243
Chinese Research on Mathematical Logic  
and the Foundations of Mathematics
WANG Hongguang*
DU Guoping**
Abstract 
This paper outlines the Chinese research on mathematical logic and the foundations of 
mathematics. Firstly, it presents the introduction and spread of mathematical logic in 
China, especially the teaching and translation of mathematical logic initiated by Bertrand 
Russell’s lectures in the country. Secondly, it outlines the Chinese research on mathemat-
ical logic after the founding of the People’s Republic of China. The research in this period 
experienced a short revival under the criticism of the Soviet Union, explorations under the 
heavy influence of the Cultural Revolution, and the vigorous development of mathemati-
cal logic teaching and research after the period of “Reform and Opening Up” that started 
in the late 1970s, and the full integration of Chinese mathematical logic research into the 
international academic circle in the new century after 2000. In the third part, it focuses on 
the unique and original results of the Chinese mathematical logic research teams from the 
following three aspects: medium logic, lattice implication algebras and their lattice-val-
ued systems of logic, and Chinese notation of logical constants, which can be used as a 
substantive supplement to the relevant literature on the history of mathematical logic in 
China. The last part is a reflection on the shortcomings of contemporary Chinese research 
on mathematical logic and the foundations of mathematics. 
Keywords: Chinese logical research, mathematical logic, medium logic, lattice implica-
tion algebra, Chinese notation
Kitajske raziskave na področju matematične logike in osnov matematike
Izvleček
Članek povzema kitajske raziskave na področju matematične logike in osnov matematike. 
V uvodnem delu ponuja vpogled v predstavitev in širjenje matematične logike na Kitaj-
skem. Še posebej pa se posveča poučevanju in prevajanju matematične logike, ki so ju 
* WANG Hongguang, Postdoctoral researcher, Department of Philosophy, 
Peking University, Beijing, China.
 Email address: hgwang@pku.edu.cn
** DU Guoping, Institute of Philosophy, Chinese Academy of Social Science 
(CAS), Beijing, China; School of Philosophy, Sichuan Normal University, 
Chengdu, China. 
 Email address: dgpnju@163.com
DOI: 10.4312/as.2022.10.2.243-266
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244 WANG Hongguang, DU Guoping: Chinese Research on Mathematical Logic...
spodbudila predavanja Bertranda Russella na Kitajskem. Nadalje povzema raziskave ma-
tematične logike po ustanovitvi Ljudske republike Kitajske. Področje je v tem času doži-
velo hitri razcvet pod vplivom kritik matematične logike v Sovjetski zvezi, temu je sledilo 
raziskovanje, na katerega je močno vplivala kulturna revolucija, živahen razvoj poučevanja 
in preučevanja pa sta se začela konec 70. let po obdobju »reform in odpiranja svetu« in se 
po letu 2000 nadaljevala v popolno vključitev kitajskih raziskav na področju matematič-
ne logike v mednarodne akademske kroge. Tretji del članka se osredotoča na edinstvene 
in izvirne dosežke kitajskih skupin, ki se posvečajo raziskavam matematične logike, in 
sicer s stališč: medialne logike, mrežne implikacijske algebre in njihovih mrežno-vredno-
stnih sistemov logike ter kitajske notacije logičnih konstant, ki jo lahko uporabljamo kot 
konkretno dopolnilo k relevantni literaturi o zgodovini matematične logike na Kitajskem. 
Zadnji del članka predstavlja razmislek o pomanjkljivostih v sodobnih kitajskih raziska-
vah matematične logike in osnov matematike.
Ključne besede: kitajske logične raziskave, matematična logika, medialna logika, algebra 
mrežne implikacije, kitajska notacija 
Introduction
The present paper is an overview of the Chinese research on mathematical logic 
and the foundations of mathematics. It mainly consists of four parts: The first 
part briefly presents the introduction and spread of mathematical logic in China, 
especially the teaching and translation of mathematical logic during the Republic 
of China (ROC) period initiated by Bertrand Russell’s lectures in the country. 
The second part is an outline of Chinese research on mathematical logic after the 
founding of People’s Republic of China (PRC). The research in this period had 
experienced a short revival under the criticism of the Soviet Union, explorations 
under the heavy influence of the Cultural Revolution (Wenhua da geming 文化大
革命), and the vigorous development of mathematical logic teaching and research 
after the “Reform and Opening Up” (Gaige Kaifang 改革开放) that began in 
the late 1970s, and the full integration of Chinese mathematical logic research 
into the international academic circle in the new century after 2000. The third 
part focuses on the unique and original results of the Chinese mathematical logic 
research teams from the following three aspects: medium logic, lattice implication 
algebras and their lattice-valued systems of logic, and the Chinese notation of 
logical constants. The last part is a reflection on the shortcomings of contempo-
rary Chinese research on mathematical logic and the foundations of mathematics. 
