Chapter 3　Tools: two negations and the tetralemma¶

This chapter assembles all the logical tools the essay will need and settles the choice of logic. The decision can be announced at once. We need only the law of noncontradiction, reductio, and standard derivations in classical logic. The tetralemma is read as rejection with presupposition cancellation; no logic departing from the classical is required. There are three tools: the distinction between two kinds of negation and the weak-rejection lemma; the reading of the tetralemma and separate treatment of its two families; and the structure of the identity-or-difference dilemma with its scope of application fixed. At the end, the first verse of the "Examination of Conditions" supplies a complete example, putting all three tools to work together. Later chapters will require no further equipment.

Assertion and rejection. Asserting ⊢A is not the only speech act available for a proposition A. There is also rejection, ⊣A: refusing to accept A without necessarily committing oneself to ¬A. Frege's classical position denies rejection an independent force; ⊣A should reduce to ⊢¬A. Bilateralism restores rejection as a primitive attitude alongside assertion. Its strong form, however, makes rejection and the assertion of a negation mutually inferable, collapsing the former back into the latter. This essay adopts weak rejection: ⊢¬A entails ⊣A, but the converse does not hold.¹ This is Lemma 1 of the essay, the weak-rejection lemma: ⊣A does not entail ⊢¬A. Here negation is internal negation, the kind that inherits the subject's presupposition. Sentences with failed presuppositions offer a straightforward witness. "The present king of France is bald" is unacceptable. Its internal negation also takes the existence of such a king for granted and is therefore equally unassertible. The only correct move here is rejection without a contrary assertion.² If negation is read externally, so that it is true whenever the embedded sentence is not true, ⊢¬A naturally becomes available again. But that sentence merely registers rejection and carries no new commitment. It does not conflict with Lemma 1.

Two kinds of negation. In Buddhist logic, the counterpart of the distinction between rejection and assertion is a division between two kinds of negation. Implicative negation (paryudāsa) negates an alternative within assertion. It retains the presupposition and commits to the complementary class. Saying "not a Brahmin" still says that the person belongs to another of the four castes. Pure negation (prasajya) performs rejection, negating without positing. "Tortoise hair does not exist" posits nothing further. It does not say that a tortoise has some other kind of hair or that there is a thing called nonexistence. This distinction is not imposed from Western learning. Saṃghabhadra himself divides negatively denoting expressions into those with a denotatum and those without one. An example of the former is "not a Brahmin," applying to a kṣatriya and others; examples of the latter are "nonexistent" and "nothing," which denote nothing at all.³ One neighbouring notion must be distinguished. Metalinguistic negation in modern linguistics rejects one expression while supplying another; pure negation supplies none. The two are close, but distinct.⁴

Presupposition and cancellation. Let π(t) mean that the subject t has a referent in the manner of E3 and D2 together. Implicative negation operates within π and inherits π; pure negation can act on π itself. Lemma 2, presupposition cancellation. When ⊢¬π(t) holds, any predication F(t) whose subject is t and which inherits π, together with its internal negation ¬F(t), is to be rejected, and only rejected. The rejections receive unified authorization from ⊢¬π(t). Modern literature already supplies procedures for testing presuppositions. This essay takes Strawson's account of presupposition and von Fintel's tests for later use in diagnosing the family of empty-reference sentences about whether the Tathāgata exists after death.⁵

The tetralemma: three readings, one principal account. Its formal appearance is as follows. S1 is F(t), S2 is ¬F(t), S3 is F(t)∧¬F(t), and S4 is neither. Two candidates for the exact notation of S4 remain; Appendix A will settle them against the original forms of expression. There are three readings. Reading A, parametrisation. Each statement carries implicit parameters, such as truth level, phase, or standpoint of dependence. Once these are made explicit, the statements no longer collide. The cost is that the parameters need independent warrant. The text warrants them only where it explicitly marks the two truths, so this becomes a supplementary account. Reading B, presupposition cancellation, is this essay's principal account. Nāgārjuna's complete sequence is ⊣S1, ⊣S2, ⊣S3, ⊣S4. A single assertion gives unified authorization for all four rejections: ⊢¬π(t). All four corners must be rejected in turn, because they exhaust the four repairs that keep π alive: asserting outright, retreating to negation, accepting both sides, and retreating to neither while retaining the referent. Only rejection of all four locates the failure in the presupposition π rather than in some predicate F. The connection with the central proposition takes one sentence. ⊢¬π(t) is precisely T instantiated at t; refuting all four statements applies T, rather than supplying another independent inference. Reading C, dialetheism and four-valued semantics. As a descriptive framework, this is tidy. As a normative logic, it requires containing explosion and redefining negation. Its cost has already led us to choose the conservative account; it remains an interlocutor for Chapter 7.⁶

