The Zhou Puzzle: A Peek Into Quantification in Mohist Logic

1 Introduction

The chapter Xiaoqu 小取, translated as Lesser Pick or Minor Illustrations, is a central document in the Mohist literature. When considering Mohist logical thoughts or ideas on argumentation, this chapter must be taken seriously.^[1] Xiaoqu begins by discussing the importance of disputation with the following lines:

The purpose of disputation is to clarify the distinctions between “is-this” and “is-not”, to examine the principles of order and misrule, to discern the patterns of sameness and difference, to investigate the relationships between names and objects, to assess the beneficial and the harmful, and to resolve confusions and doubts.^[2]

夫辯者，將以明是非之分，審治亂之紀，明同異之處，察名實之理，處利害，決嫌疑 […]

After that, it presents the patterns of argumentation, such as pi 辟 (analogy or illustrating), mou 侔 (parallelizing), yuan 援 (adducing), and tui 推 (inferring). It then focuses particularly on the pattern of mou, discussing its various cases and providing an array of concrete examples for each.

Due to the lack of a theoretical framework for those examples in the original text, providing a robust theory to interpret them and explain the difference between the two major cases, “shi er ran” (is and so 是而然) and “shi er bu ran” (is and not so 是而不然), has been a core research issue for many decades. Researchers have proposed various theories.^[3] Sun provided a detailed review and discussion in his dissertation.^[4] In a recent article, Sun and Liu introduced yet another proposal, adopting the theory of monotonicity reasoning, which offers a unified explanation for why “shi er ran” constitutes valid reasoning, while “shi er bu ran” does not.^[5] It is known that the foundation of monotonicity reasoning lies in the quantification of a sentence; namely, quantifiers at different positions of a sentence determine whether we can perform upward or downward monotonicity inference. This underscores the critical importance of quantifiers.

This article will revisit the text and examine the issue of quantification in depth. Specifically, we will discuss the passage in the Xiaoqu that ends with “yi zhou er yi bu zhou” and contains many occurrences of zhou (周). We will focus on the precise meanings and usages of zhou. Due to the confusion that zhou has created, we refer to it as the “zhou puzzle.” We will study zhou from both linguistic and logical perspectives, while also considering previous research.

The main contributions of this article are as follows:

• By examining the usages of zhou in the Book of Mozi and its interpretations in Classical Chinese, we argue that when zhou is placed in front of a verb–object construction, it acts as a universal quantifier for the object.

• We argue that “yi zhou er yi bu zhou” as a concluding statement, along with the passage in which it appears, provides an explanation for the validity of “shi er ran” and the invalidity of “shi er bu ran.” In this context, zhou in the statement “yi zhou er yi bu zhou” is used as a summary term, referring to the discussions in that passage.

• Historically, zhou has been interpreted as “distribution” in the framework of Western traditional logic by earlier Chinese scholars. However, we will argue that the complexity of the Chinese text makes this interpretation inappropriate.

• We show that our understanding of zhou as a universal quantifier and you (have 有) as an existential quantifier lays a solid foundation for the monotonicity reasoning approach. This method is expected to extend to the study of other reasoning in Mohist logic.

The article is organized as follows: In Section 2, we introduce key texts and explain certain features of Classical Chinese, along with research questions related to these texts. Section 3 examines previous research approaches regarding “zhou,” particularly the theory of distribution, and argues that simply applying this theory to the Chinese text is inadequate. Section 4 focuses on the meaning and usage of “zhou,” advocating for its legitimacy as an (object) universal quantifier. Section 5 explores how this interpretation of quantifiers can be applied to the analysis of mou reasoning and connects the findings of this article with the earlier analysis.^[6] Section 6 summarizes the article and outlines the potential future directions.

2 The Texts and the Zhou Puzzle

In the Xiaoqu chapter, mou (parallelizing) is defined as an argument pattern in which propositions are put in parallel and they proceed.