There have been several earlier works on the history of mathematical logic in Chi-
na. Lin Xiashui 林夏水 and Zhang Shangshui 张尚水 (1983) and Song Wenjian 
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245Asian Studies X (XXVI), 2 (2022), pp. 243–266
宋文坚 (2000) all outline the contributions of Jin Yuelin 金岳霖and others to 
the introduction of mathematical logic into China since the 1920s, as well as the 
work of Chinese scholars in the fields of logical calculus, set theory, recursion the-
ory, modal theory and proof theory. Chen Bo 陈波 (2019) gives a comprehensive 
introduction to the international publications and the main research progress of 
Chinese logic scholars in the fields of the history of logic (especially the history of 
Chinese logic), inductive and probabilistic logic, natural language logic, philoso-
phy of logic, informal logic and critical thinking, legal logic, and so on, the estab-
lishment and rapid development of logic research institutions in the PRC is also 
introduced. Su Rina’s 苏日娜 doctoral dissertation (2020), “History of Mathe-
matical Logic in China (1920–1966)” (Shuli luoji zai Zhongguo de fazhanshi yanjiu 
(1920-1966) 数理逻辑在中国的发展史研究 (1920–1966)) presents the intro-
duction of mathematical logic by Chinese scholars in the first half of the 20th 
century, reviews and summarizes the history and characteristics of mathematical 
logic during its initial foundation in China (1920–1949) and during the founding 
and development of the “new China” (1949–1966). Jan Vrhovski’s paper (2021a) 
examines the work of Jin Yuelin and others in the Department of Philosophy of 
Tsinghua University, and the characteristics and main progress of the teaching 
and research of mathematical logic in this context from 1926 to 1945 are summa-
rized. Du Guoping and Wang Hongguang (2020) also provide an overview of the 
introduction and research on logic in mainland China from the ROC to PRC, in 
particular, the achievements of contemporary Chinese mainland scholars in the 
fields of mathematical logic (modal logic, recursion theory, set theory, formalized 
methods and automatic reasoning, etc.) as well as philosophical logic (modal log-
ic, many-valued logic, lattice-valued logic based on lattice implication algebras, 
paraconsistent logic, etc.). 
This paper mainly focuses on the research in the period after the founding of 
PRC, especially the original work introduced in the third part, which can be used 
as a substantive supplement to the above-mentioned relevant literature. 
The Beginning of Mathematical Logic in China
The science of logic was first introduced to China in the late Ming dynasty (early 
17th century). Among the earliest works was the book Mingli tan 名理探 (De 
logica) translated by Li Zhizao 李之藻 and Francisco Furtado. This book was a 
translation of teaching materials on logic used by the members of the Jesuit order 
at Coimbra University in Portugal. Its original title was In Universam Dialecticam 
Aristotelis Stagiritae. By the early 20th century, scholars like Yan Fu 严复, Wang 
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246 WANG Hongguang, DU Guoping: Chinese Research on Mathematical Logic...
Guowei 王国维, Hu Maoru 胡茂如 and others, one after another, translated the 
most important works on Western logic into Chinese. 
In 1920, the renowned British scholar Bertrand Russell was invited to lecture in 
China for one year, in the framework of which he carried out five major series of 
lectures at Peking University, one of which was a series of lectures on “Mathemat-
ical Logic” (Shuli luoji 数理逻辑). Subsequently, in October 1921, the anthology 
Five Great Lectures by Bertrand Russell (Luosu wu da yanjiang 罗素五大演讲) was 
published by the New Knowledge Publishing House of Peking University, a pub-
lication which marked the start of dissemination of mathematical logic in China. 
In 1922, Fu Zhongsun 傅种孙 and Zhang Bangming 张邦铭 translated and 
published the book Introduction to Mathematical Philosophy (Luosu suanli  zhexue 
罗素算理哲学) by Russell. Later, some scholars gradually introduced mathemat-
ical logic into Chinese academic world: Zhang Shenfu 张申府 introduced the 
notion of mathematical logic related to Russell’s philosophy1 and his Principia 
Mathematica, Tang Zaozhen 汤璪真 as a mathematician introduced set theory 
in the context of mathematics, and Zhu Gongjin 朱公瑾 introduced Hilbert’s 
conception of “symbolic logic”. In 1926, Jin Yuelin started a course on logic at 
Tsinghua University in Beijing, in the framework of which he also taught con-
tent related to mathematical logic. One year later, in 1927, Wang Dianji 汪奠基 
published a book entitled Logic and Mathematical Logic (Luoji he shuxue luoji lun 
逻辑和数学逻辑论), which is the first monograph as a systematic introduction 
on mathematical logic and its history in China. In 1935, Tsinghua University 
published Jin Yuelin’s textbook Logic (Luoji 逻辑), in which he provided an over-
view of Russell’s systems of mathematical logic. Xiao Wencan 肖文灿 published 
a series of articles on set theory during 1933–1934, which were later collected 
and published in a volume entitled A Primer on Set Theory (Jihelun chubu 集合论
初步) by The Commercial Press in 1939. From the early 20th century on, Chi-
nese scholars like Yu Dawei 俞大维, Shen Youding 沈有鼎, Wang Xianjun 王
宪钧, Hu Shihua 胡世华, Mo Shaokui 莫绍揆 (Moh Shaw-kwei) and others 
in succession travelled abroad to study mathematical logic at foreign universities, 
and later also returned to China. In this way, mathematical logic in China under-
went a gradual development (also see Lin and Zhang 1983 for the development 
of mathematical logic in the Republican period).
