that can be found in the margins of his book. From the beginning, if not in Homer, then in Aristotle, the peculiar indeterminacy of the variable accompanies, perhaps like a shadow, the “linguistic shadows” (9) that come loose in indefinite names.


Although Heller-Roazen insists on the morphology of oútis, it would be a mistake to distinguish his analysis from the one proposed by Adorno and Horkheimer in terms of a fixed contrast between grammar and logic. To the extent that Heller-Roazen tracks efforts to circumscribe the effects of indefinite names, his book is also a history of the distinction between these two disciplines. His critical intervention on modern formal logic towards the end of No One’s Ways needs to be seen against the backdrop of this history.

Indeed, the first philosophical attempt to address expressions of the form “not-” or “non-X” already demonstrates the difficulty of accounting for them. Using “the queer term ‘not-man’ or ‘non-man’ (ouk anthrōpos)” (18) as an example, Aristotle examines the relation of such expressions to forms of negation and privation. One thesis of his, in particular, would prove to be momentous: guided by the grammatical distinction between “S is not a man” and “S is a not-man” in Greek (37), Aristotle holds that the corresponding sentences mean different things (19, 28–29). As Heller-Roazen points out, Aristotle initially hesitates to categorize an expression such as “not-man” as a noun or name (onoma) (19). Once he does so, however, he introduces a technical term that gives an indication of what may have given him pause in the first place: by calling these expressions “indefinite” or “indeterminate” (aoriston) (33), Aristotle hints at the fact that they present a stumbling block for his logic, which construes the assertion as consisting of, precisely, two well-defined terms (horoi) linked by the copula “is.”

If Aristotle says very little about the meaning of indefinite names on their own, this will change with his commentators, as Heller-Roazen shows in the first half of his book. In particular, Boethius interprets Aristotle’s “indefinite name” (onoma aoriston) as “infinite name” (nomen infinitum), arguing that an expression such as “non-man” negates one definite meaning, while introducing an infinite array of existing and nonexistent things (40–41). In one respect, however, Boethius is just as cautious as Aristotle. According to the Roman philosopher, expressions of the form “not-” or “non-X” may be regarded as names in grammar, but not in logic and philosophy (41). As Heller-Roazen explains, there are good reasons for this attempt to legislate on the use of infinite names: once attaching the negative particle to a word is understood as an act of “infinitation,” the possibility of producing expressions of this form is bound to clash with the order of philosophical terms that already “contain everything” (62). Heller-Roazen pursues this difficulty with a particular focus on the term “thing” (res), culminating in Kant’s reflections on the concept of an “object in general” (102). The second half of No One’s Ways begins where the first half left off, with Kant, but instead concentrates on his account of so-called infinite judgments. Heller-Roazen shows that assertions such as “the soul is non-mortal” (118) were to have an important career in post-Kantian thought. But however much thinkers such as Hegel (172–73.) may have pushed against Aristotle’s analysis of predication, they remained in its “shadow” (220).

The same cannot be said of Frege. As Heller-Roazen argues, philosophical reflection on the infinite name—and, by extension, the infinite judgment—is contingent upon the Aristotelian theory of the predicative statement: S is P. Frege, however, introduces a new analysis of the assertion, one that breaks it down into function and argument, rather than subject and predicate, which for their combination require the copula. Of course, Frege’s “conceptual notation” (Begriffsschrift) does account for statements such as “x is not a man,” which it analyzes as ∃(x)[¬f(x)], or “There is an x, such that not f(x).” Yet there is, among the formation rules of his logic, none that would correspond to the grammatical rules that make it possible to produce, in many languages, statements such as “x is a not-man.” Surveying discussions surrounding the development of modern formal logic from Frege to Carnap, including Carnap’s altercation with Heidegger, Heller-Roazen concludes that with the shift towards the function-argument analysis of predication, the infinite name disappears from logic and most of philosophy (220–21). Accordingly, his call to think “with languages, and not only in them, and against them” (251), is also a protest against what Quine once described as “the Procrustean treatment of ordinary language at the hand of logicians.”3


In Heller-Roazen’s history of the infinite name, the arrival of Frege’s conceptual notation thus acquires epochal significance. For Frege himself and for thinkers working in his wake, becoming clear about the defining characteristics of modern mathematical logic requires, in turn, the consideration of another way of indeterminacy: the variable. As Russell writes, “[t]he variable is perhaps the most distinctively mathe­matical of all notions; it is certainly also one of the most difficult to understand.”4

In order to approach the problem of the variable, it may be best to take another look at Heller-Roazen’s construal of the changes brought about by Frege’s conceptual notation. To begin with, if it were only a matter of illustrating the contrast between ordinary language and modern logic, Frege’s function-argument analysis would no doubt be a good case in point. The disappearance of the infinite name from logic, however, has a more specific reason: for Frege and the majority of logicians after him, “not” applies to entire propositions, rather than individual terms. Invoking the law of excluded middle, Russell, for example, argues that the negative particle can never be taken to qualify a predicate.5 The resulting syntactical rule “not-p” (where p is a propositional variable) takes leave of grammar and excludes the infinite name from the purview of modern logic. Heller-Roazen alludes to this fact when he links the vanishing of the infinite name from logic not only to Frege’s function-argument analysis, but also a more general shift from a logic of terms to a logic of propositions (221–22, 239). While Łukasiewicz was among the first to emphasize the contrast between these two strands of logic and situate them historically, Heller-Roazen draws on another distinction made by the Polish philosopher in order to deepen the comparison of traditional and modern logic. If anywhere, it is here, in Heller-Roazen’s admittedly brief discussion of the “formalistic” (221) nature of modern logic, that he approaches the fundamental change in the “position of language” highlighted by Adorno and Horkheimer.

As in Dialectic of Enlightenment, the determination of this shift turns on the question of meaning. Following Łukasiewicz, Heller-Roazen points to the use of letters in Aristotle’s syllogistic and maintains that traditional logic was already “formal” (221). By contrast, modern logic is “formalistic” inasmuch as it constitutes, in Heller-Roazen’s words, “a code of writing in which the sense of every expression is determinable on the basis of its form” (ibid.). What is surprising about this gloss on Łukasiewicz is the manner in which Heller-Roazen links form and meaning. For Łukasiewicz, the defining characteristic of a fully formalized logic is the fact that it allows one to confirm the validity of a proof “without referring to the meaning of the terms used in the proof.”6 The modern paradigm of such a system is the one set forth by Frege in Conceptual Notation: it consists of axioms from which the theorems of propositional logic can be derived by means of a small number of rules of inference. This system is “formalistic” in Łukasiewicz’ sense insofar as its axioms and rules of inference are syntactically specified. Thus, there is no need to refer to the meaning of particular terms when verifying the correctness of deductions; the only thing that matters is the logical form of the theses, which can be brought out by a combination of variables and logical constants such as and, or, or not.

