On Garfield and Priest’s interpretation of the use of the catuskoti in Mūlamadhyamakakārikā

ABSTRACT

According to Garfield and Priest’s interpretation, the positive use of the catuskoti by Nāgārjuna in Mūlamadhyamakakārikā (MMK) shows that he endorses a four-valued semantics similar to that of Belnap’s First-Degree Entailment (FDE), while the negative use of the catuskoti by Nāgārjuna in MMK indicates that what he really has in mind is a plurivalent five-valued semantics. This paper argues that their interpretation suffers from a number of problems: adequate logic, collapse of kotis, lack of literature support, and a suitable explanation of the third koti. The final part of the paper combines insights from Westerhoff, Cotnoir, Garfield and Priest together and describes the right interpretation of Nāgārjuna’s use of the catuskoti in MMK with the right semantics and the right logic of MMK. The authors argue that Nāgārjuna’s ‘rebuttal’ of all four kotis in a catuskoti should be understood as a mere denial or a mere rejection of these kotis.

KEYWORDS:

• Nāgārjuna
• catuskoti
• plurivalent logic
• MMK
• FDE

Acknowledgments

The first author Cong Wang is a lecturer from the School of Marxism, Henan Normal University, China, and the corresponding author Wen-fang Wang is a professor from the School of Philosophy and Social Development (Institute of Concept and Reasoning), Shandong University, Jinan, Shandong Province, China. For the publication of this paper, we would like to thank the support of China’s MOE project of Key Research Institute of Humanities and Social Sciences at Universities (22JJD720021) and the support of the project of National Social Science Foundation of China (23BZX123) and the support of the project of Innovation Team of Philosophy and Social Science in Colleges and Universities in Henan Province (2020-CXTD-06).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. According to Priest (2018, pp. 17–8), the catuskoti was already used as a general schema in the intellectual circles of Gautama, the historical Buddha, especially by the philosopher Sanjaya.

2. For example, MMK, VIII, 6, seems to divide the possible answers to the question about self into only three categories, while MMK, VIII, 5–15, seems to divide the possible answers to the question about nirvāna into five categories. We would like to thank the anonymous reviewer for reminding us about these verses.

3. The following example is one in which all four kotis holds. There are, however, cases in which only one of the four kotis holds in a catuskoti in MMK. For example, for the question about motion discussed in chapter II of MMK, Nāgārjuna concludes in verses 24–25 that exactly one of the four possibilities, i.e. the fourth one, holds for the things at issue. (For more discussion about this catuskoti, see Section 4.1 below.) Garfield and Priest (2009) did not tell us whether cases similar to this one in MMK should be counted as examples of the positive use or of the negative use of the catuskoti, but we believe that they will agree to classify them as examples of the positive use of the catuskoti. So the positive/negative distinction regarding the uses of the catuskoti is in fact the distinction between affirming at least one koti and denying all four kotis.

4. MMK, XVII, 8 (Garfield, 1995).

5. MMK, XXII, 11 (Garfield, 1995).

6. ‘Conventional reality’ in Buddhist philosophy refers to the reality with which we are familiar in everyday life: the world of people and stars. The objects of this reality are, however, conceptual constructions out of our minds or conceptual discrimination. By contrast, the ultimate realty is what the reality really is or what the reality ultimately comsist in.

7. By ‘exhaustive’ or ‘jointly exhaustive’, they mean that every sentence or proposition belongs to one of the four categories so that there is no need for a fifth category for sentences or propositions. By ‘mutually exclusive’, they mean that no sentence or proposition belongs to more than one of the four categories. Actually, Garfield and Priest think that the division should have one more property than merely being jointly exhaustive and mutually exclusive: none of these categories should be empty. However, we don’t think that this final request is one that must be met by a tetralemma for reasons that will be seen in Section 4.1 of this paper.

8. In what follows, ‘glb(v(A), v(B))’ refers to the greatest lower bound of {v(A), v(B)} in the Hasse diagram, while ‘lub’ refers to the least upper bound of {v(A), v(B)} in the Hasse diagram (Priest, 2018, Section 2.6). Garfield and Priest also want each FDE-model to validate every example of the T-schema and this will put one more restriction on an FDE-model. Since this last condition regarding the T-schema can be satisfied in principle, we will omit to mention it here and for any other semantics that we will meet in this paper.

9. The rationale for changing the picture of value-owners is this: if the value e stands for ‘ineffable’, it does not seem sensible to say of a sentence that it is ineffable. However, it makes perfect sense to say the same thing about a state-of-affairs.

10. The orders among t, f, b, and n are the same as those of FDE while e is incomparable to any of them (Priest, 2018, Sections 5.3 and 5.4).

11. Priest (2018) thinks that this shows that Nāgārjuna takes the ultimate reality to be both effable and ineffable, and, accordingly, Nāgārjuna is a dialetheist who believes that there are true contradictions. However, we don’t think that this interpretation of Nāgārjuna is plausible. For more about this, see Sections 2 and 4 of this paper.

