Deposited preprint

Across the Categorial Difference

Focality, Univocity, Ownness

Abstract

Across the Categorial Difference: Focality, Univocity, Ownness examines a question that has organised ontology since Aristotle: if being is said in many ways, what makes those ways belong to one field rather than form a mere list?

The study reconstructs the two principal answers given to this problem. The first is focality, or pros hen predication, which orders derivative senses of being toward a primary case. The second is univocity, which secures unity through a single concept of being and relocates plurality within its modes or differences. Through analyses of Aristotle, Scotus, Leibniz, Deleuze, and contemporary ontological pluralism, the article argues that these have appeared to exhaust the available options because both operate where categorial difference is understood through comparison, measure, or degree.

The article then proposes a third answer. A category marked by a genuinely non-scalar differentia need not be unified by a focal term or contained within a single common sense. The relevant differentia is ownness: the non-vicarious character of an obtaining as someone rather than merely something. On this basis, categorial plurality can remain irreducibly plural without hierarchy, common measure, or privileged centre.

The result develops the programme of a-metaphysical ontology and the doctrine of modal coincidence, while remaining independently assessable through explicit objections, regional controls, and falsification conditions.

Cite this as

Montaruli, Alessio. “Across the Categorial Difference: Focality, Univocity, Ownness.” Preprint, Version 1.1, 2026. DOI: 10.5281/zenodo.21684041.

This article states formally what Ethical Ontology, Volume I: The Modal Coincidence reaches through its regional trials, and takes from The Calculus of Modal Coincidence the differentia and its non-scalarity. Part One and Part Two stand independently of that work.

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Provenance. Montaruli’s doctrine of modal coincidence, the category of the who, and the programme of a-metaphysical ontology are the author’s own, developed in Positive No-Ground: The Common and the Locus of Answerability (2026a) and in Ethical Ontology, Volume I: The Modal Coincidence (2026b), and stated in Montaruli’s Calculus of Modal Coincidence (2026c), which is deposited and citable. A companion article, The Individuating Factor: Ownness, Jemeinigkeit, Haecceitas (2026e), defends the same result against a different objection. A reader who wants the doctrine rather than its argument should go to 2026c; the two volumes are unpublished manuscripts and are cited here by chapter so that inherited claims can be located within them on publication.

This study does not summarise that work. Part One and Part Two stand independently of it: they reconstruct the two devices by which the plurality of senses of being has been handled, and

argue that there have been two because both require a differentia that admits of degree. Part Three states the third answer, and what it uses from the wider account is the differentia and its non-scalarity, both of which can be assessed here. Where the argument inherits it says so and names the chapter; where it constructs, it says that too.

How this is written. Technical vocabulary is taught rather than assumed. Each term the argument needs is introduced by an ordinary example, named with its provenance, and then used. Where a term belongs to the literature it is marked as such and its source given; where a term is my own coinage, that is said plainly.

Greek and Latin are shown rather than reported, because in several places the argument turns on a word that no English rendering carries — συμβλητά against σύμμετρος, ἐλήλυθεν, nihil aliud quam. Each appears once in the original with a gloss in the same breath.

On the title. Categorial differencedoes not here name one privileged difference. What this study establishes is that there is no such thing: a category whose differentia is stated against other categories cannot stand above them. The phrase names the kind of difference a differentia makes when its contrast class is other categories rather than other species within a kind — and across rather than the, because a difference of that sort is traversed rather than held. It is not a variant of the ontological difference, and §8 gives the reason.

Version note. Version 1.1 strengthens §5.4 with two sentences: the insertion recorded there was required by the quantitative criterion of existence already fixed four propositions earlier, and is therefore a consequence rather than a hesitation. No other section is altered and no result is withdrawn.

Where a manuscript apparatus is cited, it is reported exactly. gestr. is gestrichen, struck out; erg. is ergänzt, added. What is claimed is what the manuscript does, never what its author intended.

Four citations remain to be confirmed against physical copies and are listed at the end. None bears on the argument.

§1. The problem

Being is said in many ways. That sentence has organised ontology since Aristotle wrote it, and it has received two answers.

Before either can be stated, it is worth seeing what the sentence is about, and the shortest way in is through something ordinary.

A man is healthy. His diet is healthy. The colour of his face is healthy. A gymnasium is healthy. The word is not doing the same job four times. The man has health. The diet produces it. His complexion is a sign of it. The gymnasium preserves it. Four different relations — and every one of them a relation to the same single thing.

Notice what this is not. It is not one meaning repeated: a diet is not healthy in the way a man is. And it is not four unrelated meanings that happen to share a sound: each of the four runs back to health, and none of them could be explained without mentioning it. Something in between is going on, and it has a structure.

Aristotle calls this being said pros hen: “toward one.” In English it is usually called focal meaning, a term coined by G. E. L. Owen in 1960 and standard ever since. The core case — health itself — is the focus, and the other cases are said toward it.

His claim in Metaphysics Γ.2 is that being works this way. And it is made against two alternatives he refuses in the same breath:

τὸ δὲ ὂν λέγεται μὲν πολλαχῶς, ἀλλὰ πρὸς ἓν καὶ μίαν τινὰ φύσιν καὶ οὐχ ὁμωνύμως

Being is said in many ways, but with reference to one thing and one nature, and not homonymously. (1003a33–34)

Pollachōs — in many ways. That word makes the problem; the denial at the end of the sentence makes it a problem worth solving.

For consider the two settlements Aristotle is refusing. If being were said in a single sense, there would be no difficulty at all: ontology would be one science with one subject, and the many ways would be many things said about one kind of thing. If it were said in wholly unrelated senses, there would be no science of being either — only a lexicon, a list of uses of a word with nothing behind them. The first settlement makes the problem vanish; the second makes it insoluble. Pollachōs falls between, and everything since is the attempt to say how.

It matters that the plurality at issue is not a plurality of words. Nobody has ever been confused about the verb “to be.” The plurality is in what there is: substances, quantities, qualities, relations. That is why enumerating the senses settles nothing, and why some device is needed to say why they are senses of one thing rather than a list of homonyms.

The health example is the first answer in miniature, and it can now be stated as an answer. The many senses of being are held together by reference to one primary sense; they are said toward it; and what they are said toward is substance. The plurality is real and it is ordered, and what orders it is one of its own members. §§2–3 reconstruct this answer and ask what its ordering device actually does.

The second answer refuses the question rather than answering it, and it has not appeared above at all. On this account being is not said in many senses to begin with: it is univocal, said in a single and same sense of everything of which it is said, and the plurality that remains is a plurality of modes or differences within that one sense. Nothing is ordered toward anything, because there is nothing to order. §4 takes it at its strongest.

Both answers secure unity, and they disagree about where the unity sits — in a privileged member of the plurality, or in a concept common to all of it. Neither needs the other, and between them they have held the field.

This book argues that a third answer is available; that it secures neither kind of unity and requires neither; and that it was not available before, for a reason that can be stated.

That the problem stayed live rather than antiquarian is owed to a dissertation. Brentano’s Von der mannigfachen Bedeutung des Seienden nach Aristoteles (1862) put the manifold senses back into circulation, and Heidegger recorded that it was the book that set him on the question of being. The question travels from Γ.2 to Sein und Zeit along that line, and it is travelling still.

What follows. Part One reconstructs the two answers and what each costs: the first in §§2– 3, with §3 devoted to the question of what a focal term actually does; the second in §4. Part Two argues that these two exhaust the options for a certain kind of ontology (§5) and shows this happening in contemporary work (§6). Part Three states the third answer (§7), says why it was not reached before and what access remains open (§8), and closes with objections and falsifiers (§§9–10).

§2. The first answer

The primary case, for Aristotle, is substance. The evidence for this is an inventory rather than an argument, and it repays looking at directly.

Begin again with something ordinary. A man is a thing. He stands on his own. His paleness does not stand on its own — there is no paleness walking about, waiting to attach to someone. It is always someone’s paleness.

The thing that stands on its own Aristotle calls a substance (ousia); what depends on it in this way is standardly called an accident. Older translations render the Greek as “affection of substance,” which means the same and is what you will meet in Ross.

Now read Aristotle’s list of what the derivative beings are, and watch the direction it runs:

So, too, there are many senses in which a thing is said to be, but all refer to one starting-point; some things are said to be because they are substances, others because they are affections of substance, others because they are a process towards substance, or destructions or privations or qualities of substance, or productive or generative of substance, or of things which are relative to substance, or negations of some of these things or of substance itself. (1003b5–10, Ross)

Ross’s rendering makes the structure audible: all refer to one starting-point. Paleness of a man. Recovery of a man. Blindness of a man. Every item on the list except one is named by pointing back at that one. It is not a taxonomy of independent kinds. It is an arrangement, and the arrangement has a centre.

Metaphysics Ζ.1 makes the consequence explicit:

Indeed, what was sought of old and is sought at present and always, and what is always a matter of difficulty, namely what is being? (ti to on) is this: what is substance? (tis hē ousia) (1028b2–4)

Read that twice, because the question has changed shape. What is being? has become what is substance? — and the second is narrower than the first. Every other sense of being has quietly dropped out of the enquiry: not denied, simply no longer asked about, on the understanding that answering this one will settle the rest. That is what a focal term does when it is working properly. You stop noticing the other cases, because you have been told where to look.

Two further passages fix the shape of the plurality. At Δ.7 (1017a22–23) the senses of being follow what Aristotle calls the figures of predication — the different ways a thing can be said of another: what it is, how much, what sort, where, when. These are the categories. And at Ε.2 (1026a33–b2) being is said in four ways: accidentally, as truth, according to the categories, and as potency and act.

The scholastic development of the first answer runs through analogy, with God as the primary analogate: creatures are said to be by reference to the being of God as the healthy things are said to be healthy by reference to health. It is a development of the first answer rather than a third option, and nothing here turns on it.

One difficulty inside the first answer should be recorded, because it is not this book’s discovery and its author deserves it. Pierre Aubenque argued that the science of being qua being and the science of substance do not in fact coincide, and that the resulting aporia is structural rather than a defect of execution. Aristotle’s metaphysics, he wrote, is “incapable de tout achèvement déductif” — incapable of any deductive completion — because the world it describes is one in which things come apart, and a science of that cannot be closed. His conclusion is that the impasses of first philosophy paradoxically become its greatest accomplishments.

But Aubenque argues that the unity fails; he does not offer a dispersed positive ontology in its place. That is the whole distance between his result and this book’s, and it should be conceded rather than blurred.

A remark about how Γ.2 is being read here. It is easy, at this distance, to treat the passage as the founding charter of an orthodoxy — the place where Western ontology was given its centre. It is better read as a proposal made in the open. Aristotle is working at the beginning of the domain, with the question of being still unsettled and no settled apparatus to apply; he offers pros hen as a way of keeping a science while keeping a plurality. What made it a settlement was what came after. Nothing in §3 is a charge against him, and a good deal of what follows in this book depends on his having recorded, at various places and without harmonising them, exactly what each case gave him.

§3. What focality does

Everyone agrees that a focal structure relates derivative cases to a primary one. The question is what kind of relation that is — and the available answers differ sharply in how much they claim. It can be a fact about words. It can be a fact about explanation. It can be a fact about what causes what. Or it can be a fact about who holds the right to be counted at all.

This section separates those four, because three of them are in the literature already and the fourth is what this book needs. The argument fails if the fourth cannot be established, and no position is named until it has been stated, since what is at issue is a difference between relations and not a difference between authors.

One word needs settling first, because everything turns on it.

A tenant lives in a house. She sleeps there, keeps her things there, and is in every ordinary sense the person living in that house. Her name is not on the deed. Ask by what right is this house occupied and the answer is not her name; it is the landlord’s. She is really there. The right is held elsewhere.

I shall call this holding title. The word is not standard in this literature, and I introduce it deliberately, because the existing vocabulary — dependence, priority, responsibility — has no term for it. Where it appears below it is mine, not the field’s.

The first reading: semantic

Derivative uses are understood by reference to the primary use. Healthy is understood by way of health, and someone who did not know what health was could not learn what a healthy diet is.

This is conceded without reservation. It is also where an opponent will want to keep the discussion, because at this level focality is a fact about vocabulary and carries no ontological weight at all. That “healthy” needs health to be understood tells us nothing yet about how healthy diets stand to health in the world.

The second reading: explanatory

The primary case does not merely anchor the dictionary. It explains why the derivative cases belong in one field rather than being unrelated.

To see the distinction that matters, take an English word covering two unrelated things. A river bank and a savings bank share a sound and nothing else; there is no single thing that both of them are, and no account of the one that mentions the other. Aristotle’s name for one word with unrelated definitions is homonymy — and homonymy is precisely where a science becomes impossible, since there is nothing single to have a science of. Where the definitions coincide entirely, he calls it synonymy.

The pros hen cases sit between. “Healthy” said of a diet and of a complexion is not an accident of English: the senses differ, and both run back to health. The contemporary name for this is core-dependent homonymy — Christopher Shields’s term, from Order in Multiplicity (1999), and the standard reference point for the discussion.

On his analysis, any account of a non-core instance must appeal to the account of the core instance ineliminably and asymmetrically. That is an asymmetry of account, and it already exceeds the lexical: it is what makes a unified science possible in the first place. Where the second reading holds, the derivative cases are not merely named after the primary; they are intelligible through it.

The third reading: aetiological

This is the furthest the relation has been taken, and taking it seriously is the point of the section.

Shields makes core-dependence precise with four conditions. In plain terms: the two cases share a word; their definitions overlap without coinciding; the derivative case’s being-healthy is linked to the core case’s being-healthy by some causal relation — producing it, preserving it, signifying it; and the core is responsible for the derivative’s existing at all, and not the other way round. In his own words:

a and b are homonymously F in a core dependent way iff: (i) they have their name in common, (ii) their definitions do not completely overlap, (iii) necessarily, if a is a core instance of F-ness, then b’s being F stands in one of the four causal relations to a’s being F, and (iv) a’s being F is asymmetrically responsible for the existence of b’s being F. (pp. 124–125)

One thing should be settled before going on, because the notation hides it. What are a and b? They are two things that bear the same predicate. In the health case they are a man and a diet, both called healthy — two things related through a third, since the diet is healthy for someone and the complexion is a sign of someone’s health. In the case that concerns this study they are the categories themselves: substance and quality, both called beings. So the relation runs sometimes between two dependents of one substance, and sometimes between substance and a dependent — and it is the second that matters here, since it is the being case that Γ.2 is about.