侔也者，比辭而俱行。

mou ye zhe, bi ci er ju xing.

“Parallelizing” is comparing propositions and letting all “proceed.”^[7]

Clearly, the parallel structure is a crucial feature of mou reasoning. Taking two propositions as an example, the Mohists are well aware that simply placing them in parallel does not guarantee the validity of the transition from one to another, as explicitly supported by a quote from the Xiaoqu: “The parallelism of propositions is valid only as far as it reaches.”^[8] To illustrate different cases, a diverse range of examples is presented. The first case of the examples is referred to as shi er ran (is and so) 是而然:

(1) 白馬，馬也；乘白馬，乘馬也。

bai ma, ma ye; cheng bai ma, cheng ma ye.

white horse, horse; ride white horse, ride horse.^[9]

(2) 驪馬，馬也；乘驪馬，乘馬也。

li ma, ma ye; cheng li ma, cheng ma ye.

black horse, horse; ride black horse, ride horse.

To make the structure explicit, mou reasoning can be formalized as follows:

If A is B, then CA is CB,

where A and B are nouns, C is a verb. CA and CB are “verb–object” constructions.

The second case is referred to as shi er buran (is and not so) 是而不然, and we quote two examples for illustration:

(3) 其弟，美人也；愛弟，非愛美人也。

qi di, mei ren ye; ai di, fei ai mei ren ye.

her brother, handsome person; love brother, not love handsome person.

(4) (盜人，人也)^[10]; 殺盜人非殺人也。

dao ren, ren ye; sha dao ren fei sha ren ye.

robber, person; kill robber, not kill person.

Similarly, its formal structure is the following:

If A is B, then CA is not CB.

As we can see, Examples (1) and (2) are reasoning from a positive assertion to a positive one. For Examples (3) and (4), we obtain a negative assertion from a positive one. Naturally, we can ask the following questions: is shi er ran a valid case of mou reasoning? Specifically, can we infer “riding a white horse is riding horses” from “a white horse is a horse?” Also, how do we explain the examples of shi er bu ran?

Not surprisingly, these questions also puzzled the Mohists. In the Xi-aoqu, after presenting the aforementioned examples, the Mohists engaged in an extensive discussion. Here is the core text:

Loving people: to count as “loving people”, one must love all (zhou ai) people. Not loving people: to count as “not loving people”, one need not refrain from loving all (zhou bu ai) people; not loving all (people) implies not loving people. Riding horses [i.e. horse riding]: To count as “riding horses”, one need not ride all (zhou cheng) horses; there being a horse that one is riding implies that one is riding horses. And to get to count as “not riding horses”, one must refrain from riding all (zhou bu cheng) horses. This is a case of yi zhou er yi bu zhou.

愛人，待周愛人，而後為愛人。不愛人，不待周不愛人，不周愛，因為不愛人矣。乘馬，不待周乘馬，然後為乘馬也。有乘於馬，因為乘馬矣。逮至不乘馬，待周不乘馬，而後為不乘馬。此一周而一不周者也。

As we can see, the term zhou is prominently featured in this paragraph. The phrases “zhou ai ren 周愛人,” “bu zhou ai 不周愛,” and “zhou bu ai ren 周不愛人” are interpreted as descriptions of different situations, while “yi zhou er yi bu zhou” (此一周而一不周者也) serves as the concluding statement.

Finding the precise meaning of zhou, hence understanding the statement “yi zhou er yi bu zhou” is crucial to our comprehension. In fact, this question, the zhou puzzle, has been central to historians of Chinese logic in the twentieth century, and we will discuss some existing research on this topic in the next section.

In formal logic, quantifiers are essential for constructing precise and meaningful propositions. Two fundamental quantifiers are the universal and existential quantifiers, denoted by ∀ and ∃, respectively. The universal quantifier asserts that a statement applies to all elements within a specific domain, whereas the existential quantifier indicates the existence of at least one element for which the statement holds true. In addition, natural language features a variety of other quantifiers, such as most, many, at least 2, etc., which are analyzed within the framework of generalized quantifier theory.^[11] These quantifiers dictate the scope of statements, which in turn influence the logical structure and validity of arguments. Consider a simple syllogism:

All humans are mortal. Socrates is a human. Therefore, Socrates is mortal.