Because research on the foundations of mathematics and mathematical logic is 
inextricably linked, in the following discussion it will be referred to research on 
mathematical logic. 
1 For a detailed analysis of Zhang’s critical introduction of Russell’s logic, see Vrhovski (2021b, 
229ff ).
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247Asian Studies X (XXVI), 2 (2022), pp. 243–266
An Outline of Chinese Studies of Mathematical Logic
Mathematical Logic in the Foundation Period of the People’s Republic of 
China (PRC)
In the time of foundation of the PRC, Chinese research on mathematical logic 
was influenced by the Soviet criticism of mathematical logic. However, at a dinner 
which took place in 1956, Mao Zedong told Jin Yuelin that mathematical logic 
was important, and needed to be taken care of. After it received this support from 
Mao, Chinese research into mathematical logic gradually became more active. 
In this early period Shen Youding published two important articles, in 1953 and 
1955, in which he investigated the paradox of the class of all grounded classes and 
semantical paradoxes, respectively. In 1950, Mo Shaokui constructed two new 
logical systems that could effectively prevent “paradoxes of implication”. In 1954, 
Mo proved that in a many-valued system of logic, if we do not apply any restric-
tions on the use of principle of comprehension, by the same token we can also 
construct a theory of paradox analogous to that existing in two-valued logic. In 
1957, after a one-year long campaign by the members of the IMCAS (Institute 
of Mathematics at Chinese Academy of Science), mathematical logic returned to 
Chinese universities. Hu Shihua conducted valuable research on recursive func-
tions and recursive structures in the field of recursion theory around 1960s. He 
defined a kind of kernel function class in a very concise but powerful way and 
used it to construct a universal algorithm for normal algorithms and universal 
computers, like a Turing machine. He also extended the theory of recursive func-
tions on natural number sets to sets of formulas, and established a computability 
theory, that is, the theory of recursive algorithms (Hu 1960a; 1960b; Hu and Lu 
1960). In 1963, the 3rd National Experience-Sharing Conference on Computer 
Technology was convened in Xi’an. At the conference a special group for mathe-
matical logic was organized, which was presided over by Hu Shihua. This was the 
first nationwide conference on mathematical logic held in China. Contributions 
submitted and presented at the conference involved topics such as many-valued 
logics, theory of algorithms, proof theory, the foundations of mathematics and the 
theory of automatization.
The Period of Twists and Turns (1966–1976)
Although in the ten years of the Cultural Revolution (Wenhua da geming 文化大
革命) Chinese research on mathematical logic was greatly influenced by the relat-
ed political developments, many Chinese researchers on mathematical logic still 
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managed to work despite the adversities of the time and carry on with their stud-
ies and research in the field. Subsequently, in the year 1972, the American-Chi-
nese logician Wang Hao returned to China and was received by Prime Minister 
Zhou Enlai. After that, Wang kept returning to China for several more times 
to present scientific reports to the Chinese scientific community. His visits and 
lectures brought a fresh wind into the Chinese academic world of mathematical 
logic, which gave an enormous boost to scholars in the field. However, under the 
heavy influence of the contemporary political circumstances, in this period Chi-
nese scholars’ research achievements in the field of mathematical logic were still 
rather limited. 
The Last Quarter of the 20th Century (1977–1999)
After the conclusion of the Cultural Revolution, Chinese scientific research grad-
ually returned to normal. In 1977, Wang Hao returned to China once again to 
deliver a series of six scientific reports which were later collated and published 
under the title Popular Lectures on Mathematical Logic (Shuli luoji tongsu jianghua 
数理逻辑通俗讲话; Wang 1981). These lectures were of enormous help to Chi-
nese scholars by enabling them to obtain a timely understanding of the current 
developmental situation in the international research on mathematical logic.
It was especially after China’s 1978 reforms and opening up to the world, when 
the long-suppressed research enthusiasm of Chinese intellectuals experienced an 
unprecedented growth, and Chinese science finally obtained a series of significant 
results in mathematical logic. Thus, for instance, in the two years of 1979 and 
1980 alone, Mo Shaokui published six scientific articles related to set theory and 
the theory of recursion, Zhang Jinwen 张锦文 published seven scientific articles 
about axiomatic set theory of weak predicate calculus and non-standard analysis, 
and Hong Jiawei 洪家威 published two articles on computational complexity. 