Condition (iv) is the one to remember, and the one to open first if you want to test what follows. Asymmetrically responsible for the existence of. Not a fact about definitions — a fact about existence. And asymmetric: the core does this for the derivative, and the derivative does not do it back.

The same register has been reached by other routes. Phil Corkum asks whether things have claim to their ontological status independently of standing in a relation to something else. And Michail Peramatzis separates two kinds of priority that are easy to run together.

You cannot have a smile without a face; take the face away and the smile goes with it. That is dependence for existing — in the literature, priority in existence. But there is a second kind. Even if faces and smiles had both existed forever, you still could not say what a smile is without mentioning a face. The face is in the definition, not only in the causal history. That is priority in being, and Peramatzis has argued that it, rather than the existential kind, is Aristotle’s operative notion.

Both are dependencies. Both are now well mapped. Neither is yet what this book needs, and saying why will take the rest of this section.

An interim finding: the whole family has one shape

Before the fourth reading is stated, something should be noticed about the three, because it is easy to miss when they are taken one at a time.

The relational reading of focality has already been pressed once from inside. Niels Tolkiehn has argued that Shields’s causal conditions do not establish core-dependence at all: shown that a causal relation obtains between two cases, one has not thereby shown that the case is one of core-dependence rather than of some other association. And Tolkiehn’s own repair is to replace the causal asymmetry with logical priority, on the model of Eudemian Ethics VII.2, where the primary is said to be that of which the definition is contained in all the others.

But logical priority is a relation between the definitions of two things already taken to be F. So is causal dependence. So is existential responsibility. Each of them explains why a and b belong in one category by pointing at a relation that presupposes they do.

That is not a defect in any of the three accounts taken as accounts of the relation. It is a limit on what a relation can do. No relation between derivative and core can constitute their co-membership, because every such relation presupposes it.

Shields’s own architecture records the difficulty. The causal conditions (i)–(iii) appear in the section of his chapter devoted to Cajetan’s proposal; condition (iv) is added later, in a section titled Definitional Priority. He found the causal relations insufficient and reached for a relation of priority — which is another relation of the same kind.

And there is a textual difficulty with the fourth condition worth recording, since it comes from Aristotle rather than from argument. At Metaphysics Λ.3:

ὅτε γὰρ ὑγιαίνει ὁ ἄνθρωπος, τότε καὶ ἡ ὑγίεια ἔστιν

for when the man is healthy, then also health exists. (1070a22–24)

If the healthy man is responsible for health’s existing, then on condition (iv) the healthy man is the core and health the derivative — which reverses the example the whole apparatus was built on. Tolkiehn draws this consequence and concludes that (iv) is not Aristotle’s but a later extension. Whatever one makes of that, the passage shows something the present argument needs: an existential relation cannot by itself fix which item is central. It gets the answer backwards on the standard case.

And the direction of the repair is worth noting, since it is not neutral. To make priority definitional — the primary is that whose definition is contained in all the others — is to make the primary an intelligible content that the rest carry within them. That is closer to a Form than to a this. Aristotle’s quarrel with Plato turns on refusing exactly that identification, and a repair that reinstates it has not saved the account so much as changed sides. The point is not that Tolkiehn intends this; it is that the pressure on the relational reading pushes its repairs outward in both directions — Shields toward causation, this toward definitional containment — and neither destination constitutes what the relation was asked to constitute.

The fourth reading: vicarious

Focality becomes ontological substitution when the primary case holds the derivative’s title to count as a being on the derivative’s behalf — so that the derivative appears in the ontological inventory on a warrant it does not itself hold.

Take the difference from the third reading concretely.

My parents are asymmetrically responsible for my existence. Without them I would not be here; without me they would still have been. That is exactly the relation condition (iv) describes, and it holds. Now notice what it does not give them: they do not hold my title to be a person. I am not a person on their warrant. The causal relation runs, and the title stays where it was.

A cause brings about. A proxy stands in place of. These are different relations, and nothing among Aristotle’s four causes is a relation of standing-in-place-of. A cause explains why something is there; it does not answer for it. The claim of this book is that a focal term does the second thing rather than the first.

The structure has four parts, which the tenant already gave us: a title to count as a being; a case that holds it; a case that appears by it; and an appearing-on-another’s-warrant which is neither the derivative’s own claim nor a causal consequence of the primary. Those four are what make this a shape rather than a denial. A reading specified only as not-causal would tell us as much about focality as a diver working in total darkness tells us about the sea floor when he reports that what he struck was not a cow. He has contact and exclusion; he does not have a shape, and no further list of what it was not will give him one. (The image returns in §7.1, where the same danger arrives from the other side.)

(One lexical accident should be disarmed before it is used against the distinction. Shields’s condition (iii) says that b’s being F “stands in one of the four causal relations” to a’s being F, and Tolkiehn’s formulation repeats the phrasing. That is standing-in-a-relation, not standing-in-place-of. The two vocabularies converge on a word and diverge on everything else.)

The evidence

Return to the Γ.2 list and read it as an inventory rather than as a set of correlations. The derivative cases do not enter on their own warrant; they enter as determinations of what primarily is. Ask what is there? and the answer is substance; the rest appear on that answer. Ask what a privation is, and the answer names a substance and something it lacks — so that blindness is on the list not because it has a claim of its own but because there is a man who cannot see.

Here the claim must be narrowed, because an Aristotelian’s first move is that accidents and privations are perfectly real. That is granted, and nothing in this book requires denying it. The claim is not that they are unreal but that within a focal structure their title to count as beings is held elsewhere. Focality entails substitution with respect to what licenses derivative cases as beings within one science, not with respect to whether they are real at all.

Why the fourth reading has not been stated This is a question about position, and it can be put as a test that anyone may run.

Think of someone who spends every day arguing against God. Their whole vocabulary — the arguments, the objections, the case against — requires God in order to be about anything at all. Take God away entirely and they are not victorious; they are silent. Everything they had to say was about the thing they were against.

That is the test. Ask of any position: what would it still be able to say if the thing it opposes simply vanished?

If the answer is “nothing,” then the thing has not been left behind. It is exactly where it was — still setting what can be said, still supplying the terms in which it is opposed, still the condition on which the opposing position is intelligible at all. The opposition does not merely fail to escape it. It depends on it, and the dependence is invisible from inside, because everything the position has to say is already said in its terms.

Three well-known positions give the same result. Each names an operation performed on the metaphysical principle — undoing it, weakening it, deposing it — and each needs the principle present in order to perform the operation. Self-description does not settle this. A position may describe itself as having left metaphysics and fail the test, and nothing turns on whether it says so.

Now let the test run on the case at hand, and note that it convicts nobody of carelessness.

From inside a metaphysics, that a thing’s being is grounded in another is not a deficiency in its standing. It is the ordinary way of standing: grounding relations are what an ontology is made of. So substance is responsible for the accident’s existence reads as an explanation, and the enquiry stops there. No occasion arises to ask on whose warrant the accident appears in the inventory at all — because appearing on another’s warrant is, in such an ontology, simply what it is for a derivative to be.

Only where no ground is installed does the same relation show a second face. Then a derivative that does not hold its own title is not merely dependent but represented, and the focal term is seen to be doing something other than causing.

This is not a reading that was missed. It is a reading unavailable from the position from which the relation has been analysed — which is why the vocabulary of title, warrant and proxy is absent from every account cited above, and the vocabulary of cause, dependence and responsibility is present throughout.

One retort is obvious: that this book’s own differentia is reached deictically — by pointing, the way this and that point — and so commits what it diagnoses, since a position that must point in order to reach its subject has made a fact about its access into a fact about its object.

The objection has been answered in print and is not re-argued here, but its shape must be given accurately, because it turns on three distinctions and not two. Deictic determination is the ontological determination of an occurrence as this one: what gets reached, settled independently of anybody’s reaching. Deictic reaching is the regional route by which this occurrence rather than another becomes the one at issue — a defeasible differential connection sustained within a relational and historical field, which may be perceptual, bodily, practical, addressive, mnemonic, testimonial, or archival. Linguistic deixis — the demonstrative, the this, the pronoun — is one vehicle of that reaching among several, and nowhere the paradigm. Determination is what is reached; reaching is the access; the demonstrative is a vehicle. And the reaching never constitutes what it reaches.

That last sentence is not a stipulation, and the evidence for it is the ordinary corrigibility of reaching. A nurse can mis-track an infant and be corrected by the bands; a testimony can misdirect and be corrected by a record; a recognition can fail at dusk and recover at the door. Fallibility by itself proves nothing — constitutive acts, too, can misfire under their own rules — but the shape of the correction is the evidence: a misreaching is corrected by resistance from within the field of the occurrence it was reaching for. What answers to something beyond itself does not constitute what it answers to.

And the objection has a cost it does not usually notice. Remove the first of the three terms — hold that nothing stands determined before the reaching — and the other two collapse into each other: determination becomes whatever the reaching confers, and the corrigibility just described becomes unintelligible, since a misreaching would be answerable to nothing. The objector must then either make the reaching constitutive and give up the correction, or keep the correction and concede the first term. There is no third position, because the demonstrative he might reach for is a vehicle of the access and not something prior to it.

One thing more is owed forward, and it will matter at §7. What has been specified in this section is a relation, and specified positively: a title, a case that holds it, a case that appears by it, and an appearing-on-another’s-warrant that is neither the derivative’s own claim nor a causal consequence of the primary. That specification is what allows §7 to state its differentia by excluding this relation without falling into the fault this section has warned against. A negation is empty where what it negates has no shape. The diver has only exclusion because the field he excludes from is unspecified. Here the excluded relation has been given four parts and several pages before the exclusion is performed. Determinate negation is not the diver’s predicament, and §7 is entitled to the difference because this section paid for it.

Finally, what kind of argument has this been? Not a proof from premises everyone already grants. It is a diagnosis: it says what a structure must be doing, given what it produces, and it can be wrong in the ordinary way diagnoses are wrong. Two of its assumptions carry weight and are better stated than buried: that the unity of a science requires more than family resemblance among its uses, and that what is prior in the order of being is capable of holding another’s title.

§4. The second answer

The second answer refuses the question rather than answering it.

Scotus says there is one concept of being, and it does not split — it narrows. Think of colour: red and blue are not two meanings of “colour,” they are colour narrowed. Being is like that, and it narrows all the way up to God, contracted by what the scholastics call intrinsic modes — finite or infinite, created or uncreated — without ever ceasing to be one concept.

This is univocity, from una vox, one voice.

His test for it is exact, and worth unpacking because it shows what the doctrine is for. A concept is univocal, he says, when it has enough unity in itself that affirming and denying it of the same thing would be a contradiction. The reasoning is this. If “being” meant different things in the two halves of the sentence, then saying X is and X is not would be no contradiction at all — you would merely be using a word twice, as one does with river banks and savings banks. That we do hear a contradiction is Scotus’s evidence that one concept is at work throughout.

This is not a weaker version of the first answer. It is a different device. Focal reference orders many senses toward one; univocity denies that there are many senses of being to order, and relocates the plurality to modes within a single sense. The first secures unity by a privileged term, the second by a common concept, and neither needs the other.

(That Scotus is also the ancestor of the datum on which this book’s own differentia rests — that an individual is not repeatable — deserves a sentence and no more. He reached it, and answered it with a formal individuating principle. The datum is accepted here; the machinery is not.)

Leibniz: the answer completed The second answer is not left unfinished, and it is worth seeing it carried through, because what it becomes is not what one expects.

Leibniz builds it out entirely. An infinity of substances, each expressing one and the same world from its own point of view; no interaction between them, since each is windowless; each individuated by its complete concept, so that what distinguishes any two is what is truly said of them; and the whole harmonised in advance. Every consequence drawn, none evaded.

And it does not need its theology. In the fragment known by its opening words, Ratio est in natura, Leibniz argues that whatever makes possibility demand existence also gives every possible a striving toward existence — because no reason can be found, in general, for restricting the striving to some possibles rather than others. Universality by default. He then coins a word to name the resulting state: existiturire, a desiderative built on the future participle, a hungering toward being-about-to-be. Every possible existiturit.

The manuscript records what happened next. Where the striving is first announced, Leibniz wrote in a clause founding it on the necessary being — and struck the clause out, letting the universal striving stand alone. One proposition later, where the striving is named, the clause returns and stays.

Whatever one concludes about Leibniz’s intentions, the structural point is available: remove the existentifier and the machinery still runs. Something in the field is doing what the office was installed to do. §5 will say what.

Deleuze Deleuze’s univocity is the one that matters here, because it is the settlement that most resembles what the third answer wants and arrives by an entirely different road. It should be taken at full strength.

There has only ever been one ontological proposition, he writes: being is univocal. A single voice raises the clamour of being. And then the reformulation that is his own: being is said in a single and same sense of everything of which it is said, but that of which it is said differs — it is said of difference itself.

His image for what is at stake is of dividing land. In a sedentary distribution the ground is parcelled out: fences, allotments, each holding proportioned to rank, and someone doing the assigning. In a nomadic distribution there are no fences — people spread across open ground and the ground is never divided at all. Analogy, for Deleuze, is the first of these. Univocity is the second.

The mechanism is precise and it works. Hierarchy enters ontology through analogy: if being is said in different senses across the categories, each sense is referred to a higher term, and beings get ranked by nearness to what is primary. Say instead that being is spoken in one and the same sense of all its individuating differences — his phrase for whatever makes each thing the one it is — and there is no higher term left to rank them by. Difference is no longer mediated; it is immediately in being. Dispersion is not a wish here. It is what follows once the term that would arrange the field has been denied.

What Deleuze does with Leibniz

He wrote a book about him, and it ends where the present argument says it must.

In The Fold, Leibniz’s monads are described as submitting to two conditions: closure, since each includes an entire world that does not exist outside it, and selection, since that world is distinguished from other possible but divergent worlds, and since each monad fashions its own clear zone of expression within what it includes.

Deleuze removes both. Selection goes first: harmonics lose their privilege of rank, dissonances are excused from being resolved, and divergences can be affirmed. Then closure: once a monad is in tune with divergent series belonging to incompossible monads, it can no longer contain the entire world as if in a closed circle. It is astraddle over several worlds, kept half open; monadology becomes nomadology.

And what survives both removals is a singular term. The world is still the world — now made up of divergent series, the chaosmos. The incompossibilities arrive on the same stage, where Sextus will rape and not rape Lucretia, where Caesar crosses and does not cross the Rubicon. And the book closes: we are all still Leibnizian, because what always matters is folding, unfolding, refolding.