The validity of this inference hinges on the correct application of the universal quantifier “all.” Similar examples involving other quantifiers can be easily found in natural language and in our reasoning. In our research, we will adopt a logical and analytical methodology, employing the theory of quantification to provide a comprehensive explanation of mou examples.

We conclude this section by highlighting key features of Classical Chinese grammar that are relevant to our discussion, particularly for those unfamiliar with it.^[12]

First, the particle ye 也 that often appears at the end of a sentence serves as a marker of declarative assertion or predication.

Second, verb–object structures are pervasive in Classical Chinese. In the aforementioned examples, we have seen 乘馬 (ride horse) and 愛人 (love person), similar constructions abound in ancient texts.

Third, Classic Chinese lacks explicit singular or plural noun markers unlike English, where “horses” indicates plural and “a horse” singular. In Chinese, “馬 (ma)” serves both purposes. In addition, there are no definite and indefinite articles preceding nouns. Essentially, if we were to translate Example (1) into English, it could be “a white horse is a horse, riding a white horse is riding a horse”; or “white horses are horses, riding white horses is riding horses”; or “the white horse is a horse, riding the white horse is riding a horse.” Logically, the first two interpretations are identical, and the third requires contextual cues to specify the particular white horse. The point is that these nuances are not explicitly conveyed in the Classical Chinese language.

Finally, regarding singular or plural forms, Classical Chinese notably lacks explicit quantifiers. Referring back to the earlier examples, we observe the absence of quantifiers like “all” or “some” in front of the noun ma (horse). Reasoning in Classical Chinese often depends on contextual cues and implicit understanding rather than explicit quantification. This will be our focus and will be discussed further in the article.

3 Zhou: From Distribution to Quantification

Regarding the interpretation of “zhou,” scholars broadly hold two different views. Some scholars, including Hu Shi and Chen Menglin, believe that “zhou” in the Xiaoqu corresponds to the concept of “distribution” in traditional Western logic. “yi zhou er yi bu zhou” can be simply translated as “one is distributed, and the other is not,” which pertains to the distribution of subject–predicate terms in mou-style reasoning. Other scholars, such as Zhang Jialong, adopt modern logic, arguing that “zhou” can be interpreted as a universal quantifier, while “you” represents an existential quantifier. We will present these two perspectives and argue that the former approach encounters difficulties when applied to complex Chinese sentences. Furthermore, the divergence between these views reflects the distinct theoretical frameworks that scholars adopt when studying the text.

Before examining these standpoints in detail, let us first provide a brief review of the theory of distribution. According to Henrik Lagerlund,^[13] John Buridan (c. 1300–1361) is credited as the pioneer in introducing the theory of distribution for the study of syllogism in medieval times.

For example, in a categorical statement, the universal affirmative (all S are P) distributes its subject S but not its predicate P. A term is considered “distributed” if it refers to all members of the category it denotes. Conversely, a term is “undistributed” if it refers to only some members of the category. Consider the statement “All humans are mortal.” Here, the term “humans” is distributed, while “mortal” is not. Similarly, the particular affirmative (some S are P) distributes neither its subject nor its predicate. The universal negative (all S are not P) distributes both subject and predicate, whereas the particular negative (some S are not P) distributes its predicate but not its subject.

In what follows, we provide two examples to illustrate how scholars adopted theory of distribution to analyze the logical role of “zhou” and its function in mou reasoning.