In the same period, some high-quality articles were even published in renowned 
international scientific periodicals. These included works covering Luo Libo’s 
罗里波 (also known as Lo Libo) achievements in model theory and decidabil-
ity of free groups, published in the prestigious Western periodical The Journal of 
Symbolic Logic (Lo 1983a; 1983b), as well as Hong Jiawei’s results on the theory 
of computational complexity, published in various international journals (Hong 
1982a; 1982b; 1984).
Following 1978, Chinese universities and research institutes started recruiting 
graduate students in mathematical logic. These developments caused Chinese 
mathematical logic to enter a stage of overall and comprehensive development, 
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249Asian Studies X (XXVI), 2 (2022), pp. 243–266
which saw the emergence of an uninterrupted series of high-standard research 
achievements in the field. These included works like Wang Shiqiang’s 王世强 
article on the elementary concepts, methodology and theorems of lattice-valued 
model theory (Wang 1986); Feng Qi’s 冯琦 study on the hierarchy of Ramsey 
cardinals (Feng 1990); Yi Bo’s 伊波 and Xu Jiafu’s 徐家福 article on analogy 
calculus (Yi and Xu 1993); Li Wei’s 李未 theory of the limits of formal theory 
of sequencing—open logic (Li 1993); and Ying Mingsheng’s 应明生 article on a 
logical system for approximate reasoning (Ying 1994). 
The New Century (2000—)
Since the begin of the new century, China’s community of independently edu-
cated as well as foreign-educated researchers in the field of mathematical logic 
has been constantly expanding. At present, Chinese scholars who are engaged 
in research of mathematical logic have already become an internationally influ-
ential and significant group of researchers, while Chinese research in the field 
has been completely integrated into the international developments in mathe-
matical logic. 
At the same time, a great number of Chinese scholars, such as Ding Decheng 
丁德成, Feng Qi and others, have been actively engaged in the international cir-
cles of mathematical logicians. More specifically, in the recent years Ying Ming-
sheng’s monograph Topology in Process Calculus: Approximate Correctness and In-
finite Evolution of Concurrent Programs has been published by the Springer pub-
lishing house. Moreover, the Chinese logician Zhang Yi 张羿 assumed the role 
of the editor-in-chief of the international journal Logic and Algebra, while Zhao 
Xishun 赵希顺 was named a member of the editorial board of the international 
Journal of Satisf iability, Modeling and Computation. 
Apart from this, in the last two decades a great number of international confer-
ences related to mathematical logic were initiated and convened in China, such as 
a conference on the theory and application of models of computation, and a 2008 
conference on computability, complexity and randomness. 
At the same time, China’s achievements in the international academic world of 
mathematical logic are also expanding, to a degree that it’s impossible to offer a 
complete listing of these results here. 
At present, there already exist close to 100 different Chinese textbooks on mathe-
matical logic, while at the same time a considerable amount of the latest teaching 
materials on mathematical logic from the rest of the world is being continuously 
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translated into Chinese. Courses on mathematical logic are offered at departments 
of philosophy and computer science at many Chinese universities, while at sev-
eral comprehensive universities they also educate master’s and doctoral students 
specialized in the field of mathematical logic. At relevant universities or research 
institutes undergraduate students are generally taught propositional calculus and 
first-order predicate calculus, while at the level of graduate studies they are taught 
set theory and modal logic. Finally at the level of doctoral studies they are taught 
subjects such as model theory, proof theory, theory of recursion and so on. Quite 
a few research institutions even make direct use of well-established foreign text-
books on mathematical logic.
A Few Comparatively Central Topics 
Chinese research into mathematical logic consists of a few academic groups which 
have, focusing on a specific topic in one or the other domains of study, created 
comparatively central original achievements, rich in distinguishing features and 
qualities. These are summarized as follows in the following subsections.
Medium Logic
Chinese studies on medium logic were established in the 1980s as a result of the 
long-term cooperation between Zhu Wujia 朱梧槚 and Xiao Xi‘an 肖奚安 (Zhu 
and Xiao 1984), while in the last four decades a group of young and middle-aged 
scholars engaged in research on the topic. Nowadays, medium logic has already 
evolved into a theory of logic all aspects of which, from its theory to application, 
are extremely rich in content.
The fundamental idea which gave rise to the establishment of medium logic 
was the so-called “intermediate principle”. From Aristotle onwards, a distinction 
has been made between intermediate opposite opposition and non-intermediate 
contradictory opposition. The principle of non-intermediacy posits that all an-
tinomies are non-intermediate contradictory oppositions, while the principle of 
intermediacy maintains that not all antinomies are non-intermediate contradic-
tory oppositions. The principle of intermediacy recognizes that under certain cir-
cumstances there exists the state of “both A and B”. Its philosophical basis rest on 
the intermediary state of transition that abounds in the process of transformation 
between two sides of an antinomy; its real basis, on the other hand, is in the inter-
mediary states of various kinds of objective existence, such as, for example, dusk, 
which represents the intermediate state in the change of daytime into night-time, 
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or the condition of being middle-aged, which is the intermediary stage between 
youth and old age, or semiconductors, which represent the intermediaries be-
tween conductors and isolators. 
