Chapter 7 Does it go through?¶

This is the self-audit chapter. First set its procedure, the high standard the author has laid down for the entire essay: take the strong version as far as it can be defended, and mark honestly any step that cannot be made. Finding a real flaw would itself be a gain; finding none would be quite something. This chapter proceeds accordingly. Each foreseeable mode of failure appears in its strongest form. If it can refute the argument, record the downgrade; if it cannot, record what the test has brought into clearer view. After the four modes, we engage directly with the dialetheist reading and draft a verdict for each of the three exits.

Mode one: the worry that the distinction cannot be maintained. The strongest charge does not repeat the distinction between knowledge and existence, already handled in Chapter 4. It retreats to the last step: determinacy owes nothing to statement. A thing can simply be this one, with no statable condition constituting its being this one. Medieval scholastic philosophy coined a name for this, haecceity, haecceitas. If this step holds, P no longer holds as an analytic truth, and the strong version is downgraded. The first reply is unchanged. The term "haecceity" cannot withstand its own claim. "This one" is inherently contrastive. The pointing has already picked one out from many. A "this one" that could have no "that one" opposite it has pointed out nothing. Introducing the concept of haecceity relies on the very contrast it denies.

A shortcut that must be rejected. At the second step, one apparently convenient move must be rejected on the spot. Suppose the opponent retreats further, saying that the concept is introduced contrastively but the property it names is itself primitive. We might then say that this step withdraws from statement and so falls outside D3's scope. But being primitive does not entail being unstatable. Taking identity as primitive is a position with content. It has consequences, most conspicuously the failure of the principle of the identity of indiscernibles. Modern literature has constructed a model for this: two spheres identical in all their properties yet numerically two. The position has a literature devoted to defending it and another devoted to refuting it.¹ To record it as mute realism is to record a speaking opponent as silent. This move therefore fails. The first reply reaches only the procedure for introducing a term, a level the opponent may legitimately set aside. The second step must start afresh and answer the position directly.

Separate two retreats. Retreating to the primitive and retreating to the unstatable are different retreats. Chapter 4's closing stage already distinguished four routes. For those who retreat to the unstatable, the D3 assertion loses its determination. T-middle and the clause delimiting the thesis domain take over. The middle version handles it, and this route needs no further discussion. Those who retreat to the primitive while still making statements are this mode's real opponents. There are three replies.

First: how is the primitive itself individuated? Chapter 4's question now comes down to a concrete case. Why does a's haecceity differ from b's? By their bearers? That is circular, since haecceity was posited to explain the difference between a and b. By being primitively different? Then difference already stands beside identity at the basic level, and Chapter 4's grading puts unity and difference precisely in the family both parties recognize as mutually dependent. Both routes return the primitive to contrastive dependence, exactly the level D2's first condition rejects. The distinction between this reply and the first step matters. The first step concerns the word; this reply concerns what makes the posit itself this one.

Second: haecceity theory has a party in this dispute. The fifth of the Vaiśeṣika's six categories, particularity, viśeṣa, is precisely primitive haecceity. It resides in eternal substances: the four kinds of atoms, earth, water, fire, and air, together with space, time, direction, self, and mind. Each depends on one substance, one particularity for each substance, with one job: distinguishing from one another partless atoms that are identical in all their properties. Without this, atomism cannot stand. Xuanzang's translation gives a definition of just one sentence: "It is eternal, occurs in substances, depends on one substance, and is the cause of excluding cognition of that and manifesting cognition of this."² The first three items register its identity. All the substantive content lies in the last, and the last is a pair of acts: excluding that and manifesting this. "That" is the crucial word. Qingmu's own contrast group when explaining "not constituted in dependence on other dharmas" is precisely long and short, this and that: "having no determinate nature, they exist in dependence on another." The "Examination of Time" places unity and difference in the family of mutual dependence. In defining their most fundamental primitive, the opponents use words from this very family and offer no further formulation without "that."

The Sanskrit side takes the same direction: self-differentiating, svato-vyāvartaka, differentiating its substratum and differentiating itself, with differentiation stopping here and regressing no further. This is no proof but a record. The one school that made haecceity a category, given an opportunity to supply a noncontrastive determination of it, supplied a contrastive one, and only that one. Part I, Essay 3, Chapter 6 already recorded the six categories and their symmetry with the Sarvāstivāda inventory. This case was in the book already.