If the variable thus takes center stage just as the infinite name disappears from logic, there are good reasons why Heller-Roazen would touch on the history of the variable at various points in his book. Drawing on recent research, Heller-Roazen is careful not to identify the use of letters in ancient geometry and logic with the role played by the variable in modern times (13–14, 221). Close to the arithmetical center of his book, moreover, he introduces the “unknown quantity” of algebra in his discussion of the “thing” (res) in the early modern period leading up to Kant (93–94, 105). Heller-Roazen’s subsequent discussion of the variable in modern logic, however, once more cites the “algebraic x” (222). The issue is not just that Frege, along with the concept of a function, borrows his use of the variable from analysis, rather than algebra. More importantly, the concomitant reflection on the relation of the variable to its values prevents one from assuming, as Adorno and Horkheimer do, that the formalism of modern logic must necessarily issue into “nominalism.”7

For Frege, who provides Łukasiewicz with the prime example of modern “formalistic” logic, the axioms and theorems of logic are neither tautologies that do not say anything (Wittgenstein) nor empty schemata awaiting a model-theoretic interpretation (Tarski). Rather, they are very general descriptive statements that are true for all possible values of the variables involved. Russell has a very similar understanding of the statements of logic. The question remains, however, how to conceive of the so-called free variable: that is, the “x,” “y,” or “z” that is not bound by a quantifier such as “all”. Under the auspices of what the early Russell calls a “term”—which is “anything . . . that can be mentioned” and as such “has being”—he interprets the variable “n” (which is assumed to range over the integers) as denoting “any number.”8 Whatever a free variable such as “n” denotes is, accordingly, not one definite term; but a term, albeit an indefinite one, it nonetheless is if everything that can be mentioned is indeed something that is. Russell describes this strangely indeterminate term as “any number,” and were it not for purely grammatical considerations, one would have to hyphenate this phrase and precede it by a definite article; for this anynumber is the one thing, the term, that the variable “n” denotes.

By contrast, Frege argues that while “1,” “2,” or “3” are certainly names of particular numbers, a variable such as “n” is not the name of an indefinite or indeterminate number, for the simple reason that there is no such thing.9 In his explanation of what a free variable is, Frege concentrates on the variable in the context of a function. Given his logical extension of the latter, expressions such as “x2 + 3x” and “x is a man” are both signs of functions. One gets a value for a particular argument of these functions if one replaces “x” by a proper name; viz., a numeral, the name of a human being, or the name of any other “object” (Gegenstand). The “x” itself, however, is not a name. According to Frege, this letter simply marks “the places” (Stellen) in the expression of a function where “the sign for the argument must go in.”10 In other words, “x” is merely a placeholder. Properly speaking, it “designates” (bezeichnet) nothing at all.11

After cursorily describing the variable as a sign that designates something indeterminate, Carnap, for his part, reiterates Frege’s view. Presumably, Carnap’s assertion that variables do not “mean” (bedeuten) anything was one of the reasons why Adorno and Horkheimer concluded that modern logic is nominalist.12 Remarkably enough, however, Frege also characterizes the variable as a sign—without, however, inadvertently assimilating it to a name, which would then necessitate the kind of ontological interpretation one finds in Russell. For Frege, the variable is a sign that does not designate. Moreover, the “x” does not only not designate a “meaning” (Bedeutung), it also does not express a “sense” (Sinn).13 Accordingly, it is not only unlike a numeral or a name such as “Socrates,” which, in Frege, have both a meaning and a sense; it also differs from a name such as “Odysseus,” which, assuming that Homer’s epos is a work of fiction, expresses a sense, but does not designate a meaning.14 In short, if the variable is a sign, it is a sign sui generis. As a sign, however, the “x” is not without semantic function. According to Frege, the “x” “indicates” (andeuten) objects such as numbers or human beings “indeterminately” (unbestimmt).15 With this seemingly minor adjustment, Frege displaces the question of indeterminacy from a “what” to a “how.” In the case of the variable, indeterminacy does not qualify a signified, but a mode of signification. Or, to cite a grammatical distinction that Frege employs at this point, indeterminacy enters the picture as an adverb, and not an adjective.16


The fact that the variable remains on the margins of No One’s Ways is of course no accident. However one may want to characterize the relation between the histories of the infinite name and the proverbial “x,” their trajectories can never be said to merge into one. Even when abstracting from differences of usage, their ways of indeterminacy remain distinct inasmuch as an expression of the form “non-man” is logically complex, while a variable is not. Only the former, but not the latter, involves a logical operator.

However, the presence of a logical operator does not affect the question of whether or not an expression is a name. The question is, then, whether an expression should be analyzed as a name if it lacks a determinate meaning. In this regard, the comparison with the debates surrounding the variable at the turn of the twentieth century may be instructive, the differences between indefinite names and variables notwithstanding. Russell’s analysis of the variable pro­duces difficulties that are both logical and ontological. As his notion of a term indi­cates, these problems are exacerbated by the fact that he tends to blur both the sign with its sense and its sense with its meaning. Yet even if one does take these distinc­tions into account, it is difficult to see how one could avoid asking what a vari­able refers to if one regards it as a name. It is no doubt for this reason that Frege de­velops a compet­ing ana­lysis of the variable, one in which the “x” no longer func­tions as a name. In general, Frege does of course not shy away from making far-reaching ontological commitments. When it comes to the variable, however, the entire thrust of his analysis is to avoid positing indeterminate meanings (Bedeutungen). His analysis is certainly not without cost: it forces Frege to suspend, in the singular case of the vari­able, the semantics of sense and meaning and revert to the grammatical, rather than logical, distinction between adverb and adjective. If Frege is willing to pay this price, it is because the variable, unlike expressions of the form “not-“ or “non-X”, cannot be ex­cluded from logic.

