adrian Kreutz (university of aMsterdaM) CONTRADICTION AND RECURSION IN BUDDHIST PHILOSOPHY: FROM CATUṢKOṬI TO KŌAN Introduction Everything is real and is not real, A Monk asked: Both real and not real, "What is Buddha?" Neither real nor not real. The Zen Teacher answered: This is Lord Buddha's teaching. "Three pounds of flax" MMK (18:8) Wúmén Guān: (18) Prima facie, MMK (18:8) on the left, and Wúmén Guān (18) (Ch. 無門關) on the right don't have a lot in common-but they do.1 Both are, and this is the main hypothesis of this essay, a schema for upāya (Skt. उपाय): skilful means of teaching the path to enlightenment. Wúmén Guān: (18) is a Kōan (Ch. 公案), which features predominantly in the Chinese Chan and the Japanese Zen tradition. The focus of §4 will be on the Kōan. The subsequent section, §2, will be concerned with the argument from MMK (18:8) above which goes under the name catuṣkoṭi (Skt. चतुष्कोटि), which literally means "four corners." In its simplest from, the catuṣkoti is the view that any claim can be true, false, both true and false or neither true nor false. Those are, metaphorically, the four corners (kotis). The origins of the catuṣkoṭi and the origin of Buddhism are concomitant. In fact, as 1 I am indebted to the audience at the Asian Philosophical Texts conference at the Centre for East Asian Studies (EASt), Université Libre de Bruxelles in November 2018. Special thanks go to Takeshi Morisato. Furthermore, I would like to express my gratitude for helpful comments by two anonymous referees which have improved the quality of this essay and also to Jan Westerhoff for a translational remark. 134 Asian Philosophical Texts Ruegg (1977) holds, we recognize (parts of the catuṣkoṭi) in the earliest sutras, in the intellectual circles of the historical Buddha Gautama, as in the Māluṅkyaputta Sutta, for instance. As this teaching has its origin in Brahmanism (or the rejection thereof), it is only sensible to anticipate the earliest manifestations of the catuṣkoṭi in Brahmanical texts. Yet, although the fourth koti features in the Bṛhadāraṇyaka Upaniṣad and the third koti can be found in the Samavayanga Sutra, they have never been endorsed together. So, the catuṣkoṭi seems to be a genuinely Buddhist concept. Clearly, it has found its most prominent use in the Madhyamaka school of Buddhism and the writings of Nāgārjuna2, and his Mu lamadhyamakakārikā (henceforth: MMK) and commentaries thereon. I shall focus the discussion on Nāgārjuna's catuṣkoṭi. Taking the Buddhist worldview into perspective, it is not surprising that metaphysical, epistemological and logical notions (such as the two instances above) have a practical import. At least for the practicing Buddhist, any philosophical concept should be tried in practice to decide whether it is (or not) an expedient means towards enlightenment-i.e. upāya3. Here, again, is my main hypothesis: catuṣkoṭi and Kōan have the same function in Buddhist philosophy, Madhyamaka and Zen respectively, and share a systematic structure. Both are to be considered a schema for upāya-means towards the soteriological "end-goal," which is enlightenment. Nāgārjuna's Catuṣkoṭi A lot of ink has been spilled on the catuṣkoṭi in the philosophical academic literature, compared to its (in)-significance to the Euroand Anglo centrism of Western academic philosophy of the last centuries. David Ruegg (1977) once said: "The doctrine (the catuṣkoṭi) has been described as nihilism, monism, irrationalism, 2 See Jan Westerhoff, Nāgārjuna's Madhyamaka. A Philosophical Introduction. Oxford: Oxford University Press, 2009. 3 As described in the Lotus Sutra. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 135 misology, agnosticism, scepticism, criticism, dialectic, mysticism, acosmism, absolutism, relativism, nominalism, and linguistic analysis with therapeutic value." And we can add deconstructivism, and what we will be concerned with, dialetheism and ontological non-foundationalism, to the list. It is the aim of this essay to contribute to the ongoing endeavour to make sense of this seemingly mysterious logical schema. Whether it adds just another interpretation, and the mystery remains, I cannot judge. Also, this essay will seriously question whether it, at all, makes sense to make sense of the catuṣkoṭi. This, however, is a vexed question: isn't judging something not to make sense, in the end, making sense of it? I shall not attempt to grapple with the overarching question here, although I shall allow myself a standpoint on whether there is sense to be found in the catuṣkoṭi in the conclusion. A Set of Open Questions on the Catuṣkoṭi Although a lot has been said about the catuṣkoṭi, there is as yet no comprehensive answer to the following questions: (1) What is the catuṣkoṭi, and what role does it play in Buddhist philosophy? (2) What (if there is one) is the logical from of the catuṣkoṭi? (3) What is the catuṣkoṭi's historical position in the wider history of Buddhist philosophy? In fact, I believe myself to have discovered a possible source for the controversy around the catuṣkoṭi in Western commentarial literature: The constant nullification of efforts in making sense of the catuṣkoṭi was to be anticipated for two reasons (other than the usual suspicion of Eastern thoughts being mystical, incoherent and unsystematic4): 4 For this reason, it is important to show with the tools of contemporary formalisms that those Eastern thoughts perfectly stand to reason. We can thereby contribute to an alleviation of those misconceptions. 136 Asian Philosophical Texts (i) Questions 1–3, if they have been addressed at all, have been addressed in (nearly complete) isolation. Furthermore, Western academic research has primarily been concerned with it is the second question. Hence, we see a "logic-first" approach in the research literature on the catuṣkoṭi. It is, then, no surprise that the catuṣkoṭi has been considered perplexing, mysterious and incoherent-it does not fare well with Classical Logic, as the remainder of this section is set to demonstrate. Not only in Buddhist philosophy, but in philosophy tout court, it is the metaphysics which plays significantly into the logic. Therefore, avoiding the metaphysics and purely focusing on the logic is dangerous. The catuṣkoṭi, for unknown reasons, as for the most part been detached from its metaphysical surrounding and has been considered and treated as exclusively a piece a logic- Priest/Garfield (2010) might be an exception here. (ii) The recent literature almost exclusively focuses on Nāgārjuna 's Madhyamaka, and the use of the catuṣkoṭi therein. Sure, I do too, but merely to take Nāgārjuna as the starting point. The catuṣkoṭi didn't suddenly appear in Buddhist philosophy with Nāgārjuna, just to disappear again. My hypothesis is that the catuskoti appears in various forms throughout Buddhist thought and not solely in Madhyamaka. I propose that we widen our horizons and look beyond Nāgārjuna. To give you an (albeit old, but relevant) example of what the research literature is addressing, one of the most influential papers on the subject5 is states the following: It is thus important to examine (1) how it was that Nagarjuna came to make such extensive use of the catuṣkoṭi; (2) the logical form of Nagarjuna's catuṣkoṭi; and (3) with that purpose and in what manner this "logical apparatus" was handled by Nagarjuna in exposition of his philosophy.6 5 Although the philosophical environment in which the catuṣkoṭi came up is mentioned in question (3), it has not been taken into account when answering question (2)-this is characteristic of the "logic-first" approach to the catuṣkoṭi: a misguided approach. 