Third: the Sarvāstivāda are not haecceity theorists and cannot become such. The Sarvāstivāda individuate a dharma's entity by its own characteristic. Alteration and obstruction define material form; reception defines feeling. Essay 15, Chapter 3 recorded the Mahāvibhāṣā's questions and answers aggregate by aggregate. If numerical identity is primitive and nonqualitative, a dharma's own characteristic is demoted from a condition of its entity's identity to an accompanying property. The whole inventory of seventy-five dharmas in five classes then loses its basis for individuation. What the Sarvāstivāda need is precisely that material form's being material form depends on alteration and obstruction. The posit introduced to save D2's first condition cancels the very job for which intrinsic nature was originally established.

Audit result one: derive it instead from E3. Bare haecceity with no own characteristic at all fails E3. Saṃghabhadra's criterion is "giving rise to cognition of it without depending on anything else," with material form, feeling, and so on as positive examples, pots, armies, and so on as negative ones. The entire criterion rests on own characteristics. For the test of decomposition and analysis, there is no characteristic to analyse, so the test has no application. Bare haecceity therefore falls outside the theorem's scope for a reason internal to D1: it does not count as really existent by the opponent's own criteria. The conclusion is unchanged down to the last word. T-strong's quantifier still ranges over all D3-type propositions. The route to it has changed, and the new route owes nothing to the step rejected above.

Finding and remainder. The strong version covers haecceity theorists who assign haecceity a job. Once their primitive takes on the task of distinguishing a from b, it returns to contrastive dependence. If someone insists that facts of identity come first and facts of difference supervene on them, then stops there with nothing more to say, E3 and T-middle take over. This is not handled by T-strong's semantic collision. It is still entered under the middle version. Mode one has caused no downgrade, but it has corrected the bookkeeping. From now on, the conclusions of the strong and middle versions are recorded separately. The strong version's grounds are confined to the dependence axis and the domain of conditions; the remainder belongs to the middle version. The same result is not counted twice.

Mode one's limiting case: two spheres. This mode has a modern limiting construction worth examining on its own, since it makes the conditions as favourable as possible for haecceity theory. Suppose a universe containing only two iron spheres, identical in diameter, colour, temperature, and composition. The universe is symmetrical too. A rotation of half a turn about the midpoint exchanges the spheres, so their relational properties are identical as well. They are indiscernible yet two, thereby refuting the principle of the identity of indiscernibles.³ Two propositions must first be distinguished. That principle says that qualitative indiscernibility entails numerical identity. P says that identity conditions necessarily involve difference and relations. The construction refutes the former without touching the latter. This essay can concede the former's falsity without paying a penny. Only by conflating the two propositions would one see a counterexample here.

The direction is just the reverse: this construction takes P to its maximum. Once symmetry has removed all qualitative and relational differences, nothing but difference is left at the point of individuation. The only so-called properties by which the spheres differ are impure properties containing reference, such as being identical with a particular sphere. Differing in these properties is differing numerically, not a difference beyond number. Using them to rescue that principle is circular.³ This and Chapter 4's closing passage, where only "x is x" remains, say the same thing in two ways.

The crucial point comes earlier: what makes a sphere a body? Stating this construction has already brought in space, an exterior region, and a boundary, so a dilemma arises. If space enters the inventory, the statement that the universe contains nothing but two spheres is false. The inventory contains something else, and it is precisely this that individuates by position. Position is relational. If space stays out of the inventory, "a certain distance apart" and "spherical" have no reference. No situation has been specified, so there is no counterexample to discuss. Both alternatives fall on the dependence axis. The second step of Chapter 4's three-step argument for analytic truth already says that individuation is contrast: x is individuated if and only if there is a criterion distinguishing x from non-x. Here the same principle is merely applied to the level of being a thing: the boundary is that distinction.

Someone will say that shape is intrinsic. A sphere's being a sphere depends on internal relations among its parts and owes nothing to anything outside. This objection carries a recursion. If shape is a relation among parts, real existence moves down to the parts, then further down, ending at what has no spatial parts. At that point, shape, boundary, and alteration and obstruction vanish together. The Abhidharmakośabhāṣya, fascicle 1, poses an objection and gives a reply on precisely this matter: "An atom should not be called material form, since it has no alteration and obstruction"; "No single atom abides separately; when many atoms aggregate, alteration and obstruction are established."⁴ To obstruct is to obstruct something else. Either shape is relational or the endpoint has no shape. For the Sarvāstivāda details, see Essay 15, Chapter 6.