One form, different uses: filling boxes and breaking them. The Chinese phrase 應作四句, "A tetralemma should be made," occurs 312 times in the Mahāvibhāṣā as a classificatory device. The question "Is there anyone who possesses the highest worldly dharma but does not possess the acquisition of disjunction from it?" receives the answer "A tetralemma should be made." Each box is assigned a kind of person, presupposing that the entire domain is on the register.⁷ Nāgārjuna takes up the same two-by-two grid to do something else. He rejects all four boxes to establish the failure of their common base. One form has two uses, and reading the two families as one has long been a source of confusion. The textual criterion for separating them is the closing form of expression. Tetralemmas ending along the lines of "This is called the teaching of the Buddhas" or "instruction" belong to stages of teaching, designations adapted to their hearers over time. Two passages in the "Examination of Dharmas" establish their character: the set beginning "Everything is real, unreal," and "Sometimes a self is taught, sometimes no-self is taught."⁸ Those ending along the lines of "Therefore we know there is none" or "This cannot be so" are refutational tetralemmas, this essay's proper subject. A bibliographical aside: the term catuṣkoṭi is not used in the original texts of the Mūlamadhyamakakārikā, Catuḥśataka, or Prasannapadā. The label is later.⁹

The identity-or-difference dilemma and fixing its scope. Its structure consists of x and y within a domain and a relation R between them. Fire and fuel, self and aggregates, cause and effect, agent and action all have this form. The identity horn: if x and y are identical, Leibniz's law transfers their properties and the two ends collapse. The agent is the action; what burns is what is burned. The difference horn: the difference at issue is not bare numerical difference but independent difference, written ≠. They are distinct entities that stand without depending on another. If x ≠ y, x should be obtainable apart from y, yet apart from y it cannot be obtained. Lemma 3, fixing the scope of application. What separates ≠ from ≠ is exactly the first condition of D2. If the opponent escapes by allowing things to be distinct but dependent, "not constituted in dependence on other dharmas" has already been abandoned. The ontological entity automatically drops out, and T holds directly. If independence is maintained, ≠ is ≠ for that opponent, and the dilemma closes. It applies only to object-level identity and difference free of any framework. Our own meta-level identity and difference are comparisons within a framework, as specified in Chapter 1's boundary clause; the two do not interfere. Two detailed rules apply. Identity is numerical identity. Leibniz's law is used only in extensional contexts; each use involving a cognitive context is marked separately. Comparison with the Vātsīputrīya tests the calibration of these tools exactly. Holding that the pudgala is neither identical with nor different from the aggregates and is therefore inexpressible retains the presupposition π while refusing the four answers. Nāgārjuna rejects π, leaving the four statements without a foothold. Inexpressibility occupies different levels: one puts it in the object's nature, the other in the terms' presupposition.¹⁰ A corollary drawn from this, that a verdict does not transmit identity, is given in Chapter 6.

The classical statement: the Vigrahavyāvartanī. Weak rejection and presupposition cancellation already receive a complete classical statement in the Vigrahavyāvartanī. Four verses suffice.¹¹ "If I had a thesis, I would be at fault. Since I have no thesis, no such fault arises." There is no assertoric commitment at the level of rejection. This is precisely Lemma 1. "As a conjured person acts against a conjured person, an illusory person against an illusory person, so too should rejection and what is rejected be understood." Language lacking a nature of its own still performs rejection; rejection needs no E3 basis to be effective. "What is it that you reject? What you reject is empty. There is rejection while dharmas are empty; thus your contention fails." If the opponent insists on reading rejection as assertion, the opponent's own thesis defeats itself. "Words have no nature of their own, nor does what is spoken of. Thus I am without fault and need give no superior reason." This brings the exchange to its close. One further sentence in Qingmu's commentary needs an entry of its own: "There is other because there is self." Other is established in dependence on self. If this can be established as an analytic truth of conceptual grammar, the relational premise P receives its strongest warrant. It is P's primary classical witness, formally brought into use in Chapter 4.¹²

A complete example: the first verse of the "Examination of Conditions." "Dharmas do not arise from themselves, nor do they arise from another, from both, or without a cause. Therefore we know there is no arising." The four paths of arising are read as four E3 configurations of the production relation G. Self-production is G(x,x); production from another is G(y,x), with y ≠* x; production from both combines the two paths; causeless production dispenses with a cause. Qingmu addresses each corner. Formally restated, the fault in self-production is the structural cost of a self-relation: a single dharma would have two entities, one called arising and the other what arises. Further, arising would itself arise, making arising endless. The fault in production from another is that the purely other cannot stand: "Since self is absent, other is absent too; there is other because there is self." The presupposition of independent difference defeats itself. Production from both inherits both faults. Causeless production is reduced to absurdity through its consequences: without causes, giving and keeping precepts should lead to hell, while the ten evil deeds and five heinous offences should lead to heaven. The moves and criteria all match. Rejection is performed at each corner, with unified authorization from ⊢¬π(what arises). The ending, "Therefore we know there is no arising," leaves no doubt that this belongs to the refutational family. The two Chinese characters 無生, "no arising," register rejection; they do not make a new assertion. The full account of this distinction belongs to the level clauses in Chapter 5. Meanwhile, not one instance of dependently originated arising at E2 is harmed. Seeds may still be said to sprout. This is the point of connection with the preservation theorem, addressed in Chapter 6.¹³