12. And we learn that Garfield tends to agree with it (in a private communication during Dialetheism and Related Issues in Analytic Asian Philosophy: An International Workshop, Kyoto University, Japan, June 23–24, 2017).

13. Priest (2018, pp. 79–80) uses a relational semantics to describe his semantics for p-FDEe. Though his description of the semantics for p-FDEe is different from what we will give below, they are nonetheless equivalent.

14. We don’t actually think that Nāgārjuna does use the classical RAA in MMK. After all, one is bound to commit to ‘¬A’ after s/he draws a contradiction from ‘A’ by the classical RAA, but Nāgārjuna often goes on to argue that ¬A is also not the case after he rejects A. We are inclined to think, rather that Nāgārjuna is actually using a meta-rule essentially different from, though very similar to, the classical RAA; that is, we are inclined to think that Nāgārjuna is actually using the logic of p-Ke for the negative use of catuskoti. For more on this problem, please see Section 4.3 of this paper.

15. Of course we can say that Nāgārjuna is simply wrong when he takes these inference patterns to be valid, but this explanation doesn’t seem very plausible.

16. Neither the classical RAA nor the one very similar to it that we mention in footnote 14 and Section 4.3 is valid in Jc Beall’s proposed logic system (Beall, 2009), and we think that this is an interpretative problem for taking Jc Beall’s move here.

17. See Priest (2010, p. 32) and Beall (2009, pp. 104–5) for the proofs.

18. Neither the classical RAA nor the one very similar to it that we mention in footnote 14 and Section 4.3 is valid in Priest’s (2006), and, again, we think that this is an interpretative problem for taking Priest’s move here.

19. Besides, there is a worry about whether one can still make every instance of the T-schema true. For this problem, see Field (2008, chapter 7).

20. How rich is rich enough? It does not have to be very rich: Godel’s fixed-point lemma proves that any language that can talk about its own syntax or about elementary arithmetic is bound to have self-referential sentences.

21. A liar sentence A is a sentence that says of itself that it is not true so that ‘A ↔ ¬T〈A〉’ is true.

22. For the proof of this claim, see Priest (2010, p. 32) and Beall (2009, pp. 104–5).

23. The proof of this claim is simple. Let A be a liar sentence such that both ‘A ↔ ¬T〈A〉’ and ‘T〈A〉 ⋀ F〈A〉’ are true (Priest (2006 will agree with both). Since T〈A〉 is true, it follows from the T-schema and ‘A ↔ ¬T〈A〉’ that ¬T〈A〉’ is true. Since F〈A〉 is also true, it further follows that ¬(T〈A〉 ⋀ F〈A〉)’ is true.

24. This is confirmed by our Buddhist colleagues in Taiwan. It should also be mentioned that Tsongkhapa (AD 1357–1419) attempts to bring together Dharmakīrti’s and Dignāga’s system of logic together with Nāgārjuna’s Madhyamaka and this may show that the ‘big break’ suggested here is not genuine at all for Tsongkhapa. Again we would like to thank the anonymous reviewer for reminding us about Tsongkhapa.

25. Either of the classical form or of the form that we mention in footnote 14 and 18 and will discuss in Section 4 of this paper.

26. Simply replace t in the Hasse diagram with <1, 0>, f with <0, 1>, b with <1, 1>, and n with <0, 0>. The resultant diagram will represent the order among Cotnoir’s four truth values.

27. Negation toggles <1, 0> and <0, 1> similar to negation in FDE, but it also toggles <1, 1> and <0, 0>.

28. So the designated values include both <1, 1> and <1, 0>.

29. There is a further problem with Cotnoir’s suggestion: there is no obvious reason why negation should switch 0 to 1 and 1 to 0 for the second element of a value <x, y>; i.e. it is not obvious why negation should change the ultimate value of a proposition.

30. Westerhoff (2009, Section 4.1) uses these descriptions interchangeably. The prasajya/paryud¯asa distinction seems to be first proposed by Bhāviveka (500–570 CE) when interpreting MMK. The non-implicational/implicational distinction seems to be terminologically more standard nowadays than the rest, though Westerhoff prefers to describe the distinction as the presupposition-canceling/presupposition-preserving distinction. This distinction is also called the ‘external/internal’ distinction in modern Buddhist literature. Note that Westerhoff warms us not to confuse the prasajya/paryud¯asa distinction with the exclusive-negation/choice-negation that could be found in modern three-valued logic, but should take the latter distinction to be a special case of the former.

31. Westerhoff uses the word ‘deny’ to introduce the distinction when he says: ‘the difference … is the difference between negations carrying with them the presuppositions implied by the propositions they negate, and those that deny these presuppositions’ (Westerhoff, 2009, p. 70, emphasis added), but we think that it is more appropriate for him to use the word ‘negate’ here. Anyway, in order not to confuse it with a mere denial as a purely illocutionary act or a pure speech act (see Section 4.2 of this paper), we will use the word ‘negate’ henceforth.