The distinction, and what each answer pays

Deleuze secures dispersion by making being univocal, and the plurality migrates inward, into differences within one sense. The third answer keeps the plurality of senses — being is modal, and a regional manner may be truly predicated of plural occurrences — and refuses the focal term anyway.

Deleuze pays for dispersion with univocity. The third answer pays with a differentia. Both are coherent. They are two settlements of one problem, and each declines what the other keeps.

One point of contact deserves a sentence and no leaning. Deleuze’s polemic names Aristotelian analogy, and Owen’s core-dependence is precisely the reading on which the categorial case is not mere analogy — so the apparatus §3 analysed is not the apparatus he engages.

And the sentence this section earns: the second answer does not need its God, and the routes out of it keep what mattered. Deleuze keeps the unity of the stage; the immanent reading of Leibniz keeps the quantity of essence. What is removed is the office. What survives is the measure.

What that means, and why it was not avoidable, is the business of Part Two.

Notes to Part One [§1] Owen, “Logic and Metaphysics in Some Earlier Works of Aristotle,” in Düring and Owen, eds., Aristotle and Plato in the Mid-Fourth Century (Göteborg, 1960). The coinage is his; the paper is universally credited with it.

[§2] The 1003b5–10 list is given in Ross’s translation (Oxford, 1908; the basis of the Revised Oxford Translation). Two variants circulate in the last clause — negations of some of these things (Ross 1908) against negations of one of these things in the Internet Classics Archive text; the difference does not bear on the argument. Ross is the edition of record for the Metaphysics throughout this study, being the basis of the Revised Oxford Translation and the only rendering covering all of Γ, Δ, Ε, Ζ, Ι and Λ, from each of which this study quotes. Kirwan’s Clarendon volume is noted where a rendering is disputed and is followed nowhere. — Aubenque, Le problème de l’être chez Aristote (Paris, 1962; 2nd edn 1966); the concluding chapter is available in English as “Science Regained,” trans. Clayton Shoppa, in Greenstine and Johnson, eds., Contemporary Encounters with Ancient Metaphysics (Edinburgh, 2017), 119–137.

[§3] Shields, Order in Multiplicity: Homonymy in the Philosophy of Aristotle (Oxford, 1999); the definitions at pp. 106, 119 and 124, the last added in the section on definitional priority. — Corkum, “Aristotle on Ontological Dependence,” Phronesis 53 (2008): 65–92. — Peramatzis, Priority in Aristotle’s Metaphysics (Oxford, 2011), 13. — Tolkiehn, The Notions of Homonymy, Synonymy, Multivocity, and Pros Hen in Aristotle, Dissertationen der LMU München 37 (Munich, 2020), §6.1.1.5. — The reflexive objection is answered in the author’s The Individuating Factor, §7.3, and in Ethical Ontology, Volume I, ch. 11.

[§4] Scotus, Ordinatio I d.3 p.1 qq.1–2; the univocity texts in Wolter, Duns Scotus: Philosophical Writings (Indianapolis, 1987), 19–22. — Ratio est in natura, Akademie-Ausgabe N. 2680, ed. Henrik Wels, §§5–6 with apparatus; also Gerhardt, Phil. Schr. VII, 289–291, and Couturat, Opusc. et fragm., 533–535. The fragment is treated at length in the author’s Positive No-Ground, ch. 6. — Deleuze, Difference and Repetition, trans. Patton (New York, 1994), 35–36 = Différence et répétition (Paris: PUF, 1968), 52–54; the univocity section runs Patton 35–42 = PUF 52–61. — The Fold: Leibniz and the Baroque, trans. Conley (Minneapolis, 1993), 137 and the neo-Baroque passage.

On the possible and the virtual. Deleuze distinguishes the virtual from the possible on the ground that the possible is opposed to the real and undergoes “realisation,” while the virtual possesses full reality and undergoes actualisation; and he charges the possible with being retroactively fabricated in the image of what resembles it, so that existence becomes a brute eruption subject to all-or-nothing (Difference and Repetition, 211–212). The contrast requires the possible to be inert. Leibniz’s is not. His possible is prior rather than derivative, determinate throughout, and existiturit — and existence there is not a leap but the outcome of striving within a topology of compossibility. Deleuze does not name Leibniz in the passage, and the point is not that he misreads him; it is that his own principal source does not supply the inert possible the contrast needs. The pull is relocated rather than removed: it is on the basis of the virtual’s reality, he writes, that existence is produced.

On the reflexive question. Deleuze asks whether, when we speak the univocal, it is not still the equivocal that speaks within us — but he asks it of Aristotle, as a fracture in that position, and concludes that nothing of the kind occurs there (Difference and Repetition, 33). The question is not turned on univocity, and The Fold does not treat univocity at all.

Part Two follows: §5, Only Two Devices; §6, The Contemporary Demonstration.

PART TWO

Why There Were Only Two

§5. Only two devices

Any ontology that recognises a plurality of categories owes an account of why they are categories of being, rather than a list of things that happen to have been put together. Without such an account the plurality is not a plurality of anything — it is an enumeration, and enumerations do not have subjects. This is the demand that generated the first answer and the second, and it does not disappear when either is refused.

Two devices have met it. The first installs a privileged term and orders the rest toward it. The second denies that there are many senses to order, and relocates the plurality inside a single sense. What has not been available is a third — and this section argues that its unavailability is not an accident of who has worked on the problem, but a consequence of how the categories in question have been marked off.

A category is marked by a differentia. And a differentia may mark a difference of degree or a difference in kind. The distinction is old and is usually treated as a matter of precision. Here it decides the structure of the whole ontology, and the reason can be set out in Aristotle’s own vocabulary.

5.1 Comparison, measure, and where measure comes from

Three things need distinguishing, and the Greek keeps them apart more carefully than the English does.

The principle at issue is stated in terms of comparison. At Physics VII.4, discussing whether motions of different kinds can be set beside one another, Aristotle formulates the rule that governs the whole subsequent discussion:

εἰ μὴ ὁμώνυμα συμβλητά

if not homonymous, comparable. (248b6–9)

συμβλητός comes from συμβάλλω, to throw together — to set alongside, to bring into one view. It is the wide word. Two things are συμβλητά when they can be placed side by side and something said about how they stand to one another. And the chapter that states the rule is saturated with the vocabulary of homonymy: seven of the nine occurrences of ὁμώνυμον in the whole Physics fall in VII.4, while synonymy appears nowhere in it.

The rule is standardly rendered as a principle about commensurability, and Shields gives it that way: unless two things are F synonymously, it is impossible to compare them in terms of F-ness. But “commensurable” is a translation choice. Aristotle’s word there is the wider one.

He has a narrower word, and he keeps it for geometry. In Metaphysics Δ.7 — the chapter that enumerates the senses of being, and gives the senses following the figures of predication at 1017a22–23 — the strict word for sharing a measure appears exactly once, thirteen lines later:

οὐκ ἔστιν ἡ διάμετρος σύμμετρος, ὅτι ψεῦδος

“the diagonal is not commensurable” — because that is false. (1017a35)

The diagonal and the side. σύμμετρος, with-measure: the classic theorem, produced here as the stock example of a falsehood, in the chapter that lists the ways being is said. When Aristotle wants the strict notion of sharing a measure, he is doing mathematics.

And he says where the notion comes from. In Metaphysics Ι.1, working out what it is to be one, he arrives at measure and immediately locates it:

μάλιστα δὲ τὸ μέτρῳ εἶναι πρώτῳ ἑκάστου γένους καὶ κυριώτατα τοῦ ποσοῦ· ἐντεῦθεν γὰρ ἐπὶ τὰ ἄλλα ἐλήλυθεν.

…but especially to be the first measure of each kind, and most properly of quantity; for from here it has gone to the others. (1052b18–20)

ἐλήλυθεν is the perfect of ἔρχομαι: has come, has gone. Aristotle is not saying that measure applies analogically elsewhere. He is saying it travelled — that the notion belongs first to quantity and moved out from there into the other categories. And he says what a measure does, in the next line: μέτρον γάρ ἐστιν ᾧ τὸ ποσὸν γιγνώσκεται, measure is that by which the quantity is known (1052b20). What a measure answers is how much, which is what ποσόν means.

So the three notions stand in an order. Comparison is the widest: setting alongside. Measure is narrower and openly quantitative. And measure has migrated outward from quantity into the rest. The question this section presses is how far the migration reaches, and whether it arrives at a difference of manner at all.

5.2 What a scalar differentia does

More here, less there, all alike, all full. Every one of these is a way of speaking about portions — and a portion is a portion of something. To say that one case has more of what marks the category than another is already to have assumed a stuff that divides.

You can have more water. You cannot have more Tuesday.

That is the whole of it, and it can now be stated as a principle. A scalar differentia makes its category a quantity. Its instances become portions of what the differentia measures; portions stand to one another in an order of more and less; and that order is exactly what a focal term formalises. The primary case is the portion by which the others are measured.

Notice what this does not say. It does not say that anyone decides to install a primary case, or that hierarchy is chosen. The focal term does not impose the ordering. It names one that the differentia has already produced. Which is why refusing hierarchy as a preference never works, and why a maximum need not be honoured, or noticed, in order to operate. Wherever the differentia is scalar, the question which case has most of it is available — and a question that is available has an answer.

5.3 The objection

An objection has recently been pressed which, if it holds, removes the ground from under this section. It should be stated at full strength.

Focal terms do admit comparison. Aristotle himself says that health is more good than healthy things. So gradability and focal structure are not opposed but companions — and if that is right, a category can perfectly well admit more-and-less while being organised toward a primary case, and nothing whatever follows from non-scalarity. On this view what blocks comparison is not degree but homonymy: you cannot compare a riverbank with a savings bank, because there is no single thing being compared.

The objection is well made, and the reply is not that it is wrong.

That focal terms admit comparison is granted — indeed §3 predicts it. Where a primary case holds the derivatives’ title, the derivatives stand to it in an order, and orders admit comparison. The present claim runs the other way.

And the source of the objection concedes the direction. Donald Morrison, whose reading the objection follows, states the exception precisely: normally one is not allowed to compare across ambiguity, but when the items to which the ambiguous predicate is applied are related to each other as prior and posterior, then comparison is allowed.

Read the structure of that. The ordering licenses the comparison. The comparison does not produce the ordering. Where a prior-and-posterior relation obtains, the exception fires and comparison is permitted despite the multiplicity of senses. Where no such order forms, the exception has nothing to fire on, and Aristotle’s general prohibition stands.

So the present claim is not that comparison is impossible wherever a focal term exists. It is that comparison is a quantitative relation, and categorial difference is not a quantity. Difference of manner is not division of a mass. A category marked by a non-scalar differentia does not admit the question which case has more of it, and so supplies nothing for comparison to order.

One deflection remains. It might be said that all of this is a fact about adjectives rather than about the world — that gradability belongs to language and not to what language is about.

The present argument does not need to settle that. It needs one conditional: if the instances of a category stand in more-and-less with respect to what marks the category, then there is an order among them, and an order is what a focal term formalises. Whether the order is called semantic or metaphysical does not make it absent. And an opponent who calls it merely semantic owes an account of how health is more good than healthy things remains a claim about health and healthy things, rather than a report about the word “good.”

5.4 The demonstration: one text, two definitions

The argument to this point is structural, and structural arguments are easy to doubt. What follows is a case in which both structures appear in one text, in the same grammatical form, written by one hand — and in which the critical apparatus records the moves between them.

The text is a working sheet of four manuscript pages, provisionally dated after September 1698, known by its opening words: Ratio est in natura. Twenty-four numbered propositions. It survives in Leibniz’s draft, was printed by Gerhardt and by Couturat, and was reprinted by Heidegger in the second volume of his Nietzsche.

What this section owes and what it adds should be separated before the reading begins, since the fragment has been worked before and not for this purpose.

The reading of Ratio est in natura on which everything below rests is not new. The universal striving and the coinage existiturire; the conflict of possibles settling into a maximal series; the drop gathering sponte naturae and what it shows about determination without a selector; perfection as quantitas realitatis; order defined at §15 as a relation among the many, doing one work, with no ground beneath it and no purpose above it — all of this is established in an earlier study of the fragment, which reads §15 as exceeding the office erected at §§2–4 to secure it. That reading is inherited here entire, and this section adds nothing to it.

What is constructed here is two things. The first is the manuscript apparatus — the revision of §11’s definitional clause, the struck founding clause at §5, and the word added at §20. Two cautions belong with it. The clause struck at §5 is not struck from the fragment: it reappears at §6, qualifying not the striving but the naming — it can be said that every possible strives to exist, in so far as it is founded… A clause moved from the assertion to the naming is a more delicate operation than one removed, and it is what the apparatus shows and the printed text cannot. And §11’s three beginnings are not distinctive: §9 has three as well. The evidential weight lies in the definitional revision and in the insertion at §20, not in the count of false starts. The second is the structural reading of the two definitions against each other: that a scalar definition and a non-scalar definition stand four propositions apart in identical grammatical form, that they give the field incompatible structures, and that the non-scalar one does not survive its own text. The earlier study found a strain in the fragment and named it. This section claims that the strain has a resolution recorded in the ink, and that the resolution went the way §5.2 predicts.

The fragment’s characteristic gesture is definition by nihil aliud quam — “nothing other than.” It occurs five times: of cause (§2), of perfection (§11), of order (§15), of pleasure (§18), and of justice (§20). Three of these matter here.

The scalar definition, with the superlative in the same sentence

(11) Existit ergo perfectissimum, cum nihil aliud perfectio sit, quam quantitas realitatis.

Therefore the most perfect exists, since perfection is nothing other than quantity of reality.

The superlative is inferred, not posited. Ergo — therefore. And the ground of the inference is given in the same sentence: perfection is an amount. Once perfection is quantity of reality, the most perfect exists the way the largest number in a finite set is the largest — by arithmetic, not by decision.

The apparatus shows the sentence being built. Leibniz began the proposition three times: first Sequitu[r] — it follows; then Porro — furthermore; and finally Existit — it exists. He moved from an inferential connective to a flat existential assertion. And he revised the definitional clause too, from nihil aliud enim perfectio est — a causal for with the indicative — to cum … sit, a subordinate since with the subjunctive. The definition was moved out of an aside and into the logical structure of the inference.