In medium logic we use P and ╕P to express contrary antithetical notions, while 
the symbol ～ is used to designate a fuzzy negator, which is to be interpreted and 
read as “partially”. If an object x satisfies ～P(x)∧～╕P(x), i.e. that it partially pos-
sesses the property P and at the same time also possesses the property ╕P, then x 
is referred to as the intermediary object of opposite antinomy (P, ╕P). 
The system of medium propositional logic MP consists of two single-variable 
conjunctions: ╕(opposite negator), ～ (intermediary); and one binary conjunction 
→ (implication). It further defines the single variable conjunction: ﹁A=df A→～A. 
The inference rules of MP consist of:
(∈)  A1, A2, …, An├Ai (i=1, 2, …, n);
(τ)  If Γ├Δ, Δ├A, then Γ├ A;
(﹁)  If Γ, ﹁A├B, Γ, ﹁A├﹁B，then Γ├A;
(→-)  A→B, A├B，A→B, ～A├B;
(→+)  If Γ, A├B，and Γ, ～A├B, then Γ├A→B;
(Y)  A├┤﹁╕A, ﹁～A;
(Y～)  ～A├┤﹁╕A, ﹁A;
(Y╕)  ╕A├┤﹁A, ﹁～A;
(╕╕+)  A├ ╕╕A;
(╕╕-)  ╕╕A├ A;
(╕→)  A, ╕B├┤╕(A→B);
(～～) A→A├┤～～A.
The system of medium propositional logic MP* represents the system of medium 
propositional logic MP expanded by the binary connective “≺”, which is called 
a “truth degree operator” and read as “the degree of truth-value is not stronger 
than”. In addition, it is also enlarged by the following three inference rules:
(≺) A≺B├┤(A→B)∨(～A∧～B)
(～≺) ～(A≺B)├┤(～A∧╕B)∨(A∧～B)
(╕≺) ╕(A≺B)├┤A∧╕B
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The elementary semantic of conjunctions in the system of medium propositional 
logic MP* is:
Apart from the above-described system of medium propositional logic MP*, the 
system of medium logical calculus ML also includes the system of medium propo-
sitional logic MP, the systems of medium predicate logic MF and MF*, systems of 
medium predicate logic with identity ME and ME* (Xiao and Zhu 1985a–1985e; 
Zhu and Xiao 1985a; 1985b). 
Today, medium logic has already developed into a very broad field of research, 
which consists mainly of the following already established and advanced research 
contents or directions: 
(1) Medium system of algebra (Wu and Pan 1990);
(2) Medium system of modal logic (Zhang and Zhu 1995, etc.);
(3) Medium system of axiomatic set theory (Zhu and Xiao 1988, etc.);
(4) Medium proof theory (Zou 1988);
(5) Medium theory of forcing (Zhu et al. 1996);
(6) Medium system of reasoning with incomplete information (Deng 
1994);
(7) Medium programming language MILL and its interpretation sys-
tem (Song and Zhu 1994);
(8) Medium systems of theory and practice of automatic reasoning 
(Zhang and Zhu 1994a–1994c, etc.); 
(9) Numeralization of medium truth-degree operators and their appli-
cations in computers (Hong et al. 2006; Hong et al. 2007).
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Lattice Implication Algebras and Their Lattice-Valued Systems of Logic
Nonclassical logic constitutes one of the foundations of artificial intelligence. The 
lattice-valued system of logic based on lattice implication algebra is a kind of 
nonclassical logic. Since 1993, Xu Yang 徐扬 and other Chinese logicians have 
been conducting research on lattice implication algebras, lattice-valued systems of 
logic based on lattice implication algebras, and imprecise inference and automatic 
inference based on these systems of logic. 
Lattice Implication Algebra
To set up a new system of logic, Xu Yang proposed a lattice implication algebra 
(Xu 1993), which represented a kind of nonclassical logical algebra combining 
lattices and implication algebra. 
Definition 1. Lattice implication algebra is an algebraic system ℒ = (L, ∨, ∧,' , →, 
O, I ), where
(1) (L, ∨, ∧, O, I ) is a bounded lattice, while O and I represent its least and 
greatest elements, respectively;
(2) ' : L → L is an inverted order involutory mapping;
(3) →: L × L → L is a binary operation and for any x, y, z ∈ L, there exist
① x → (y → z) = y → (x → z);
② x → x = I;
③ x → y = y' → x' ;
④ If x → y = y → x = I，then x = y;
⑤ (x → y) → y = ( y → x) → x;
⑥ (x ∨ y) → z = (x → z) ∧ ( y → z);
⑦ (x ∧ y) → z = (x → z) ∨ ( y → z).
