Chapter 3: Tools — Two Negations and the Tetralemma¶

This chapter assembles at one stroke the logical tools the rest of the part will use, and the choice of logic is settled here as well. The settlement may be announced in advance: what this part requires is the law of non-contradiction, the method of reductio, and the standard derivations of classical logic, and nothing besides; the tetralemma is read as rejection under presupposition cancellation, and no logic that departs from the classical is called for. There are three tools: the division of negation into two kinds, together with the weak-rejection lemma; the reading of the tetralemma, together with the separation of its two families; and the skeleton of the identity-difference dilemma, together with the scope-lock. The chapter closes with MMK I.1 as a full worked sample, running the three tools through in one pass. No chapter after this one forges new instruments.

Assertion and rejection. For a proposition A the speech acts available are not exhausted by assertion, ⊢A. There is also rejection, ⊣A: refusing to accept A, without thereby undertaking any commitment to ¬A. Frege's classical position is that rejection is no independent force and that ⊣A is to be reduced to ⊢¬A; bilateralism promotes rejection to a primitive attitude standing beside assertion; and in its strong form bilateralism lets the two entail each other, whereupon rejection collapses back into the assertion of a negation. This part adopts weak rejection: ⊢¬A entails ⊣A, and the converse does not hold.¹ This is Lemma 1 of the part, the weak-rejection lemma: ⊣A does not entail ⊢¬A, where the negation is the internal one, the kind that inherits the subject's presupposition. It has a plain witness in sentences whose presupposition fails. "The present king of France is bald" is unacceptable; its internal negation takes it equally for granted that there is such a king, and so cannot be asserted either. The one correct move here is to reject without counter-asserting.² Read the negation externally, as true whenever the embedded sentence is not true, and ⊢¬A does come back; but that sentence is then no more than the registration of the rejection, carrying no fresh commitment, and it stands in no conflict with Lemma 1.

The two negations. The distinction between rejection and assertion has, on the side of Buddhist logic, its counterpart in the division of negation into two kinds. An affirming negation (paryudāsa) performs an alternative-selecting denial inside assertion: the presupposition is retained, and a commitment to the complementary class is taken on. To say "not a brahmin" is still to say that he belongs to one of the other three of the four classes. A pure negation (prasajya) does the work of rejection, negating and positing nothing: to say that the hair of a tortoise is non-existent is to posit nothing whatever in the saying — not that the tortoise has some other hair on it, and not that there is a thing bearing the name non-existent. The distinction is not an import from Western learning. Saṅghabhadra himself divides expressions of negative signification into those that signify something and those that signify nothing: the former, such as "not a brahmin," run over to the kṣatriya and the rest; the latter, such as "non-existent" and "no thing," signify nothing at all.³ One near neighbour has to be marked off from these. Metalinguistic negation, as modern linguistics describes it, refuses a form of words and supplies another form of words in its place; a pure negation supplies none. The two are close and are not the same.⁴

Presupposition and cancellation. Write π(t) for: the subject term t denotes in the manner of E3 and D2. An affirming negation operates within π and inherits π; a pure negation may act upon π itself. Lemma 2, presupposition cancellation: when ⊢¬π(t) holds, every predication F(t) that takes t for its subject and inherits π, together with its internal negation ¬F(t), is to be rejected and no more than rejected; and the several rejections are authorized uniformly by the single ⊢¬π(t). For the testing of presuppositions the modern literature has ready instruments; this part takes Strawson's account of presupposition and von Fintel's test, to be applied later in diagnosing the family of empty-subject sentences about whether the Tathāgata exists after his decease.⁵