The propositions on either side confirm what kind of field this is. The prevailing series is the one through which the most exists — Series omnium possibilium maxima (§9). Its illustrations are extrema: among lines the straight, among angles the right, among figures the most capacious, the circle or sphere; and, as liquids gather of themselves into spherical drops, so in the universe the most capacious series exists (§10). A maximum forming sponte naturae, without anyone choosing it — which is the picture of what §5.2 states as a principle.

The non-scalar definition, four propositions later

(15) Est enim ordo nihil aliud quam relatio plurium distinctiva. Et confusio est, cum plura quidem adsunt, sed non est ratio quodvis a quovis distinguendi.

For order is nothing other than a relation that distinguishes a plurality. And confusion is when many are indeed present, but there is no ratio for distinguishing anything from anything.

A relation that distinguishes admits no most. There is no more-distinguishing and less-distinguishing in the way there is more and less reality; and the companion clause makes the alternative binary — either there is a ratio for telling one thing from another or there is not. The next proposition applies it operationally: atoms are abolished, and with them any body in which there is no ratio for distinguishing one part from another (§16).

Two definitions, four propositions apart, in identical grammatical form, giving the field incompatible structures. Perfection is an amount, and an amount has a maximum. Order is a relation, and a relation that distinguishes has none.

The absorption, and it is an insertion

(20) … cum Justitia nihil aliud sit, quam ordo seu perfectio circa Mentes.

… since justice is nothing other than order, or perfection, concerning minds.

Seu makes them equivalent. The relation-word and the quantity-word, set side by side as alternatives for one thing.

And the apparatus records that the second of them was added. Leibniz first wrote quam ordo circa Mentes — order concerning minds — and then inserted seu perfectio. The quantity-word was written in beside the relation-word.

That insertion is what this section has been arguing, visible in the ink. The non-scalar definition does not survive its own text. Five propositions after order is defined as a relation that distinguishes, perfection is written in beside it — and perfection has already been defined as quantitas realitatis, from which a superlative follows.

And the insertion was not an accident of drafting. It was required by what precedes it. At §9 the criterion of existence is already quantitative — what exists is the series through which the most exists, series omnium possibilium maxima — and at §11 the measure is named. Once a world is the one selected by a quantity, whatever cannot be said in quantitative terms has no standing in the selection. Order, defined at §15 as a relation among the many that distinguishes them, had no part in that selection and could not survive the accounting. The manuscript shows the mechanism, and then shows the mechanism operating, four propositions apart.

One remark about the kind of evidence this is, because the kind is rarer than the finding. Almost everything an argument of this sort can offer is reconstruction: a claim about what an architecture required, inferred from what it produced. §3 is reconstruction. §5.2 is reconstruction. §6’s four cases are reconstruction, however careful. Reconstruction can be right, and can be shown wrong in the ordinary way, but it always leaves the opponent the reply that the architecture might have been doing something else. This is not reconstruction. A clause was struck and re-placed. A definition was rewritten. A word was written in beside another. The operation described is not inferred from the fragment’s results; it is recorded in the fragment’s own margins, by the hand that performed it. And the method is the edition’s own: the Akademie editors date this piece by Leibniz’s mid-writing correction of his neologism, existurire to existiturire, comparing it with the same correction made at a slight remove elsewhere. Apparatus evidence is not a device borrowed for this argument. What is claimed is what the manuscript does, never what its author intended — and that is the strength of it, since intention is where an opponent would otherwise take the argument, and there is nothing here for that argument to take hold of.

None of this is a defect in Leibniz. He is the philosopher least willing to leave a consequence undrawn, and the fragment is evidence precisely because he draws them all. The point is structural. In a field organised by degree throughout — the maximal series, the most capacious figure, the most distinct thinkability (§14), the greatest variety in the least space (§21), the greatest quantity of perfection (§23) — a relation that merely distinguishes has nowhere to stand. It is absorbed by the term that does.

One complication should be stated rather than smoothed. The fragment grades distinctness one proposition before it defines order non-scalar: the prevailing series is the one through which the most distinct thinkability arises (§14). So the definition at §15 is not a quantity and the use at §14 makes it one. That is the same shape as a difficulty in Aristotle which §5.6 takes up, and it is the shape this section is about: the definition does not grade; the use does; and the making is the movement to be explained.

5.5 The same thing happening now

The demonstration is historical. The pattern is not.

Consider the differentiae actually used to mark off a category of beings that answer, choose, or are owed something: rational animal; self-conscious being; person as bearer of rights; system of sufficient complexity.

Each is scalar. Each therefore partitions. And each produces the same periphery, arranged by how much of what the differentia measures a case has: the newborn, the animal, the sleeper, the cognitively impaired, the dying, the machine.

Most readers will have met every one of those debates. What is worth noticing is that they all have the same shape — and the shape is not a coincidence of the particular theories. It is identical across theories that disagree about everything else, which is what makes it structural rather than incidental.

Note what is not being claimed. Not that these theories are wrong about their peripheries, nor that their maxima are badly chosen. Only that the ordering was not something they decided.

5.6 A distinction Aristotle draws himself

Categories 5 denies gradation within a kind while granting it between kinds, in the same breath:

Substance, it seems, does not admit of a more and a less. I do not mean that one substance is not more a substance than another (we have said that it is), but that any given substance is not called more, or less, that which it is. For example, if this substance is a man, it will not be more a man or less a man either than itself or than another man. (3b33–4a9)

It is tempting to draw a stronger conclusion from this than the texts will bear: that a core is never among the things it measures — health is not healthy, substance is not more or less substance, and both follow from the core standing outside the ordering it anchors.

Aristotle denies the general form. At Ι.2 he writes that the one in colours, at any rate, is a colour, e.g. the white (1053b29). The measure of colours is a colour.

The narrower version is the true one, and it is stronger. Colours admit of more and less; white is an extreme of a bounded range. Where a genus is scalar, its measure sits within it, at an end — which is exactly what one expects if measuring is a quantitative operation and measure has migrated out of quantity. And where a genus is not scalar, there is no range, so no extreme, so no measure of that kind is available at all.

The two passages are not in tension. Ι.2 shows what the measure-structure looks like where there is a quantity; Categories 5 records that in the case of substance there is not. Health is not healthy because health is not the top of a range of healthinesses. Substance admits no more and less because substance is not the top of a range of beings.

This is an inference across three texts — Ι.1 on measure as first of quantity, Ι.2 on the measure of colours, Categories 5 on substance — and it is offered as an inference rather than as a reading of any one of them.

It also disposes of an objection the third answer will meet. If a category has no maximum, is it a category at all? Aristotle’s own primary case admits of no more and a less, and is a category.

5.7 The white, twice

One more thing follows from Ι.2, and it reaches forward to the second answer.

Aristotle’s measure of a scalar genus is white. And when Deleuze explains how univocal being can include its individuating differences without being changed by them, he reaches for the same illustration, at Difference and Repetition 36: univocal being includes its differences just as white includes various intensities, remaining essentially the same white.

Intensities. The image chosen to show how the univocal holds a plurality without hierarchy is a bounded scale — and the surrounding clauses run on equal and not equal: being is the same for all the modalities though they are not the same; equal for all though they are not equal; said in a single sense though they do not have the same sense.

Two things follow, and neither requires polemic.

The classical answer and the modern one illustrate themselves with one example, and it grades. If this section’s argument is right, that convergence is not a coincidence. Where a plurality is to be held together without a focal term, and no non-scalar differentia is available, what one reaches for is a scale.

And the criterion is visible in the same sentences. Sameness appears in every clause and is not among the modalities. It is what distributes them, not one of the things distributed.

Deleuze had the problem exactly: how to hold a plurality without hierarchy. What was not available was a differentia that could disperse without a common measure. The white analogy is what one reaches for in its absence.

5.8 The conditional, and a prediction

If a differentia were non-scalar — if what it marks either obtains or does not, with no more and no less — the category would not be a quantity, nothing would partition, and no primary case could form by which the others were measured. Whether any such differentia exists, and what it would be, is the business of §7.

What this section establishes is narrower and, if it holds, sufficient. The two classical devices are not two choices among many. They are the two options that remain when a difference in kind is unavailable. The pros hen was not, on this account, a preference of Aristotle’s that a bolder thinker might have declined. It is what is left when categories are marked in a way that admits degree.

And that yields something checkable. Any ontology recognising a plurality of ways of being, and lacking a non-scalar differentia, will either install a focal term or fail to say why its plurality is a plurality of being. There should be no third outcome.

The next section watches it happen.

§6. The contemporary demonstration

The prediction is being tested now, in both directions, by a field with no stake in the present argument.

Ontological pluralism in analytic metaphysics dates from 2009–10, with a book-length statement in Kris McDaniel’s The Fragmentation of Being (2017). Its ancestry is explicitly Heideggerian. And its form is worth learning, because it is the form the whole discussion takes.

In modern logic, “there is” is written with a single symbol — the existential quantifier — and to say that something exists is to say that it can be the value of a variable which that symbol binds. One symbol; one way of existing. Ontological pluralism proposes more than one: perhaps one quantifier for abstract things and another for concrete ones, with no single “there is” covering both. Jason Turner’s formulation is that a logically perspicuous description of reality will use multiple quantifiers which cannot be thought of as ranging over a single domain.

Notice what this pluralism is about. It is a claim about the logical vocabulary that a complete description of reality would require. It is not a claim about what marks one kind of being off from another. That absence is not a criticism; it is the point of this section. The condition §5 identifies is simply not in play, which is precisely why the prediction applies.

6.1 The first direction: analogy is taken

McDaniel’s 2010 paper is titled “A Return to the Analogy of Being,” and the author glosses the title in a footnote: on most versions of ontological pluralism being is taken to be an analogous feature — hence the name.

It runs on Aristotle’s own health example, cited to Metaphysics IV. Healthy is said of an organism, of a circulatory system, of a foodstuff, each in virtue of something different — the organism flourishing, the part functioning properly, the food contributing to an organism’s flourishing — and the whole is unified without being a mere disjunction. And the inheritance is stated directly: being is not a genus, and the modes of being are each more fundamental than being simpliciter, which is offered as a reading of Aristotle.

Then the ordering arrives. In the ontological scheme of that paper’s fourth section, attributes exist in substances: the mode of being of an attribute is two-placed, that of a substance one-placed. And the modes are ranked. The mode of being of substances is prior to the mode of being of attributes — to grasp fully how an attribute is, one must be acquainted with how a substance is. The view, McDaniel says, captures the intuitions of Aristotle and Aquinas.

The most precision-minded plural ontology available, needing an account of why the many ways of being are ways of being, returns to the analogy — takes Aristotle’s own focal example as its paradigm, puts substance in the primary position, and announces it in a title.

The same body of work supplies the scalar reading directly. On McDaniel’s account, the degree of being enjoyed by an entity is proportionate to the naturalness of the most natural mode of being it enjoys. Degrees of being, in a plural ontology, from the field’s leading figure, and a chapter devoted to them.

6.2 The second direction: analogy is refused, and the price is paid

Turner’s multiple quantifiers dispense with any central term. And the charge that follows is that the disagreement is not real.

If the pluralist’s two quantifiers translate without loss into the monist’s one quantifier plus two predicates — is abstract, is concrete — then the pluralist has not described a different world. He has written the same description differently. A notational variant, not a rival theory.

A further argument presses from the side of logic. Applying the standard criterion for whether a piece of vocabulary belongs to logic at all — invariance under relabelling of the domain — Griffiths and Paseau conclude that the test favours the monist: the pluralist’s restricted quantifiers do not pass it, and if the test is loosened enough to admit them, it admits is abstract and is concrete as logical too, which the pluralist rejects.

Both charges are usually read as technical, and they are not. They are two registrations of one fact about how the position is stated.

Look again at the formulation. A perspicuous description of reality will use multiple quantifiers which cannot be thought of as ranging over a single domain. The single domain has to be there for the cannot to bite. The plurality is specified by what it is not ranging over — and that is a fact about the monist’s device, stated in the monist’s terms, with a number attached.

So the translation goes throughbecause the pluralist never said anything the other vocabulary could not carry. He said not one domain, and not one domain is expressible as a distinction between two predicates. The notational-variance objection is not an accident of translation manuals. It is the discovery that a position stated as a denial can always be re-expressed in what it denies. And the logicality argument finds the same thing from the other side: loosen the test enough to admit the pluralist’s quantifiers and it admits the monist’s predicates too, because there was never a difference of content for the test to detect.

The question the objection is really pressing is this: how does this plurality describe itself independently of the grammar it opposes? The answer available in the literature is that it does not. It describes itself by quantifier count — which is an arithmetic of the opposed term, not a differentia of its own. §5’s condition is not merely absent here. Its absence is what the objection has found.

So the map has four positions rather than two. Focal term: Aristotle, and now McDaniel. Univocity: Scotus, and after him Deleuze. Bare quantifier-plurality with no device to unify: Turner, facing the homonymy charge. And the third answer.

§5’s claim is not a prediction about what would happen if its condition went unmet. It is a description of what is happening where it is unmet.

6.3 Two systems further afield

Both confirm the pattern, and the objection to each is Aristotle’s rather than this book’s.

Badiou Things do not arrive in ones. A heap of sand is not one thing until somebody counts it as one, and the counting is done to the sand — it is not something the sand was.

Badiou builds an ontology on that. The One is never found, only performed, in what he calls the count-as-one; mathematics is ontology; and beneath every situation sits the empty set, which he names the proper name of being.

The first half of that is Ι.2’s question and Ι.2’s answer. Is there a principle of oneness, something essentially one that accounts for everything else’s being one? Aristotle says no, against the Pythagoreans and against Plato: to be one is to be indivisible and to be a measure, and there is no One-substance behind the measuring.

They part at the second step, and the second step is this book’s. Having denied the One, Aristotle distributes the measure regionally — the one in colours is a colour — so that each genus has its own and no single one inhabits substance, quantity and quality alike. The parallel he draws is the good, which occurs in every category with no category-transcendent good-maker.

Badiou denies the One and then supplies a term that does its work — and he does it deliberately. The void is unique: not several voids but one, and its mark is a proper name precisely because a proper name requires a unique referent. It is universally included, a subset of every set, present in every presentation — what he calls the omnipresence of the void, and its errancy in all presentation. And it is named, in a chapter heading, the proper name of being.

He derives the unicity from the denial, and says so: it is because the one is not that the void is unique. He calls the conclusion remarkable, and it is. The distinction he takes care to draw is between the one and unicity — and having drawn it, he takes the second from the refusal of the first.