To the extent that Heller-Roazen charts the career of the particle “not-“ or “non-“ within a largely Aristotelian tradition, his book remains, as its subtitle indicates, An Essay on Infinite Naming. After having circumscribed the closure of the logico-philosophical formation in which expressions of the form “not-“ or “non-X” were analyzed as names, however, Heller-Roazen announces a sequel to his investigation, one that will turn to fields such as law, literature, and linguistics. The new volume will take Aristotle’s example of “non-man” as a paradigm for the examination of the human being in relation to language—or, more precisely, his or her “voice” (12, 258, et passim). One may surmise, however, that even after philosophy it will  remain a question for thinking whether there could be, with regard to the twin conception of “man” and “non-man,” an avenue for the development of a notion of indeterminacy that is, if not adverbial, then non-adjectival. Maybe Heidegger can be seen to gesture in this direction when he defines, in the text that was to become the target of Carnap’s polemic, the human being as “the placeholder of the nothing” (Platzhalter des Nichts).17 For what is perhaps most significant about Hei­deg­ger’s formulation is not his reification of “not” (nicht), for which he was duly criticized by Carnap, but his use of the term “placeholder,” which only gained prominence in logical discussions after Frege.

  1. Max Horkheimer and Theodor W. Adorno, Dialektik der Aufklärung: Philosophische Fragmente, in Adorno, Gesammelte Schriften, ed. Rolf Tiedemann (Frankfurt/M.: Suhrkamp, 1997), 3:78. English in Horkheimer and Adorno, Dialectic of Enlightenment: Philosophical Fragments, trans. Edmund Jephcott (Stanford: Stanford University Press, 2002), 47. Throughout, I modified translations to allow for a more literal reading.

  2. Horkheimer and Adorno, Dialektik der Aufklärung, 79; Dialectics of Enlightenment, 47. The characterization of the variable as a sign that designates “something indeterminate” can be found in Rudolf Carnap, Abriß der Logistik: Mit besonderer Berücksichtigung der Relationstheorie und ihrer Anwendungen (Wien: Springer-Verlag, 1929), 3. Horkheimer had referred to Carnap’s discussion of functions and variables a decade before the publication of Dialectic of Enlightenment. See Max Horkheimer, “Der neueste Angriff auf die Metaphysik,” Zeitschrift für Sozialforschung 6 (1937) 4–53, at 37. English in Max Horkheimer, Critical Theory: Selected Essays, trans. Matthew J. O’Connell et al. (New York: Continuum, 1999), 132–87, at 169. Needless to say, Carnap and the Vienna Circle remained a target throughout Dialectic of Enlightenment.

  3. Willard van Ornam Quine, “Mr. Strawson on Logical Theory,” in The Ways of Paradox and Other Essays, rev. ed. (Cambridge: Harvard University Press, 1976), 137–57, at 149.

  4. Bertrand Russell, Principles of Mathematics (London and New York: Routledge, 2010), 89.

  5. See Bertrand Russell, The Philosophy of Logical Atomism (London: Routledge Classics, 2010), 43. In an earlier take on the same issue, Russell had arrived at a different conclusion. See Russell, “On Denoting,” Mind 14 (1905) 479–93, at 490.

  6. Jan Łukasiewicz, Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic (Oxford: Clarendon, 1957), 16. Just as in the case of term logic and propositional logic, Łukasiewicz associates “formal” logic and “formalistic” logic with the Peripatetics and the Stoics, respectively. According to Łukasiewicz, Stoic logic is the true progenitor of modern logic, whose inventor is Frege, “the greatest logician of our time.” See Łukasiewicz, “On the History of the Logic of Propositions,” in Selected Works, ed. L. Borkowski (Amsterdam: North-Holland Publishing, 1970), 197–217, at 215.

  7. Horkheimer and Adorno, Dialektik der Aufklärung, 79; Dialectics of Enlightenment, 47.

  8. Russell, Principles of Mathematics, 44–45 and 91. Russell’s interpretation of the variable at one point attracted the interest of Agamben. See Giorgio Agamben, La comunità che viene (Turin: Giulio Einaudi, 1990), 47–52. English in Giorgio Agamben, The Coming Community, trans. Michael Hardt (Minneapolis: University of Minnesota Press, 1993), 70–77. Agamben’s investigation into “any” (qualunque) in Russell needs to be seen in the context of his ongoing attempt to recover what he describes as Plato’s “idea of language.” For a commentary on early articulations of this project, see the characteristically rich introduction by Daniel Heller-Roazen, “To Read What Was Never Written,” in Giorgio Agamben, Potentialities: Collected Essays in Philosophy, ed. and trans. Daniel Heller-Roazen (Stanford: Stanford University Press, 1999), 1–23, esp. 5–14.

  9. See Gottlob Frege, “Was ist eine Funktion?,” in Funktion, Begriff, Bedeutung: Fünf logische Studien, ed. Günther Patzig (Göttingen: Vandenhoeck & Ruprecht, 1986), 81–90, at 83–84. English in Gottlob Frege, Collected Papers on Mathematics, Logic, and Philosophy, ed. Brian Mc Guinness, trans. Max Black et al. (Oxford: Blackwell, 1984), 285–92, at 287.

  10. Gottlob Frege, “Funktion und Begriff,” in Funktion, Begriff, Bedeutung, 17–39, at 22; English in Frege, Collected Papers on Mathematics, Logic, and Philosophy, 137–56, at 141.

  11. Frege, “Was ist eine Funktion?,” 87; “What Is a Function?,” 290.

  12. Carnap, Abriß der Logistik, 3.

  13. On the distinction between “sense” and “meaning,” see Gottlob Frege, “Über Sinn und Bedeutung,” in Funktion, Begriff, Bedeutung, 40–65; English in Frege, Collected Papers on Mathematics, Logic, and Philosophy, 157–77. Frege uses the verbs “to mean” (bedeuten) and “to designate” (bezeichnen) interchangeably.

  14. On the example of “Odysseus,” see Frege, “Über Sinn und Bedeutung,” 46–48; “On Sense and Meaning,” 161–63.

  15. Frege, “Was ist eine Funktion?,” 84; “What Is a Function?,” 288.

  16. Frege, “Was ist eine Funktion?,” 84; “What Is a Function?,” 288.

  17. Martin Heidegger, “Was ist Metaphysik?,” in Gesamtausgabe, vol. 9 Wegmarken, ed. Friedrich-Wilhelm von Herrmann (Frankfurt a.M.: Vittorio Klostermann, 1976), 105–22, at 118; English in Heidegger, Pathmarks, ed. William McNeill, trans. David Farrell Krell (Cambridge: Cambridge University Press, 1998), 82–96, at 93.