The tetralemma: three readings, one adopted. The formal appearance of the tetralemma is this: S1 is F(t), S2 is ¬F(t), S3 is F(t)∧¬F(t), and S4 is the denial of both, for which there are two candidate notations; the collation of these against the phrasing of the original is settled in Appendix A. There are three readings. Reading (a), parameterization: each limb carries a hidden parameter (which truth, which temporal division, which correlate, and the like), so that once the parameters are unfolded the limbs do not collide. The price is that the parameters must be independently warranted, and the textual warrant extends only as far as the places where the two truths are expressly marked; the reading is accordingly demoted to an accompanying one. Reading (b), presupposition cancellation, is the one this part adopts. Nāgārjuna's complete gait is ⊣S1, ⊣S2, ⊣S3, ⊣S4, and the four rejections are authorized uniformly by a single assertion, ⊢¬π(t). All four corners must be swept, because the four limbs exhaust the four repairs by which π might be kept alive: plain assertion, retreat to the negation, taking up both sides at once, and withdrawal to the denial of both while still keeping the reference. Only when all four corners have been rejected does the failure locate itself in the presupposition π, instead of falling upon one predicate F or another. The interface with the central proposition takes one sentence: ⊢¬π(t) is precisely the instance of T at t, and the exhaustive refutation of the four limbs is T's working face, not a second machine. Reading (c), dialetheism with a four-valued semantics: as a descriptive framework it is tidy, but as a normative logic it must tame explosion and redefine negation, a price the conservative option has already ruled out. It is kept for Chapter 7 as an interlocutor.⁶

One form, two uses: grid-filling and grid-burning. "A fourfold answer should be made" occurs three hundred and twelve times in the Mahāvibhāṣā, where it is a classificatory device. Question: is there anyone who possesses the supreme mundane dharma and does not possess the acquisition of disjunction that goes with it? Answer: a fourfold answer should be made. Each of the four cells is assigned a kind of person, and it is presupposed that the whole domain is on the register.⁷ Nāgārjuna takes up the same two-by-two grid and uses it to clear mines: all four cells are rejected, and what is thereby shown is that the base they share has failed. One form and two uses; and reading the two families as one has long been among the chief sources of confusion. The textual criterion for separating them lies in the closing formula. Those that close with "this is the dharma of the buddhas," or with a note of instruction, are pedagogical tetralemmas, devices adapted over time to the hearer; the set beginning "all is real, or not real" in MMK 18, the chapter on the examination of dharmas, together with "the buddhas sometimes teach a self, sometimes teach non-self" in the same chapter, attest to that character out of the text's own mouth.⁸ Those that close with "therefore one knows there is none," or with "that is not so," are refutative tetralemmas, and they are the proper subject of this part. A note on the history of the literature in passing: catuṣkoṭi as a technical term is not used in the text of the Mūlamadhyamakakārikā, the Catuḥśataka, or the Prasannapadā; the name is of later coinage.⁹

The identity-difference dilemma and the scope-lock. The skeleton: x and y within a domain, and a relation R holding between them. Fire and fuel, self and aggregates, cause and fruit, agent and act — all have this form. The identity horn: if x and y are identical, then by Leibniz's law properties transfer, the two ends collapse together, and the agent is the act, the burner the burnt. The difference horn: what is in play here is not bare numerical distinctness but independent distinctness, written ≠ — a separate substance, and one established without depending on the other. If x ≠ y, then x ought to be obtainable apart from y; and apart from y x is not obtained. Lemma 3, the scope-lock: what lies between ≠ and ≠ is exactly the first condition of D2. Should the opponent escape by pleading that they are different and yet dependent on one another, he has already surrendered not established in dependence on anything other than itself, the entity drops out of the running of its own accord, and T carries directly. Should he hold on to independence, then for him ≠ simply is ≠, and the dilemma closes upon him. The dilemma burns only the frame-free identity and difference of the object level; the identity and difference we ourselves employ at the meta-level are comparisons within a frame, as the boundary clause of Chapter 1 provided, and the two do not interfere with each other. Two details of the rule: identity is taken as numerical identity, and Leibniz's law is used within extensional contexts only, each case involving an epistemic context being marked separately. The Vātsīputrīya position gauges the instrument exactly. To hold that the person is neither identical with nor different from the aggregates, and thence that it is inexpressible, is to preserve the presupposition π while refusing to answer among the four limbs; Nāgārjuna rejects π, whereupon the four limbs lose their footing of themselves. The inexpressible sits one storey apart in the two cases: in the one it is lodged in the nature of the object, in the other in the presupposition of the words.¹⁰