Which is why this is an exhibit and not a lapse. The denial of the One does not by itself disperse anything. Aristotle saw that the denial must be followed by a distribution — the one in colours is a colour — so that each genus has its own measure and none inhabits all. Where that distribution is not made, the denial can itself produce the replacement, and here it produces one by argument.

Harman Object-oriented ontology is explicitly flat, and its author is precise about what the flatness is. He denies holding that all objects are equally real; his claim is that they are equally objects.

Which requires object to have a differentia — and it has one, announced in a chapter title. Objects are denied in only two ways: reduced downward to something more basic, which is undermining; or dismissed upward in favour of their qualities and relations, which is overmining. Both fail. What survives is the object as irreducible to its own pieces and equally irreducible to its outward relations with other things — a dark crystal in a private vacuum. That double irreducibility is what makes them all equally objects, and everything counts as an object on its warrant: diamonds and rope and neutrons, and equally armies, square circles, pixies and utopias.

And it is stated in two negations. Not reducible downward; not reducible upward. Which is the fault §6.2 has just described — a position specified by what it declines, and therefore restatable in what it declines.

So a categorial difference is doing the work in an ontology that says there are no categorial ranks, and the difference is a pair of refusals. And the form of that objection is Aristotle’s standard one against unitary positions. In Physics I.2–3 against Parmenides and Melissus: you cannot say all is one without saying in what way it is one, and saying that requires the distinctions the thesis denies. In Γ.4 against the denier of non-contradiction: he need only say something determinate, and in saying it he uses the principle.

The position cannot be stated in a vocabulary it permits.

Two systems, one diagnosis. Badiou takes a unique term from the denial of the One. Harman takes a differentia from the refusal of two reductions. Neither supplies the distribution that the denial requires, and in both a positive-looking result is drawn out of a negation.

6.4 The rival conclusion in this book’s own source

One position remains, and it should be met rather than left for a reader to find, because it comes from the book this study has relied on most.

In the final chapter of Order in Multiplicity, Shields argues that Aristotle fails to establish the homonymy of being. The argument runs: all beings are commensurable; comparison requires synonymy; therefore beings are synonymously beings; therefore there is no plurality of senses of being to explain.

That is a rival conclusion from the same premise this section has been using, about the same case.

The reply is not that beings fail to be commensurable. That answer would take his question on his ground, and the question is what is at issue.

The reply is that the premise does not apply. Comparison is a relation among portions, and a difference of manner is not a portion. Where a category is not a quantity, it is not that the measure fails to reach — it is that there is nothing of the kind a measure takes. On that reading Shields’s first premise is not false but inapplicable, and the argument for synonymy does not start.

And on one point the disagreement is narrower than it looks. Shields notes that Aristotle’s degrees-of-reality thesis is of dubious coherence, and gives the reason in a footnote: degrees belong most naturally to graded or functionally determined kinds, and being does not seem to be one — for to the question is a more than b? we are inclined to answer with a question, is a more what than b?

That is the test this study runs at §7.2, and Leibniz’s test at §5.4, arriving from a third direction. On whether being admits degrees, we agree, and agree on the reason.

Where we part is on what survives that agreement. He holds that commensurability stands without the degrees thesis, on independent grounds: the difference of account required for non-synonymy pertains to exist taken by itself; ontic dependence does not entail account dependence; and cross-categorial existence claims are possible, since to deny existence to one category is to deny precisely what is claimed for another.

That is the argument this section must meet, and the reply is the one already given. Where a category is not a quantity, the measure does not fail to reach; there is nothing of the kind a measure takes. And cross-categorial denial is not commensuration. To say that there are relations and not points is to deny of one what is claimed of another; it is not to place them on a scale.

The point worth recording, and worth being generous about, is therefore not a concession but a location: the weight rests on an account of exist taken by itself, which is exactly what §5 says a plural ontology cannot get without a differentia. Order in Multiplicity remains the most careful treatment of core-dependence available, and the disagreement is narrow and locatable.

6.5 What this section has shown

Where a field is genuinely non-scalar, a single term across it is not a discovery. It is an installation.

Analogy, returned to, is an installation acknowledged as such and defended — which is the most honest of the four. The void is an installation not noticed as one. Withdrawal is a differentia deployed by a position that denies needing one. And the fourth case is what happens when all three are refused and nothing is available to replace them: the refusal is made, and it turns out to have said nothing that the position it refuses could not say for it.

Four careful attempts, in the absence of one condition. Three install a term; the fourth is re-absorbed by the term it declined to install.

§5 said what makes such a term unnecessary. §6 has shown it being installed anyway, four times, by people working carefully, in the absence of the condition §5 identifies.

What that condition would take, if it could be met, is the business of Part Three.

Notes to Part Two [§5.1] Physics VII.4, 248b6–9. The concentration of ὁμώνυμον in that chapter is noted by Comay del Junco (below). — Metaphysics Δ.7, 1017a22–23 and 1017a35. — Ι.1, 1052b18–20 and 1052b20; the extension to the other categories at 1052b24–25. Laura M. Castelli, Aristotle: Metaphysics Book Iota, Clarendon Aristotle Series (Oxford, 2018), is the commentary of record for Ι. — Shields’s formulation of the comparison principle: Order in Multiplicity, 262.

[§5.3] Elena Comay del Junco, “Aristotle on Comparison,” Oxford Studies in Ancient Philosophy 61 (2022): 103–142, following Donald Morrison, “The Evidence for Degrees of Being in Aristotle,” Classical Quarterly 37 (1987): 382–401, at 398. Comay del Junco’s blocker for comparison is homonymy, and she does not treat Categories 3b33–4a9. — On the semantic construal of gradability, her note at 103. — The disputed reading of μᾶλλον at Metaphysics Ζ.1, 1028a25–26 admits three constructions: the intensity reading, which degrees-of-being requires; the metalinguistic reading, which Owen adopts (1960: 186); and Morrison’s own. The reading the objection needs is contested at its origin.

[§5.4] Ratio est in natura, Sämtliche Schriften und Briefe, Akademie-Ausgabe N. 2680, ed. Henrik Wels; manuscript LH IV 8, Bl. 100–101; provisionally dated after September 1698 on the strength of the self-reference to De ipsa natura cap. 13 (Acta Eruditorum, September 1698, 436–438). Also in Gerhardt, Philosophische Schriften VII, 289–291, and Couturat, Opuscules et fragments inédits, 533–535; reprinted in Heidegger, Nietzsche II (Stuttgart, 1961), 454–457. Citations here are to the Akademie text, since the argument depends on its apparatus. — The apparatus at §11 records (1) Sequitu (2) Porro (3) Existit, and the revision of the definitional clause from nihil aliud enim perfectio est to cum … sit. — The apparatus at §20 records seu perfectio erg. — ergänzt, added. — The apparatus at §5 records the founding clause (prout fundatur in Ente necessario actu existente) as gestr. — gestrichen, struck out; it returns at §6 and stays. — The fragment is treated at length, as a modal trial of a different question, in the author’s Positive No-Ground, ch. 6.

[§5.6] Categories 5, 3b33–4a9, in Ackrill’s translation. (Ackrill and not Ross here is not an inconsistency: the rule is one translation per work, and Ross did not translate the Categories.) — Metaphysics Ι.2, 1053b29.

[§5.7] Deleuze, Difference and Repetition, trans. Patton, in the univocity section of ch. 1.

[§6] Kris McDaniel, “Ways of Being,” in Chalmers, Manley and Wasserman, eds., Metametaphysics (Oxford, 2009), 290–319; “A Return to the Analogy of Being,” Philosophy and Phenomenological Research 81.3 (2010): 688–717 — the health example at 695–696 with n. 27 citing Metaphysics IV; the gloss on the title at n. 31, 697; the reading of Aristotle at n. 29, 697; the priority of substance over attribute at §4.2, 705–706, with n. 48 citing Metaphysics IV, De Ente et Essentia, and the Commentary on the Metaphysics. Reprinted as ch. 2 of The Fragmentation of Being, and available from the author at krismcdaniel.com; The Fragmentation of Being (Oxford, 2017), ch. 7, “Degrees of Being,” 195–222. — Jason Turner, “Ontological Pluralism,” The Journal of Philosophy 107.1 (2010): 5–34. — Bruno Whittle, “Ontological Pluralism and Notational Variance,” Oxford Studies in Metaphysics 12 (2021): 58–72. — Owen Griffiths and A. C. Paseau, “Ways of Being and Logicality,” The Journal of Philosophy 120.2 (2023): 94–116. — Badiou, Being and Event, trans. Feltham (London, 2005): the count-as-one at Meditations 1 and 3; Meditation 4, “The Void: Proper name of being”; Meditation 5, “The Mark ”, for the unicity of the void and the derivation it is because the one is not that the void is unique; Meditation 7 §3, “The Void and the Excess”, for universal inclusion and the omnipresence of the void. — Harman, The Quadruple Object (Winchester, 2011): the equally objects formulation in the Introduction; the differentia at ch. 1, “Undermining and Overmining”, §D; withdrawal as a feature of the real at chs. 3–4, and the non-withdrawal of sensual objects at ch. 2. — Shields, Order in Multiplicity, ch. 9, “The Homonymy of Being”: the two premises at 264; the degrees-of-reality material and the independent argument for commensurability at 265–266, with n. 374 at 265; the conclusion at §9.10.

Part Three follows: §7, The Third Answer — with a prospect after it, which establishes nothing and may be passed over; §8, Why It Was Not Reached; §9, Objections; §10, Falsifiers.

PART THREE

The Third Answer

§7. The third answer

The differentia is ownness, and the character by which it differentiates is non-vicariousness. Both terms belong to a wider account of which this study is one part, and both are marked as mine rather than the field’s.

One thing about that dependence should be exact, since §8 will turn on it. What is used from the wider account is the differentia and its non-scalarity, and both can be assessed here, on this page, by a reader who has read none of it. What is cited from it is a record of trials — regions entered, instruments restricted, verdicts returned and withheld — and a record cannot be re-run in an article. Wherever it is cited it is named as citation, and nothing in the argument rests on it.

7.0 Three levels, and two offices

Before the differentia is stated, three things must be held apart. Running them together is the standing error in this area, and §1 has already given us the material to keep them separate. Return to the ward. Two infants, two bands.

A band is of a certain kind. Plastic, printed, from a batch — and everything in that description can be had again, which is what makes it a description. Call this the suchness: the repeatable how, said truly of many. It is what a curriculum can transmit and a factory can reproduce.

This band is this one. Not the other, though nothing tells them apart. Call this the thisness: that this obtaining is this one rather than that. §1 established that the bands have it — they are two, and no inventory of their properties makes them so.

And the infant is someone. The band is not. This is the third thing, and §1’s whole problem was that the first two do not deliver it: an answer to which one is not yet an answer to someone rather than merely something.

The plain series, then:

Suchness says what manner. Thisness singles out a something. Ownness discloses someone.

The second point must be pressed, because everything below depends on it. Thisness is region-neutral. It answers which occurrence is this in any region whatever — of bands, of storms, of coastlines, of persons. A category of being is not marked off by it, because nothing is left outside it. Whatever else the differentia of this category is, it cannot be thisness, or the bands are in the category.

So the differentia is the third thing:

Ownness is the differentia: the whole obtaining is someone rather than merely something.

One guard travels with the word and must be given in the same breath, because English will otherwise supply a proprietor. Own here does not mean possessed. There is no occurrence first and an owner later, no inner tenant to whom a life is assigned. The manner is ownerless; the obtaining is own; nothing stands behind it as owner — and the obtaining is own precisely because nothing does.

And ownness has a differentiating character of its own, which is what does the marking-off. Ask what it is about ownness that distinguishes anything, and the answer is that this own obtaining cannot be another’s — that it is stated against a possible substitute who cannot substitute. That character is non-vicariousness.

These are not two names for one thing, and they are not two differentiae. They stand in an order: the category is marked off by ownness; ownness marks off by being non-vicarious. §7.4 will need the first and §7.3 the second, and the sections use different words because they work at different levels.

Two scopings belong here and are stated once for the whole section.

Neither is a principle. That a differentia is constitutive of the difference it splits is not a further claim; it is what the word says. Nothing archic follows. Neither grounds any case, produces none, or stands beneath any: non-vicariousness is the mode by which ownness is the differentia it is, as ownness is the character by which the category is the category it is. A mode is not a principle.

And the character attaches to no part of what it marks. It qualifies the whole obtaining and nothing within it — not the occurrence, not the manner, not any aspect analysis distinguishes. This is not fastidiousness. An occurrence possessing ownness would be a substrate wearing the differentia as a property, and the bearer this whole account refuses would have been readmitted in the differentia itself. §7.3 will need this, and it is stated now so that it is not invented there.

7.1 The differentia is disclosed, not defined

Now the difficulty a reader has been holding since §3.

§3 met this danger from one side; it arrives here from the other, and the same image serves. A differentia specified only by what fails to happen to it stands to its object as the diver in the dark stands to the sea floor: he reports that what he struck was not a cow, and he is right, and he has told us nothing. Contact and exclusion, and no shape.

There the danger was that focality might be characterised only as not-causal. Here it is that the differentia might be characterised only as not-substitutable. Both would be denials wearing the grammar of definitions. And §6.2 showed what such a denial costs when made in earnest: a plurality specified as not ranging over a single domain is specified by the device it declines, and can therefore be restated in that device’s terms.

So a definition is wanted. And here the argument must say something that will look, at first, like an evasion, and is the reverse of one.

No definition is available, and none is possible, and the reason is not a limitation of the present account.

Suppose a content were supplied — a property, a mark, a structure, such that to have it is to be someone. Then that content is a such. It is articulable, and what is articulable is repeatable in principle: that is what makes it articulable. And the whole problem of §1 was that the repeatable does not deliver the this, still less the someone. A definitional content would therefore be one more item on the side of the ledger already found insufficient — insufficient in exactly the way everything on that side has been, which is that a substitute could have it too.

The demand for a definition is thus not a demand this account fails to meet. It is a demand whose satisfaction would defeat the account it was meant to complete.

But that leaves the diver where he was, unless something stands in a definition’s place. Something does, and it is what a definition could not have been: a case placed beside a case.

The pair An original painted canvas burns. The most exact copy is then made — pigment for pigment, craquelure for craquelure, indistinguishable to every eye and every instrument. The original painted object has not been restored. Nobody thinks it has. What was worked by that hand in that year is gone, and the copy’s making is a different making, however perfect its product.

So the canvas has everything one might have supposed the mark of a who to be. It is deictically singular — this canvas and no other. It is non-restorable — the most exact substitute conceivable fails to bring it back. And it is indelible in having-been — that it was painted, then, thus, is henceforth permanently the case, and nothing later can unmake it.