  • Daniel Heller-Roazen

    Daniel Heller-Roazen


    Response to Hardtmann

    Markus Hardtmann’s generous and penetrating reading of No One’s Ways shines a bright light on the book’s central project. At the same time, it frames the book’s investigation with respect to a fascinating question that it leaves largely unexplored. His response thus affords me a welcome occasion to return to my argument and to set it in a new context. Hardtmann has begun by recalling Adorno and Horkheimer’s brief yet stimulating remarks on the Homeric episode with which No One’s Ways opens. I have read Odysseus’s ruse as offering a fable for the power of naming and un-naming by non-, an “infinite” power to which thinkers after Homer turn. Adorno and Horkheimer make a different claim for Odysseus’s play. They maintain that it puts an end to “mythical fate,” breaking “the spell of the name” and effecting a change in “the historical situation of language.” For them, the hero’s cunning announces the general “formalism” that characterizes “bourgeois society”; more specifically, it clears the way towards “the schema of modern mathematics.”1 Hardtmann has drawn out the principle that supports the formalism of this “schema.” Evoking Carnap, Hardtmann stresses that modern mathematics and symbolic logic make a crucial use of “the variable as a sign that designates ‘something indeterminate’ (etwas Unbestimmtes).”

    Setting Adorno and Horkheimer’s allegorical reading of the Homeric passage against my own, Hardtmann has identified a “story” at “the margins” of No One’s Ways, which he has brought to light: “from the beginning, if not in Homer, then in Aristotle, the peculiar indeterminacy of the variable accompanies, perhaps like a shadow, the ‘linguistic shadows’ that come loose in indefinite names.” The claim is provocative and raises questions important issues, which pertain both to Adorno and Horkheimer’s account of the Homeric passage and to philosophical method.

    I will only briefly treat the first. It seems to me difficult not to raise questions about Adorno and Horkheimer’s reading of the Odyssey, even if one does not contest their notion of “mythical fate.” I take the claim that Odysseus “discovers that an identical word can mean different things” to be a simplification, if not a distortion, since it is precisely the identity of “the word” as such that the hero’s play contests. Ou tis and Outis are not one word, nor are they merely two words. The play of ou and ou tis (like that of mē tis and mētis) consists less in the running together of two nouns or names for “different things” than in the running together of the multiple, formally diverse syntactic elements by which these two “names” are representable in grammar. To grasp the hero’s ruse, one must hear the negative particle and the pronoun, on the one hand, and the pseudo-proper name, on the other, both together and in separation. A negative, indefinite pronoun (“Not one . . .”) must be perceived as being homophonous with a new, spurious proper noun (“Not-one”); heterogeneous parts of speech must be understood in their identity and difference.

    Adorno and Horkheimer argue that by his ruse, Odysseus makes himself the author of “two contradictory actions”: “obedience to his name and repudiation of it.” Yet once again, this claim seems to me to reduce the complexity of the performance. Outis is unmistakably linked in sound to Odysseus. Whether or not one accepts that the name Outis is a truncated form of Odysseus, as has been argued, it is difficult to speak here of any univocal “repudiation.”2

    These points concern the reading of the Homeric text. Hardtmann’s response, however, also raises a substantial theoretical question. It is that of the extent to which “the schema of modern mathematics,” in Adorno and Horkheimer’s term, is discernible in the poetic scene. Hardtmann has argued persuasively that “No One,” being “a pseudo-name,” is “not unlike the variable in modern mathematics and symbolic logic,” which “refers ‘to all sorts of contents.’” I have indicated some reasons why I would hesitate to concur. Outis seems to me more equivocal than the category of “pseudo-name” suggests. It seems to me, moreover, that in general, naming by non- results in something other from a “pseudo-name,” because of the tension that arises between the sense of any name and that of the term fashioned when a “non-” is joined to it. Yet to dwell on the expression Outis alone would be to lessen the full philosophical force of the question that Hardtmann has posed: What is the relation between the sense and function of the non- and the sense and function of the variable?

    Hardtmann has noted that the variable remains at “the margins of No One’s Ways,” adding—rightly, to my mind—that it is no accident that this is the case: “However one may want to characterize the relation between the histories of the infinite name and the proverbial ‘x,’ their trajectories can never be said to merge into one. Even when abstracting from differences of usage, their ways of indeterminacy remain distinct inasmuch as an expression of the form ‘non-man’ is logically complex, while a variable is not.” These terms seem to me compelling, yet I am inclined to offer a different reason for the divergence of the two subjects Hardtmann has distinguished. Infinite naming is an operation effected in language—and in languages; the same is true for infinite judgment. In their inquiry into these subjects, philosophers run up against facts of grammar, which they clarify, even as they draw from them the matter for their inventions. By contrast, the variable belongs to formula languages. It is not heard “in the voice,” as Aristotle might say; rather, it is invented, or instituted, by a rule. While the non- is therefore subject to the rules of grammar, “the variable” is a symbolic entity of which multiple and incompatible logical definitions may legitimately be offered.

    For this reason, I hesitate to reason in terms of “the variable.” Under what conditions may one reason in terms of “the variable,” with the singular, definite article? Hardtmann has given an illuminating account of the differences between several modern logicist interpretations of the variable’s senses. I would go still further. Can one speak of “the variable” when referring at once to the “unknown quantity” of algebra, Kant’s doctrine of the transcendental object and the unknown values on which Frege and Russell’s logics are to bear? To answer in the affirmative seems to me tacitly to deny one major difference. It is the difference between a reasoning about something unknown and a reasoning that reckons with something unknown only on a single condition: that it be “bound” to a specified quantity and that its existent being, therefore, be posited and quantified in advance of any reasoning. Broadly speaking, this is the difference between traditional and “modern” logics, that is, between pre-quantificational and quantificational logics, whose famous first examples are to be found in the “algebras of thinking” formulated by Peirce and Frege.

    This difference seems to me fundamental. From it, I would draw a consequence: the “variables” of discussed Frege and Russell have little more than their name in common with the variously “unknown” terms of classical, Scholastic or Kantian philosophy. Yet there are reasons to think that, even within the field of modern quantificational logic, there may also be no single concept of “the variable.” Quantification has been treated in several ways; an example is the hesitation between “substitutional” and “objectual” understandings of quantification, which is already discernible in Frege.3 The variables that are bound by quantifiers may, therefore, also be defined in conflicting ways. Hardtmann has shown that, while sharing certain fundamental principles, Frege and Russell disagree about the status of variables as names. Hardtmann has also offered a fascinating sketch of the reasons for Frege’s refusal to admit that x stands for anything at all. These remarks all suggest to me the intractable difficulties that any definition “the variable” in general would be forced to confront.