All lattice implication algebras make up a proper class, which possesses many 
favourable properties, such as: 
(1) (L, ∨, ∧) is a distributive lattice;
(2) x ≤ y iff x → y = I;
(3) x → O = x', I → x = x;
(4) If x ≤ y, then z → x ≤ z → y, y → z ≤ x → z;
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(5) x → y ≥ x' ∨ y;
(6) If ∀x ∈ L, x ∨ x' = I, then (L, ∨, ∧,' ) is a Boolean algebra;
(7) (x → y) ∨ (y → x) = I;
(8) (x → y) → y = x ∨ y;
(9) z → (x → y) ≥ (z → x) → (z → y);
(10) (x → y) → (z → y) = ( y → x) → (z → x);
(11) x ∨ y = I iff x → y = y.
If L is a finite chain, then there exists in L only one such implication →, so that 
it can become lattice implication algebra. Finite lattice implication algebra can be 
decomposed into a Cartesian product of finite chains. Over the interval [0, 1] we 
can define infinitely many kinds of lattice inference algebras. 
A subset J of lattice inference algebra L is called its filter. If I ∈ J and at the same 
time x, x → y ∈ J, there are y ∈ J. A filter of L can be generated from a subset of 
L. It can also be used to define implicative filters, generated filters, prime filters, 
ultra filters, I-filters, associative filters ( Jun 2001), fantastic filters, involution 
filters, obstinate filters and so on. Filters of lattice implication algebra can be 
mutually defined with congruence modulo relations. Apart from that, there also 
exist many results in filter-related methods of fuzzification, dual structures of 
filters (LI-ideal, ILI-ideal (Liu et al. 2003), WLI-ideal), constitutive categories 
of lattice inference algebras, and relations between lattice and other non-classi-
cal logical algebras.
Lattice-Valued Systems of Logic Based on Lattice Inference Algebras  
(Xu 1993)
Based on lattice inference algebra, Xu Yang and others established a lattice-valued 
system of propositional logic LP(X) with lattice inference algebra as its truth-val-
ue range (Qin and Xu 1994; Xu and Qin 1993) and lattice-valued first-order sys-
tem of logic LF(X) (Xu et al. 1997), and on the basis of these further established 
a lattice-valued system of propositional logic Lvpl (Xu et al. 1999) and lattice-val-
ued first-order system of logic Lvfl (Xu et al. 2000). For these systems of logic, 
they further researched their semantics, grammatical structures, and correlation 
properties, and presented their reliability, completeness, compatibility, deduction 
theorems as well as other important conclusions. Below, we will describe the main 
concepts and conclusions using the example of Lvpl.
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255Asian Studies X (XXVI), 2 (2022), pp. 243–266
The formulae of Lvpl are based on propositional variables and constant formulae 
by means of logical connectives, while the set of all its formulae  also constitutes 
the lattice inference algebra (briefly referred to as  ). An assignment is a homo-
morphic mapping of  onto a lattice inference algebra. n-valued inference rules 
have the form (rn, tn), in which rn is a n-valued partial operation on  , and tn is 
an n-valued truth-value operation within ℒ. The domain of rn is marked as Dn( rn), 
whereas  is subset of the set of all rules of n-valued inferce, . L-type of 
fuzzy power set defined over  is .
Definition 2. Let X be any element of , let (r, t) be n-valued inference 
rule in  , and let a be any random element in the valuation field L. 
(1) In Dn (r), if,  then we refer to X of (r, t) as α - I type 
closed.
(2) In Dn (r), if,  then we refer to in X of (r, t) as α - II-type 
closed.
If for any rule (r, t) in R, X of (r, t) is α - I (α - II) type closed, then we call X 
of R as α - I (α - II) type closed. 
Definition 3. Let , and α represent any element in the valuation 
field L. If for any element T in , T of R is α - i type closed, then  R  of  is 
α - I type reliable, in which i = I, II. 
Definition 4. Let X be any element in , , let p be a formula in 
, and let α, β, θ be any element in the valuation field L.
(1) The semantic of definition of X entails two different forms of p:
① 
② , Y of R is α-i type closed}, 
i = I, II.
(2) If we map P I (P II): (n)   × L ((n)={1, 2, …, n})
i |→ ( pi , θi )
fulfilling the following conditions:
① ( pn , θn ) = ( p , θ )
②  or
③  or
④ there exist i1,...,ik ≤ i and the rule (r, t) of , so that if we make
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256 WANG Hongguang, DU Guoping: Chinese Research on Mathematical Logic...
then we call (Pi, (n), X, (p, θ) − (α, β)) to be a (α, α) - I type proof of 
value degree θ from X to p, i = I, II. 
Theorem 1. (Reliability – completeness) If truth-value operations of R satisfy 
bounded semi-continuity, then for any formula p and i = I, II, there is
Definition 5. Let δ be any element in the valuation field L, let T be an ele-
ment of , let p be a formula in . If T(p') = (T(p))', X(p) → T(p) ≥ δ, then 
we say that T satifies X in the type δ - i, at the same time we also call X as 
satisfiable of the type δ - i, i = I, II.