And it is no one. Nobody stands before the ash and asks who? The question does not arise, is not suppressed, and would be a misunderstanding if raised.

Now return to the ward — and to a different pressure from §1’s. There the two infants were indiscernible, and the question was what makes each of them this one. Here there is one, and a substitute. Let the substitute preserve appearance, voice and manner; then memory and dispositions; then narrative continuity; then social embedding, so that family, colleagues and institutions receive her without friction. Every structure the major theories have proposed as criterial is present and functioning.

Was the one who was here restored? The inventory does not settle it, and cannot: prepare two such substitutes simultaneously and they cannot both be she, though each has everything the other has.

Line the two cases up. Both singular. Both non-restorable. Both indelible in having-been. Every mark that could be listed, matched — and the two cases are not the same case, and no one has ever confused them. A work is destroyed there. Someone is gone here.

That difference is not read off a further property, because there is no further property: the match was maximal by construction. It is legible in the intentional form of the loss itself — in what the loss is a loss of, which is given in the losing and not inferred from an inventory.

That is the shape the diver lacked. He had contact and exclusion. Here there are two cases matched at every point that could be listed, and one divergence, and the divergence is exhibited rather than described. This is what disclosure supplies where definition cannot: not a predicate, but a demonstration one can be walked to and can refuse.

Which is why it can fail A disclosure that could not fail would be a stipulation with better manners, and this one is offered under conditions that defeat it. A purported disclosure of ownness is defeated where:

• it tracks role, status, possession, or convention rather than the case itself; • it disappears under reframing without resistance from the region; • structurally symmetrical cases receive unequal treatment without relevant reason.

These cut, and they cut here. Take the canvas again and suppose its owner reads his own grief as the loss of someone — the presence in the frame, the studio’s companion. The first condition convicts the reading: the refusal of the copy is sufficiently explained, in this case, by possession, by fidelity to the original making, and by the conventions of originality, with no remainder directed at anyone. What is left standing is a true grief over the historical singularity of a making. Which is what the pair said it was.

And one guard, which the pair must carry wherever it goes. It is one member of a convergent constellation — substitution, loss, death, address, attribution — and not a test to be applied. Non-restoration by itself discloses no one: it may disclose the historical singularity of a making, a deed, or an object instead, and in the canvas it does exactly that. The contrast is demonstrative, never definitional, and never sovereign.

One clarification is owed forward, because §7.3 will say something about the character and a reader will call it the definition just refused. A how is not a what. What is refused here is a content — a mark, a property, a structure, which a substitute could also have. What §7.3 supplies is a relation: this manner, this occurrence, complete and unconsumed. A substitute can have every such. No substitute can have this coincidence, because the completeness is with this occurrence.

What can be said positively, and where Nothing above is a definition, and the section does not end by smuggling one in. But something can be said positively about the character in the regions where it has actually been tried, and it should be said, because a differentia with nothing positive to show in any region would be idle.

In the finite-living regions — the human and the feline, which are the regions where these trials were run — non-vicariousness has a positive articulation:

This life cannot be somebody else’s living.

The restriction is binding and is not a hedge. A definition through life would settle the machine, the civic, the sacred and the divine negatively in advance, by wording rather than by trial, which is precisely the vice this book has spent two Parts describing. The unrestricted statement is the one given at §7.0 — the whole obtaining as someone rather than merely something — and that is what travels. The life-sentence does not travel; it stays where it was earned.

And the boundary so drawn is provisional in a way worth stating. It is drawn at living, on an account of living this argument has not renewed. It marks where the trials have gone, not where the character stops.

7.2 It is not scalar

The pair now does its second piece of work, and it is the piece §5 has been waiting for.

Look at what the two cases match on. Singularity: both, fully. Non-restorability: both, in full rigour. Indelibility of having-been: both, permanently. Every one of these could in principle admit degree — one might imagine a case more nearly restorable than another — and on every one of them the pair is level. The match was built that way.

Now look at where they diverge. A work destroyed; someone gone. And ask the question §5 taught us to ask of any differentia: which of these has more of it?

The question does not receive an unhelpful answer. It receives none. There is no quantity of which a work destroyed is a smaller portion and someone gone a larger one. The divergence is not an amount of anything, so there is nothing in it to partition, nothing to order, and no primary case for the others to be measured by. §5.2’s principle runs only where a differentia is scalar; here it does not run.

Notice what has happened to the argument’s standing. Non-scalarity is not being asserted, and it is not being defended by intuitions about what sounds odd. It is exhibited in the same pair that supplied the differentia — which is the only form of evidence a disclosed differentia can offer, and it is offered.

Against the assimilations This must be defended against being quietly replaced, because the danger is precise and the history of this category is a history of replacements.

Ownness is not autonomy. Not reflexivity. Not intelligence. Not moral standing. Not complexity. Not capacity for self-relation.

Every item on that list is scalar. Several are traditional differentiae for exactly this category — and §5.5 showed what each produces: the same periphery, arranged by how much of what the differentia measures a case has. If ownness is any of them in disguise, this study fails at once, and fails in the manner it has spent two Parts describing.

The distinction is not subtle, and the pair makes it visible. Every item on that list is a capacity: something a case does, exercised well or badly, developed or diminished, present at maturity and absent in decline. Now apply any of them to the pair. They do not discriminate it — the canvas has none of them and the substitute has all of them, and the divergence the pair exhibits is not the divergence any of them would mark. The pair turns on none of the capacities, which is why the capacities are not what it discloses.

The test in Leibniz’s form §5 supplied the sharpest available test of a proposed non-scalar differentia, and it should be run here rather than deflected.

What happened to ordo nihil aliud quam relatio plurium distinctiva was that a quantity-word was written in beside it — seu perfectio, added in the margin, five propositions after the relation had been defined, and perfection had already been defined as quantitas realitatis, from which a superlative follows in the same sentence. The non-scalar definition did not survive its own text. The question to put to any proposed non-scalar differentia is whether the same can be done to it: whether some term of the form quantitas x could be written in beside it as an equivalent, without loss.

Here there is none, and the reason is now visible rather than promised. To write in quantitas x one must be able to say what x is — what the cases have more and less of. The pair is that demand made concrete: two cases, matched at every listable point, diverging in one respect. Name the quantity of which that divergence is a difference. The candidates that present themselves are the capacities, and the previous paragraph has shown that the pair does not turn on them.

That claim is the one an opponent should attack, and §9 states what an attack would have to produce.

7.3 Two faces, and what follows within the category

The character faces two ways From the character, uniqueness comes in two directions at once.

Turned toward the manner, it refuses that the coincidence recur. The manner is open and may obtain again — feline living obtains in the next cat, and truly obtains, and the household that loves the second cat is not deceived. What may not obtain again is this coincidence of that manner with this occurrence. Call this face non-repeatability.

Turned toward the mode, it refuses that anyone stand in this place. Call this face non-replaceability.

These are not two doctrines but one character seen from two sides, and the way they were found is worth recording, because a tidier presentation would destroy the evidence. They were reached separately. The substitution cases press the second: a substitute is prepared, everything transferable supplied, and the one who was here is not restored. The recurrence cases press the first — and they cannot be pressed at all until the commonality of the manner is in view, since one cannot refuse the recurrence of a coincidence before noticing that the manner recurs. The two results stood apart for a long time and were treated as separate findings. That they are one character is what the differentia supplies; it is not what the trials were designed to show.

§7.3 runs on the second face. §7.4 runs on what the first requires.

The refusal has a term, and the term is another Take the second face and look at what it is made of. No one can stand in this place. In the regions where the character has been tried, the same face has a fuller wording: this life cannot be somebody else’s living.

Read the words that cannot be removed. A refusal of substitution is empty where no substitute is possible — it refuses nothing, because there was nothing to refuse. So the refusal is stated against a possible substitute; and a possible substitute is another of the same category. Other whos are therefore carried inside the formulation, and are not added to it by a further argument.

And one thing more should be said, because the formulation depends on it and it is easily passed over. What makes a substitute possible is that the manner was not used up in obtaining here. One manner is truly said of each, and undiminished by any of its obtainings — feline living obtains in the next cat, wholly, and this cat’s having lived took nothing from it. So another could have stood here; so the refusal has something to refuse; so the character is statable at all. Exhaust the manner and non-vicariousness is not false but unsayable, since a refusal of substitution where no substitute is possible refuses nothing.

And the non-exhaustion is modal, not a count. Consider the last cat — felinity with no further occurrences anywhere. Her ownness is not diminished, because the manner remains sayable of another whether or not another obtains. A reading on which the refusal weakens as occurrences grow scarce, and fails at the last, has let the actual decide the possible — which is the collapse §5 spends its length diagnosing, arriving here under a different name.

The status of that observation decides how the point can be attacked, so it should be exact.

This is read off, not inferred. Somebody else’s is not an illustration of the formulation and not a consequence drawn from it: it is a term the sentence cannot do without. There is no step here, and therefore no step to dispute. An inference can be attacked at the inference. A reading-off must be attacked by showing that the term is dispensable in the formulation — that the same thing can be said without it — and it cannot be said without it.

This reverses the usual burden. Where an argument moves from premise to conclusion, an opponent may grant the premise and deny the passage. Here there is no passage. What is offered is the observation that the character was never statable except against another, and the opponent’s task is to restate it otherwise.

No instance can be the focal instance Now §3’s analysis applies, and the application is direct.

A focal instance within the category would be one whose being is the sense of which the others are said — one that holds their title to count as instances, so that they appear on a warrant it holds. That is what §3 argued a focal term does: not cause the derivative cases, not explain them, but answer for them.

There are two reasons this cannot obtain here, and they are independent, so an opponent must defeat both.

The first is the constraint of §7.0. The character attaches to no aspect of what it marks. It qualifies the whole obtaining and nothing within it. Now ask what it would take for one case to hold another’s title. A title held elsewhere must be lodged somewhere: the landlord of §3 holds a deed, and the deed is a thing, in a drawer, transferable, held by one party rather than another. What would be lodged, here? Not the obtaining, which is not the sort of thing that can be somewhere else. Not an aspect of it, since the character attaches to no aspect. There is nothing detachable to be held, and no holder-position for it to be held in — not because the position is occupied, and not because it is far away, but because the constraint that keeps the bearer out of this account keeps the deed out with it.

The second is the term of the refusal. For case A to hold case B’s title is for A to stand in B’s place with respect to B’s claim to be one of these at all. That is substitution in exactly the respect the character refuses — not substitution in performance, or role, or function, all of which succeed and are conceded throughout this book, but substitution in the one respect the character was stated against. A focal instance within this category is therefore not merely absent. It is what the differentia says there is not.

So:

No instance of the category can be its focal instance.

And it is worth walking the alternative far enough to see where it arrives, because a structural claim is stronger when the position it excludes has been followed to its destination.

Suppose the refusal ran the other way — suppose that what a case’s own obtaining settled were not the non-recurrence of this coincidence but the non-recurrence of the manner. Then this obtaining would be in charge of what remains possible for anything else: it would decide whether feline living can obtain in the next cat. It would hold the title of every future obtaining of that manner — which is precisely the structure §5 exists to diagnose, generated at the level of instances by a category that was supposed to have escaped it. And the point is not that a case is too small to own its manner. Ownership has no application here: a reading on which a case would own the manner but is prevented has reinstated the proprietor and merely denied him his estate.

The plurality is dispersed by construction: many, none primary, none exemplary, no paradigm against which the rest are read as approximations.

And the register of that result should be stated plainly, since it will otherwise be heard in the wrong one. This is not a resolution to treat cases equally, and it is not a moral position wearing ontological dress. It is a structural fact about a category marked in this way, and it holds whether or not anyone approves of it — and, equally, it obliges no one to anything. Nothing normative follows from it here or anywhere in this book; what an ethics may make of it, it makes on its own warrant and by its own route.

7.4 What follows for the category

The same argument runs at the other level, and to run it we need the other thing the formulation carries.

The differentia is stated against other ways of being Return to the unrestricted statement: the whole obtaining is someone rather than merely something.

Read the words that cannot be removed here. Rather than merely something is not a flourish. A differentia differentiates, and what it differentiates from is a contrast class; strike the contrast class and nothing has been differentiated. So the formulation carries, inside itself, what obtains in another way — a stone, a river, a tool, the band on the infant’s wrist.

And the contrast class is categorial, not specific. This is the point on which everything in this section turns. The who is not marked off from other species within a kind, as a man is marked off from a horse within animal. It is marked off from what obtains in another way. The band is not a defective who or a distant relative of one; it is a something, obtaining as somethings obtain, and the differentia is stated against that.

Read off again, and not inferred, exactly as at §7.3: rather than merely something is a term the sentence cannot do without, and the opponent’s task is to restate the differentia without it.

So plurality is carried twice, in opposite directions — and the two directions now have their reason. They are read off formulations at two different levels. Inward, at the character’s: a refusal wants a possible substitute, and a possible substitute is another who. Outward, at the differentia’s: a differentia wants a contrast class, and this one’s is other ways of being. A refusal wants a substitute; a differentia wants a contrast class; and the two wants are not the same want.

One guard, before the obvious misreading A reader who has just been told that the differentia carries plurality will conclude that plurality is the differentia.

It is not, and the two claims are not close. Imagine an obtaining as plural as one likes — many manners, many relations, many histories, a maximum of internal multiplicity — with nothing own about it. That is not a who. It is a rich something. Plurality is not what marks the category; it is what the marking cannot be stated without.

Co-inclusion is not identity. Plurality is a condition of the statement, and never the character stated.

The category cannot stand above the others Could this category be the primary sense of which other ways of being are said — the focal term for the categories, as substance was?

No, and for the reason §7.3 gave, at the second level. To be the sense of which another way of being is said is to hold that other’s title to count as a way of being — to stand in its place with respect to its claim. The character refuses standing-in-place-of, and refuses it as such: it does not refuse it only among whos, as though the exclusion were a local rule about one category’s internal affairs. What it excludes is the relation. Excluded within, it is excluded between.

The category therefore stands among the ways in which being obtains, and not above them.

And the consequence that is the largest thing this book claims

A category whose differentia is stated against other categories cannot be seen apart from them. To see the who as a category is already to have seen that being obtains in more than one way.