    At the end of his response, Hardtmann has wondered whether, in aiming to move beyond the various accounts of “non-man” discussed in No One’s Ways, I may draw resources from Heidegger and his account of human being as “the placeholder for the Nothing.” If I am reluctant to grant as much, it is for the reasons I suggest in No One’s Ways: in insisting on the phenomenological priority of the “not” of anxiety to the “not” of the proposition, Heidegger covers over the phenomenon that seems to me most in need of study—the equivocations of the “not” in language and languages. Heidegger has presupposed the univocity of “the Nothing.” This seems to me contestable. In place of “the Nothing,” I would reason in terms of the multiple and heterogeneous “nothings” that are familiar to languages, whose logic and metaphysics remain to be developed. In assuming that one may speak coherently of “the Nothing,” and in assuming, moreover, that this “Nothing” possesses a single “place” that human being is to “hold,” Heidegger fails to do justice to Outis and the challenge that he poses. Here I would raise several questions. “For” what non-being does No One stand? Is it one or many? Does it truly “hold” a position (Stelle), as Heidegger suggests? Might it not also shift it? If so, in what ways, and to what consequence?

    In the book on which I have been working since finishing No One’s Ways, I have chosen to respond to such questions by taking a cue from languages and grammar, rather than philosophy or logic. I have found a guiding indication in one of the meanings of non- in contemporary English usage: that of deprecation, even ruination. As I have written elsewhere in these responses, my guiding hypothesis is that “non-man” bears the relation to “man” that “non-entity” bears to “entity,” or that “non-starter” bears to “starter.” Each time, the non-thing appears as the named thing, according to some failure to be that calls out for commentary. I have sought to explore some of these modes of this failure to be. How persons can be missing, how they can be made to be missing, how they can persist in not being there — these seem to me to be some crucial questions in need of study. I have drawn my objects in my new project from the domain of myth, law, literature, and religion. The philosophy and logic that they suggest is rarely more than implicit. I am well aware that in setting out to treat such various objects philosophically, there is the risk that analysis may lose itself among disparate and unrelated details. Yet my conviction has been that an historical and philological investigation can nonetheless draw from exemplary cases some crucial theoretical terms and orients. They may lead further in the study of the acts and senses by which we name ourselves, and by which we name what – and who – we believe ourselves not to be.

    1. Adorno and Horkheimer, Dialektik der Aufklärung, 78–79; Dialectic of Enlightenment, 47–48.

    2. See Konrat Ziegler, “Odysseus—Utuse—Utis,” Gymnasium 69 (1962 396–98; Jürgen Wöhrmann, “Noch einmal: Utis—Odysseus,” Gymnasium 70.6 (1963) 549.

    3. The literature is immense. The problem is famously discussed by Willard Van Orman Quine, Ontological Relativity and Other Essays (New York: Columbia University Press, 1969), 91–113.



Particles of Non-Being in Heller-Roazen and Lacan

This work recalls in multiple ways Ann Smock’s incomparable What Is There to Say? Not only does this phrase capture my sense of inadequacy before such a magisterial revisioning of the history of philosophy as the history of the “non-” in its negative, positive, privative, and infinite variations, but both works arrest us at the juncture of speech and non-speech, logical agility and inertia. No One’s Ways is a stunning intellectual history of the trajectory of the grammatical and ultimately logical particle “not” or “non-,” from Aristotle and Scholastic thought to the early decades of the twentieth century, with a major focus on Kant and thinkers like Salomon Maimon and Hermann Cohen who revisited Kant’s structural frameworks, and concluding with the divergence between logical positivism and phenomenology. The connection to Heller-Roazen’s earlier investigations of linguistic forgetfulness, sensation, and number is palpable, but this work stands apart as something of a culmination and synthesis of the sequence. Its discussions of predication, transcendentals, the thing (res), and Kantian thought are mesmerizing and exceed the tight focus of the “essay.” The recurrent terms and motifs—unity and disjunction, positive and negative, even the true and the false—are presented for the most part in a manner that is neither oppositional nor dialectical, but according to a different and less categorizable logic of nominatory exposition. Indeed the mode of this essay’s presentation echoes something of the concluding formulation a propos of Heidegger that “we are confronted with the question of Being and non-Being not in understanding . . . but in naming” (255). Without naming itself as such, this unorthodox study presents a transhistorical lineage of Western epistemological encounters with, or predications of, the ontological.

In overcoming a certain speechlessness, and a wish to linger on Heller-Roazen’s intrepid forays into the heights of Scholastic predication, I am drawn above all to respond to one of the concluding challenges, that these investigations be extended to other domains such as anthropology, law, psychoanalysis. . . . To cut to the chase, while in no way engaging directly with psychoanalysis, this work has grasped and offered, with more lucidity than I have previously encountered, the ostensible key to Lacanian ontology, as it is lodged in the particles ne and . This might be simplified to the question of how non-being appears in language.

Such an ontological problematic strikes this reader as difficult to reconcile with the work of the psychoanalytic clinic, but in any case it is the small particles of non-being that, more than anything else, raise these Lacanian stakes. They might be termed particles of negation, a term used by Freud (Verneinung) and commonly applied to Lacanian psychoanalysis. But it is Heller-Roazen’s coup de grâce to delineate, at both linguistic and logical levels, why a seeming negation may not be one—it may be privative, indefinite, or even affirmative, but is generally not “lacking” in any straightforward sense. And this same formulation can be made, albeit more contentiously, for Lacanian thought, if not psychoanalysis more generally.

