In examining the use of tetralemma in the MMK, I found it important to separate purely formal logical considerations, historical precedents in use, and finally strategic applications in developing the overarching philosophical critiques in MMK. Obviously, we are hard-pressed to examine the tetralemma from a purely structural point of view without any recourse to the philosophical maneuvers in the text. However, coming to grips with what type of logics are at play can reveal much about the philosophical thrust of the MMK.
The four positions/alternatives may be read:
(1) With the rules of classical logic in mind. This means we would have to make sense of the third alternative (P & ~P). The obvious contradiction can be maintained if we :
(a) Hedge the statement with qualifications (like “P is conventionally real, but ultimately not,” or more generally speaking, “In one sense P, but in another sense ~P).
Although the most charitable gloss on the third alternative may employ something of the hedged or qualified sort, the Australian logician, Graham Priest, and our beloved J. Garfield (I think I mean this without irony…but I’m not yet sure, perhaps, P & ~P)– they both challenge us to accept the statement full force and unhedged (see Tillemans essay in “Pointing at the Moon”). In other words, a “paraconsistent” reading gets worked out through Priest and Garfield.
An important facet of their work is to show that in speaking of a totality (think of Russell’s paradox concerning the set of all sets who are not members of themselves, and Cantor’s “diagonal” proof) we are often lead to unavoidable contradictions—however, we may be able to formalize and thereby make sense of such contradictions under localized conditions (see footnote to Tillemans’ article, and have fun!!).
(b) By quantifying the alternatives, we may read the third alternative as ‘There is some X such that P holds of it & there is some Y such that Not-P holds of it.’
In a quantificational scheme, we can claim the statement above without commitment to the identity of x and y. In other words, x and y might be the same, but they might be different. If they are different, then of course there is no problem asserting that “Kevin Talks Way too Much” and “Ryan does not talk way too much.” We have to ask ourselves why Nagarjuna, or any Buddhist thinker employing the tetralemma, would make such an uncontroversial and placid statement.
On the other hand, if we claim that x and y are identical, then we might make sense of it via mereological analysis. This analysis would claim that part of X is H and part of X is ~H. So ‘Part of Kevin has hair (his hairy legs, for example) and part of Kevin has no hair (the male-patterned bald spot just north of his widow’s peak). This may give us interesting reads for issues like causation, for example. One might want to claim that X causes part of itself, and X does not cause some other part of itself (whatever that might mean).
(c) We may read the alternatives as Tillemans does by quantifying while employing a referential semantics. What does this mean? When we quantify (i.e. use predicate operates like ‘there exists a y such that P of y’ or ‘for all x, P holds of x’) we assume the variables (x and y) are stand-ins for objects in a domain of discourse. So those variables refer to objects. However, if the domain is “empty,” in other words if what we speak of actually lacks the singular independent objects we think we refer to, then our references fail. So Tillemans argues that we are dealing with an empty domain of discourse, and thus Nagarjuna denies all four alternatives.
It’s important to note Westerhoff’s critique of this maneuver. First of all, any denial of an existential claim makes a positive statement as well. For example, if I deny, ‘there exists an x such that P holds of x,’ then I am simultaneously claiming ‘for All x it is not the case that P holds of x.’ So in this way, Nagarjuna is making a positive claim about the domain of discourse (namely, that P does not hold of any x).
Also, and this is a big problem in general with reading the third alternative in a purely straightforward semantics of classical first order logic, the third alternative and the fourth alternative are redundant (and thus useless for our purposes). We want to prevent such an uncharitable reading of the third and fourth alternatives, because we assume these bright thinkers were not unaware that these statements read in one way are equivalent. However, with Tillemans’ reading, the third and fourth alternatives remain redundant.
(c) Finally, Westerhoff’s reading.
The first thing to note is that Tillemans’ reading has merit in so far as it does not need to radicalize the logic of the tetralemma (and as we all know, conservative moves are most prized).
However, Westerhoff is able to avoid strange readings of the tetralemma, and also keep all four alternatives disjoint and therefore substantive (in their content—-but not with “svabhava” substance…that would be very bad!).
For the rest of this post, I will simply give my outline of Westerhoff’s analysis, and offer a few points that I may or may not have covered in the presentation. I hope this is helpful.
