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Chinese Logic: An Introduction

zao-wou-ki-1

[LaTeX version here; Chinese version here]

Introduction

As late as 1898, logic was seen by the Chinese as “an entirely alien area of intellectual inquiry”: the sole Chinese-language textbook on logic was labeled by Liang Qichao (梁启超)—at that time a foremost authority on Western knowledge—as “impossible to classify” (无可归类), alongside museum guides and cookbooks (Kurtz, 2011: 4-5). This same textbook had previously been categorized by Huang Qingcheng (黄庆澄) as a book on ‘dialects’ (方言). The Chinese word for logic (luójí 逻辑) itself is, according to the Cihai (《辞海》/ Sea of Words) dictionary, merely a transliteration from the English—the entire Chinese lexicon had no word resembling it (Lu, 2009: 98). Hence it never occurred even to specialists that this esoteric discipline might have close affinities with the roots of Chinese philosophy, from the I Ching (易经) to the ancient Chinese dialecticians (辩者), as well as the famous paradoxes of Buddhism.

With the advent of computers, “there is now more research effort in logics for computer science than there ever was in traditional logics” (Marek & Nerode, 1994: 281). This has led to a proliferation of logical methods, including modal logic, temporal logic, epistemic logic, and fuzzy logic. Further, such new logical systems permit multiple truth values, semantic patterns based on games, and even logical contradictions. In light of these possibilities, research in ‘Chinese logic’ aims to reinterpret the history of Chinese thought by means of such tools.

This essay consists of three parts: the mathematics of the I Ching, the debates within the School of Names, and the paradoxes of Buddhism. The first section will, through examining the binary arithmetic of the I Ching, provide an introduction to basic logical notation. The second section will explore Gongsun Long’s famous bái mǎ fēi mǎ (白马非马) paradox, as well as the logical system of the Mohist school. The third section will explain the seven-valued logic of the Buddhist monk Nāgārjuna by way of paraconsistent logic.

yijing-lattice

1. The I Ching (易经) & Binary Arithmetic

The I Ching is one of the oldest books in history. Throughout the world, there is no other text quite like it. Its original function was for divination, giving advice for future actions; yet, after centuries of commentary, it has taken on a fundamental role in Chinese culture. In part, this is because its commentaries became (apocryphally) associated with Confucius, thereby establishing it as a classic.

Its survival of the ‘burning of books and burying of scholars’ (焚书坑儒) in 213-210BC has magnified the I Ching’s importance. Historically, the Zhou dynasty was marked by hundreds of years of war and dissension. Finally, Qin Shi Huang united the nation in 221BC, to become China’s first emperor. According to the standard account, in order to unify thought and political opinion, Emperor Qin Shi Huang ordered that all books not about medicine, farming, or divination be burned. And so, the vast majority of ancient Chinese knowledge has been lost to history. Yet, since the I Ching was about divination, it avoided sharing the same fate. In a sense, then, the I Ching has come to represent the collective wisdom of ancient China—it embodies their entire philosophical cosmology.

Confucius’s interest in the I Ching is well known. In verse 7.16 of the Analects, he says: “If some years were added to my life, I would give fifty to the study of the Yi [I Ching], and then I might come to be without great faults.” Curiously, this appears at odds with the rest of his philosophy. After all, the Analects elsewhere says: “The subjects on which the Master did not talk, were—extraordinary things, feats of strength, disorder, and spiritual beings.” (7.20). That is, Confucius had no interest in oracles. Hence we can conclude that for Confucius, the main content of the I Ching was not divination, but philosophy.

The core tenet of the I Ching is deeply metaphysical, namely: the complementarity of Yin (阴) and Yang (阳). Yin represents negativity, femininity, winter, coldness and wetness. Yang represents positivity, masculinity, dryness, and warmth. Accordingly, the gua (卦) or fundamental components of the I Ching’s hexagrams, are two lines: ‘⚋’ for Yin, ‘⚊’ for Yang.

The trigrams, made up of three lines, have 8 combinations (2³ = 8), and so are called the bagua (八卦), where (八) means 8. The bagua and its associated meanings are: ☰ (乾/天: the Creative/Sky), ☱ (兑/泽: the Joyous/Marsh), ☲ (离/火: the Clinging/Fire), ☳ (震/雷: the Arousing/Thunder), ☴ (巽/风: the Gentle/Wind), ☵ (坎/水: the Abysmal/Water), ☶ (艮/山: Keeping Still/Mountain), ☷ (坤/地: the Receptive/Earth). The I Ching’s commentaries revolve around 64 hexagrams of six lines (2⁶ = 64 combinations). There are multiple ways of ordering the hexagrams: the most well-known is the King Wen (文王) sequence, but the most important for our purposes is the Fu Xi (伏羲) sequence.