At more than one juncture, Lacan broaches the ontological function of the pleonastic “ne,” the addition of a seemingly unnecessary negative particle after certain expressions of fear or doubt. The example he returns to is “Je crains qu’il ne vienne” (I’m afraid he will come), which takes a solo “ne” without the usual accompanying “pas,” and in this fashion introduces no negative meaning to the phrase. In The Ethics of Psychoanalysis, Lacan at once highlights the “negative” dimension of the particle “ne” and at the same time its non-negating function:

The negative particle “ne” only emerges at the moment when I really speak, and not at the moment when I am spoken, if I am on the level of the unconscious. And I think it is a good idea to interpret Freud in a similar way when he says that there is no negation at the level of the unconscious. Given that immediately afterwards, he shows us that there is indeed negation. That is to say, in the unconscious there are all kinds of ways of representing negation metaphorically. There are all kinds of ways of representing it in a dream, except, of course, for the little negative particle “ne,” because the particle only belongs to discourse. (64)

Not unlike the vicissitudes of the “non-” in No One’s Ways, here the “ne” takes on a unique affirmative quality precisely because of its status as a discursive entity. It also provides the alibi for Lacan’s persistent non-renunciation of Freud. On the one hand, Lacan affirms Freud’s oft-cited proposition that there is no negation in the unconscious. Then he counters that Freud himself shows negation to be present, albeit metaphorically (Lacan will emphasize to great effect in his anecdote of the sardine can in Seminar XI that he is “not speaking metaphorically” [95–96]). Furthermore, it is possible to represent negation in a dream, but the dream can not represent the particle “ne” per se, which rather counterintuitively escapes negation by not allowing metaphoric approximation. In this sense, the “ne” alone approaches a non-negatable reality, “the moment when I really speak, and not . . . when I am spoken.” Perhaps we could say the pleonastic “ne” exists with no useful grammatical function but more nearly an ontological one, as something like a being-particle. This would seem a non-Kantian reformulation of what Heller-Roazen recounts as eighteenth-century German Scholasticism’s “two determinations of a ‘thing’ (res): ‘reality’ (realitas) and ‘negation’ (negatio)” (111). In Lacan, however, rather than being distinguished categorically, the real and negation find a rare coincidence in the pleonastic “ne.”

Lacan’s emphasis on the expressive aspect of Freudian negation seems compatible with Heller-Roazen’s demonstration that articulating negation in language paradoxically undermines the being of negation as such. “Thus the Verneinung, far from being the pure and simple paradox of that which presents itself in the form of a ‘no,’ isn’t just any old ‘no.’ There is a whole world of no-saying (non-dit), of interdiction (interdit), since it is in that very form that the Verdrängt, which is the unconscious, essentially presents itself. But the Verneinung is the most solid beachhead of that which I would call the ‘intersaid’ (entre dit) in the same way we say ‘interview’” (64–65). Heller-Roazen has uncovered in the history of philosophy something like a whole world of no-saying, which to be sure is not any old no. But does the “no” in No One’s Ways operate in the same fashion as the Lacanian unconscious and its unique access to being’s particles? Perhaps the two approaches are most proximate when Heller-Roazen evokes the “zero of affirmation” as a “thinking beyond judgment” in the beautiful passages on Maimon (151). We might ask if this is something akin to what Jacques-Alain Miller famously identifies as the zero representing the “not-identical with itself” which the “discourse of logic summons . . . and then rejects as the pure negative . . . in order to constitute itself as that which it is” (99, my emphasis). Like Heller-Roazen, Miller emphasizes the logical process, the effective vanishing of the subject in the field of the signifying chain. Yet what would it mean to refocus our attention on the ontological constitution, that being is positively named in quite absolute terms through this process of exclusion or suture?

The preceding psychoanalytic examples share with the series of thinkers and works surveyed in this extraordinary intellectual history the fact that in almost all cases, except possibly Hegel and Frege (for opposing reasons), the negating particle is not fully a negative but retains something of the affirmative by dint of its very enunciation or naming. This distinction pertains in the brief discussion of Cohen’s use of the ancient Greek particle , as the “qualified” negation in contrast to the definitively negative ou. Lacan both echoes this and takes in a quite different direction.

The Greek is central to Lacan’s analysis in the Ethics seminar of the self-willed vanishing of Oedipus at the end of Oedipus at Colonus. At issue is an ever-proud Oedipus unreconciled with the son who Antigone will later try to bury, and empowered to bestow blessings and curses on Athens and Thebes. Beyond this, his power extends to an inexplicable ability to absent himself from the world, escorted only by the Athenian king Theseus: “But by what doom / Oedipus died, not a man alive can say, / only Theseus, our king. / No blazing bolt of the god took him off, / no whirlwind sweeping inland off the seas, / not in his last hour. No, it was some escort / sent by the gods or the dark world of the dead, / the lightless depths of Earth bursting open in kindness / to receive him. That man when on his way, / I tell you, not with trains of mourners, / not with suffering or with sickness, no, / if the death of any mortal ever was one, / his departure was a marvel!” (lines 1779–891, p. 381). It is in fact the Chorus at an earlier moment that pronounces the confounding term mē phunai [μὴ φύναι], “not to be,” rendered in its assessment of Oedipus’s inhuman fate: “Not to be born is best” (line 1388, p. 358), and which Lacan perceptively if incorrectly attributes as “Oedipus’s last word.” He further connects the in mē phunai to the pleonastic “ne,” characterized as “the remains [in French] of that which means μὴ in Greek, a word that does not signify a negation” (Lacan, Ethics, 305–6). Notably both not being born and Oedipus’s dis-appearance are something other than the general negation of death.

But here the something other is at once the splitting that for Lacan constitutes the subject and a subjective excess that allows for both terms of Blanchot’s non-coincident first and second deaths (the death I will and the death that comes over me from the outside) to coincide in the final choice of Oedipus. Lacan highlights both the absoluteness of death’s negation and the excess that doubles it: “That’s the choice with which a human existence such as Oedipus’s has to end. It ends so perfectly that he doesn’t die like everybody else, that is to say accidentally; he dies from a true death in which he erases his own being. . . . It’s a beautiful attitude, and as the madrigal says, it’s twice as beautiful on account of its beauty” (306). We might ask whether this “true death,” this perfect and excessively beautiful negation heralded by the , is still on the order of the Aristotelian “non-man” with which No One’s Way’s begins its riveting story of the indefinite name that is not a negation (18–19). Is there an outside to Heller-Roazen’s system, and does Oedipus mark a form of definite naming that is a pure negation?


Works Cited

Blanchot, Maurice. The Space of Literature. Translated by Ann Smock. Lincoln: University of Nebraska Press, 1982.

Heller-Roazen, Daniel. No One’s Ways: An Essay on Infinite Naming. New York: Zone, 2017.

Lacan, Jacques. Seminar VII: The Ethics of Psychoanalysis 1959–1960. Translated by Dennis Porter. New York: Norton, 1992.

———. Seminar XI: The Four Fundamental Concepts of Psychoanalysis. Translated by Alan Sheridan. New York: Norton, 1981.

Miller, Jacques-Alain. “Suture.” Translated by Jacqueline Rose. In Concept and Form, vol. 1, Selections from the Cahiers pour l’analyse. London: Verso, 2012.

Smock, Ann. What Is There to Say? Lincoln: University of Nebraska Press, 2003.