Hu Shi is one of the pioneers who applied Western logical theory to the analysis of Chinese texts. According to him, the term “zhou” in the Xiaoqu is identical to the concept of “distribution.”^[14] He substantiates this claim using the verb–object phrase cheng ma (乘馬, ride horse), systematically recasting it as a categorical proposition: “That which is ridden is a horse.” To adhere to the Mohist interpretation of cheng ma, he then performs a subject–predicate inversion, yielding a particular affirmative proposition: “Some horses are ridden.” Through this logical pivot, Hu asserts that the Mohist term buzhou (不周) functions in the same way as the concept of “undistributed,” signifying that the term ma fails to denote every member of the domain of horses.

Chen Menglin also adopted the theory of distribution, but he rewrote verb–object phrases like ai ren (愛人, love person) or cheng ma (乘馬, ride horse) as statements such as “Some people are loving people” or “Some people are riding horses,” respectively.^[15] This way, the theory could be readily applied. He stated that ai ren and cheng ma, when used as predicates in particular affirmative statements, are undistributed. However, when explaining mou reasoning, ai ren had to be understood as “loving all people,” which led to a contradiction. He then argued that, due to the Mohist philosophy of universal love (兼愛, jian ai), ai ren should be treated as distributed. As one can see, in Chen’s work, verb–object phrases are treated as wholes, func tioning as predicates that are distributed. This already marks a departure from the traditional application of the theory of distribution to categorical statements. Modern logicians, such as Russell, have also criticized this point, as the verb in a verb–object construction is essentially relational.^[16] Unfortunately, Chen did not seem to have noticed this distinction. We will return to this point later.

One possible reason scholars particularly favor identifying distribution with zhou lies in the history of translation. In the early twentieth century, many logic books in foreign languages were translated into Chinese, introducing Chinese scholars to the theory of distribution among other logical theories. In these translations, the technical term “distributed” was rendered as “zhou yan” (周延) in Chinese. The term zhou yan already contains zhou and refers to the members of a term’s extension. Similarly, in the phrase zhou ai ren (loving all people), zhou functions as an adverb, indicating that the action ai (love) applies to every person in the domain of people. This resemblance is likely why many scholars interpret zhou in the Xiaoqu as synonymous with zhou yan.

Identifying “zhou” with “zhou yan” yields significant criticism. A representative critic is Zhang Binfeng, who argued that “zhou” essentially conveys the meaning of “all around (zhou bian 周遍),” functioning as an adverbial to indicate that an action extends to all objects. Therefore, “zhou” does not necessarily imply distribution in the subject or predicate of a categorical statement.^[17]

As we have seen, the grammatical complexity of examples in the Xiaoqu makes it inappropriate to strictly apply the theory of distribution in a simplistic manner when analyzing the logical function of “zhou.” This becomes clear through two key observations.

First, phrases such as cheng ma and ai ren are simple verb–object constructions that lack explicit subjects. This characteristic undermines the compatibility of Hu Shi’s transformation with the grammatical structure of the Mohist text, as his analysis assumes a level of syntactic regularity of categorical statements that is not present in these examples. In addition, the term “zhou” in the Mozi does not function as an adjective or determiner to directly modify nouns (for further discussion, see Section 4). This discrepancy challenges Hu Shi’s assertion that “zhou” signifies the entire extension of terms like ma, as his interpretation lacks robust textual grounding.

Second, statements such as “cheng bai ma is cheng ma” (“riding white horses” is “riding horses”) involve both the subject and predicate as verb–object phrases. These constructions logically express binary relations – such as “cheng” – and are better understood as relational propositions with an omitted subject. Strictly applying Chen’s approach to such phrases risks category errors, since labeling them as “distributed” introduces considerable ambiguity. For instance, interpreting “cheng ma” as “distributed” could imply multiple distinct meanings, including “riding all horses,” “riding every horse in every manner,” or even “everyone riding every horse in every manner.” This inherent ambiguity suggests the limitations of distribution theory within the traditional term logic framework when applied to ancient Chinese texts.