This is why the answer of this book is not a modification of the first answer or a variant of the second. The differentia does not merely fail to install a focal term; it cannot be seen at all by an account that has only one way of being to work with, because what marks it off is stated against ways of being that such an account does not admit. The refusal of the focal term is not a further commitment added to the differentia. It is what the differentia is, seen from the outside — and §8 will ask why that made the third answer unavailable rather than merely unchosen.

A note on where this stands. The consequence is registered in the wider account of which this study is one part, and argued there nowhere: it was entered so that a later work would find it already recorded rather than have to introduce it. It is argued here. Nothing in Parts One or Two rests on it, and nothing earlier in this section does either; it is a result of §§7.3–7.4 and stands or falls with them.

7.5 Three consequences

These are consequences rather than commitments, and they should be read as reports. Each says what follows from a category being marked in this way; none is a policy, and none is a permission.

The first: every instance is full Where there is no exemplar, nothing falls short of one. A case is not a diminished version of a better case, because there is no better case for it to be diminished from. Whatever a given instance is, it is the whole obtaining of its own occurrence.

Recall what §5.5 exhibited. The differentiae actually in use to mark off the beings that answer, choose, or are owed something — rational animal, self-conscious being, person as bearer of rights, system of sufficient complexity — are each scalar; each therefore partitions; and each produces the same arrangement, in which cases stand at a greater or lesser distance from a maximum. §5.5 listed what collects at the far end of every one of them: the newborn, the animal, the sleeper, the cognitively impaired, the dying, the machine.

Under this differentia that arrangement does not form — and the claim must be stated with its scope, because it does not reach the whole of that list.

Where the character has been disclosed, no case falls short of another, because there is nothing for a case to fall short of. Not because a rule forbids it, but because nothing is being measured: there is no quantity, so no order of more and less, so no centre, so no distance from one.

Where it has not been disclosed, nothing follows in either direction. And it is essential to see why, because the temptation runs the other way.

A focal ontology excludes by measurement: a case too far from the maximum falls off the end, and the ordering does the excluding. Remove the ordering and that particular mechanism of exclusion is gone. But a mechanism of exclusion is not a mechanism of admission, and removing the first does not install the second. A category with no centre does not thereby have open doors; it has no doors, which is a different thing and admits no one.

The absence of a periphery licenses no admission. A category with no centre excludes nothing by having no centre, and admits nothing either.

This is the same discipline the argument has run since §5, applied to itself. Non-disclosure licenses openness, not negation — and the mirror is exact: non-arrangement licenses openness, not admission. An account taking the second half and dropping the first would be helping itself to whatever it had not yet examined, and would deserve everything a critic could do to it.

The list at §5.5 is accordingly not homogeneous under this differentia, and the wider account’s record is cited here rather than relied on: some of those cases were entered and disclosed; one — the machine — was entered and returned no verdict, standing open in both directions; and others were never entered at all. The first group is where the arrangement does not form. For the second and third, this section says nothing whatever.

And what does not follow from it Nothing normative follows, here or anywhere in this section. Every instance is full is a report about a category’s structure, and a reader who has just come through §5.5’s list of peripheries will be tempted to hear it as a verdict about how the cases at those peripheries ought to be treated. It is not one, and the temptation should be refused rather than exploited.

This account maps what obtains; it issues nothing. That finite cases are not arranged as approximations to a paradigm is a structural fact and holds whether or not anyone approves of it — and it commands no one to do anything, because a structure is not an instruction. Whatever is owed to anyone is owed on its own warrant and by its own route, and the route does not run through this section.

The second: each region returns what it has An enquiry proceeding under this differentia can try one region at a time and take what the trial gives, becauseno result has to be expressed as a distance from a paradigm.

The difference is practical rather than decorative. Where a paradigm is installed, a region disclosing something unlike the central case can only report a deficiency: so much of what the primary case has, and no more. Here a region that discloses something unlike the others discloses that — a difference, in its own terms, at whatever strength its evidence carries.

And the borders fall where the trials leave them. A border named is a finding rather than a failure, and this is why a non-verdict can be reported as a result instead of concealed as an embarrassment: an enquiry that had to place every region on a scale would have had to place that one too, and would have placed it by default at the bottom, which is a verdict nothing had earned.

The third: the field stays open A further category of being is not, on this account, a candidate for a summit that is not there; nor an approximation to a category already in hand; nor a province to be annexed by the largest one. It is another way in which being obtains, standing on its own differentia and disclosed on its own terms.

The doctrine is in this sense generative rather than closed: it leaves room for categories it has not itself articulated, and the room is real, because nothing admitted to it would need measuring against what is already there.

And here the guard of the first consequence must be repeated, because the third invites its violation more strongly than anything else in the section. That the field is open is a fact about the absence of an arrangement, and not a policy of admission. Nothing enters this field by finding it hospitable. What would admit a category is what has admitted every category the argument has: a differentia of its own, disclosed in a region where the phenomenon shows itself, at the strength its trials carry. The room is empty, not welcoming.

That third consequence is the one most likely to be read as a boast, and it is a report rather than a claim about capacity. Where nothing is arranged around anything, there is nothing for a new arrival to be arranged around — which says what will not happen to an arrival, and says nothing at all about arrivals.

Prospect: a programme this result opens

The argument of this study is complete at §7.5. What follows establishes nothing and is offered as a prospect: a direction the result opens, stated so that a reader can see where it leads and decline to follow. Nothing earlier depends on it, and nothing later in this book uses it. A reader with no interest in what comes next may go straight to §8.

One thing must be said before it is described, because the previous section has just guarded against exactly this misreading. What is at issue here is a way in which obtaining is timed, not a case admitted to the who-category. The two questions are distinct; §7.5’s guard governs the second and not this one; and nothing below admits anything to anything.

A tide does not simply happen in time. It happens in the manner of a tide — answering a body at a distance, on a rhythm that is its own. An animal timed to that rhythm has a third way of being timed, which is neither the tide’s nor the moon’s. This is what it would mean to say that a region has its own temporalisation: not a position within one universal time, but a way of being timed.

If that is right, the regions such a route discloses would be distinguished not only by what obtains in them but by how obtaining is timed in them — plural regional temporalisations, interconnected without any one of them serving as the measure of the others. And the absence of a clock among them would be the same result as the absence of a focal term among the categories, for the same reason: a privileged temporality would be a primary case in the temporal register, and the differentia forbids it there as elsewhere.

That is a programme and not a finding. Its articulation is not the work of this study; it is stated here because a result that opens a direction should say so, and because a reader is owed the chance to test the direction rather than discover it later as an assumption.

§8. Why it was not reached, and what remains open

If the argument holds, a question follows immediately. A non-scalar differentia requires no apparatus that was unavailable before now. It requires only that one be found. Why was one not found?

The answer is that a differentia of this kind must satisfy two conditions at once, and the conditions pull apart.

The first is access. The differentia must be found rather than posited — disclosed in a region where the phenomenon actually shows itself. The second is standing before the tradition. It must be statable in the register where focal meaning and univocity are the live options, so that it can be shown to replace them rather than to proceed alongside them in a private vocabulary.

8.1 Where the phenomenon does not appear

Ontologies that begin from entities satisfy the second condition and fail the first.

They are unmistakably in the register of the tradition — that is where the question was formed, and §6 showed how carefully it is being worked. But nothing in an entity, or in the quantifier that ranges over it, exhibits the character §7 describes. One may inspect an entity for as long as one likes and never come upon it — and the reason is exact rather than atmospheric. The character is stated against another: against a possible substitute who cannot substitute, and against what obtains in another way. It is not a feature of a thing’s constitution but a refusal that has a term, and a relation with a term is not found by inspecting the term alone. An ontology whose primitive unit is the entity, and whose question is what a complete inventory of entities would have to quantify over, is looking in the only place where such a character cannot show.

This is why the pluralism built there reaches differences that are structural or scalar, and must then import a device to unify them. It is not a failure of rigour. The phenomenon is not in that field — and the field is not defective for lacking it, any more than a scale is defective for not weighing colours.

8.2 Where it does appear

And this fixes what the second condition requires of an access. If the character is a refusal with a term, then a route that will reach it must be one on which the term is already present — where another is not an object encountered but a relatum of the structure being examined. That is a strong condition, and it disqualifies more starting points than the previous section’s did: not only inventories of entities, but any access whose primitive situation contains one party.

The claim that a phenomenon may be foreclosed from one starting point and reached from another can be offered as an exhibit rather than as an argument, and it has been offered as one in print.

It is not a claim about any author. It is a claim about starting points, and it is testable because the route can be inspected.

That route ran through a phenomenology of being-with — through the structure the existential analytic named as basic and left underived. Heidegger held that being-with others is not added to a self that was already there: it is as basic as being oneself, given at the same moment and not derived from anything more basic. His word is gleichursprünglich, usually rendered equiprimordial. What he did not do was derive it. He named it as basic and left it there — and left underived precisely the structure in which the term of the refusal is given, which is why the derivation had to be taken up from the perimeter where the shared field fails.

The derivation was taken up from the other side, at the perimeter where the shared field fails:

the withheld greeting; the greeting that does not land; the animal that responds and cannot be greeted; the speech with no one answerable behind it.

And that work was written before the present diagnosis was formulated— which forecloses the suspicion that the access was chosen because it would deliver the result. The route was walked for other reasons, and the differentia was found on it.

8.3 Aristotle’s position, which is not a failing

Something should be said here about the author on whom Part One most relied, because the shape of this section applies to him and the application is easy to get wrong.

Aristotle reaches these limits repeatedly and records them accurately. He denies a trans-categorial One. He denies a single good. He denies gradation within a kind while granting it between kinds. He says measure is first of quantity and has gone to the others. He notices that health is not healthy.

Every one of those is a limit found at the phenomenon and written down. He is not inconsistent across the corpus; he is reporting what each case gives.

What he does not do is unify them. And that is not an oversight. Seeing that measure travelling out of quantity is a general fact about categorial difference requires a vantage on the whole domain of the metaphysical question — and he is at its opening. The unification requires a completed domain; he is opening one.

This is why Part One read Γ.2 as a proposal rather than as the founding of an orthodoxy, and why Aristotle functions throughout this study less as the origin of a difficulty than as its most reliable witness: not yet defending a position, and so with no reason to smooth the limits he found.

8.4 What this does not claim

Here the argument must be careful, because it is easy to convert an account of one access into a denial of others, and nothing above licenses that.

Consider an ontology of an element or a region — of water, of the river, of the moon. Such an enquiry may disclose a genuine way of being with its own manner of temporalising, and there is nothing whatever defective about it. Its difficulty is only that, arriving after substance has been installed as primary, it enters a field already organised and yields a phenomenon with no statement of what it is a differentia of. Its results may be true and cannot be adjudicated against the answers already in play.

But that difficulty is a matter of where an enquiry enters, not of what it studies.

An elemental ontology entered from the early Ionians satisfies both conditions, and the reason is worth stating precisely.

The Ionians did not ask a material question. They asked a modal one: how does being arrive to be, and what is the way of being of that arrival? Thales answeredlike water; Anaximenes like air; Anaximander like the boundless. The like carries the whole of it — not that everything is made of water, but that to be is to arrive as water arrives. That is a question about manners of obtaining, asked from the everyday, before any schema, and asked in the register in which the present question is asked.

What turned it into a contest was the presupposition that there must be one such way. Aristotle’s own summary records the shape it had acquired by his time:

ὅσοι μὲν οὖν ἕν τε τὸ πᾶν καὶ μίαν τινὰ φύσιν ὡς ὕλην τιθέασι… δῆλον ὅτι πολλαχῶς ἁμαρτάνουσιν.

Those who suppose the all to be one, and some single nature, as matter… evidently err in many ways. (A.8, 988b22–24)

The doxography — the later reports through which we know them, almost nothing of their own writing having survived — preserves rivals disputing which, and the dispute was read ever after as a competition of candidates for a single first principle. The reading stuck so well that the question of whether there had to be one stopped being asked.

Freed from that presupposition, the Ionian question-form is not a defective anticipation of the archic answer. It is an open access, and one that has not been walked. Two of their regional accounts survive well enough to learn from, and what is instructive is that they disclose their regions differently — which is what a plurality of ways of being looks like before it is required to reduce to one.

So the claim of this section is narrow, and its narrowness is deliberate:

This is not a claim that no other access exists, and it is not a claim that other regions of phenomenality are closed. It is a claim about which access has been walked, and why the walking was possible.

Being-with satisfies both conditions and has been walked. The elemental access satisfies both and stands open. That the differentia was reached by one route rather than the other is a fact about a history, not a restriction on a field.

And this is where the historical claim of the study can be stated, since it is now a consequence rather than a premise. So far as I have been able to establish, this is the first ontology to be plural, regional and dispersed without a focal term and without univocity — and it is so not because a bolder refusal was made, but because a differentia became available that had not been available before, on a route that was walked for other reasons. An ontology whose central result is that no category organises the others toward itself would in any case be poorly served by a method claiming to be the only way in.

§9. Objections

Five, each stated in the objector’s voice before it is answered.

“Deleuze has already done this.”

He affirms dispersion, denies hierarchy, rejects analogy as the accomplice of representation, and says being is said of difference itself. Whatever the present differentia adds, the position it produces is already occupied.

It is not, and the difference is not one of emphasis. Univocity secures dispersion by denying that there are many senses of being: what is plural on that settlement are differences within a single sense. The present answer keeps the plurality of senses — being is modal, and a regional manner may be truly predicated of plural occurrences — and refuses the focal term anyway.

Two settlements of one problem, each declining what the other keeps. Deleuze pays with univocity; this pays with a differentia. And §5.7 gave the reason the difference matters: without a differentia that disperses without a common measure, what one reaches for is a scale, and the image of white including its intensities is what arrived.

“The differentia is haecceitas renamed.”

Scotus held that what makes this individual this one requires a positive explanation, and named the ingredient that supplies it. You have the same datum and a new word for it.

The debt should be stated before the disagreement, and it is larger than the disagreement. This argument stands to Scotus in three places, and only one of them is a refusal.

The datum is his, and it is accepted entire. Two indiscernibles are two. He does not explain the twins away, does not make their duality a function of our failure to discriminate, and does not treat the question as confused. §1’s ward is his scene before it is anyone’s, and §1 took it from him.

The semantic discipline is his, and it is accepted too. A term applied across heterogeneous cases must bear genuinely the same sense, or comparison dissolves into equivocation and there is nothing to have a science of. That demand is learned from him and is at work wherever the differentia is said to hold across regions that differ in everything else.