A limit that must not be overstepped. This essay does not claim to have defeated the construction in its original aim. Its target is the principle of the identity of indiscernibles, whereas this essay needs only one thing: the construction cannot establish a nonrelational kind of individuation, because its very intelligibility depends on a relational structure. That point is secured, and that point alone is needed.

Another question this essay records as unresolved. What makes those two spheres conceivable is a metric. This metric is a third party brought in to demonstrate indiscernibility, and the essay has not determined what it itself is. If settling this third party's status were in turn to change the spheres' inventory of properties, whether indiscernibility itself holds would also have to be asked anew. This question belongs to the metaphysics of space, outside this essay's scope. Here we only record it as unresolved and do not use it to strengthen the conclusion. For this essay's purposes, the case has a one-sentence outcome: it reinforces P; it is no counterexample to P.

Settle the numerical level too. The point reached by modern discussion at this level coincides with this essay's axis. Quine distinguishes three grades of discernibility, and the spheres can be distinguished by weak discernibility: one sphere is a certain distance from the other but not that distance from itself. This relation is symmetric and irreflexive.³ Weak discernibility is precisely the level of contrastive dependence graded in Chapter 4. Note that it does not rescue that principle, which requires absolute discernibility. What survives is individuation, and it survives only relationally.

One supporting argument follows. Suppose a plurality of really existent things with the same own characteristics. By D2's third condition they have no separate origins; by its second, no separate histories. They are also qualitatively indiscernible. Only two resources for individuation remain: relations and primitive haecceity. The former directly violates the first condition. The latter, by the definition quoted above, still "excludes that" and so still violates the first condition. The last two conditions create a need for individuation, while every available resource violates the first condition. This argument does not use P once and so does not depend on P's analyticity. It has greatest force against actual historical opponents. Those willing to say only "there are just two," with nothing further to say, still come under the middle version, in the same entry as the remainder above, without double counting.

Mode two: the suspicion of realism about relations. The charge: P speaks constantly of difference and relations. You have merely given relations the place of entities. If relations really exist, nothing substantive has changed; if relations too are empty, does P defeat itself? The reply has four parts. First, P is a claim about determination: statements of identity conditions necessarily use relational vocabulary. It does not claim that relations are E3 items. Second, the emptiness of relations is a natural extension of the same inference, not an expedient. A relation standing on its own without depending on its two ends could relate to neither, as Chapter 3's difference alternative proved. A relation with intrinsic nature fails to be a relation. All relations therefore fall at the level of E2, and E2 relations are exactly all that P needs: determinations, frameworks, vocabulary. Third, a deeper worry must be settled. If individuation depends entirely on a framework, has idealism been smuggled in? Framework-relative individuation does not mean that things depend on mind. A framework is a public practice, not a private consciousness. As for what ultimately lies behind the framework, that question asks for an E3 item. The question itself falls within the scope of application and is rejected by the rule.

Fourth, take the charge in its strongest form. P says that identity conditions must involve relations, and the reply just given places all relations at the level of E2. Taken together, does this mean that everything's being what it is passes through a network of language and names? Is E3 closed to every candidate in advance, breaking the scope clause's promise to leave existence assertions untouched? A boundary must be drawn here. This essay posits no inference from the level of conditions to the level of existence. That identity conditions involve relations at the level of language and names is a fact about determination. Whether a thing depends on language and names for its establishment is a matter for the E3 tests, decided by decomposition and analysis and by giving rise to cognition, not by the vocabulary used in its determination. No such inference exists between the two levels, and the essay has never made one. Everywhere the theorem concludes that a conjunction defeats itself, nowhere that a thing does not exist.

The ensuing symmetrical question is the crucial one. If we deny this inference, the opponent seems able to use the same denial to fend off the core derivation. The answer lies in the structure. The core derivation contains no inference across levels to deny. D2's first condition itself stands at the level of conditions: "not constituted in dependence on other dharmas" says that its being what it is is independent. P asserts necessary dependence at exactly the same level. Pledge and conceptual grammar collide on one level; contradiction needs no crossing between levels. If the opponent rewrites the pledge at the level of existence to avoid the collision, the first condition has been abandoned. What is posited is no longer intrinsic nature in D2's sense, and under the scope clause this essay has no dispute with it. For candidates claiming that these two levels are one in their own case, the simplicity clause is handled in Chapter 8's first route. Finding: mode two has caused no downgrade and has yielded a further piece of evidence. That relations too are empty is forced by this inference itself, not posited to rescue P.