Sophocles. The Three Theban Plays. Translated by Robert Fagles. New York: Penguin, 1982.

  • Daniel Heller-Roazen

    Daniel Heller-Roazen


    Response to Eleanor Kaufman

    The extraordinary generosity of Eleanor Kaufman’s response to No One’s Ways consists not only of its estimation of the book’s contributions (whose excess I am tempted to try to capture by an allusion to Kaufman’s own work, The Delirium of Praise), and not only of its reconstruction of the book’s argument, which perfectly condenses what I sought to do, even as it casts my project in a new light. What I take to be most precious are the suggestions that Kaufman makes for ways to develop my inquiry, both in the book’s terms and in those of what she has called “Lacanian thought, if not psychoanalysis more generally.” As Kaufman is the first to indicate, “Lacanian thought” is barely treated in the book. Yet one of her most stimulating indications is that my project bears essential affinities with the account of being and language that Lacan developed. Kaufman goes so far as to write that, “while in no way engaging directly with psychoanalysis, this work has grasped and offered . . . the ostensible key to Lacanian ontology, as it is lodged in the particles ne and . This might be simplified to the question of how non-being appears in language.” These formulations suggest that the project of No One’s Ways could be presented in at least two distinct, yet complementary vocabularies: in the discourse of general metaphysics and logic, first, and, second, in “Lacanian thought.” Here I would like to approach the second of these discourses by means of the first.

    What Kaufman calls “the question of how non-being appears in language” exactly names the central subject of the book, while alluding also to its complexity. There are several reasons for the difficulty of the “question.” First, it is massive in extension, stretching from word to word, discourse to discourse, language to language. It is also massively complex in its details; their analysis demands the instruments of several disciplines, from linguistics to rhetoric, logic, and philosophy. Yet it is also impossible to confront this question seriously without facing two other challenges. The first involves the event of “appearance”: What does it mean for non-being to “appear”? “Non-being” would seem by nature what does not show itself: a limit for any phenomenology. To be sure, linguists, logicians, and philosophers have located manifestations of non-being in speech, turning their attention to the variegated class of “negative” expressions: predicative or sentential operators, such as “not” and “no” word-parts such as “un-,” “im-,” and “non-,” verbs such as “to lack” and “fail to . . .” Yet it would be easy to show that such a focus is, at least in part, a matter of convention, and it is probable that any strict delimitation of the set of “negative” expressions in natural languages could be rightly contested for one reason or another. 1

    A further difficulty is also worth rendering explicit. Can one be certain that “non-being appears in language”? The syntax of that formulation suggests that non-being is something transmitted by language: a content stored, like any other, “in” it. But this presupposition may be unfounded. What if non-being were not “in” language but of it, that is, essentially inseparable from it? One finds modern instances of this thesis in Hegel, Mallarmé and Lacan. I share this position. It is one of the basic hypotheses of No One’s Ways that non-being belongs to speaking. Each of the book’s major subjects seems to me to confirm this premise in its own way. In the Aristotelian doctrine and its reception, the doctrine of naming opens onto a theory of naming in non-naming, or naming “indefinitely,” and the relation between name and being, designation and existence, is suspended, if not sundered. Inexistence—indeed, inexistence of several kind—thus becomes intelligible in being nameable. In Scholastic philosophy, the account of the categorial determinations of things runs up against the necessity of admitting terms for things that are not, according to multiple modes of non-being. In the Kantian and post-Kantian systems of reason, finally, the logic of judgment requires a space for variously infinite, null, self-suspending, and generative judgments, all of which obligate thinking to articulate forms of inexistence. In one of the last chapters of the book, I evoke Lacan’s mi-dire in this sense. By virtue of the belonging of non-being to language, all dire is a mi-dire, not only a “half-speaking,” or a “murmuring,” as the French mi-dire suggests, but also a speaking by means of the (mi, as it is pronounced in modern Greek). Languages are in this sense “of non-being.”

    If I were to present my undertaking in Scholastic terms, I would say that I pursued a “me-ontological” project in No One’s Ways. It owes much to Aristotle and to Heidegger. Aristotle poses the question of metaphysics as that of the many ways in which Being “is said.” For this reason, he distinguishes the meanings of Being, such as being by substance, being by accident, being as true and false, and being as actual and potential. In what sense, he asks, are these all senses of “Being”? As is well known, he excludes the possibility that Being is genus, with species; but he does not resolve the question of the meaning of Being, deciding on whether it is analogous, equivocal, or univocal. For him, Being thus remains, as Pierre Aubenque has shown, a “problem.”2 In Being and Time, Heidegger seeks to pose this problem anew, orienting his inquiry with respect to a single being, which he is the first to define: the being that we are, which he calls “Dasein.” Heidegger justifies the “ontico-ontological” priority of Dasein in metaphysics by arguing that we, Dasein, have a unique access to the question of Being: we always already have some understanding of Being, precisely insofar as we understand our own Being.

    I recall all this to introduce what I take to be an implicit point of departure of No One’s Ways. Non-being “is said” in many ways: there is the non-being of what is inexistent (according to several forms: the impossible, the imaginary, the unrealized, and so on); the non-being of the contrary; the non-being of the contradictory; the non-being of what is spurious. . . . In what sense, I have wondered, are these all ways of “non-being”? What is the “meaning of non-Being”? This is a question of me-ontology. Classical philosophy retreats before the prospect of me-ontology, with good reason. A me-ontology requires that non-being “be” defined, yet this seems an incoherent demand; non-being appears, by “definition,” to elude determination. The strictures against me-ontology in philosophy are as at least old as Parmenides, who advises against following the path of “what is not.” A twentieth-century echo of such warnings can be heard in Carnap, whose polemic against Heidegger I discuss in No One’s Ways. On the measureless cusp of ontology and me-ontology, one might locate Frege, to whose account of the zero Kaufman alludes in evoking J. -A. Miller’s essay on “the logic of the signifier.” Yet there are also examples of what I would call me-ontological projects in the history of philosophy. Some may well be parodic; Gorgias thus aims to demonstrate in his treatise On Non-Being that “there is only non-Being.” Other projects seem undoubtedly serious; perhaps the most eminent example is the “Table of Nothing,” in which Kant aims, in the Critique of Pure Reason, to distinguish the varieties of non-Being. Hermann Cohen, as I have argued, struggles with a related question in his logic of origin. One might well add Lacan to this series. From seminar to seminar, writing to writing, he distinguishes a panoply of “nots” and “nothings”: the “nothing” that is the objet a, the nothing in “meaning nothing,” or “wanting to know nothing,” the “nothing” discernible in the “not” of the negated, the repressed, the disavowed, the “not-all”….