Quick Outline of Westerhoff
I. Problematizing the Catuṣkoṭi
a. Equivocal use of the catuṣkoṭi throughout Buddhist philosophy and MMK in particular.
i. [MMK 18: 8]: Positive tetralemma, where Nāgārjuna claims the Buddha’s teaching affirms the reality (of things?), the non-reality of (things), both, and neither. Note Garfield’s “interpolation” analysis in keeping with Tsong kha pha and Tibetan interpretations: X is conventionally real, ultimately not-real, both conventionally real and ultimately not-real, and nothing is ultimately real or completely non-existent. [Let’s carefully assess footnote 93 in Garfield’s MMK, p.250 and footnote 6 in Pointing at the Moon, p. 97]
ii. [MMK 22: 11, 12, 14]: Negative tetralemma, where all alternatives are denied.
iii. [MMK 25: 22, 23, 24]: Does Nāgārjuna deny the Law of Excluded Middle (LEM) here? If this is a presupposition-negation, viz., that substantial (utterly independent) entities must exist in order to ascribe to them finitude or infinitude, then we need to think about whether or not denying LEM (for mathematical intuitionists like Brouwer or anti-realist philosophers like Dummett) works in the same way. Or is this more in line with Tillemans “substitutional” characterization, where Nāgārjuna’s move is akin to quantifying over an empty domain.
• Consider the difference between resisting an unwarranted or undefended presupposition versus making an epistemic claim that a proof for certain types of entities, properties, or states of affairs cannot in principle be delivered (even if a negative reductio proof is at hand), and thus resist universal application of bivalence and LEM. If Nāgārjuna’s project is not simply resistance to unwarrantable claims, but a full-scale denial of the clean division between self and other, identity and difference, then verses like 22 and 23 would not so much be denying LEM (for it may quite clearly hold at the conventional level of described reality) but simply indicate the inappropriateness of certain category assumptions.
• In one sense, rejecting LEM and bivalence makes a stronger point: it shows that in principle we must accept an epistemic deficit regarding certain types of cases, which defeats the rule that for all X, either Ax v ~Ax. In the Tibetan-Garfieldian discourse, intuitionist anti-realism defeats the rule even at “the conventional level,” which the Buddhist thinker might resist. On the other hand, Nāgārjuna attacks the very basis at which we can strictly draw the line between any specific identity or difference between allegedly individual entities in a “domain of quantification.” In keeping with Tillemans, we might say that for Nāgārjuna the domain of quantification is empty. So he wants to show that every aspect of our most stubbornly substantialist assessments are either incoherent (and thus wrong, placing Nāgārjuna in the error theorist camp) or simply “not-evident” or truly supported beyond some sort of inherent substantialist dogma (the Pragamatic reading of Nāgārjuna).
b. Competing Views on the Logics in Play in Each of the Four Alternatives
iii. Substitutional (Nominal), and Referential
iv. Denial of Law of Excluded Middle
vii. Paraconsistent/Dialetheism logic
• Brief exposition on iii. – vii
II. Two Kinds of Negation
• Grammatical distinction concerning scope of a negation with respect to either a verb or a noun:
(1) Prasajya (non-implicative propositional negation): ‘This is-not a pleasant pineapple!” Denying this statement does not commit me to the pineapple being unpleasant, moody, or world-weary. Think of Westerhoff’s “colored numbers.” Deny that seven is yellow does not commit you to the claim that it is some other color.
(2) Paryudāsa (implicational term-negation): “This is a non-essay.” Strange sentence, I know, but in essence it means that what I have before me is some piece of writing of some sort or another. When I deny that it is an essay, I imply that it is some sort of writing and is a member of the set of written things. It’s a written outline of sorts, but it’s certainly not sweet, sour, dejected, or moody.
• Westerhoff maps this Sanskrit grammatical distinction onto a similar distinction between:
(a) Exclusion Negation (Rejects what is denied without proposing any sortal specifications—The pineapple is-not pleasant, nor do I assume it must instantiate some other kind of mood or humor; I make no commitments concerning what sort of things are ascribable to the pineapple).
(b) Choice Negation (Denying it’s an essay, I still assume it’s some piece of writing—in other words, it’s either writing or it isn’t writing: those are the proper choices at hand given the type of thing I am speaking of).
• Westerhoff argues that this distinction makes sense of contradictory statements riddled throughout the MMK (see 18:10), particularly those that seem to both affirm and deny an ascription to some thing.
➢ We might ask ourselves what this accomplishes for reading the MMK. In one sense, the overall picture developed throughout the MMK is radical enough that puzzlement remains unavoidable (and if this is flippantly denied, then we should wonder why Nāgārjuna wasted the ink to defuse views so easily abandoned). On the other hand, Westerhoff rescues the text from an oversimplifying air of clichéd paradox or superficial mysticism. He attempts to rationalize Nāgārjuna, for better or for worse, and show that many of Nāgārjuna’s points might be made without straightforwardly denying the law of contradiction. I think Westerhoff and Tillemans are engaged in similar ventures, because both won’t to avoid hard claims of paraconsistency.
