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## Chinese Logic: An Introduction

###### [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.

#### 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 *bā* (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.

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:

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):

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):

- The Creative [乾/☰] is the union (⊻) of complements.
- The Joyous [兑/☱] is the union (⊻) of the Arousing [震/☳] and Abyss [坎/☵].
- Fire [火/☲] is the union (⊻) of the Arousing [震/☳] and Stillness [艮/☶].
- The Gentle [巽/☴] is the union (⊻) of the Abyss [坎/☵] and Stillness [艮/☶].
- Arousing [震/☳] is the intersection (∧) of the Joyous [兑/☱] and Fire [火/☲].
- Abyss [坎/☵] is the intersection (∧) of the Joyous [兑/☱] and Gentle [巽/☴].
- Stillness [艮/☶] is the intersection (∧) of Fire [火/☲] and the Gentle [巽/☴].
- 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’.