diagram-of-i-ching-hexagrams-owned-by-german-mathematician-and-philosopher-gottfried-wilhelm-leibniz

diagram of the I Ching’s hexagrams owned by Leibniz

In the 17th century, the mathematician Gottfried Wilhelm Leibniz attempted to develop a system of arithmetic using only the numbers 0 and 1, called binary arithmetic. Binary arithmetic is in base 2: its key point is that any integer can be uniquely represented as a sum of powers of two. For example, 7 = 4 + 2 + 1 = 1×(2²) + 1×(2¹) + 1×(2⁰), and since each of the coefficients is 1, therefore the binary representation of 7 is (111). Conversely, 5 = 4 + 1 = 1×(2²) + 0×(2¹) + 1×(2⁰), where the middle coefficient is 0, so that 5 in binary is (101). For larger numbers, we simply include larger powers of two: 2³ = 8, 2⁴ = 16, etc.

Leibniz corresponded with various Christian missionaries in China, and had received a poster containing the Fu Xi sequence. To his astonishment, by letting ⚋ = 0 and ⚊ = 1, the Fu Xi sequence of 64 hexagrams exactly corresponds with the binary numbers from 0 to 63! Using the trigrams as a simplified example, from top to bottom we read: ☱ = (110) = 1×(2²) + 1×(2¹) + 0×(2⁰) = 4 + 2 = 6, ☵ = (010) = 0×(2²) + 1×(2¹) + 0×(2⁰) = 2, and so on. Thus, according to the Fu Xi and binary sequence, the bagua are ordered as: ☷, ☶, ☵, ☴, ☳, ☲, ☱, ☰.

Further, since we can treat the trigrams as numbers, we can also perform on them arithmetic operations such as addition and multiplication. To do this involves modular arithmetic, which for pedagogical purposes is occasionally called ‘clock arithmetic’. Its main feature is that it is cyclical: after arriving at the base number (‘mod n’, in our case: mod 2), we start up once again at zero. So in mod 2 arithmetic, 1 + 1 = 0: we only use the numbers 0 and 1. In the same way, a 12-hour clock only involves the numbers 1 to 12, and so is ‘mod 12’; hence, 15:00 is the same as 3:00, and so on. Therefore, the mod 2 addition of the I Ching’s trigrams can be represented by the following table:

logic-table1

Note that this is equivalent to the ‘⊻’ (exclusive or) operation in Boolean logic. (Boolean logic simply uses 0 for ‘false’ and 1 for ‘true’.) This logical point of view comes most in handy for defining multiplication, since binary multiplication is equivalent to the logical ‘∧’ (and) operation (Schöter, 1998: 6):

logic-table2

The advantage of logic over modular arithmetic is that we can define complements (¬). For example, Fire (☲/101) and Water (☵/010) are complementary, and so are Sky (☰/111) and Earth (☷/000). The use of logic is actually quite helpful in analyzing the trigrams’ associated meanings. Using the slightly different terminology of lattice theory (Schöter, 1998: 9):

  1. The Creative [乾/☰] is the union (⊻) of complements.
  2. The Joyous [兑/☱] is the union (⊻) of the Arousing [震/☳] and Abyss [坎/☵].
  3. Fire [火/☲] is the union (⊻) of the Arousing [震/☳] and Stillness [艮/☶].
  4. The Gentle [巽/☴] is the union (⊻) of the Abyss [坎/☵] and Stillness [艮/☶].
  5. Arousing [震/☳] is the intersection (∧) of the Joyous [兑/☱] and Fire [火/☲].
  6. Abyss [坎/☵] is the intersection (∧) of the Joyous [兑/☱] and Gentle [巽/☴].
  7. Stillness [艮/☶] is the intersection (∧) of Fire [火/☲] and the Gentle [巽/☴].
  8. The Receptive [坤/☷] is the intersection (∧) of complements.

In a beautiful essay, Goldenberg (1975) uses a branch of mathematics called group theory to unify the above points. A group is an algebraic structure with two operations (e.g. addition and multiplication). It turns out that the I Ching’s hexagrams satisfy many of the conditions for a group, which are as follows. 1) Closure: any operation between two hexagrams produces a new hexagram. 2) Associativity: in arithmetic operations, the order of the hexagrams does not matter, e.g. (☵ + ☴) + ☳ = ☵ + (☴ + ☳) = ☲. 3) Identity Element: there exists a hexagram (the identity element) such that an operation with it and any other hexagram produces that same hexagram, e.g. ☷ + ☱ = ☱, as well as ☰ × ☱ = ☱. 4) Inverse: for every hexagram, there exists another hexagram, such that an operation combining them produces the identity element; here, for the addition operation, every hexagram is its own inverse, e.g. ☶ + ☶ = ☷. Note, however, that there does not exist a multiplicative inverse. Further, addition and multiplication both satisfy the property that a ⋅ b = b ⋅ a, so that the hexagrams are commutative. So while the hexagrams’ lack of a multiplicative inverse precludes them from being a group, since they satisfy the remaining properties they are thus a ‘commutative ring’.