What is refused is the machinery, and that is what the objection is about. Haecceitas — or, in the Ordinatio’s own words, the differentia individualis, the ultima realitas entis — is a principle installed to individuate. It is introduced because something was thought to need individuating: a common nature, real, of a unity less than numerical, indifferent of itself to being in one or in many, which must be contracted to this one by a positive individual difference formally distinct from it. The apparatus is subtle, it is built against every crude objection, and it answers a question.

Here nothing individuates, because nothing was assembled to be individuated. That is one claim and not two denials, and the distinction matters because the objection assumes it is two. The manner and this occurrence are not independently complete terms subsequently joined; there is no seam where a joining occurred and no components a joining would need. Once nothing is assembled, there is no constituent to contract — and equally no bearer for a constituent to be lodged in. A thisness added to something otherwise general is not what this account has and not what it lacks: it is a repair to a structure this account does not build.

And the differentia grounds nothing, which is the second half of the same point. It produces no case, stands beneath none, explains none into being. A differentia that grounds nothing cannot be the constituent Scotus requires, and the difference is not one of ontological weight. It is not a lighter haecceitas, not the same office filled more modestly. It is not that office.

So: alliance on the datum, convergence in semantic discipline, divergence on the individuating machinery — and the objection, which alleges a renaming, has identified the one of the three places where the two accounts do not meet.

“You said no definition was possible and then gave one.”

§7.1 argues at length that the differentia can have no definitional content. §7.3 then tells us what the character consists in. One of these is not doing what it says.

What was refused was a content; what is given is not one. A content — a mark, a property, a structure — is a such: articulable, therefore repeatable, therefore something a substitute could also have, which is why §1 found that side of the ledger insufficient. What §7.3 supplies is not a determination of anything. It says where the non-possibility falls: the manner stays open and may obtain again, and this coincidence of it with this occurrence may not.

The first would be transferable. The second is not, because the completeness is with this occurrence and nowhere else. A how is not a what, and the demand that it be cashed into one is the demand §7.1 refused.

“An ontology with no focal term is several ontologies, not one.”

You have refused the device that makes a science of being possible. What you are left with is a set of regional enquiries with nothing in common but a name.

This assumes that the only available unity is unity by a primary term — which is the assumption under examination, and the one §5 argued to be a consequence rather than a choice.

But there is a deeper premise beneath it, and it is the one to deny. The objection assumes that naming plurally requires something that holds the names together — that unless the regions are gathered under one head, the naming of each is somehow incomplete.

Nothing requires that. A naming stands on its own disclosure: its own differentia, stated against its own contrast class, warranted by its own trials and defeasible by its own defeaters. It does not draw its warrant from a whole it belongs to, and it is not better warranted when a second region is named beside it. The demand that naming wait upon a unified theory of being is not a demand that naming makes. It is a demand the focal picture makes, and it goes when the picture goes.

A second reply is available and should be declined, because it is worse than it looks. One might say that the enquiry is one enquiry because it proceeds by one method — regions tried singly, results stated at the strength the trials carry, borders named where the carrying stops, losses recorded rather than absorbed. All of that is true of how the work goes. But a method that held the results together would be a primary term in procedural dress, and §7.4’s argument would convict it: whatever holds a plurality’s members in their title to belong is a focal term, whatever it is made of.

The word itself is the caution. Method is μετά and ὁδός — a going-along-the-way. It names how one proceeds, not what one’s results belong to. A path is not the ground of the country it crosses, and a road that gathered the regions it ran through would have stopped being a road.

So: one enquiry, in the ordinary sense that it is one going. Not one science because one thing holds it.

“This is the pros hen renamed.” You have a differentia that every instance shares and that no instance lacks. That is a common nature toward which the cases are said, whatever you call it.

The test is §3’s, and it is the test the whole study turns on. A focal term discharges the derivative cases’ title: they appear on its warrant.

Ownness discharges nothing. It holds no case’s title, and no case appears by it — and §7.3 gave two independent reasons, either sufficient. There is nothing to lodge: the character attaches to no aspect of what it marks, so there is no detachable item a holder could hold. And the holding is what the character refuses: for one case to hold another’s title is to stand in that other’s place with respect to its claim to be one of these at all, which is substitution in exactly the refused respect.

A common nature is a such — repeatable, articulable, said of many. The differentia is not a such, and the trouble the objection is really registering is that it is hard to see what else something could be. §7.1 answers that: it is not a content at all, and no content could do its work, since a content would be repeatable and the repeatable was what §1 found insufficient. What stands in a content’s place is a pair of cases and a divergence between them, which is why the objection cannot be met by producing a better definition and is met instead by pointing.

And the two do different things to a field. A category of being not yet articulated is, under a focal term, a candidate for a place in an order already established: it must be shown to stand toward the primary case, and its standing is settled there, in advance, by the structure it is entering. Under this differentia nothing is settled in advance, because there is no order for it to enter. That is not hospitality, and §7.5 says so: a category with no centre admits nothing by having no centre. What would admit a new category is what admitted every category this study recognises — a differentia of its own, disclosed where the phenomenon shows itself. The difference between the two structures is what each does to an arrival’sburden, not to its chances.

“The differentia is scalar after all.” You claim that it admits no more and less. But every candidate differentia has looked non-scalar to its author, and every one has turned out to grade.

This is the serious objection, and it would end the argument. If the character admits of more and less, then §5’s route reopens: the category becomes a quantity, its instances become portions, a primary case forms, and this study will have described a focal ontology while claiming to have escaped one.

The objection is also right about the history. Rational animal did not look scalar to those who used it. Neither did self-consciousness, nor personhood, nor sufficient complexity. Each looked like a threshold and each turned out to be a slope, and an author’s confidence that his own differentia is different is worth nothing at all.

So the reply must be something other than confidence, and §7.2 supplied it: the claim is exhibited in a pair, and the pair is what an objector must break.

Set the two cases beside each other again. The burned canvas: singular, non-restorable, indelible in having-been. The ward under maximal substitution: the same. Everything listable, matched. And one divergence — a work destroyed, someone gone.

What the objection must produce is a case intermediate between those two, in which the divergence itself admits degree. The requirement should be stated precisely, because two things that look like such a case are not one.

Not a case that is somewhat of each in different respects. These are easy to produce and irrelevant: a violin played by four generations, a house a family was raised in, a working animal buried by the people it worked with. In each, something is destroyed and, arguably, someone is gone, and the two verdicts attach to different aspects of one situation. That is a compound case, not a graded one. The divergence is not partially present in it; it is fully present twice, about different things.

And not a case in which the verdict is unclear. These are more tempting and must be refused more carefully, because they feel like evidence of a slope. Take whichever case the reader finds hardest — the argument need not nominate one. In such a case the disclosure has not settled, and that is a fact about the trial: about what has been reached, with what instruments, in a region whose evidence may be thin. An unsettled verdict is not a partial character. To grade the differentia the objector needs a case in which the disclosure has settled and has settled at a fraction — where someone is disclosed as somewhat gone in the way the canvas is a work destroyed, with the somewhat belonging to the disclosure and not to our purchase on it.

Two further forms the objection may take, in the vocabularies the two Parts supplied.

In Aristotle’s terms: what would show the character scalar is that its instances stand to one another as prior and posterior — which is what Morrison’s exception requires, and what licenses comparison across non-synonymy. If some cases are prior and others posterior, comparison is permitted, an order exists, and a focal term formalises it.

In Leibniz’s terms: that some term of the form quantitas x could be written in beside it as an equivalent, in the way seu perfectio was written in beside relatio plurium distinctiva. If there is such a term, this account will be absorbed exactly as that definition was — and §7.2 has stated what the objector must supply to write it in: not a quantity that some cases have more of, but a quantity of which the divergence the pair exhibits is a difference.

That the comparative is somewhat more someone-gone than that one fails to describe a possible case is what this argument holds. It is a claim, it is where the argument is weakest, and §10 prints it as such.

§10. Falsifiers

The argument can fail in six ways. They are printed here in descending order of danger.

First. A demonstration that focality is only semantic, explanatory, or aetiological, and never substitutive — that the primary case brings about, grounds, or explains the derivative cases without ever holding their title to count as beings. §3 fails, and the study with it. This is the load-bearing premise, and Shields’s fourth condition is where to begin.

Second. A demonstration that the divergence exhibited in §7.1’s pair is an artefact of the intentional form of the loss rather than of what is lost — that a work destroyed against someone gone records a difference in how the two are mourned, in what the mourner brings, in the practices that shape grieving, and not a difference in what the two cases are. §7 fails, and §7.2’s non-scalarity with it, since both now rest on the pair. This is the sharpest available attack on the argument, and the shape it would take is clear enough to be worked: show that the two intentional forms can be redistributed across the two cases by a change in the mourner rather than in the case, and the pair stops exhibiting anything about the obtaining.

Third. A demonstration that the differentia is scalar — meeting the condition §9 states: a case in which the divergence itself admits degree, with the somewhat belonging to the disclosure rather than to our purchase on it. §7.2 fails; the category becomes a quantity; partition and paradigm return.

Fourth. A plural ontology, already in the literature, that preserves a genuine plurality of senses of being with neither a focal term, nor a univocal concept, nor bare quantifier-plurality. The historical claim fails outright. It is not enough that a position describes itself as having no privileged term; what would count is a position that survives the test of §3.

Fifth. A demonstration that the manner’s non-exhaustion is a count rather than a modal fact — that the character weakens as occurrences of the manner grow scarce and fails at the last one. §7.3 fails, and with it the account of why a substitute is possible at all.

Sixth. A non-scalar categorial differentia derived from an ontology that begins with entities rather than with address. §8’s account of the bottleneck fails, though §7 may survive it: the third answer would remain available, and only the explanation of why it was not given earlier would be lost.

Two of the claims made here are absences, and they are reported as such. I have found no instance of the substitutive reading of focal reference in the literature, and no instance of a positive dispersed ontology proposed in answer to the aporia of the first answer. Neither is a proof that none exists, and the searches behind both were curtailed rather than completed.

The claim of this study is narrow, and it fits in a line:

Against Aristotle, it rejects the primary case. Against univocity, it refuses to save plurality by collapsing it into one sense.

Notes to Part Three

[§7] The doctrine in which this differentia is developed is set out in Ethical Ontology, Volume I: The Modal Coincidence, and stated formally in The Calculus of Modal Coincidence, DOI 10.5281/zenodo.21497200. Nothing in the present argument depends on that account: what is used here is the differentia and its non-scalarity, and both can be assessed independently. — The reflexive objection to a deictically reached differentia is answered in The Individuating Factor, §7.3, and in Ethical Ontology, Volume I, ch. 11: the distinction is threefold — deictic determination, which is what gets reached; deictic reaching, which is the access, in its several regional forms; and linguistic deixis, one vehicle of that reaching among others — and the reaching never constitutes what it reaches.

[§8] The Individuating Factor, §4.2, DOI 10.5281/zenodo.21608145, states the route “as an exhibit rather than a credential,” and notes that “it is not a remark about the author.” The four perimeter figures are drawn from Koinonia: A Phenomenology of Being-With. — On gleichursprünglich: Heidegger, Sein und Zeit, §26. — Metaphysics A.8, 988b22–24; the section is verified, the sub-line follows from A.8 beginning at 988b21. A third occurrence of the one-nature formula, of the Eleatics, stands at A.5, 986b10–12. — For the Anaximander fragment and the Milesian material, Kirk, Raven and Schofield, The Presocratic Philosophers, 2nd edn (Cambridge, 1983). The textual problems in Anaximander should not be smoothed: it is disputed where the report of Simplicius and Theophrastus ends and Anaximander’s own words begin, and the word ἀλλήλοις — “to one another” — is absent from some manuscripts, which materially changes whether the reciprocity holds among the beings or between the beings and the apeiron.

[§9] Scotus, Ordinatio II, d. 3, p. 1: the common nature at q. 1, nn. 32–34 (Vatican VII, 403–405; Spade, 63–65); the rejection of qualitative individuation at q. 4, nn. 87–89 (432–434; Spade, 79–80); the individual difference and the formal distinction at qq. 5–6, n. 188 (483–484; Spade, 107), where the ultima realitas entis and the formally distinct entity of singularity against entity of nature are stated. The Ordinatio does not use haecceitas; the word is Scotus’s own but rare, and occurs elsewhere. On the relation to Scotus more broadly, The Individuating Factor treats the matter at length. — Morrison’s exception: “The Evidence for Degrees of Being in Aristotle,” Classical Quarterly 37 (1987): 382–401, at 398. — The Leibniz test refers to Ratio est in natura §§15 and 20, discussed at §5.4 above.

End.

Open items

Four, all library errands rather than questions. None bears on the argument.

1. Owen’s page for the coinage of “focal meaning.” The 1960 Göteborg volume is not

digitised and no source quotes the page. Cited here without one, which the attribution does not require. Three printings exist: Göteborg 1960, 163–190; Logic, Science and Dialectic (Cornell, 1986), 180–199; and Barnes, Schofield and Sorabji, eds., Articles on Aristotle, vol. 3: Metaphysics (Duckworth, 1979), 13–32, which is the most widely held.

2. KRS pages for the Anaximander discussion, should §8.4 acquire a footnote. 3. Harman’s page within the introduction of The Quadruple Object. 4. Bekker sub-lines for Physics I.2–3, and for the signifying something argument within

Metaphysics Γ.4, cited in §6.3. Γ.4 begins at 1005b35 and the defence of non-contradiction runs to 1007a20; the sub-line for the specific move is what remains. A.8, 988b22–24 stands: the section is verified, and the sub-line follows from A.8 beginning at 988b21.

One item marked approximate in earlier drafts is now exact: Deleuze’s white analogy, at Difference and Repetition 36. One closed by consultation of the text: McDaniel, whose article corrected §6.1 rather than confirming it.

Note on the two absence-claims

Two claims made in this book are absences, and §10 reports them as such. I have found no instance of the substitutive reading of focal reference in the literature, and no instance of a positive dispersed ontology proposed in answer to the aporia of the first answer.

Neither is a proof that none exists. The searches behind both were curtailed rather than completed, and the second in particular would repay a further pass through the reception of Aubenque in Italian and French scholarship — Berti’s response defends the unity of first philosophy rather than embracing dispersion, but the downstream literature has not been exhausted.

The reason for reporting them this way rather than more confidently is the same reason §10 exists. A claim about what nobody has said is testable only if it is stated in a form that lets someone test it, and I have found no instance invites the search that nobody has forecloses.

• Articles The other deposited preprints and their identifiers.
• Volume I The book this result belongs to, free in full.