Together, these points indicate that both Hu and Chen recognized that “zhou” is related to quantification. They attempted to rewrite phrases or sentences in Chinese texts as categorical statements to apply the theory of distribution. However, it is clear that this theory is inadequate for addressing the complexities of the Mozi text.

As an alternative approach, Zhang Jialong offers an interpretation of “zhou” as a first-order universal quantifier, while “you” is seen as an existential quantifier. His paper demonstrates several valid inference patterns in the natural deduction system.^[18] This application of first-order logic provides a fresh perspective on the analysis of “zhou” within Mohist logic, particularly in explaining mou reasoning.

Numerous studies have examined mou reasoning through the lens of modern logic,^[19] many of which address the issue of “zhou.” While it is beyond the scope of this article to comment on every proposal, we would like to highlight a significant observation: most of these proposals effectively explain the “shi er ran” examples, but when addressing the “shi er bu ran” case, they often resort to different strategies, such as intensional approaches.

For instance, some scholars^[20] argue that the transition in Example (3) is blocked because “love” has different meanings in different contexts. This inconsistency motivates us to seek an explanation based on a unified theory.

4 Linguistic Interpretation of Zhou in the Mozi Text

To identify the precise meaning of zhou in the Xiaoqu, we will examine the usages of zhou in a slightly broader scope, specifically within the Mozi text, and consider its interpretation in other contexts.

In Xu Shen’s Shuowen Jiezi 說文解字,^[21] the character 周 (zhou) simply means “dense”; hence, “dense and thick” is considered to be the original meaning of zhou. From this, it extends to meanings such as “thorough” and “comprehensive.” The commentary by Duan Yucai explains: “zhou extends its meaning to thoroughness and comprehensiveness.”^[22]

In pre-Qin texts, when zhou is used as an adverb, it often denotes an action extending to all objects involved. For example, in the Book of Changes ·Xici: “Knowledge covers all things, and Dao (the Way) governs the world.”^[23] Here, zhou itself has the meaning of “extending to all things.”

In what follows, we will focus on the usages of zhou in the Book of Mozi. According to our statistics, the word “zhou” appears 68 times in the Book. Excluding instances where it does not relate to the meaning of “all,” such as in personal names, place names, textual errors, or phonetic loan characters, the remaining instances are shown in the following table.

In the aforementioned table, we have listed the various parts of speech of zhou in context, its meanings, and examples where zhou occurs. We will translate the key examples and usages in our discussions. Here are some findings:

• In Examples 2–5, zhou acts as an adverb, indicating that the action extends to all involved objects. In particular, Example 4 includes examples from the Xiaoqu chapter, such as “zhou ai ren” (love all people). These can be seen as (object) universal quantifiers.

• In Example 7 and 8, as an adverb, zhou modifies the action’s location, meaning “everywhere.” The meaning of “周鼓” (zhou gu) is “to beat drums everywhere,” similarly, “周置” (zhou zhi) means “to place them everywhere.” In Example 9, zhou is combined with the verb 垣 (yuan) and together they have a fixed meaning “to build a wall all around in a circle.”

• Interestingly, we have not found any expressions like “zhou white horses” (zhou bai ma 周白馬), where “zhou” directly modifies a noun in adjectival usage (or as a determiner).

• Finally, in Example 1, in “yi zhou er yi bu zhou,” zhou functions as a summary term, more specifically, it is used in a concluding statement to summarize the cases of “the quantification of the object term within verb-object constructs,” as discussed earlier in that passage.

Revisiting the Xiaoqu passage, the Mohists convey several key ideas about quantification in statements. When they say “ai ren” (love person), the intended meaning is “zhou ai ren” (love all people), where “zhou” acts as a universal quantifier, modifying “ren” in the verb–object phrase. Conversely, “cheng ma” (ride horse) does not imply the necessity to ride all horses. Instead, it translates to “you cheng yu ma” (there is a horse to ride), with “you” serving as an existential quantifier.