The classical statement: the Vigrahavyāvartanī. The weak-rejection lemma and presupposition cancellation have a complete classical statement already in the Vigrahavyāvartanī, and four verses will serve.¹¹ "If I had a thesis, I would thereby be at fault; since my thesis is nothing, I incur no fault": at the level of rejection there is no assertoric commitment, and this is Lemma 1 exactly. "As a conjured man to a conjured man, as an illusory man to an illusory man, so is it with the negating and the negated": language without a substance of its own performs the work of rejection all the same, and the force of a negation requires no backing from E3. "What is it that you negate? What you negate is empty; dharmas are empty and yet there is negating — so your objection fails": if the opponent insists on reading rejection as assertion, what he destroys is the form of his own thesis. "Speech has no substance of its own, and what is spoken of has none either; thus I am at no fault, and need adduce no superior reason": the close. One further sentence from Piṅgala's commentary must be filed on its own: where there is self-being there is other-being. Difference is established in dependence upon self-being; and if that sentence can be established as an analytic truth in the grammar of the concepts, the relationality premise P receives the strongest legislation available to it. It is the foremost classical witness for P, and is formally brought into service in Chapter 4.¹²

The full worked sample: MMK I.1. Dharmas do not arise from themselves, nor do they arise from another, nor from both, nor without a cause; therefore one knows there is no arising. The four routes of arising are read as four E3 configurations of a generating relation G: arising from self is G(x,x); arising from another is G(y,x) with y ≠* x; arising from both takes up both routes at once; arising without a cause dispenses with the cause and arises. Piṅgala proceeds corner by corner, and his treatment, restated formally, runs as follows. The fault of arising from self is the structural price of a self-relation: one dharma then has two substances, the one the arising and the other that which arises; and further, the arising has an arising of its own, and the arising is thereby without end. The fault of arising from another is that a pure other cannot be set up: since there is no self-being there is no other-being either, for where there is self-being there is other-being, and the presupposition of independent distinctness destroys itself. Arising from both inherits the two faults. The fault of arising without a cause is drawn by reductio upon consequences: were there no cause, then giving and keeping the precepts and the rest ought to fall into hell, and the ten evil deeds and the five heinous crimes ought to be reborn in heaven. Gait and criterion agree at every point. Each of the four corners is met with rejection; the uniform authorization is ⊢¬π(that which arises); and the close, "therefore one knows there is no arising," puts the passage beyond doubt in the refutative family. The words "no arising" are the registration of a rejection and not a fresh assertion, and the full unfolding of that distinction is the business of the hierarchy clause in Chapter 5. Meanwhile at the level of E2 the arising of dependent origination loses not one item: the seed still sprouts, and one still says so. This is the interface of the Preservation Theorem, which Chapter 6 opens.¹³

The list of tools, handed over. Lemma 1, weak rejection: ⊣A does not entail ⊢¬A, the negation being internal. Lemma 2, presupposition cancellation: ⊢¬π(t) uniformly authorizes the rejection of all four limbs. Lemma 3, the scope-lock: the dilemma bites only upon those who hold to independence. Two criteria besides: the two families are separated by the closing formula, and parameterization is warranted only where the two truths are expressly marked. Not one of these tools steps outside classical logic; and to the question whether Buddhist logic, on this use of it, departs from the classical, the answer this part gives is that it does not.¹⁴ Chapter 4: the proof begins.

[The Chinese text closes with a restatement of this chapter in plain language — the same content in a different register, the academic half governing wherever the two diverge. That restatement is not yet translated; nothing in the argument is missing from what stands above.]

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