Mode three: the suspicion of an inductive leap. The charge: twenty-seven chapters are a finite sample. Leaping from twenty-seven reductios to all D3 propositions is induction, not proof. The reply: this objection mistakes the essay's architecture. The strong version rests on no enumeration of cases. It rests on Chapter 4's analytic argument. The conflict between P and D2's first condition follows from the concepts. Its universal force comes from conceptual grammar, not the number of examples. The chapters serve as demonstrations and checks: the same inference is applied unchanged at each historical site. These are corroboration, not premises. Even the original's universal statement, "Never has there been a dharma not born of causes and conditions," quoted in Chapter 6, is used only to record the text's intention, not as an argumentative premise. There is only one place for a possible exception: a candidate claiming the full strength of D2 yet escaping the derivation's constraint. But the derivation uses only D2's first condition and P, both universal conceptual truths. Such a candidate's only route is to deny that it is subject to identity conditions, returning to mode one's remainder, already handled. Finding: mode three dissolves. Between weak and strong there is no induction, only analysis and its historical corroboration.

Mode four: the status of the truth predicate. The charge: if everything is nominally designated, then truth too is nominally designated. Is your theorem still true? This is the deepest of the four. The reply distinguishes levels. Truth at the ordinary level is judged within a framework. "To say, in accordance with conventional principles, that pots and the like exist is true, not false." Both parties agree, so no defence is needed. The theorem's truth is analytic truth, and its framework is conceptual grammar and logic themselves. This framework has one property no other framework has: it is irreplaceable. Any framework capable of individuating, asserting, and reasoning already uses identity and difference, so this inference is valid within it. The theorem's truth is thus inescapable within every framework, not truth seen from nowhere. This gives the final formalization of ultimate truth: not a higher domain of objects, but an invariant of all frameworks. Chapter 6 said that the two truths distinguish levels of assertion. This chapter adds the other half: conventional truth is truth within a framework; ultimate truth is invariant truth across all frameworks.

A difficulty of self-application must be tested here too. If the theorem is true in every framework, it seems to exclude nothing and thus, by this essay's own equation of content with exclusion, to say nothing. Resolving this difficulty requires three dimensions and two kinds of saying nothing. Exclusion has three possible objects: situations, frameworks, and claims. In the dimension of frameworks, the theorem indeed eliminates nothing. That is what an invariant means. But the law of noncontradiction likewise eliminates no workable framework, and no one therefore says that it says nothing. A norm's work lies in the dimension of claims. Within each framework, the theorem forbids a family of assertions that have actually been advanced. It closes an area that was once epistemically open in fact. The history of debate across two hundred fascicles is evidence that this area was once open.

As for silence in the dimension of situations, two kinds of saying nothing must be distinguished. The theorem's saying nothing fulfils its duty. A meta-constraint's job is to issue rules, not describe situations, just as the law of noncontradiction has nothing to say about situations. The bearer of intrinsic nature's "x is x" fails its duty by saying nothing. Identity conditions are meant to distinguish x from non-x, and this distinguishes nothing. One sentence gives the criterion: what says nothing yet cannot be discarded is a rule; only what says nothing and can be discarded is empty talk. "Cannot be discarded" means that discarding it loses the framework. Strictly speaking, what cannot be discarded is the set of items the theorem makes explicit: identity and difference, individuation, and the law of noncontradiction. The opponent's framework has always relied on them to work. This was exactly the assertion version's original charge. "Can be discarded" means that the framework keeps working without it. The essence clause is precisely a dispensable component, as part (a) of the preservation theorem has demonstrated on the spot. The two silences belong to different duties. The opponent cannot invoke silence at the level of rules to excuse the opponent's own idle spinning.

As for anyone wanting to posit truth as an E3 item, a truth-entity or correspondence with reality, Chapter 5's clause (c) already provides for it. Reified truth falls within the scope of application and is handled by the rule. Finding: mode four has caused no downgrade. The invariant formulation above is the second audit result.