Definition 6. Let τ be any element in the valuation field L. If
then we call X as τ' - I type compatible of (α, α, ), i = I, II.
Theorem 2. (Compatibility) If X is satisfiable of the type δ - i, then X of (α, α, ) 
is compatible of type  - i, i = I, II.
Theorem 3. (Deductive theorem) Let (r20, t2) be an inference rule if  R , let p 
and q be any two formulae within , and σ and θ be any elements within the 
valuation field L. 
(1) If i = I, , then ;
(2) If i = II, , then ;
(3) If , then .
Where, r20 (p, p→q) = q，  = {T | T is a homomorphic mapping of  into L}.
Basing their work in lattice inference algebra, they also researched the corre-
sponding imprecise inference (Xu et al. 2000) and resolution automatic reasoning 
(Xu et al. 2000; Xu et al. 2001; Xu et al. 2003; Xu et al. 2011) of the lattice-valued 
system of logic. 
Chinese Notation
An appropriate symbolic notation can enable us to express logical thought in a 
clearer and more efficient manner, and subsequently to construct tools of infer-
ence. When it comes to logical constants (propositional connectives, quantifiers, 
,
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257Asian Studies X (XXVI), 2 (2022), pp. 243–266
modal operators etc.) in formal languages in particular, which constitute the core 
content of research on inference, constructing appropriate methods of symbolic 
notation can enable clearer and more precise expression, and subsequently also the 
expression of inference rules for logical constants. 
The Chinese method of notation represents a kind of symbolic notation for log-
ical constants as proposed by Du Guoping 杜国平 and others. In this kind of 
notation, by using only a pair of parentheses “( )” we can express each and every 
kind of logical constant (Du 2019a, 2022). 
The most commonly used notation methods for logical constants include the 
infix expression method, Polish notation and reverse Polish notation. 
The infix expression method places binary propositional conjunctions “disjunc-
tion”, “conjunction”, “entailment” and “equality” between their two linking sym-
bols p and q, so that corresponding expressions are formed as “p∨q”, “p∧q”, “p→q” 
and “p↔q”. In that regard, the infix method is identical to the common use of 
mathematical symbols +, -, ×, and ÷. When formulae become complex enough, 
the infix method must draw support from symbols such as parenthesis and others 
to express the priority of different combinations between symbols, all in order to 
avoid ambiguity. For example, with the use of parenthesis, the formulae (p∨q)→r 
and p∨(q→r) are able to express different meanings. 
The Polish notation is an independent form of symbolic notation invented by 
the Polish logician Jan Łukasiewicz, which uses different capital letters to express 
logical connectives, placing these connectives in front of the propositions which 
they are connecting. For this reason, this notation method is also known as prefix 
expression method. Its concrete working method resides in using expressions like 
“Np”, “Cpq”, “Kpq”, “Apq”, “Epq” to express “negation”, “entailment”, “conjunction”, 
“disjunction” and “equality”, respectively (Łukasiewicz 1966, 22–30). One of the 
special advantages of Polish notation resides in the fact that it does not have to use 
parenthesis nor is it able to produce ambiguities. Its expressive efficiency is higher 
than that of the infix expression method. 
The reverse Polish notation is also referred to as the suffix expression method. 
Its working method is similar to that of Polish notation, with the only difference 
being that the connective is placed behind the proposition that it is connecting.
Chinese notation is different from the above three notations. In contrast to the 
Polish notation, it uses other kinds of symbols, while it only uses a pair of paren-
theses to express each and every kind of logical constant. Thus, for example, it 
treats parenthesis “( )” as a ternary symbol. “(ABCx)” can thus be used to express 
all propositional connectives and the quantifiers ∀ or ∃. Furthermore, parenthesis 
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258 WANG Hongguang, DU Guoping: Chinese Research on Mathematical Logic...
“( )” can also be regarded as a quaternary symbol. Thus, by using “(ABCxD)” we 
can express all propositional connectives, quantifiers ∀ or ∃, and modal operators 
□ or ◇. 
For example, if we define (ABCx) as [﹁A∨﹁B]∧∀x[B→C], then (ABBx) is 
[﹁A∨﹁B]∧∀x[B→B]. This is further equivalent to [﹁A∨﹁B], which is one 
of Sheffer functions, and can therefore define all propositional connectives in-
cluding negation “﹁”, disjunction “∨”, etc. In addition to that, (C﹁CCx) is 
[﹁C∨﹁﹁C]∧∀x [﹁C →C], which is further equivalent to ∀ x C, which in this 
way defines the universal quantifier. 
Secondly, if (ABCxD) is defined as [﹁A∧﹁B]∧∃x[B→C]∧□ [C→D], then (ABBxB) 
is [﹁A∧﹁B]∧∃x[B→B]∧□[B→B], this is equivalent to [﹁A∧﹁B], which is a 
Sheffer function, as a result of which it can be used to give the definition of all 
propositional connectives, including the negation “﹁”, conjunction “∧”, and so on. 