The tools, delivered. Lemma 1, weak rejection: ⊣A does not entail ⊢¬A, with negation read internally. Lemma 2, presupposition cancellation: ⊢¬π(t) gives unified authorization for rejecting all four statements. Lemma 3, fixing the scope of application: the dilemma has force only against those who maintain independence. Two further criteria accompany them. The closing form of expression distinguishes the two families, and parametrisation is warranted only where the two truths are explicitly marked. None of these tools goes beyond classical logic. To the question whether Buddhist logic departs from classical logic in this use, the essay answers that it does not.¹⁴ Chapter 4 begins the proof.

A plain-language restatement follows, under the same rule as before: change only the language and add no content. Where the two diverge, the upper half governs.

Start with two ways of not accepting. A jury that says the charge is not established has not said that the person is innocent. Those are two different things. A doctor who says a test cannot count has not said that its result is negative. Those are different too. Rejecting a sentence differs from asserting its opposite. Keeping those acts apart is this chapter's first tool. One kind of sentence shows the gap especially clearly: "The present king of France is bald." You cannot say yes, but you cannot say no either, because the sentence that says no also takes the existence of such a king for granted. The right response to such a sentence is to push the whole thing back, without taking the opposite side. Here "the opposite side" means saying that this king is not bald. Another response is available: "There is no such state of affairs at all." Of course that can be said, but it still just pushes the statement back. It adds nothing further.

Buddhism distinguished these two ways of not accepting long ago, calling them two kinds of negation. One is implicative negation. Saying that someone is not a Brahmin still posits that the person belongs to another of the four castes. The other is pure negation. Saying that there is no tortoise hair posits nothing further. It does not say that a tortoise has some other kind of hair or that there is a thing called "there is none." This division was not imported from the West either. Saṃghabhadra's own book says that some negative expressions have a referent and others refer to nothing.

The tetralemma is precisely where this distinction is most easily overlooked. There are four boxes: exists, does not exist, both exists and does not exist, neither exists nor does not exist. The Mahāvibhāṣā uses it as a questionnaire, filling it out more than three hundred times. Each box contains a kind of thing, and the arrangement works well. Nāgārjuna takes up the same questionnaire for another purpose. He rejects all four boxes and signs his name in none. All four must be rejected because they share one base. Each assumes that there really is a solid thing, standing in its own right, to talk about. Once all four have been rejected, it is the base that collapses, not one particular box. Rejection is not a reverse assertion, so the four rejections do not clash with one another. Ordinary logic handles the whole sequence; no deviant new logic is needed. We must also distinguish the families. One family of tetralemmas ends with "This is called the teaching of the Buddhas." These are teaching steps that lead people upward stage by stage, not refutations. The distinction rests on that closing sentence.

Now the dilemma of identity and difference. Fire and firewood: say they are one, and what burns and what is burned become one and the same thing. Say they are two solid things that each stand in their own right, and fire ought to burn on its own without firewood. This dilemma seeks out only one kind of person, someone who regards both things as standing in their own right. Ordinary talk of two things, compared as two within a framework, is outside its scope. So this is no trick for picking quarrels. It is a test. Anyone caught by it is covertly using that conception of two things standing in their own right.

The verses in the Vigrahavyāvartanī are the home of these tools. Someone asks Nāgārjuna: you say everything lacks intrinsic nature, but does your own sentence have intrinsic nature? Are you not contradicting yourself? The answer is precisely this chapter's first tool. I have no thesis, so I have no fault. Refusal is not a reverse assertion. I push your claims back one by one without owing you a claim of my own. An illusory person can block an illusory person. Language does not itself have to be real to do its work.

Finally, put the three tools to work together with the opening verse of the Mūlamadhyamakakārikā. Things do not arise from themselves, from others, from both together, or without a cause; therefore we know there is no arising. Four ways of being born, four boxes, rejected one by one. For something to produce itself, there must first be two selves, one producing and one produced. For something to arise from the purely other, the trouble is that other can be other only if the self side is established first. The purely other cannot stand. Production from both inherits the faults on both sides. If arising were causeless, cultivating good and doing evil should have no consequences. There would be nothing to prevent those who give and keep the precepts from going to hell and the utterly wicked from going to heaven. Once the four boxes have been rejected, what is removed is the base, a solidly existing thing that is produced. Ordinary arising loses nothing. Seeds still sprout and crops still yield their harvest. The two Chinese characters 無生, "no arising," mark rejection, not a new claim. Chapter 5 explains the full import of this sentence.

There are just these three tools. Refusal is not a contrary assertion. Rejecting all four boxes removes the base. The dilemma has force only against those who insist on that base. The next chapter begins the proof.

References¶