An interlocutor: dialetheism. Priest, Garfield, and others in this line read the Mūlamadhyamakakārikā as proposing true contradictions at the limits of thought. Saying the unsayable is one example, and Nāgārjuna is a precursor of this discovery.⁵ This essay respects that reading as a reasoning opponent. The pressure it sees is real: a statement about limits seems to have to cross those limits. We differ over the solution. They solve it through true contradictions, at the cost of taming explosion and redefining negation. Logic itself must change. This essay solves it by distinguishing levels, with level clauses and rejections distinguished by type, leaving logic intact.

Two reasons support the conservative proposal. Cost: with the presupposition-cancellation reading available, no verse forces us to accept a true contradiction. The third statement of the tetralemma remains an object of rejection, never an asserted truth. Text: sorting the closing forms of expression in Chapter 3 shows that the refutational family of tetralemmas ends in rejection, while the family of stages of teaching ends in adaptation to the audience. Neither family ends by declaring contradiction true. The book's position is therefore this: the conservative proposal is the correct reading; the dialetheist reading is recorded as a complete alternative held in reserve. Should the distinction of levels fail somewhere, it is an available replacement. No such place has yet been found. One boundary must also be clear. Dialetheism is a logical proposal subject to norms. It has nothing to do with the option of overturning the table discussed in Chapter 2. It remains among those who reason, an opponent, not someone who forfeits the contest.⁶

The three exits: draft verdicts. Exit one, analyticity holds. The four modes have been tested, and none has caused a downgrade. The strong version holds at analytic strength, grounded in the dependence axis both parties acknowledge and the domain of conditions. Residual cases are recorded separately under the middle version, as in mode one. Three audit results accompany it: the delimitation of the thesis domain, the invariant formulation, and Chapter 4's finding that independence itself is a property of contrastive dependence. Exit two, downgrade to the transcendental. Its trigger would be someone's demonstration of noncontrastive individuation. This chapter has been unable to construct such a demonstration, and even its description cannot avoid contrast. This exit has not been triggered. Exit three, downgrade to the empirical. That requires both earlier alternatives to fail and is further away. This book holds that the verdict falls at exit one. The formal announcement, together with the premise audit table, belongs to Chapter 9. The author's opening standard can now be settled: a serious search for flaws found one, repaired it on the spot in mode one, and produced a repair stronger than the earlier reply. Three other things also emerged more clearly than expected.

What this chapter delivers. Findings on four modes of failure; one correction through self-audit; three audit results; a record of the dialogue with dialetheism and its position in reserve; draft verdicts for the three exits. Chapter 8 takes the theorem beyond its domain to test it. Chapter 9 announces the verdict.

A restatement in plain language follows the same rule as before: change only the language, add no content, and let the first half govern any discrepancy.

This chapter finds fault with our own work. The author set the rule: go as far as the strongest version will go, and honestly mark the step where it stops. Finding a flaw counts as a gain; finding none is what would be impressive. Each of the four challenges below appears at its strongest.

The first challenge is the opponent's last step. A thing, he says, can simply be this one, this one without a reason that can be stated. It owes you no account. Our first reply is unchanged: "this one" is inherently a term of comparison. Pointing already picks one out of a group. If there were no "that one" in the world, the three Chinese characters 这一个, "this one," would point to nothing.

One easy reply must be rejected on the spot. Someone might say that if he retreats further, claiming that the word is introduced through comparison but the thing it names is itself primitive, he has withdrawn from reasoning. Our theorem can no longer reach him and need not do so. But primitive does not mean unstatable. Taking a thing's identity as primitive is a position with content. It has consequences. The most conspicuous is the failure of the principle of the identity of indiscernibles. Modern literature has constructed a model expressly for this: two spheres identical in all their properties yet numerically two. There are writings defending it and writings refuting it. To record it as a silent realism is to record a speaking opponent as silent, so this reply is wrong. The first reply reaches only the way a word is introduced, a level he may legitimately set aside. The second reply must start afresh and answer him directly.

First separate the two retreats. Retreating to "this thing's identity is primitive, but I keep speaking all the same" differs from retreating to "I simply will not speak." Chapter 4 has already separated four routes. The latter goes to the middle version and needs no further discussion. The former is this challenge's real opponent. There are three replies.

One: ask how that primitive thing is itself distinguished. He uses it to distinguish a from b. Then ask what makes a's primitive and b's primitive two different ones. If the answer is a and b, we have gone in a circle. This thing was meant to explain why a and b differ. If the answer is that the two are primitively different, difference stands alongside identity at the basic level, and unity and difference are precisely what both sides put in the family of mutual dependence. Both routes return this primitive to contrastive dependence, exactly the grade that "not constituted in dependence on other dharmas" pledges to reject. The distinction from the first reply matters: that reply concerns the word; this one concerns what makes the thing itself this particular thing.