➢ In fact, it may be helpful to look at several passages in the MMK to see what might motivate Westerhoff’s reading. Consider the final chapter, (27: 17). Here, Nāgārjuna claims, “If one part [of a person] were divine and one part were human, it would be both permanent and impermanent. That would be irrational” (Garfield, p.81—sorry for destroying the form of the verse).
➢ The word translated by Garfield as “irrational” is ‘yujyate’ in Sanskrit. This plays off a distinction (Kalupahana points out in his commentary) between what is “proper” as spoken (‘yujyate’ as “what makes sense” or “what is coherent and meaningful”) and what is “evident” or “known” (‘vidyate’). Although Nāgārjuna certainly claims that, “What language expresses is nonexistent [non-substantial]” (28:7), he readily makes use of the principle of non-contradiction to point out the error of other views. How would he not?
➢ That brings up another issue: why does the principle of non-contradiction remain unscathed even under talk of śūnyavāda? Nāgārjuna contends that all things are empty, including such seemingly inviolable conditions of conceptual form. But notice that any straight affirmation in the MMK of both A and ~A, or the denial of A or ~A, often get explained away or hedged by commentaries that qualify the statements in a way that still respects classical logic. Claiming we are constrained by the ineluctable conventions of language seems to scratch at some fundamental mental bedrock (and calling these “conventions” doesn’t quite get at what these forms do, because they seem culturally universal—so this human convention would be opposed to what other convention?). As Kantian as this may come off, again the weight and profundity of the MMK only resounds if we come face to face with the difficulty of simply casting aside what appear to be ineluctable bedrocks or foundations. Equally important, the tools Nāgārjuna must use for his analysis rely on these “bedrocks.” If we make ample use of such tools, but somehow find these have no ultimate basis outside convention, does this amount to a question of how we may truly think both sides of a limit?
• Simple example of prasajya and paryudasa functioning through exclusion and choice negation, respectively:
(1) X is neither identical to what causes it nor is it different from what causes it.
• We can plainly see the contradiction that arises here. If we were to formalize this, it would roughly amount to: ∀x (Ix ∧ ~Ix). Westerhoff uses the prasajya and paryudāsa to make sense of this contradiction.
(2) ∀x −(Ix ∧ ∼ Ix) *Note, the use of ‘−’ indicates prasajya negation.
• For (2), the first negation denies that the concept “identical-with x” or “not identical-with x” applies to any object x. The negation within the parentheses functions like a standard choice negation (paryudāsa), while the negation outside the parentheses functions non-implicatively (prasajya) by denying any positive assertion regarding the concept of strict identity or difference.
• This is a fair point, because Nāgārjuna simply questions the implicit assumption underlying the application of concepts like identity and difference. As we know, those concepts allegedly require a commitment to independent, thoroughly un-relational svabhavic entities. If the MMK continually points out the shortcomings of such a presumption, then certainly Nāgārjuna would aim to deny the underlying assumption rather than get caught in the snares of antithetical views.
I should also add this (and pardon me for the redundancies):
Very Brief Outline, Westerhoff
I. Two Kinds of Negation
a. Prasajya: non-implicational propositional negation or exclusion negation
b. Paryudāsa: implicational term-negation or choice negation
II. Rejecting a False Dichotomy (For all x coming into being depending on a particular object…)—MMK 18:10
a. (∀x)(∀y) (Dxy ==> ~[(x = y) & ~(x = y)] )
• Simplify ~[(x = y) & ~(x = y)] with metavariables ~(P & ~P). By DeMorgan’s law (~P & ~~P) and Double negation:
(~P & P) —-contradiction
b. Westerhoff (and Matilal, Epistemology, Logic, and Grammar in Indian Philosophical Analysis. Mouton, The Hague, 1971), resolve the contradiction
by analyzing the first negation as prasajya and the second as paryudāsa.
• So ¬ (P & ~P).
• This means Nāgārjuna denies the underlying assumption that a genuine dichotomy is applicable in this case. One may challenge his reluctance, but at least Nāgārjuna’s statement is coherent.
c. Any problems here? Although Nāgārjuna’s statement gets rescued from contradiction, when we examine the n-place predicate relation “Depends on (D)” in (a.) above, we are certainly left with odd results.
• For example, when we claim x depends on y we usually specify relative to a domain of quantification whether or not bound variables are identical to each other or not. We can definitely claim that x depends on itself, but what’s interesting in this case is that the consequent of the conditional rules out any identity-specifications for the two variables.
• Of course, this is just Nāgārjuna’s point. For the consequent of the conditional (analyzed in Matilal and Westerhoff’s manner) tells us that the identity relationship in this instance is really only a false dichotomy.