6 Ranil Dion Gunaratne, The logical form of the catuṣkoṭi: A new solution (Philosophy East & West, 1980), 214. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 137 In the same paper, we find-and this, for many reasons, is astonishing-two further positions on how to see the catuṣkoṭi which the author considers necessary to get "the overall picture." (D) The catuṣkoṭi was used by him (AK: Nāgārjuna) as a dialectic which progressively leads one to truth. (E). The catuṣkoṭi was used as an instrument of meditation. It is clear that these positions need not be mutually exclusive. Limitations of space prevent any consideration of (D) and (E) here, although I think that both of these are possible interpretations of Nāgārjuna's use of catuṣkoṭi and the consideration of them is necessary to get the overall picture of Nāgārjuna a's effort in the (MMK).7 It is exactly positions (D) and (E) that I want to advocate in this essay. I find it peculiar how the Gunaratne came up with these positions, as there is, as far as I am aware, no trace of it in Nāgārjuna's corpus. It is even more surprising why the author, being unsatisfied with the (hitherto) noncomprehensive treatments of the catuṣkoṭi did not continue research on those two possibilities, already guessing them to be indispensable for the attaining of a cohesive concept. The decision to disregard the ontological, soteriological and historical framework has (unsurprisingly) led to an unrewarding treatment of the catuṣkoṭi, detached from its philosophical context8. This essay is supposed to take up Gunaratne's legacy. It is, perhaps for those two reasons, given above, that Tillemans (1999,189) held that "within Buddhist thought, the structure of argumentation that seems most resistant to our attempts at a formalization is undoubtedly the catuṣkoṭi"-a thought which stretches back to Poussin's (1917) paper (probably the earliest philosophical treatment of the catuṣkoṭi in the West) and manifested its position as the "Buddhist dilemma." Why a dilemma? In fact, if at all, the catuṣkoṭi is a tetralemma as it has four positions as opposed to two. Why the catuṣkoṭi 7 Ranil Dion Gunaratne, 215. 8 As a metaphysical example, try examining the natural behaviour of a dolphin in a swimming pool where it is not surprising that the results of the study will not be true to the studied object (we cannot expect the dolphin to behave naturally in captivity). We need to put the dolphin back to the ocean to study its natural behaviour. 138 Asian Philosophical Texts is posing a tetralemma becomes apparent through logical formalization. This is what I shall turn to now. The Logic of the Catuṣkoṭi The reason why the catuṣkoṭi is considered resistant to formalization is the following: in Western orthodoxy9, we encounter the principle of tertium non datur: everything is either true, or false-c'est tout. The catuṣkoṭi inflates (while denying) this principle to a quintum non datur: It sates four exhaustive and mutually exclusive possibilities for any proposition: (1) either it holds, (2) it does not hold, (3) it both holds and does not hold, (4) it neither holds nor does not hold-those are the four kotis (corners). It seems (but it is not) easy to spell out the basic schema of the catuṣkoṭi in classical Boolean terms. Here is a first (bad) try. Let "everything" from our example in MMK (18:8) be expressed by 'A'. The four corners of the catuṣkoṭi become visible before our eyes: Positive catuṣkoṭi (1) A (2) ¬A (3) A∧¬A (4) ¬(A∨¬A) A (formal) model for the catuṣkoṭi has to maintain the mutual exclusivity and exhaustive nature of the kotis. The reason for the exhaustivity, other than wanting to be charitable to the coherence and logical abilities of the authors who have employed the catuṣkoṭi, lies in the way the (negative) catuṣkoṭi, 9 As well as classical Indian Nyaya thought, by the way. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 139 Negative catuṣkoṭi (1) ¬A (2) ¬¬A (3) ¬(A∧¬A) (4) ¬¬(A∨¬A) which denies all four possibilities, is commonly employed as an argument that is supposed to undermine all possible ways a predicate can be attributed to something-a kind of reductio argument to reveal the deficiency of the concept in question. Ma dhyamikas have called this method prasaṅga. The concepts the MMK is dealing with are causation, motion, self, identity and others. In MMK (18:8), specifically, the thesis is that everything is real. The four kotis are supposed to exhaust all of the logical space and to mention every possible way "reality" can be attributed to the object which is "everything." Hence, the argument sees the anti-thesis that 'everything' is not real, the conjunction of thesis and anti-thesis, and the disjunction of thesis and anti-thesis. A denial (as in the negative catuṣkoṭi) of every possible way in which "realness" can be attributed to "everything" is then a prasaṅga-argument against the possibility of ascribing "realness" to "everything." The conclusion the Mādhyamikas drew from this is that nothing is "real." While this alone might sound mystic to the Western philosopher, it should become comprehensible as a philosophical argument once the metaphysical fundament on which it is based is made explicit. Consider Priest (2010) here: The central concern of the MMK is to establish that everything is empty of self-existence (svabhāva), and the ramifications of this fact. The main part of the work consists of a series of chapters which aim to establish, of all the things which one might plausibly take to have svabhāva (causes, the self, suffering, etc.), that they do not do so.10 10 Graham Priest, The Logic of the Catuṣkoṭi (Comparative Philosophy, 2010): 12. 