Two: this position has a real party in this dispute. The Vaiśeṣika make it their fifth category, called particularity. It resides in eternal entities, the atoms of the four elements, together with space, time, direction, self, and mind, one for each entity. Its only job is to distinguish from one another partless atoms that are identical in all their properties. Without it, atomism falls apart. Xuanzang's translation gives it a definition of just one sentence: "It is eternal, occurs in substances, depends on one substance, and is the cause of excluding cognition of that and manifesting cognition of this." The first three items register its identity. All the substantive content is in the last, a pair of acts: blocking out that one and marking out this one. "That" is the crucial word. When explaining "not constituted in dependence on other dharmas," Qingmu gives long and short, this and that as his ready examples. They have no determinate nature, he says, and need something else to exist. The "Examination of Time" also puts unity and difference in the family of mutual dependence. The opponent defines his most fundamental primitive with words from this very family and gives no further formulation without "that." The Sanskrit determination takes the same direction: self-differentiating. It differentiates what it resides in and differentiates itself. The search stops here and goes back no further. This counts as no proof but as a record: the school that actually made this step a category, given an opportunity to describe it without comparison, supplied a comparative account, and only that one.

Three: the Sarvāstivāda never belonged to this camp and cannot become members of it. They distinguish a dharma's entity by its own characteristic: alteration and obstruction define material form; reception defines feeling. Essay 15 recorded the passage of the Mahāvibhāṣā that asks the question aggregate by aggregate. If a thing's numerical identity is primitive and has nothing to do with what sort of thing it is, its own characteristic is demoted from an identity condition to an accompanying property. The whole table of seventy-five dharmas in five classes then loses its basis for individuation. What the Sarvāstivāda need is precisely that material form's being material form depends on alteration and obstruction. The posit summoned to save that pledge cancels the very job for which intrinsic nature was established in the first place.

The route to that audit result therefore changes. A bare "this one" with no own characteristic cannot pass the test for solid reality. Saṃghabhadra's criterion asks whether it can independently serve as an object of cognition. His positive examples are material form and feeling, his negative ones pots and armies. The whole criterion rests on own characteristics. For the test of decomposition, there is nothing at all to take apart. So it falls outside the theorem's jurisdiction for a reason on the real-existence side: by the opponent's own criteria it does not count as really existent. Not one word of the conclusion changes. What changes is the route to it, and the new route owes nothing to the step rejected above.

This challenge has not forced a downgrade of the theorem, but it has made our accounting more precise. If an opponent states a determinate nature, the result belongs to the strongest version. If he takes identity as primitive and assigns that primitive item the task of distinguishing A from B, the result also belongs to the strongest version. If he insists that identity facts come first and facts of difference follow, and says no more, the E3 route and the middle version take over; this remaining case is recorded under the middle version, not as a semantic collision established by the strongest version. An opponent who refuses to say anything is likewise handed over to the middle version. The two versions keep separate accounts, with no result counted twice.

This challenge has a modern limiting example worth examining on its own, since it makes the conditions as comfortable as possible for the opponent. Imagine a universe containing only two iron spheres, identical in diameter, colour, temperature, and composition. The universe is symmetrical too. Rotating half a turn about the midpoint exchanges the spheres exactly, so even their relations to other things are identical. The spheres cannot be distinguished, yet there are two. This refutes the old principle that what cannot be distinguished is one and the same.

First separate two statements. The old principle says that things indistinguishable by their properties are numerically one. Our premise says that making clear how a thing is itself cannot avoid difference and relations. The example refutes the first statement without touching the second. We can acknowledge the first as false without paying a penny. Only by mixing them up would someone see a counterexample here.

In fact, the direction is the reverse. This example takes our premise to its maximum. Once symmetry removes all differences in properties and relations, all that remains to make the spheres two is their not being each other. Nothing else is left. The only so-called properties in which they can count as different are statements such as being the same as a particular sphere. But differing in such a statement is differing numerically, not a difference beyond number. Using it to rescue that principle goes in a circle.