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“There is no economic world.”

There is no economic world. There is only an abstract economic description. It is wrong to think that the task of economics is to find out how the economy is. Economics concerns what we can say about the economy…

This thesis (adapted from Niels Bohr, the father of quantum theory[1]) is, to anyone not thoroughly debauched by philosophy, clearly nonsensical—the sort of postmodern tripe that embodies everything wrong with ‘theory’. Yet, it is quite the opposite. François Laruelle argues that any notion of ‘world’—as a priori/mnemotechnic cognitive mapping—is a product of philosophical thinking; in fact, he often uses the words ‘philosophy’ and ‘world’ interchangeably. Therefore, if the corpus of economics has a ‘world’, this implies that any worthwhile statements it makes are translatable into philosophy, which thus becomes privileged as a meta-discourse in relation to the ‘regional knowledge’ of economics. Such a role has been traditionally claimed by Marxism, as well as obliquely by disciplines such as psychoanalysis, whose proponents believe that they can have knowledge of the economy by imposing their concepts a priori upon whatever data is at hand (regardless of whether said theorist knows minutiae such as the difference between stocks and bonds…). To subvert this hierarchy—to argue that economics is properly non-philosophical, thus eliminating all grounds for the use of postmodern tripe—the thesis that ‘there is no economic world’ becomes essential. This paper presents a unified theory of economics and philosophy, arguing that economics consists of nonknowledge rather than knowledge (episteme/technē), that economics operates through unwriting or deconceptualizing the material of the other social sciences, and that economic models should not be viewed as attempts to represent the world, but as a radically non-Bayesian method of framing events in their contingency.

§1. World versus ‘World’

There is a famous story involving the British analytic philosopher A.J. Ayer and the French continental philosophers Georges Bataille and Georges Ambrosino, in a midnight conversation in January 1951 (Bataille, 2001: 111-3). Ayer introduced the simple proposition that “the sun existed before man,” which as a scientific realist he saw no reason to doubt. Ambrosino, a physician steeped in French phenomenology, insisted that “certainly the sun had not existed before the world.” Bataille, on the other hand, was agnostic. As he wrote afterwards (111):

This is a proposition that indicates the perfect non-sense that a reasonable proposition can assume. A common meaning must have a meaning within all meaning when one asserts any proposition that in principle implies a subject & an object. In the proposition: there was the sun and there were no humans, there is a subject without an object.

The easy way out of this dilemma (or as Bataille put it, this “abyss between French philosophers and English philosophers”) is to say that while Ayer was talking about the sun (as a well-defined scientific object composed of various elements, etc.), Ambrosino and Bataille were talking about ‘the sun’ (as ideal representation of the Real).[2] While Ambrosino had taken a purely idealist position, Bataille’s stance is much more interesting: he had, in fact, hit upon a problem that would later become known as the ‘arché-fossil’. This idea would be central to Quentin Meillassoux’s attempt to philosophize in a way that avoids what he calls ‘correlationism’—that is, the idea that “we only ever have access to the correlation between thinking and being, and never to either term considered apart from the other” (2008: 5), with ‘thinking’ and ‘being’ meant in the sense of ‘models’ and ‘objects’. In more visual terms, Meillassoux is searching for a way of doing philosophy that doesn’t just involve the imposition of a ‘grid’ of concepts (or ‘syntax’) upon the mass of data comprising the world—as has been the norm in philosophy since Kant’s Critique of Pure Reason.[3] An arché-fossil is any sort of scientific object or datum describing the state of the universe prior to the existence of subjects (e.g. humans) that could experience it—or, recalling the above anecdote: the arché-fossil describes the state of the world prior to ‘the world’. After introducing this concept, Meillassoux goes on to outline the ‘mechanics’ of why this idea is so immediately absurd to philosophers in the phenomenological tradition. The existence of ‘ancestral’ data implies (15):

  • that being is not co-extensive with manifestation, since events have occurred in the past which were not manifest to anyone;
  • that what is preceded in time the manifestation of what is;
  • that manifestation itself emerged in time and space, and that consequently manifestation is not the givenness of a world, but rather an intra-worldly occurrence;
  • that this event can, moreover, be dated;
  • that thought is in a position to think manifestation’s emergence in being, as well as a being or a time anterior to manifestation;
  • that the fossil-matter is the givenness in the present of a being that is anterior to givenness; that is to say, that an arché-fossil manifests an entity’s anteriority vis-à-vis manifestation.