32. There is certainly a grammatical difference between prasajya negation and paryud¯asa negation; as pointed out by Westerhoff (2009, p. 68): in prasajya-negation the negative particle connects with a verb, while in paryudāsa-negation it connects with a noun. However, we agree with Priest in thinking that the grammatical difference does not entail a semantic difference. As we will argue in Section 4.2 of this paper, the apparent semantical difference can be explained as a scope ambiguity with the same semantical negation.

33. Notice that what we are denying here is that the ‘negation’ or the ‘refutation’ of the four kotis by Nāgārjuna in a negative catuskoti should be interpreted as prasajya negation as explained by Westerhoff to be the negation of a presupposition; we are not denying that it should be explained as an illocutionary act such as a mere denial. The difference between these two will be explained in Section 4.2 of this paper. If the reader agrees with us about the distinction that we will make in Section 4.2 but insists that prasajya negation in Sanskrit expresses only a mere denial, then perhaps what we should conclude is that it is more likely that Westerhoff is mistaken regarding the use of prasajya in Sanskrit.

34. MMK, XVIII, 8.

35. So an A-type catuskoti is one in which the third koti is an explicit contradiction. This feature is lacking in a B-type or a C-type catuskoti.

36. Yet, it may be asked: why should Nāgārjuna even want to raise the third and the fourth kotis if they are empty? The explanation might be that they are raised merely for rhetorical, heuristic, or ad hominem reasons.

37. MMK, I.1 (Garfield, 1995).

38. MMK, XII.1 Garfield (1995).

39. So the third koti of a B-type catuskoti is, like that of an A-type catuskoti, the conjunction of the previous two kotis, but it is not, unlike that of an A-type catuskoti, an explicit contradiction.

40. Westerhoff seems to realize this when he says, ‘Here we can distinguish two varieties [of the third koti]. In the first case Nāgārjuna rejects it because its claim is … contradictory. … In the second case Nāgārjuna rejects the third alternative since it would combine the difficulties facing the first and second alternatives [though it is not contradictory]’ (Westerhoff, 2009, p. 82).

41. This is the abbreviation of a formalization of the sentence ‘There is a unique F which is not G’. In this sentence, the scope of the negation ‘not’ is the predicate ‘is G’’.

42. This is the abbreviation of a formalization of the sentence ‘It is not the case that there is a unique F which is G’. In this sentence, the scope of the negation phrase ‘it is not the case that’ is the sentence ‘there is a unique F which is G’.

43. Or a mistake. However, we don’t think that it is plausible to attribute such a mistake to Nāgārjuna; here, we adhere to the famous principle of charity proposed by Davidson (1984, Chapter 13).

44. This view is close to what Maltilal says (Matilal, 2005, Section 5.6).

45. We take the phrase ‘illocutionary act’ to be synonymous with ‘speech act’, though we know that these two phrases may be understood differently by some philosophers.

46. Therefore, speech acts are not propositional attitudes. Because they are different, our suggestion is actually a disjunctive one: either the word ‘not’ in a negative catuskoti in MMK indicates a speech act of denial or it indicates a propositional attitude of rejection or both. we don’t know which disjunct of the disjunction is true, yet we believe that either disjunct is enough to lend its support to our conclusions in this paper.

47. Philosophers influenced by Frege are likely to take the illocutionary act denial as the assertion of a negative sentence or proposition. However, not all philosophers agrees that this is the only illocutionary act that can be associated with denials. As we have already noted in footnote 45 and will note again in the next footnote, the idea of a mere denial, a denial without the accompanied assertion, as a possible illocutionary act is also embraced by Maltilal (2005) and Field (2008). In the same vein, Restall (2005) argues that there are cases of people who appear to reject or deny A without accepting its negation, while Rumfitt (2017) also proposes that we distinguish expressing a falsehood from failing to express a truth as two kinds of speech acts and to distinguish accepting as false from rejecting as untrue as two kinds of propositional attitudes.

48. Field (2008, pp. 73–74) makes a similar distinction between (merely) rejecting a proposition ‘p’ and accepting its negation ‘not-p’, though he does not say anything about (mere) denial. Our distinctions between negation, (mere) denial and (mere) rejection are actually inspired by his insights.

49. Other often used means to express one’s denial include shaking one’s head and waving one’s index finger when one hears ‘p’. These have often been used by a Zen master.

50. Most of them are also valid in the logic for the negative catuskoti that we propose below, i.e. p-Ke. So Nāgārjuna is actually quite free to use them in any kind of catuskoti.

51. In terms of truth tables, the three semantic rules can be explained by the following tables.

Table

52. The logic of p-Ke is actually the same as the three-valued logic weak K₃, but we will not give the proof of this claim here.

53. For details about the Prāsaṅgika interpretation of Nāgārjuna’s rebuttal and our defense of it, please see Chen and Wang (2020).

Additional information

Funding

This work was supported by the China’s MOE project of Key Research Institute of Humanities and Social Sciences at Universities [22JJD720021]; Innovation Team of Philosophy and Social Science in Colleges and Universities in Henan Province [2020-CXTD-06]; National Social Science Foundation of China [23BZX123].