The real point comes earlier. What makes a sphere count as a body? As soon as the example is stated, space, the region outside, and the sphere's surface have come in with it. This yields a dilemma. If space counts, the claim that the universe contains nothing but two spheres is false. There is something else on the inventory, and that very thing distinguishes the spheres by position. Position is relational. If space does not count, neither how far apart they are nor what shape spherical is has any reference. The situation has not been stated at all, so there is no counterexample to discuss. Both ends fall on the side of dependence. The second of Chapter 4's three steps already says that identifying a thing is distinguishing it from what is not it. Here the same statement is merely applied to being a thing: the surface is that distinction.

Someone may say that shape is the thing's own affair, a relation among the parts inside a sphere, owing nothing to the outside. This opens a downward trail. If shape is a relation among parts, the real thing lies in the parts. Follow the parts down to the particle with no further parts to divide. At that particle, shape, surface, and the ability to block things all vanish together. The Abhidharmakośabhāṣya, fascicle 1, posed this objection itself, and its reply is clear too. An atom should not count as material form because it blocks nothing. Yet no single atom stays by itself in one place. Only when many atoms gather does blocking hold. Blocking must block something else. So either shape is relational or there is no shape at the end of the trail. The Sarvāstivāda details are in Essay 15, Chapter 6.

One limit must be observed. We do not claim to have defeated this example in its original aim. It seeks to refute the old principle. We need only one thing: the example cannot establish individuation that does not rely on relations, because understanding it first requires the support of a relational structure. That point is secured, and it is all we need.

Another question this essay does not pursue. What lets us imagine those two spheres is a metric, the apparatus for measuring distances. This metric is a third party brought in to demonstrate indistinguishability, and the essay has not judged what it itself counts as. If settling this third party's identity were in turn to change the spheres' inventory of properties, whether they remain indistinguishable would have to be asked again. That question belongs to the metaphysics of space, outside this essay's scope. Here we record it without judgement or using it to add weight to the conclusion. For this essay's purposes, the account is one sentence: the example reinforces our premise; it is no counterexample to it.

Settle the numerical level as well. The place modern thinkers have reached here coincides with our axis. Discernibility has three grades, and the spheres can be distinguished at the weakest: this one is a certain distance from that one but not that distance from itself. The relation is symmetric but does not point back to itself. The weakest grade is exactly the level of contrastive dependence graded in Chapter 4. Note that this cannot rescue the old principle, which requires the strongest grade. What survives is individuation, and it survives only relationally.

There is a supporting argument too. Suppose there are several solidly real things with the same own characteristic. By intrinsic nature's third condition they have no distinct origins, and by the second no distinct histories. They cannot be distinguished by their properties either. All that is left to distinguish them is relations or a primitive "this one." The former directly violates the first condition. The latter, by the definition quoted earlier, still "blocks out that one" and also violates it. Intrinsic nature's last two conditions create the need to distinguish them, while every available means violates the first. This inference never uses our premise and so is unaffected by its strength. It is strongest against actual historical opponents. Someone willing to say only "there are just two" and nothing else is still handed over to the middle version, the same entry as before, counted only once.

The second challenge: have you given relations the place of entities? No. If a relation too were solidly real and established on its own, it could not reach its two ends and would fail to be a relation. Chapter 3's dilemma derives this on its own. So relations too are empty, and empty relations do just the work needed. Determinations, frameworks, vocabulary are all at the ordinary level and suffice. Someone asks further what ultimately lies behind the framework. This again asks for something solidly real. The question itself lies within the theorem's jurisdiction and is dismissed by the rule.

Now put this challenge at its strongest. You say that anything with identity conditions involves relations, and that all relations are at the ordinary level of names. Put the two together and anything's being itself must pass through names. No one can obtain the title of solid reality. Have you broken your promise to leave existence itself untouched? Here we draw a boundary. We posit no inference from identity conditions to existence. That a thing's identity conditions cannot avoid relations at the level of names concerns its determination. Whether its existence depends on names is governed by separate, established criteria. Test whether analysis disperses it and whether it can serve as an object of cognition, not what words its determination uses. There is no such step between the two sides, and we have never taken it. The theorem always concludes that the conjunctive claim collapses on its own, never that a particular thing does not exist.