• But consider the “Depends on” relationship. If we cannot determine whether or not x and y are identical or distinct, then we certainly have trouble making sense of the two-place predicate used here. In other words, Nāgārjuna does not only resist the dichotomy of identity, but he must equally deny the ultimate reality of this relation, because it entails a clear distinction between the relata. This may not pose a metaphysical-ontological problem to some, but dependence seems to be the Buddhist category par excellence. What implication does this have for karma? For the ethical project of Buddhism?
• So we see that just in this statement, we are confronted with the question of whether or not Nāgārjuna puts forward any type of positive thesis, or if even dependence remains purely empty. Can Tilleman’s substitutional approach help us out of this problem?
III. Negative Tetralemma, Focus: (3) and (4)
A. Form [Using MMK 22:11]: Śunyam iti na vaktavyam aśūnyam iti vā bhavet
ubhayaṃ nobhayaṃ ceti prajñapty arthaṃ tu khathyate
(1) A “Empty”
(2) ~A “Not empty”
(3) A & ~A “Both”
(4) ~(A v ~A) “Neither”
B. Distinctness of (3) and (4)
• We must immediately recognize that (3) and (4) are equivalent by this rendering, and that is a problem, because we want to maintain that all four alternatives are disjoint and not simply redundant, muddled notions.
• When we reject all four alternatives we get for (4): ~~(A v ~A)…a very negative statement (and very bad joke)! Nevertheless, this reduces to A v ~A, precisely not what Nāgārjuna aims to declare.
• SOLUTION: (Westerhoff) interprets (4) as ¬¬(A v ~A), where the outer negations are prasajya and the inner is paryudāsa.
• Westerhoff argues that (¬¬ A ⇔ A) does not apply in this instance, because a double prasajya does not reduce to a positive assertion.
• We can briefly discuss why Westerhoff does not think Nāgārjuna therefore rejects the Law of the Excluded Middle. He argues that although Nāgārjuna denies the applicability of certain properties (or perhaps all properties) to objects, he by definition affirms prasajya negation to the ascription of such properties to the objects.
• This seems like a very slippery move. The objection against this move is really only another form of questioning whether or not “the emptiness of emptiness” is a self-refuting proposition. Is Westerhoff meaning to say that Nāgārjuna cannot help but apply at least one property to what he is denying? Or do we make a category mistake in assuming negation operates like a property?
C. Illocutionary Negation
• Westerhoff wants to avoid assuming Nāgārjuna denies LEM, but must make sense of how Nāgārjuna’s denial of (¬¬ A ⇔ A) can avoid that trap.
• Westerhoff introduces the notion of illocutionary negation, which subsumes the notion of prasajya and paryudāsa negation. Illocutionary negation entails negating an illocutionary prefix of an assertion to a proposition. This means, one declines to assert a proposition (which of course, does not entail the assertion of its opposite). This tool may be employed for several reasons:
i. Non-commitment to a category of predication for some object
ii. Lack of evidence for the assertion
iii. Problems of meaning and translatability
• While (i) incorporates the prasajya/paryudāsa negations, (ii) and (iii) show that illocutionary negation is the larger category.
• Also, we may note that (ii) seems to get at Kalupahana’s reading of Nāgārjuna as a pragmatist-empiricist denying the reality of our substantialist categories, and instead asserting the empirically grounded thesis of dependent origination.
• Be that as it may, by employing illocutionary negations for
(¬¬ A ⇔ A), we avoid the problem of defying LEM, because declining to assert a proposition that in turn asserts that we decline to assert another proposition does not then assert this third proposition.
• Discuss Tillemans’ objection, and Westerhoff’s counter.
• Discuss further objection concerning redundancy of illocutionary denial of first and third alternatives. Bring in points concerning the use of ‘&’ as classical truth-function operator.
D. Tillemans’ Objection
• Briefly put: using illocutionary locutions to refuse to assert would leave Nāgārjuna without any sort of substantial negation, and surely he wants to negate these propositions in some form or another.
• Westerhoff’s reply: not a problem, because Nāgārjuna wants to deny the existence of svabhāva underlying presupposed in the four positions—so he is “negating” in some form or another.
E. Problems with the Third Alternative
1. Richard Robinson’s Quantificational Take: (3) is not a problem if we understand the possible interpretations of a quantified form, where some P may be said of some x and ~P of the same P (think mereological ascriptions, or different valued utilities under different descriptions)
2. Examples of Nāgārjuna rejecting (3):
i. Contradiction (25:14)—rhetorical maneuver, because contradiction of opponent’s view not a viable option, and so Nāgārjuna will reduce view to this untenable form
ii. Rejection of conjunction of two false claims
3. Third alternative fits into a schema of the tetralemma used as a heuristic device for assessing positions actually propounded by other schools (e.g. Garfield’s reading of I : 1)