DCP_0086 by Phillip Stearns

The notion of the arché-fossil underscores the tension between the world and ‘the world’. From the perspective of ‘the world’ there is either ‘world’ or ‘non-world’, whose boundary is set by the existence of an experiencing subject. Yet, by carbon-dating a meteorite (for example), it is possible to state that the ‘non-world’ and the world existed simultaneously (or: co-extensively), and moreover, that the evidence for this is given to us within ‘the world’. Philosophically, this is clearly unacceptable. Yet, it sheds some light upon an old Daoist koan:

庄子:“如果把天下就藏在天下,就不会被丢失,这是一般事物的通理”

“Hide the world in the world and the world will never be lost—this is the eternal truth.” ~Zhuangzi[4]

Zhuangzi is the same person who, upon waking up from a dream that he was a butterfly, wondered if he was actually a butterfly dreaming that he was a man. The anecdote is no doubt as popular as it is because of its stark opposition of ‘world’ (dream) and world (reality). A dream, after all, proceeds according to an internal logic where any sort of (arché-)hints that it is a dream, e.g. words on a page changing the second time you look at them, somehow don’t count. The most absurd events may occur in the most bizarre of settings, but any sense of contingency (the idea that it could be otherwise) is lost. If we take the lack of contingency in dreams as a principle, however, the very fact that Zhuangzi can ask whether he’s a butterfly or a man proves he isn’t dreaming! Zhuangzi’s query creates a false partition—with ‘dream’ and ‘non-dream’ as the only members of the state space—and is thus self-defeating: nonknowledge is in fact the most useful kind of knowledge he can have. So in order to avoid a performative contradiction, Zhuangzi must accept that the principle can’t be psychologically necessary. This gives rise to a fundamental contingency, where in order to make a convincing case that he is a butterfly, Zhuangzi has to argue that the current rules of psychology (and perhaps even of nature) would have to be able to be other than they are—the same position as Meillassoux!

For Meillassoux, this division of world and ‘world’ is the problem, and ought to be gotten rid of; Zhuangzi’s stance is similar, though his method eliminates this opposition in an entirely different way—which is the same as that of economics. Anyone accustomed to think in philosophical terms may be inclined, on reading the following sections, to suppose that the argument rests on a tacit assumption of this dyad. If such a supposition is found helpful, there is no harm in the reader’s adopting it as a temporary working hypothesis. In fact, however, no such division is made.

§2. Econo-fiction

To verify the claim ‘oil prices are manipulated by the USA’, a researcher could (in theory) physically go to each stage of the oil production/distribution process, from oil wells to spot or futures markets, to various nodes along logistical networks, to gas stations, etc. In the above claim, ‘oil price’ is well-defined as a variable; moreover, its role as subject of the sentence makes the former claim ‘economic’ in its genre. (Cf. the political statement ‘the USA manipulates oil prices’, with its focus on agency.) ‘USA’ is of course vague, but suffices for the problem at hand. The verb ‘to manipulate’ reifies (in this context), but is in principle observable. Our researcher could measure the ‘value added’ in each stage as it is expressed in price, then perform an (unavoidably qualitative) analysis of how fluctuations in the magnitude of this value-added (with respect to production costs, etc.) can be causally traced to the USA. In this context, economic methods would not per se be needed, only mercantile arithmetic. Economics is often thought of as simply an armchair version of our poor researcher’s task (implying that an ideal model is one that is just as complex as the real world). Yet, in the above statement economics acknowledges not the subject, verb, or object, but the preposition ‘by’: in a sort of econo-fiction, it shows the numerical properties that make ‘manipulation’ meaningful.

Economics can be defined as the science of non-discursive social relations, with a broad definition of ‘discourse’ such that one could equally say ‘non-conceptual’.[5] In fact, economics takes place through a process of deconceptualizing the findings of business, finance, and politics. As soon as you think you can understand an economic notion (e.g. an algebraic relation) intuitively and talk about it lucidly, economists develop a way to formalize it (via econometrics and so on) so as to make it entirely untranslatable into normal language. John von Neumann once remarked: “in mathematics you don’t understand things. You just get used to them.” This is exactly what Bohr was saying! By continually deconceptualizing its former results economics systematically prevents itself from creating a ‘world’. As in Roland Barthes’ famous formulation, the task of economics is to inexpress the expressible.[6]

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水仙

the curse of narcissus

水仙死的时候,田野里的百花充满了悲伤,乞讨河给他们几滴水,让他们为水仙哭泣。

河回答说:“假使每滴我的水都是泪水,也没有足够的泪为他哭泣。我爱了他。”

花儿们说:“怎么能不爱上他?他是那么俊美。”

河问:“ ……他俊美吗?”

花儿们问:“谁知道得比你更清楚?他每天都躺在你的岸边,你的水面映出了他的俊美。”

河潺潺地说:“我爱他,在他俯身我的时候,我看见了他的眼中我自己的美。”

奥斯卡·王尔德

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