The question that follows is sharper. If we do not accept this step, the opponent seems able to refuse it too and fend off our main argument. The answer is that the main argument contains no such step to refuse. The opponent's pledge, to be itself without relying on other things, was made at the level of identity conditions in the first place. Our truth of conceptual grammar, that identity conditions must involve difference, concerns exactly that level. Pledge and conceptual grammar collide on one level; contradiction needs no crossing of levels. If he moves the pledge to the level of existence to escape, the first pledge is lost. What he posits is no longer the original kind of intrinsic nature, and under the scope clause we have no dispute with him. If someone claims that the two levels are one in his case, Chapter 8's first route handles that case specifically.

The third challenge: twenty-seven cases cannot yield an induction about everything in the world. The answer: we were never counting cases. The inference follows from the meanings of words. Its universal force comes from conceptual grammar, not sample size. Those cases are demonstrations and checks, not the evidential foundation. There is only one route to a real exception: claim to be exempt from identity conditions. That returns to the first challenge, already handled.

The fourth and deepest challenge: if "true" too is a nominal designation, is your theorem still true? The answer has two levels. Ordinary truth is judged within a framework. A pot is a pot, truly and not falsely; both sides agree. The theorem's truth is a truth of conceptual grammar, and this framework has one capacity the others lack: it cannot be avoided.

A difficulty of our own can be handled here as well. If the theorem is true in every framework, does it exclude nothing and say nothing? Answer in three dimensions. Exclusion has three objects: situations, frameworks, and claims. The theorem indeed eliminates no workable framework. That is called an invariant. But the law of noncontradiction eliminates no framework either, and no one calls it empty talk, because its work concerns the third object. In each framework, it really prohibits a family of claims that people have actually made, closing a region that was once really open. The two hundred fascicles of debate are evidence that its doors were once open.

As for silence at the level of situations, distinguish two kinds of having nothing to say. The theorem fulfils its duty by having nothing to say. A meta-constraint's job is to issue rules, not describe situations, just like the law of noncontradiction. The opponent's "I am I" fails its duty by having nothing to say. Identity conditions are meant to distinguish things, and this distinguishes none. One sentence gives the criterion: what has nothing to say yet cannot be thrown away is a rule; only what has nothing to say and can be thrown away is empty talk. "Cannot be thrown away" means that the whole framework collapses without it. Strictly, what cannot be thrown away is what the theorem specifies: identity and difference, individuation, and the law of noncontradiction. The opponent's own framework has always run on these too. "Can be thrown away" means that the framework keeps working without it. The essence clause is precisely dispensable, as Chapter 6's preservation half demonstrated on the spot. The two silences belong to different duties. No one can use a rule's silence to excuse their own idle spinning.

Anyone who identifies things, makes assertions, or reasons already uses identity and difference, so this inference is valid within that person's framework. The theorem's truth therefore follows wherever one goes among frameworks. There is no need to look down from the clouds. We can now give the final word on ultimate truth: not a higher world, but the invariant all frameworks share. Conventional truth is truth within a framework; ultimate truth is truth across frameworks. Whoever makes truth an entity falls under Chapter 5's third rule.

Then we talk directly with a school of accomplished thinkers. They say there must be true contradictions at the boundary. Saying the unsayable is a living example, and Nāgārjuna saw it long ago. We respect them as reasoning opponents. The pressure they see is real. We differ over the method. They accept true contradictions at the cost of changing logic itself. We distinguish levels and leave logic intact. Two reasons favour our approach: it costs less, and no verse in the original forces us to accept a true contradiction. The third statement of the tetralemma has always been something rejected, never a truth accepted. So the conservative proposal is the correct reading. Their account is recorded as a complete alternative held in reserve. If the distinction of levels ever fails, a replacement is ready. So far that day has not come. One further line must be drawn: these thinkers reason logically and follow rules. They are not of a kind with the person overturning the table in Chapter 2.

Draft verdicts for the three exits: the first exit holds. None of the four challenges has forced a downgrade, and three things have become clearer into the bargain: the theorem governs claims that are stated; ultimate truth is an invariant of frameworks; independence itself is a property that exists through contrastive dependence. The threshold for the second exit is to construct a way of identifying things without comparison. This chapter cannot construct one; even describing it needs comparison. The third exit is further away. Chapter 9 will pronounce the verdict formally, laying out the premise inventory alongside it. The author's opening standard comes out like this: we searched seriously, found a flaw in the easy reply to the first challenge, and repaired it on the spot. The repair is stronger than the earlier reply, and three further things became clearer into the bargain.

The next chapter takes the theorem beyond its domain to test it. Chapter 9 announces the verdict.

References¶