    The method of Being and Time has remained an important source for me, insofar as I have sought to define the question of Being and non-Being by reference to one being among others: the being that we are. Yet, as I write in the concluding passage that Kaufman cites, my thesis is that our primary access to being is not in understanding, as Heidegger states, but in speaking—and more exactly, in naming. In this respect, my perspective seems to me consonant with Lacan’s: where human beings are concerned, ça parle, “it” or “something speaks,” “there is speaking,” even when the speaking remains unknown and perhaps unknowable to us. We are naming beings because we are speaking beings, because, in other words, ours is a “speaking being,” even when taciturn. As Lacan suggests in his last period, our être is parlêtre.3 “Understanding” comes later.

    In No One’s Ways, the crucial example of this speaking before understanding, or speaking without understanding, is naming and judging by non-: we avail ourselves of this possibility, without knowing clearly what it may involve and what we may thereby be meaning. From Aristotle to Deleuze, philosophers have turned their attention to this possibility, elaborating theories in which logics of the “non-” play crucial roles. Yet only rarely have philosophers lingered on the force of “indefinite,” “infinite” “privative” and “negative” particles in ordinary speech. Since its beginnings, psychoanalysis, by contrast, has proceeded otherwise. The meanings of a particle such as un-, the force of a “not,” and the vanishing of certain apparent contraries are all matters that fascinated Freud. Lacan developed Freud’s attention to details of ordinary speech in a new direction, being committed to two fields of analysis in which few psychoanalysts before him had a serious interest: linguistics and logic.

    The discussion Kaufman cites from Seminar VII is a perfect example of Lacan’s attachment to the logic of grammatical detail. In constructions such as “I fear that he may come,” French demands that a sentence that translates literally as “I fear that he may not [ne] come” (je crains qu’il ne vienne), where the “ne,” elsewhere a particle of negation, plays a non-negative role. As an English counterpart, one might consider the sentence: “I wonder whether he may not have said too much,” which does not mean, “I tend to think that he did not say too much,” but “I tend to think that he did say too much.” The “not” in “I wonder whether he may not have . . .” is not negative, but rather interrogative; Deleuze might have called it “problematic.” Here Lacan’s insight is somewhat obscured in the English translation of Seminar VII, which calls the French ne “pleonastic.” The term translated as “pleonastic” is in French discordantiel: “discordant” would be more apt. Lacan is using a technical term introduced by two grammarians, Édouard Pichon and Jacques Damourette.4 Their idea is that this French “ne” expresses the “discordance” between the “fear” and the “coming” in “I fear that he may come.”

    Kaufman has drawn attention to the way in which Lacan’s conclusion in this passage could well apply to the terrain discussed in No One’s Ways. Lacan states: “There is a whole world of no-saying, of interdiction, since it is in that very form that the Verdrängt, which is the unconscious, essentially presents itself.” Kaufman’s observation seems to me exact. Perhaps the congruence she has indicated is even more far-reaching than the English translation of Lacan suggests. When Lacan writes of a “world of non-dit,” he may mean “a world of no-saying,” as the translation has it. Yet the primary meaning of le non-dit is “the unsaid,” and I take Lacan to be claiming, “There is a whole world of the unsaid.” His thesis, in other words, bears on the un- as much as on the no. One could also take a further step, rendering the non-dit as the “not-said”: “There is a whole world of the not-said…” The proximity between his “world” and that of No One’s Ways is then maximal.

    At the close of her response, Kaufman considers the Greek (“not” or “non”) on which Lacan lingers in this Seminar, and more specifically the phrase mē phunai, “not being born,” as it appears in the choral ode in Oedipus at Colonus. Kaufman observes that Lacan “connects the in mē phunai to the pleonastic ‘ne,’ characterized as ‘the remains [in French] of that which means μὴ in Greek, a word that does not signify a negation.’” In conclusion, she writes: “We might ask . . . whether this is still on the order of the Aristotelian ‘non-man’ with which No One’s Way’s begins its riveting story of the indefinite name that is not a negation. Is there an outside to Heller-Roazen’s system, and does Oedipus mark a form of definite naming that is a pure negation?” Kaufman herself indicates some of the ways in which Lacan’s reading of Sophocles submits the dramatic text to unexpected turns, which complicate any reading of “Oedipus” as a single character. The terms in which I have reasoned do not allow me to define a negation as “pure”; my inclination is once again to insist on the plurality of diverse and even incommensurate “negations.” In untangling “this ” from any other, and in relating this “non” to the “non-man” with which No One’s Ways begins, we distinguish among the meanings of non-being, as between words and word-parts, in single languages and across them. We are, that is, conducting a me-ontology. However “indefinite” it may seem, non-being shows itself to be of many kinds.

    This is also why I am reluctant to try to define “the order of the Aristotelian ‘non-man.’” In the book I am now finishing, which is in a certain respect a continuation of No One’s Ways, I explore various figures of “non-persons”: beings of whom one may neither state “It is a person,” or “It is not a person” and who thereby testify to the real possibilities of being “non-man.” What Aristotle himself intended by “non-man” remains obscure. I take his name to be a spur to thinking precisely because of its opacity. It has long called out for commentary. For this reason, too, I would retreat from any claim to having proposed a system or erected an edifice with an inside and outside. I would rather set the cases that I studied on a one-sided surface, perhaps like some of the figures that interested the late Lacan. There, the examples of thinking with the non– might appear as a series of permutations and configurations: some continuous, some discontinuous. I take them to illustrate some of the ways in which philosophy has—and has not yet—attended to the “speaking being” that is the insistent presupposition of our reason.

    1. See Antoine Culioli, “La Négation: Marques et opérations,” Pour une linguistique de l’énonciation: Opérations et représentations, vol. 1 (Paris: Ophrys, 2000), pp. 91-114.

    2. Pierre Aubenque, Le Problème de l’être chez Aristote (Paris: Presses universitaires de France, 1966).

    3. Jacques Lacan, Joyce le symptôme II (Paris: Navarin, 1987).

    4. See Jacques Damourette and Édouard Pichon, “Sur la signification psychologique de la négation en français,” Journal de psychologie normale et pathologique (1928) 228–54. On Lacan and Damourette and Pichon, see Michel Arrivé, “Ce que Lacan retient de Damourette et Pichon: L’Exemple de la négation,” Langages 30.124 (1996) 113–24.