140 Asian Philosophical Texts To do what Priest describes, every possible way something can be has to be ruled out eventually, to illuminate its emptiness of self-existence (lack of svabhāva)-this is what is meant by saying that something is not "real" in a Buddhist context: is lacks self-existence. The quadruple-wise exclusivity, i.e. that none of the kotis expresses something that one of the other kotis already expresses, comes as a natural consequence of wanting to establish the exhaustivity of the kotis. Only if none of the kotis is equivalent to any of the other kotis; and only if each koti establishes a distinct possibility, can the logical space can be exhausted, and the catuṣkoṭi used as a prasaṅga–argument. Evaluation of the Proposed Formalization That the catuṣkoṭi, as formulated above, does not go hand in hand with classical logic is no surprise. In fact, it collapses in a classical framework: (4) is equivalent to (3) by De Morgan, and (3) entails both (2) and (1). Priest (2010) convincingly refutes a number of influential attempts to capture the spirit of the catuṣkoṭi in a bivalent framework and puts forward his formalization of the catuṣkoṭi in (plurivalent) First-Degree-Entailment which allows for things to be both true and false, and neither true not false- just what koti three and koti four are saying. This is not the point to go into the logic, but it is (nearly) undeniable that once the Law of Non-Contradiction and Explosion are given up, the mutually exclusive and exhaustive nature on the kotis can be preserved.11 One formality which is going to be crucial for a thorough 11 Unfortunately, the framework has a problem on its own, which, in the modern debate on non-classical logic, is known as the Recapture Problem. The paraconsitent logic in which the catuṣkoṭi is formulated makes classical principles such as modus ponens invalid, but in certain situations those classical inferences are taken to be valid. (this is the classical Recapture Problem). The MMK might be such a situation, where Nāgārjuna, on the one hand, is using the catuṣkoṭi in the paraconsitent and paracomplete framework, but is on the other hand using classical inferences such as modus ponens, hence, Nāgārjuna is facing a Recapture Problem. For a detailed analysis of the logic and a proposed solution to the problem, see Adrian Kreutz, Recapture, Transparency, Negation and a Logic for the catuṣkoṭi (Comparative Philosophy, 2019): 8. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 141 understanding of the subsequent discussion is the notion of a status-predicate, which is a meta-linguistic tool to express the alethic status of a truth carrier. In the search for an adequate model for the (Nāgārjuna's) catuṣkoṭi Priest (2010) expresses the four kotis with the following set of status-predicates: S={T, F, B, N}. Let T be the truthpredicate "is true," F is "is false," B is "both true and false," N is "neither true nor false." T(A) is the proposition "A is true," where (A) is a name for A To define B, N Priest first defines F as F(A)= T(¬A). Here is how we define the four kotis. 1) T(A)= T(A)∧¬F(A) 2) F(A)= ¬T(A)∧F(A) 3) B(A)= T(A)∧F(A) 4) N(A)= ¬T(A)∧¬F(A Bear in mind that koti three and koti four are genuine possibilities in the logic Priest is using, although they surely appear paradoxical, they are not-in fact, what (partly) motivated the development of those logics was the quest to deal with paradoxes, such as the liar paradox. The catuṣkoṭi can now be expressed in the following way: Catuskoti:T(A)∨F(A)∨B(A)∨N(A) Can I, at this point, already motivate my main hypothesis, that the catuṣkoṭi is an instance of upāya?12 There are clear indications in both Nāgārjuna and Candrakīrti which prompt the suspicion that both were seeing a connection between the study of reality from several perspectives (perhaps the catuṣkoṭi) and a change in attitude of soteriological/psychological importance (perhaps enlightenment). Consider the final verse of the MMK (27.30), for example, which claims that the purpose of Buddhist teaching is the abandonment of all views-which is what the negative catuṣkoṭi announces. Also, there is Candrakīrit's metaphor of 12 Thanks go out to an anonymous referee for making me aware of this. 142 Asian Philosophical Texts a purgative drug in his commentary on MMK (13.8), where emptiness is compared to a drug that purges other things before purging itself as well. Also, we find the point that a prasaṅga argument is always in a dialectical context in opposition to another view (it is upāya applied to philosophical argumentation as it were). Are these considerations decisive? I am not sure. I shall therefore leave the realms of Indo-Tibetan Buddhism and follow the development of the catuṣkoṭi further to the East. Things will become clearer there, yet also somewhat cryptic. Going Beyond Nāgārjuna We have familiarized ourselves with Nāgārjuna and shall skip the epoch of his commentator's and arrive in China (500AD), where Madhyamaka Buddhism and Daoism merged into one.13 It is here that the role of the catuṣkoṭi as upāya becomes perspicuous. The school of Sānlùn (Ch.三論宗), which translates as "Three Treatises"14, absorbed Madhyamaka philosophy and with it the catuṣkoṭi. In Jízàng's (Ch: 吉藏) Erdi Zhang (Ch. 二諦章) we can rediscover the catuṣkoṭi, albeit in a modified form.15 The Erdi Zhang With this framework at hand, let us now turn to the role of the catuṣkoṭi in Jízàng's "Erdi Zhang" (Ch. 二諦章; Eng. "Essay on the Two Levels of Truth"), which commences with: 13 See Wing-Tsit Chan, A Sourcebook in Chinese Philosophy. (Princeton NJ: Princeton University, 1963), 360. 14 The "three treatises" from which the Chinese version of Madhyamaka took its name are: Mūlamādhyamakakārikā-Zhōnglùn (Ch. 中論), Nāgārjuna's Dva daśanikāyaśāstra-Shíèrménlùn (Ch. 十二門論) and Āryadeva's Śatakaśāstra-Băilùn (Ch. 百論). It later came to Japan under the name Sanron, but disappeared during the Nara period, or was most probably absorbed into Shingon. 15 Shortly after this paper has been given at the APT Conference, Priest (2018,7) also published on the hierarchy of truth-values and expressed it with a recursive model. He is not, however, drawing the same conclusions from it as I do, as will be explicit in §3.3 and §3.4. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 143 The three kinds of 'Two Levels of Truth' all represent the principle of gradual rejection, like building a framework from the ground. Why? Ordinary people say that dharmas, as a matter of true record, possess being, without realizing that they possess nothing. Therefore, the Buddha propounded to them the doctrine that dharmas are ultimately empty and void.16 What Jízàng describes in this opening paragraph of his most influential text is a hierarchy of truth-levels, built from the "ordinary people's" idea that dharmas possess being (i.e., in Madhyamaka vocabulary, that things have svabhāva). In line with Madhyamaka thought he holds instead that no dharma (i.e. no phenomenon, may it be physical or psychological) possesses being (i.e. everything is devoid of svabhāva), and only emptiness (i.e. sunyata) is to be ascribed to all dharmas-nothing has (intrinsic) self-being, everything is empty. We can read the "ordinary people's" view that dharmas have self-being, taking the status-predicates into account, as T(B), and so we have the first koti. Its negation, therefore, is the Buddha's doctrine that dharmas are ultimately empty and void-¬T(B)17-which is our second koti. Jízàng's following analysis then, sustaining the thought that what is described here are the corners of the catuṣkoṭi, supports the hypothesis that the catuṣkoṭi's position in Buddhist philosophy is upāya (skilful means). When it is said that dharmas possess being, it is ordinary people who say so. This is worldly truth, the truth of ordinary people. Saints and sages, however, truly know that dharmas are empty of nature. This is absolute truth, the truth of sages. The principle is taught in order to enable people to advance from the worldly to the absolute, and to renounce the truth of 16 ibid. 360. 