Furthermore, the passage clarifies the use of negation bu (不). It explains that “bu ai ren” (not loving people) does not necessitate “zhou bu ai ren” (not loving all people). However, from “bu zhou ai” (not loving all), one can directly infer “bu ai ren” (not loving people), illustrating the nuanced understanding of negation and quantification in Mohist logic.^[24]

What is worth noting is that the verb–object phrases have exactly the same structure, yet “ai ren” (love person) is interpreted universally, while “cheng ma” (ride horse) is interpreted existentially. In this case, we agree with Chen Menglin’s explanation that this distinction is related to the Mohist philosophy of “universal love,” which posits that one should love everyone unconditionally. This contrasts with the Confucian stance. Although there is no explicit explanation for cheng ma, it is reasonable to assume that it does not mean “I ride all horses” but rather implies “I ride horses.” This likely reflects a convention in ordinary language.

However, in this passage, the use of “zhou” in “yi zhou er yi bu zhou” warrants special attention. Unlike its typical placement before a verb–object phrase, “zhou” appears after the numeral “one” (yi 一). The phrase “yi zhou er yi bu zhou” has a contrastive structure, indicating that one case involves “zhou,” while the other does not. When we integrate the previous explanations of “zhou” within the same passage, the sentence becomes clear. It suggests that in the context of “ai ren” (love person), it is “zhou ai ren” (love all persons), whereas in the context of “cheng ma” (ride horse), it is not “zhou cheng ma” (ride all horses). Here, in the concluding statement “yi zhou er yi bu zhou,” “zhou” functions as a summary term, referring back to the previous discussions.

To conclude, by examining the usage of zhou in the Book of Mozi, we have identified its precise meaning in the Xiaoqu chapter. When placed in front of a verb–object (VN) structure, zhou acts as an object universal quantifier. Through the explanations of “ai ren” (love person) and “cheng ma” (ride horse), we understand that the former means “zhou ai ren” (love all people), while the latter means “you cheng yu ma” (there is a horse to ride). This also clarifies the meaning of the statement “yi zhou er yi bu zhou.” This insight turns out to be crucial for understanding the mou reasoning pattern.

5 Zhou in Mou Reasoning

Building on our interpretation of zhou from the previous section, we can now provide a unified explanation for both the “shi er ran” and “shi er bu ran” examples. The framework we will employ is natural logic and monotonicity.^[25] Natural Logic is a branch of logic and semantics that aims to model reasoning in natural language more closely than traditional formal logics like first-order logic. It emphasizes the intuitive and inherent logical structures found within natural language expressions, aiming to bridge the gap between formal logic and linguistic semantics.

The prevalence of verb–object structures in Classical Chinese aligns well with developments in natural logic, where more complex sentence structures are managed through an extended framework known as relational syllogism.^[26] This approach allows us to handle the intricacies of natural language reasoning as presented in the Mozi text, providing a comprehensive analysis.

To illustrate, we will first review two rules in natural logic that we will use in our analysis, namely, the principles of right-upward monotonicity and right-downward monotonicity. The former allows us to make inferences from a specific proposition to a general proposition, while the latter allows us to infer from a general proposition to a specific one. Consider the example:

If all boys eat some apples, then all boys eat some fruit.

The inference is made from “apple” to “fruit,” from specific to general (upward). The inference can be formalized as the following:

if Q ₁ N ₁ V Q ₂ N ₂ and N ₂ ⊆ N 2 ′ , then Q ₁ N ₁ V Q ₂ N 2 ′ ,

where Q ₁ and Q ₂ are quantifiers, N ₁ and N ₂ are nouns, and V is a verb.

This inference is valid if the subject quantifier Q ₁ is a positive quantifier (some or all) and the object quantifier Q ₂ is some.

Similarly, the following example is an instance of the right-downward monotonicity principle:

If all boys eat all fruit, then all boys eat all apples.

The inference is valid if the subject quantifier Q ₁ is positive (some or all) and the object quantifier Q ₂ is all.