Aside from that, (C﹁CCxC) is equal to [﹁C∧﹁﹁C]∧∃x[﹁C→C]∧□[C→C]. 
This is equivalent to ∃ x C, by which we have defined the existential quantifier. 
Moreover, (C﹁C﹁CxC) is [﹁C∧﹁﹁C]∧∃x [﹁C→﹁C]∧□[﹁C→C], which is 
equivalent to □C, by which a definition was given for the “necessary” modal oper-
ator (Du 2019b; 2019c; 2020; 2021a; 2021b).
Chinese notation is inspired by Sheffer functions and the related ideas by Zhang 
Qingyu 张清宇. Sheffer functions employ a simple symbol, | or ↓, to denote 
the logical functions ﹁C∨﹁D or ﹁C∧﹁D, which specifies the propositional 
connectives as one symbol (Mendelson 2015, S21–23). On the basis of Sheffer 
functions, we have presented our specifications of the common logical constants. 
Apart from that, Chinese notation method is also greatly inspired by Zhang 
Qingyu’s proposal of not using propositional connectives and instead using only 
parentheses and the nullary connective “T” to express the ideas of propositional 
connectives and quantifiers (Zhang 1995; 1996; 1997). 
Chinese notation is an integral notation method. Because the left and right pa-
rentheses are used in pairs, their scope is clearly defined. Within themselves both 
parentheses have the capacity to express logical constants as well as the capacity 
to express the linking priority of symbols. 
Some Reflections on Past Developments 
Today, the development undergone by Chinese mathematical logic and foun-
dations of mathematics in the last 100 years has already become an important 
and integral part of Chinese research on foundational theories. In its major 
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259Asian Studies X (XXVI), 2 (2022), pp. 243–266
developmental plans for science and technology, such as “Outline of the 14th 
Five-Year Plan for the Development of National Economic and Social Develop-
ment and 2035 Long-Term Objectives of the People’s Republic of China”, the 
Chinese state places significant emphasis on research into foundational theories, 
which also includes mathematical logic, and has put forward numerous major 
research topics which are related to mathematical logic. In that way, Chinese re-
search on mathematical logic has entered a time of favourable circumstances and 
great opportunities in the context of national strategic development. 
On the other hand, looking back at the developmental trajectory of the past 100 
years, Chinese mathematical logic encountered the following problems which are 
worth taking into further consideration:
(1) One of the comparatively central problems is the still pending pro-
gress in establishing influential scientific institutions. Thus, institu-
tions like the Research Laboratory for Mathematical Logic at the 
Chinese Academy of Sciences, the Department of Mathematics at 
Nanjing University as well as the Institute of Logic and Cognition at 
Sun Yat-sen University have been in the past or still are China’s most 
important research institutes for research into mathematical logic. 
If China wants to become a technological and scientific superpow-
er, then it must accelerate the move to make mathematical logic an 
integral part of national research into foundation theories, by found-
ing more internationally influential centres of scientific research that 
would specialize in this important scientific discipline. 
(2) The problem of the relatively diffuse nature of research areas, and 
the still pending strengthening of scientific teams focusing on spe-
cific fields of research. As a field of theoretical research, mathemat-
ical logic is still basically in the state of having to struggle on its 
own. While Chinese mathematical logic is characterized by relative-
ly focused research based on teacher-student relationships, Chinese 
academia still has not seen the formation of a group of experts that 
would garner international acclaim and influence in one specific area 
of such studies. 
(3) Individual scholars tend to struggle on their own, and the number 
of academic exchanges is still insufficient. Therefore, China needs 
to work at establishing its own internationally influential scientific 
journal for mathematical logic, in order to advance and increase the 
academic exchanges among Chinese researchers in the field. 
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260 WANG Hongguang, DU Guoping: Chinese Research on Mathematical Logic...
(4) Exchanges with the international academic world also need to be 
increased. China requires a substantial increase in the organiza-
tion of international conferences related to mathematical logic, so 
as to boost young scholars’ engagement in international academic 
exchanges. 
(5) The problem of insufficient interdisciplinary research in science. 
Because of the disparities that exist among the humanities, natural 
sciences, and technology, different research groups for mathematical 
logic were formed in the areas of philosophy, mathematics, and com-
puter science and artificial intelligence. Since these three groups of 
researchers still lack mutual exchanges and cooperation, there is an 
urgent need to address this issue in order to give rise to an atmos-
phere where the humanistic direction of research would receive equal 
attention as the scientific research in the field. 
(6) The lack of significant, original results. Looking at the overall state 
of Chinese research on mathematical logic, we can notice that there 
is a profusion of results in secondary, follow-up research but at the 
same time, a great scarcity of ground-breaking original results, espe-
cially major, internationally leading scientific achievements. How-
ever, we are firmly convinced that soon after the promulgation of 
China’s strategy emphasizing interdisciplinary scientific research, 
this situation will greatly improve! 
The f irst draft of the manuscript was translated into English by Jan Vrhovski.
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