17 Which is equivalent to F(B), i.e. the ultimate falsehood that B (that dharmas have self-being). Which is equivalent to the ultimate truth T(¬B), i.e. that ultimate truth that dharmas have no self-being. As we will see later, formulating the ideas of Jízàng with the status-predicate T (and its negation) only avoids confusion. 144 Asian Philosophical Texts ordinary people and to accept that of sages. This is the reason for clarifying the first level of twofold truth.18 We can interpret Jízàng as saying that it is an ordinary truth that dharmas have being-T(B)-but it's the absolute truth that they don't (i.e. a falsehood that they do), i.e. ¬T(B). Still being on the first level of twofold-truth (there are many more to follow), let me construct a diagram that depicts the metaphysical hierarchy of levels of truth.19 1st Level of Truth T(B) ¬T(B) Ordinary Truth Absolute Truth The highlighted part in the quote above is, I guess obviously, referring to upāya, which, in this case, consist in the negation of ordinary belief. The negation of the ordinary belief that dharma's have being, we can stipulate, is the first step on the path towards enlightenment. Jízàng's essay continues: Next comes the second stage, which explains that both being and non-being belong to worldly truth, whereas non-duality (neither being nor non-being) belongs to absolute truth. It shows that being and non-being are two extremes, being the one and non-being the other. From these to permanence and nonpermanence, and the cycle of life and death and Nirvana are both two extremes, they therefore constitute worldly truth, and because neither-theabsolute-nor-the-worldly, and neither-thecycle-of-life-and-death, nor Nirvana are the Middle Path without duality, they constitute the highest truth.20 We are being told that T(B) and ¬T(B) are merely worldly truth, this is, something that "ordinary people" would hold-which is no surprise, 18 Ibid. 360, emphasis added. 19 In Nāgārjuna's words, Jízàng describes "a truth of mundane conventions (saṁvṛti-satya) and a truth of the ultimate (paramārtha-satya)" as in Westerhoff, Nāgārjuna's Madhyamaka, 2. 20 Ibid. 360 A. Kreutz Contradiction and Recursion in Buddhist Philosophy 145 as both propositions are expressed in language, and language is usually considered to be an aspect of the ordinary realm. Speaking of realms here; whence the compartmentalization of reality into different realms? Madhyamaka Buddhism distinguishes between conventional (ordinary) reality, which is the realm of language, thought, and the (from the ultimate perspective) erroneous view that things have own-being, and ultimate reality in which nothing has own-being and language and thought lack their descriptive power. Ordinary truth, then, are truth about the conventional reality, whereas ultimate truths are truths about ultimate reality. To hold both T(B) and ¬T(B) is being grounded in duality, which has to be overcome on the way to enlightenment. But we have to be aware that, for Jízàng, T(B) and ¬T(B) is not a metaphysical duality, as they both correspond to different metaphysical realms. This would come down to saying that "car A is red" and "car B is blue" is a duality, where it is clearly not. The conjunction of both, thus, is a worldly truth, as thought and language (in which this duality is formulated in) is part of only one realm, the conventional realm. It is thus dualistic thinking that is conventionally false, and ultimately not-false. This dualistic thinking, as part of the Buddhist practice, has to be overcome. The gist of Jízàng's treaties, then, is that overcoming dualities is seeking enlightenment, and enlightenment is the state of non-duality21. In other words: the duality does not rest in the world, but in the (cognitive) apprehension of it. It is Jízàng who hints at it, and Zen Buddhism (with which we will later be concerned with) that makes it explicit. From a semantic point of view, since we are now evaluating the status of the statements on the first level of truth, we are working on a 'second' and higher level of truth (i.e. in a metameta-language), and so I shall introduce a second-level status predicate 'Ʈ' which semantically works just like T but ranges over the first level status-predicates instead of propositions. The worldly truth on the second level-Ʈ(T(B) ∧ ¬T(B))- is koti 21 The problem with language, again, is that non-duality itself refers to a duality in that the "non" provokes an opposite concept from which non-duality is demarcated, which is duality. It is therefore better to refer to emptiness, rather than non-duality. 146 Asian Philosophical Texts (3), which, since ordinary and erroneous can be overcome, just like T(B) can be overcome by negating the duality. We establish koti (4), which in Jízàng's framework reads, by negating the worldly duality, as follows: ¬Ʈ(T(B) ∧ ¬T(B)) . 2nd Level of Truth Ʈ(T(B) ∧ ¬T(B)) ¬Ʈ((T(B) ∧ ¬T(B))) 1st Level of Truth T(B) ¬T(B) Ordinary Truth Absolute Truth The Sānlùn Catuṣkoṭi Moving this schema into a vertical position, we see the four corners of the catuṣkoṭi before us: The Sānlùn catuṣkoṭi (1) T(B) (2) ¬T(B) (3) T(T(B) ∧ ¬T(B)) (4) ¬T((T(B) ∧ ¬T(B))) Each koti is established by a recursive method of conjoining ordinary and absolute truth on the nearest lower level to yield the ordinary truth on the higher level and negating the ordinary truth in the same level to yield the absolute truth. Yet again, the systematic thinker will recognize that Ʈ(T(B) ∧ ¬T(B)) and ¬Ʈ(T(B) ∧ ¬T(B)) are dualities, which, so the Buddhist doctrine, shall be overcome. Jízàng noticed this himself and opts for an again higher level, intended to overcome the duality between duality and non-duality. But does that help? At every level of truth, we return to a higher-level duality which contradicts the Buddhist doctrine of non-duality, and so on ad infinitum22... The production of new levels of truth will never come to a halt. 22 This has also been recognised by Chan, ibid. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 147 Let the following be a schematic model of the recursion,23 where Cx stands for conventional truth, Ux for ultimate truth, the x for the respective truth-level, > for the step from one conventional to ultimate truth "horizontally" on one level to the other via negation, and » for the "vertical" transgression to the next higher truth-level. I call this a truth-transfer (TT). TT: C0>U0 » C1>U1 »...Cn>Un »... It is interesting to see that TT, without being explicit on its potentially infinite nature, can already be found in the MMK, also in verse 18:8, where Nāgārjuna affirms all four alternatives: "All is so, or all is not so, both so and not so, neither so nor not so. This is the Buddha's teaching." As Westerhoff points out, the commentarial literature on this verse, especially Candrakirti, interprets this verse as referring to the graded nature of the Buddha's teachings (anusasana)24. Candrakirti's comments reflect the idea that "all is so" is taught to ordinary disciples, "all is not so" to those while informing then about the doctrines of momentariness and impermanence, and so on. Returning to the main threat of the analysis of Jizang: what shall we make of this now, knowing that the four corners of the catuṣkoṭi are just the beginning of an infinite and never terminating process of overcoming non-duality? In Garfield (2015) and Fox (1995) we find some noteworthy comments on Jízàng's levels of truth, and the observation that genuine nonduality can never be found: This is why I claim that either of your two truths, i.e., interdependent nature and discriminative nature; or two truth that is not two, or firmlyestablish truth on the one hand, and 'Non-two and Non-non-two'; i.e., three non-nature or non-firmly-established truth on the other hand, is merely my conventional truth, whereas 'Forgetfulness of words and annihilation of thoughts' is really ultimate truth.25 23 Not a mathematical model of the recursion, of course. 24 Jan Westerhoff, Nāgārjuna's Madhyamaka. 89. 25 Jay Garfield, Engaging Buddhism. Why It Matters to Philosophy (Oxford: University Press, 2015), 257. 148 Asian Philosophical Texts Yet, there is also Fox elaborating on Jízàng, establishing that, apparently on the fourth level, ... all of these distinctions (on the lower three truth levels) are deemed conventional, and the authentic discourse regards that any point of view cannot be said to be ultimately true, and is useful only so far as it is corrective in the above sense.26 A sudden stop of the recursion at whatever level, however, seems arbitrary, and unsupported by the other texts of Jízàng, so, let us not take the third or fourth level to have any significant role. Let us call genuine non-duality (forgetfulness of words and annihilation of thoughts), or rather the experience or insight into it; . Jízàng, as the sources above suggest, gives no precise answer as to when insight into is achieved (i.e. how many levels of truth have to be crossed)-sometimes its three, and sometimes four-but let us assume that after n-progressions through higher and higher truth-levels, insight into can be achieved. We can add it to our model in the following way: nth Level of Truth ... N ... N ... 2nd Level of Truth Ʈ(T(B) & ¬T(B)) ¬Ʈ((T(B) & ¬T(B))) 1st Level of Truth T(B) ¬T(B) Ordinary Truth Absolute Truth TT(N): C0>U0 » C1>U1 » ...Cn>Un »... The Bóxiè Xiànzhēn In a way, then, the recursion's cul-de-sac, the soteriological "end-goal" is external to the catuṣkoṭi, as there is no bridge of logical necessity between the recursion (everything that happens before N and 'N' itself. This supports the argument that the 26 Allen Fox, Jizang (New York: Harper Collins), 87. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 149 catuṣkoṭi is more than a merely argumentative framework to refute philosophical enemies but a schema of the (Bodhisattva's, or Arhat's) path to the soteriological "end-goal," while it is also coherent with the idea that the catuṣkoṭi does not defend any position on its own. One could say that what the (positive) catuṣkoṭi is (implicitly) defending is the Buddhist doctrine of non-duality, although the defence never fully terminates: each level of truth, then, is upāya for the nearest upper level of truth, which again is upāya for the nearest upper level of truth- genuine non-duality, via this process, can never be achieved. So, the catuṣkoṭi doesn't defend a position on its own. It is, if we believe Rogacz (2015), Jízàng who disagrees: This pragmatical approach (the levels of truth) provides us to the central concept of "refutation of erroneous views as the illumination of right views," bóxiè xiànzhēn (驳谢现真), which was enunciated in the "Profound Meaning of the three Treatises." As we remember, Prāsan gikas claimed that Mādhyamaka is only a negative method of refuting views, but Sva tantrikas believed that it has also its own, undoubted view. Although Jízàng cannot have been a witness of this dispute, he subverted the salience of this argument: refutation of erroneous views is always the illumination of right views, and vice versa. All beliefs are empty because they depend on their rejections. Two opposite beliefs (statements) share the same premises and the horizon of possible continuations. Tetralemma is transcending these artificial oppositions, such as nothingness/absolute, false/truth, samsara/nirvana, and so on. ... In this perspective, the doctrine of emptiness seems to be the reinterpretation of the doctrine of expedient (skr. upāya, ch. fāngbiàn, 方便) means.27 We have now discovered that the role of the catuṣkoṭi in Buddhist philosophy exceeds its usage as a purely argumentative tool. We have also witnessed how the catuṣkoṭi has lost its distinctive four-valued form. We don't find the distinction into positive and negative catuṣkoṭi in Jízàng anymore. In a sense, they have melted into one. We also don't call it the catuṣkoṭi anymore, but bóxiè xiànzhēn: the concept of the "refutation of erroneous views as the illumination of right views," (Ch. 驳谢 27 Dawin Rogacz, Knowledge and Truth in the Thought of Jizang, (The Polish Journal of the Arts and Culture, 2015), 232. 150 Asian Philosophical Texts 现真, Eng. Refute the Truth). What is not clear now, however, is whether the bóxiè xiànzhēn (i.e. the catuṣkoṭi) is defending a position on its own, or not. On the one hand, we have the formalization which strongly suggests that the bóxiè xiànzhēn does not, in fact cannot, defend a position on its own. On the other hand, we have Rogacz's interpretation of Jízàng, according to which Jízàng's takes the bóxiè xiànzhēn to defend a position on its own. I shall put forward a different interpretation of Jízàng's comments and argue against Rogacz. For Jízàng, as the title of his essay suggests, the bóxiè xiànzhēn is not only intended to refute other philosophical positions, but also for an illumination of the right view. For Rogacz's interpretation of Jízàng, the recursion (i.e. the refutation of all possibilities) is equivalent to . (i.e. "the right view") Hence, not defending any possibility is defending . But how can this be right as there is no logical inference from the first position to , at all? If the logical apparatus is right, the bóxiè xiànzhēn does not defend a position on its own, as there is no logical connection between the kotis and , whatsoever. We are desperately searching for an argument in either direction in Jízàng's work. We can approach this problem with considerations on upāya. The bóxiè xiànzhēn understood as upāya alleviates the problem. The idea is simple: a means must not ultimately terminate in a goal, and might only be but one of many means necessary to reach a goal. In other words: a means doesn't have to be on its own sufficient to reach a goal, and so the recursion that is the "refutation of erroneous views" might be necessary, but doesn't have to be, on its own, sufficient for "the illumination of right views." From the viewpoint