Our next task is to apply the monotonicity rules to the examples of mou, for both “shi er ran” and “shi er bu ran.” Here, we provide an quick illustration, with an emphasis of the role of zhou in mou reasoning.^[27]

Let us analyze the examples step by step, starting with the subject X:

White horse is horse,

(X) rides white horse,

(X) rides horse.

Once we put the subject X back, the verb–object phrase transforms into a complete sentence with a full subject–verb–object structure. For each position of a subject and object, there are possible quantifiers we can apply. According to our interpretation of “cheng ma” (ride horse), we should place “some” in front of “ma” (horse), namely:

White horse is horse,

(X) rides some white horse,

(X) rides some horse.

For the quantifier in front of X, it does not matter in this case; we could use either “all” or “some,” as long as they are positive. If the quantifier in front of X is “none,” it would not work, but “none” seems much less probable in this context. For convenience, we place “some” in front of X. The example then becomes:

White horse is horse.

(Some X) ride some white horse,

(Some X) ride some horse.

Making the English grammatically correct, the example is now:

A white horse is a horse.

(Some X) ride some white horses, Therefore,

(Some X) rides some horses.

According to the principle of right-upward monotonicity, we can infer the third sentence as a conclusion from the first two sentences. Note that this interpretation is slightly different from treating “cheng bai ma cheng ma ye” (riding a white horse is riding white horses) as one sentence. By splitting it into two parts, each forming a complete sentence, we can see the structure of well-formed monotonicity reasoning.

A similar approach can be applied to the “ai ren” example. By placing a universal quantifier in front of the object “ren,” it becomes the following:

His brothers are beautiful men,

(Some X) love all his brothers.

Therefore, (Some X) love all beautiful men.

Since the quantifier is “all,” it is the right-downward monotonicity rule that should apply. However, we cannot infer from “brothers” to “men.” That is the reason why this inference is not valid; hence, it belongs to “shi er bu ran.” As shown earlier, by using monotonicity rules, we are able to provide a unified explanation for the examples of the two cases of mou reasoning.^[28]

6 Conclusions and Future Directions

The concept of zhou within Mohist logic has been the subject of intense scholarly debate, with extensive existing research. Through linguistic analysis, we discerned two interrelated usages of zhou: one as an adverb that determines the quantification of the object term within verb–object constructs, the other as a summary term in “yi zhou er yi bu zhou.” These insights help us identify the quantifiers in a sentence and establish a solid foundation for the logical analysis of mou. By applying natural logic and monotonicity rules, we have successfully offered a unified analysis of the examples present in mou reasoning. Crucially, we have remained faithful to natural language reasoning.

For future work, in this article, we only considered zhou, but there are more quantifiers in the Mohist text. For instance, jian 兼, jin 尽, and ju 俱 are also considered universal quantifiers. In other texts, additional quantifiers, such as fan 凡, jie 皆, and fan 泛, are being used.^[29] It would be interesting to study those and compare them. In particular, in Harbsmeier’s work, he distinguished two sorts of quantifiers depending on the objects they modify: subject universal quantifiers and object universal quantifiers. He regretfully commented that this distinction has not received much attention. Our research might be able to contribute more to this issue. In addition, we would like to look into the existential quantifiers, including huo 或, too.

In medieval logic, there are extensive discussions about descending and ascending inferences, see the work of Copenhaver, Normore, and Parsons,^[30] which have clear connections to monotonicity reasoning. Parsons has already studied these concepts in his book.^[31] However, we aim to further investigate these issues and compare them with the corresponding notions in classical Chinese logic. By examining the similarities and differences between these two traditions, we hope to uncover deeper insights into the nature of logical reasoning and its historical development. Specifically, we will explore how medieval scholars approached the concepts of ascending and descending inferences and contrast this with the ways classical Chinese logicians, such as the Mohists, dealt with related logical phenomena. This comparison will help us understand the historical development of logic, and it may also shed light on the unique contributions of each tradition to the broader field of logical theory.