of upāya, for (i.e. "the illumination of right views) to be brought about, the "refutation of erroneous views" has to be brought about, yet the "refutation of erroneous views" is on its own not enough to bring about . It is then wrong to think of the "as" in the "refutation of erroneous views as the illumination of right views" in terms of logical equivalence. Rogacz's use of the term "vice versa" is clearly misleading. It is the "illumination of right views" which is always the "refutation A. Kreutz Contradiction and Recursion in Buddhist Philosophy 151 of erroneous views," but not vice versa. Hence, the bóxiè xiànzhēn does not defend a position on its own, neither did its predecessor, the catuṣkoṭi. The Paradox The problems do not end here. The right view for Jízàng is that of the middle way between dualities, inherited from Nāgārjuna, which we express with. He also writes that attachment to the doctrine of emptiness is misguided, hence it should be overcome. It is like water able to extinguish the fire, if the water itself could ignite, what would be used to extinguish it? Nihilism and eternalism are like fire and emptiness can extinguish them. But if someone insists on adherence to emptiness, there is no cure which could help him.28 This comment is paradoxical only if we were thinking that the bóxiè xiànzhēn terminates in , and does not go on, as expressed in TT(). What the comment suggests, though, is that , although it represents some kind of qualitative change, does not force the bóxiè xiànzhēn to a halt, as ascribing to all things is onesided and doctrinal, and thus needs to be overcome29. One could argue in the following way: 'N' is the soteriological end-goal of Buddhist practice with which upāya (i.e. an ongoing refutation of erroneous views), prima facie, seems to become redundant- most means to reach a goal seem to be useless, once the goal is reached. The existence of a car, for examples, loses its immediate significance once the destination is reached. Thinking about having to find a parking place, we see that a means can even become obstructive. As soon as it has satisfied its immediate purposes, it loses its value. 28 Huseh-Li Cheng, Empty Logic: Madhyamaka Buddhism from Chinese Sources (Motilal Banarsidass, 1991), 49. 29 The relates to footnote (14). Yet, the argument here is slightly different: Whereas the linguistics of non-duality provokes a duality, ascribing emptiness (and only this) to a phenomenon is one-sided, regardless of whether there is a dual or not. This one-sidedness is doctrinal and therefore to be rejected. 152 Asian Philosophical Texts So here is a problem: 'N' both does, and does not bring the bóxiè xiànzhēn to a halt. We either have, TT(N): C0>U0 » C1>U1 » ...Cn>Un » ...N where brings the recursion to a halt. Or we have, TT(N)*: C0>U0 » C1>U1 »N » ... Cn>Un » ... where the recursion does not halt. According to Jízàng (he has no stance on that), either is a viable option. To dissolve this looming paradox, let us now follow the development of Buddhism (and with it the catuṣkoṭi) further to the East. Towards Kaku-an Shi-en Jízàng is at the same time the greatest and the last philosopher of the Sānlùn school. Yet, the development of Mahayana Buddhism did not stop, and with it, the development of the catuṣkoṭi did not either. I shall put forward the idea that what is known as a Kōan in the Chan/Zen tradition bears so many similarities to the catuṣkoṭi that it is hard not to recognize a systematic connection. The situation in which Jízàng has left us is, from the perspective of Zen, no longer paradoxical, as I shall attempt to illustrate. The Kōan A monk asks: "Does a dog have Buddha-nature?" The Zen Teacher says: "Mu" Wúmén Guān: (1) We now find ourselves in the teachings of Chan/Zen Buddhism. What to make of this little dialog is difficult to say. The answer is baffling, and, at first sight, sense cannot be made of it. The Zen Teacher'sanswer is "Mu" (Jap. 無), which translates as "nothingA. Kreutz Contradiction and Recursion in Buddhist Philosophy 153 (ness)."30 The answer turns into something intelligible, only if we think a little more carefully about question. The question, in fact, goes wrong in two ways. First, it is the Nirvanasutra that explicitly states that all sentient beings have Buddha-nature, it is their fundamental nature, so a dog has Buddha-nature, too. The monk, we can guess, should have known this. It is the question, now, that seems to be illposed and based on an inadequate concept of being. But this is not the reason for the teacher's enigmatic answer. So, what is the answer supposed to tell us? The question, as it is posed, is supposed to be answered with an affirmation-negation linguistic device, i.e. simply with "yes" or "no." But the latter presupposes that the question is well-formed, which, taking into account the metaphysics of being in Mahayana Buddhism of which Zen, in which context the question is posed, is a branch-it is not. But how then, does the answer refer to the ill-posed nature of the question? Let me try a different approach: The above, Wúmén Guān: (1), is known as "Joshu's Dog" (Ch. 趙州狗子), and often ridiculed in popular culture. Just like Wúmén Guān: (18), it is a Kōan (Jp. 公案). Kōans feature heavily in the teaching and practice of Zen Buddhism. The aim of grappling with, and meditating over a Kōan is to overcome conceptual thinking-exactly the thinking that has instigated the monk's question. Therefore, it is not the monk's ignorance of the fact that, according to Mahayana tradition, every sentient being has Buddha-nature, but his conceptual thinking which manifests itself in a yes/nodichotomy, in which a concept either applies or does not apply. The Kōan practice (grappling with Joshu's Dog for example) is intended to overcome conceptual thinking, not to teach doctrines, such as that of Buddha-nature. Now, I argue, it is the dichotomy in conceptual thinking that the Zen teacher's answer is pointing at, as a hint for the monk to question his thought process (not necessarily his knowledge of the sutras). 30 Chao-chou Ts'ung-shen (Jap. Joshu) is the teacher referred to. 154 Asian Philosophical Texts Abbreviated Catuṣkoṭis A possible answer to "Joshu's Dog" could be stated as a negative catuṣkoṭi. The act of denying all kotis comes down to denying all possible ways a dog could or could not have Buddha nature. As the case of Buddha-nature is certainly an exceptional case, let us rethink the Kōan in a Nāgārjunian manner. In the context of the MMK, the Kōan would have been stated as: "does a dog have svabhāva?" The Nāgārjunian answer, which we are already familiar with, is a denial qua negative catuṣkoṭi of all the possible ways a dog could have svabhāva. Ultimately, the dog's nature is śu nyata (emptiness), Nāgārjuna would say. It is not a big conceptual leap from 'emptiness' (śu nya, Skt. शून्यता) to 'nothing-ness' (Mu, Ch. 無). In effect, both Nāgārjuna's and Joshu's answer to the monk's question are, conceptually, equivalent31. It is only that Nāgārjuna gives a profound logical apparatus with which all the possible ways a dog could have own-being, i.e., svabhāva, is denied. It is this logico-ontological apparatus in the background that has vanished from the Zen tradition, but given the historical connection, we can assume that it still resonates somewhere among other implicit principles that have been inherited from India. I shall, therefore, put forward the thesis that the Kōan is an abbreviated catuṣkoṭi. Identifying the Kōan with upāya is not as reckless as my initial thesis that the catuṣkoṭi should be considered upāya, as it is widely agreed that the Kōan's role in Chan/Zen is that of a meditational object. We have the Rinzai school of Zen which focusses heavily on the Kōan as a means to gain enlightenment (Jap. satori). Zen, furthermore, makes a distinction regarding how enlightenment occurs. According to the Sōtō school, 31 I have to mitigate this parallelism: whereas the Indian Buddhist's an-atman (no own being) and Nāgārjuna's sunyata (emptiness) are epistemological concepts-referring to the fact that things are only forms (as superimpositions), "Asian nothingness" (tōyōteki mu)," referring to the Kyoto School's concept of nothingness through satori (and therefore the intellectualisation of Zen), refers to a lived experience of reality as a way arising naturally out of nothingness, and should therefore rather be considered a pragmatic concept. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 155 enlightenment comes gradually, sometimes glossed at as "silent enlightenment," as it is seen as a process of discovery through Zazen (sitting meditation). On the other hand, we have the Rinzai school which advocates "sudden enlightenment." The Kōan and Zazen, both are a means of reaching satori. Hence, satori is for Zen what we have, so far, referred to as 'N'. We have now gone the full circle to support the main hypothesis of this essay: Catuṣkoṭi and Kōan play the same role in Buddhist philosophy and share a systematic structure. Both are to be considered a schema for upāya means to the soteriological "end-goal," which is enlightenment. Although I cannot claim that the verification of my hypothesis is airtight (it lacks a lot of historical exegesis), I still hope that it makes sense on the grounds of the material provided. The similarities and practical use of concepts (Kōan and catuṣkoṭi) is too apparent not to draw this connection. That Kōan and catuṣkoṭi share a systematic structure still needs to be elicited. We have left Jízàng with the paradox that both terminates and does not terminate the upāya, and I have promised a dissolving of this paradox in the Zen tradition, so the next section shall model the remainder of the catuṣkoṭi, once it reached Japan. For this, I shall consult another staple of the Zen tradition: the jūgyūzu. It is here that the schematic structure which is missing in the Kōan is made overt. The Zen catuṣkoṭi (aka. the Jūgyūzu) In fact, what is left over of the catuṣkoṭi in the Zen tradition is perhaps too sparse for it to still be an instance of the catuṣkoṭi. in any case, it is drastically abbreviated. It is merely the very scaffold of the bóxiè xiànzhēn that has found its way into Zen literature in the form of ten pictures of a man and an ox and their corresponding verses: the jūgyūzu (Jp. 十牛図, Eng. Ten Ox Herding Pictures). 156 Asian Philosophical Texts The pictures and verses of the jūgyūzu above, as it is widely conceded recognized, are similes of the path to enlightenment, with the ox as a symbol for meditation (Kōan or Zazen) practice. The ox is actually ubiquitous in Buddhist literature. It features in texts as early as the Maha Gopalaka Sutra as a symbol for meditation (perhaps because wrestling with one's mind is as strenuous as wrestling with an ox), and is one of the many iconographics that has survived the Mahayana's journey from India to the East. According to D. T. Suzuki (unpublished), the Oxherding pictures and verse made their first appearance in China around the 12th century. The most well-known version of the jūgyūzu is the one by the Chinese Chan master Kaku-an Shien (Ch. 廓庵師遠), who was also the first to contribute verse to the pictures as well as an introduction. 尋 牛 見 跡 見 牛 得 牛 牧 牛 騎 牛 忘 牛 人 牛 返 本 入 鄽 "Ten Bulls" by Tokuriki Tomikichiro 帰 家 存 人 倶 忘 還 源 垂 手 A. Kreutz Contradiction and Recursion in Buddhist Philosophy 157 It is also with Kaku-an Shi-en that the Ox Herding becomes a jūgyūzu (jū, Jp. 十, Eng. ten), instead of a hachigyūzu (hachi, Jp. 八; Eng. eight). It is the blank space at which the hachigyūzu halts, whereas the jūgyūzu incorporates two further pictures. This detail is of outmost significance for our ongoing discussion. To see the relations and the similarities between bóxiè xiànzhēn and jūgyūzu, we need to know the meaning behind the pictures, and we need to refer back to the recursive schema (TT), which is the scaffold of the bóxiè xiànzhēn. To remind you, the reader, of (TT), here is the schema again: nth Level of Truth ... N ... N ... 2nd Level of Truth Ʈ(T(B) & ¬T(B)) ¬Ʈ((T(B) & ¬T(B))) 1st Level of Truth T(B) ¬T(B) Ordinary Truth Absolute Truth TT(N): C0>U0 » C1>U1 » ...Cn>Un »...N Also, we haven't settled whether (TT) is the correct analysis, or whether (TT*) is what the bóxiè xiànzhēn expresses. TT(N)*: C0>U0 » C1>U1 » N » ... Cn>Un » ... The jūgyūzu depicts a character, a boy in the case above, and his search and taming of an ox as a metaphor for on his path to, and through, enlightenment. So, it already suggests that there is a post-enlightenment state, such as TT(N)* describes it. In the first picture, we meet him alone in the wilderness, lost and confused, but searching. He knows about the conventional truths and is unsatisfied with them, it is interpreted. Let this conventional belief be T(B), and his stage on the way towards enlightenment be represented by C0. The boy goes on and finds the traces of the ox in picture three and four, which serve as a metaphor for finding the sutras and 158 Asian Philosophical Texts inquiring into the Buddhist doctrines. He then learns about the erroneous nature of conventional truth and, for the first time, has a glimpse of the ultimate truth, ¬T(B). Let this stage be represented by U0. Here we have the first two kotis, and the first level of Jízàng's truth-hierarchy complete. However, the boy is unable to distinguish conventional from ultimate truth, and his mind is still confused as to truth and falsehood, and so the boy believes in both T(B) & ¬T(B), and is yet not ready to discriminate them-this is the third koti. This stage, C1, depicts the boy catching and taming the ox in the fourth and fifth picture. Picture six shows the boy riding the ox. He has successfully tamed the beast and is able to let loose the line. He has realised that the duality of truth and falsehood needs to be overcome. This is stage U1 of Jízàng's hierarchy and the fourth and final koti ¬Ʈ((T(B) & ¬T(B))). As the series continues, the ox has disappeared, and the boy is back at home. The next picture is an empty circle, which I shall represent as 'N'. It's the manifestation of an ineffable reality which is beyond the dualities of language and thought. This is where, for the boy, the illusion of self has vanished, and he experiences non-duality. Hence not only ox, as it is now useless (as the discussion on what it means to be a means as already revealed; meditation is a means and not in itself sufficient, yet necessary for enlightenment32), but also the boy, is gone. This represents enlightenment. This is where the hachigyūzu ends. The boy has found enlightenment, and the ox, which is a metaphor for the boy's upāya, like Wittgenstein's ladder, has become redundant. Let the hachigyūzu (OX8) be represented by this schema, which shall remind us of the bóxiè xiànzhēn in its TT(N)-reading: OX8: C 0 » U0 » C1 » U1 » 32 Some traditions of Zen would even reject that. Formal Meditation, such as Zazen, might not be necessary, enlightenment can come unexpected. But even if formal meditation is not required, some kind of quasi-meditative process is involved. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 159 This has been the first option of two readings of the bóxiè xiànzhēn, which has led us to a paradox in Jízàng for which we couldn't find a solution within his own canon. If the hachigyūzu is a successor of the bóxiè xiànzhēn, the process is supposed to terminate with reaching enlightenment, and the recursion has become obsolete. However, the jūgyūzu above includes two further pictures after the enlightenment. What to make of those? The ninth picture shows a serene landscape with the boy still gone. We can interpret it as a return to the ordinary world, which, although the boy has found enlightenment, has not vanished. The boy, too, has not vanished either. He returns to the world in the last picture of the jūgyūzu. It hence appears as though the journey continues, and as with Jízàng's hierarchy, has not brought the process to a halt. The jūgyūzu and the hachigyūzu seem to contradict themselves in that in the latter is the end of the series, whereas in the former it is not. But to think in this way is to overlook a significant detail in the iconography: the ox is still missing, and upāya has become pointless after enlightenment. Although the boy has returned to the ordinary, he has not returned to a pre-enlightened state which would again necessitate and overcoming of dualistic thinking and thereby, analogously, keep the recursion going. For the boy, the illusion of a self and that of a world with svabhāva (to serve the Nargarjunian term) has vanished, and conventional reality is seen as what it is-a conceptual superimposition on ultimate reality. The jūgyūzu (OX10) shall hence be represented in the following way, where the superscript 'C' represents the post-enlightened perspective on conventional reality, knowing that the states-of-affairs in C are merely conceptual impositions on ultimate reality N: OX10: C 0 » U0 » C1 » U1 » » CN It is important to note that the process does not stop with CN, but it is the post-enlightened state CN in which the boy will remain throughout his post-enlightened life33. What a lucky boy. 33 In fact, several comments of Hakuin suggest that experiences of enlightenment can be had over and over again, and only through ongoing practice, the postenlightened state could be maintained and cultivated. This would question the 160 Asian Philosophical Texts Solving Jízàng's Paradox Jízàng's paradox can now be resolved if we take the distinction into preand post-enlightened states into account. In both cases of the bóxiè xiànzhēn (TT(N) and TT(N)*) the recursion stops, it is only in the latter that the post-enlightened state is added. With this in the background, we now know how to interpret Jízàng's comment that a dogmatic belief in is erroneous and has to be overcome. Not by applying the recursive method of the bóxiè xiànzhēn to , but by returning to the starting point with a postenlightened perspective. A famous Chan/Zen saying articulates this thought: Thirty years ago, before I practiced Chan, I saw that mountains are mountains and rivers are rivers. However, after having achieved intimate knowledge and having gotten a way in, I saw that mountains are not mountains and rivers are not rivers. But now that I have found rest, as before, I see mountains are mountains and rivers are rivers.34 Here, talk of the mind being still and at peace in Nāgārjuna (MMK dedication and 25.24) is also worth mentioning35. For Nāgārjuna, compulsive philosophical questioning (such as asking questions in form of the four corners) would be stopped when the mind stops grasping at philosophical theories and simply accepts those as conventional. Nāgārjuna calls this prapañca upaśama (Eng, "pacification (upaśama) of mental proliferations (prapañca)"36). Hence, the idea of quietening the mind of philosophical questioning is integrated in Madhyamaka thought. Candrakīrti commented on it in the Prasannapadā, saying that complete repudiation of upāya. Still, one could guess that post-enlightened upāya is different from pre-enlightened upāya, which in consequence would not question the account given above. 34 Urs App, Master Yunmen: From the Record of the Chan Teacher Gate of the Clouds (New York: Kodansha International Press), 1994. 35 Thanks to an anonymous referee for making me aware that we can return to Nāgārjuna at this point. 36 Thanks go to Jan Westerhoff for this translation. A. Kreutz Contradiction and Recursion in Buddhist Philosophy 161 For the Nobles when they see dependent-arising as it really is, that very dependent-arising is called "the calming of manifoldness (prapañca upaśama)-in the sense that there is the calming of instances of manifoldness in it. And because (it) is entirely without the misfortunes of birth, old age, death and so forth owing to the ceasing of (any) dealing with (the dichotomies): cognition and cognizables in view (of the fact) that mind and mental factors do not arise in it, it is (ultimate) welfare.37 The leap from Nāgārjuna to Candrakīrti to Zen is a minute one. Conclusion In seeking a comprehensive account of the catuṣkoṭi, this paper has been working on logical, ontological, historical, and soteriological aspects in unison. The fact that the logical aspects of the catuṣkoṭi have been unhinged from their ontological fundament has prevented any fruitful modelling and obstructed further insight into the historical development of this fascinating piece of Buddhist philosophy. I hope that with this study of the development of Madhyamaka thought and with the help of the formal modelling applied thought the paper it is now clearer what the catuṣkoṭi is, and how it functions in the Buddhist canon. As the confluence of the last chapters suggests, we should not consider the catuṣkoṭi as exclusively a phenomenon of Madhyamaka thought, but think of the Madhyamaka's catuṣkoṭi as an instance of a much broader category, which is a schema of, and for, upāya, which includes the bóxiè xiànzhe and the jūgyūzu (and perhaps many, yet unexplored others). 37 Compare, Anne McDonald, In Clear Words. 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