27. LOGIC AND CORRECT MINDSET ANY PEACE-MAKING LEADERS MUST ACQUIRE

Chủ nhật - 05/05/2019 22:39
 
LOGIC AND CORRECT MINDSET ANY PEACE-MAKING LEADERS MUST ACQUIRE

by Can Dong Guo*
*
. Instructor, Academy of Wisdom and Enlightenment, Canada.


ABSTRACT
Global leadership for sustainable peace is a colossal task for great minds who can think outside the box. The complexities of global conflict are multifaceted. Traditional Buddhist mindfulness training such as various meditation techniques are designed to discipline the mind to concentrate and focus. Such disciplines fall short of providing political leaders the necessary mind set to comprehend complex problems and see many solutions. I expand the traditional definition of mindfulness” to reveal the hidden teachings on logic systems buried in various sutras over millennium. In particular, non-duality logic and quadratic category logic are expounded. Conflict arises when the mind fails to perceive both sides of the story. This mental block can be overcome by non-duality logic. Quadratic category logic opens up the mind to unseen possibilities by considering and conceiving contexts in 4 categories: yes, no, both yes and no, both not yes and not no. The gold mine of Buddhist logic is unearthed here to serve as critical thinking tools.

Non-duality logic and Quadratic category logic began in the Cūla-Mālunkya-sutta when Malunkyaputta posted 10 questions which Shakyamuni declined to answer; henceforth known as the 10 inexpressible but 14 such questions exist in later Agama Sutra. The 10 are: (1) the world is eternal or (2) it is not eternal, (3) the cosmos is finite or (4) it is not finite, (5) soul is the same as body or (6) soul is one
thing and body another thing, (7) the Tathāgata exist after death , or


(8) he does not exist after death , or (9) he both (at the same time) exist and not exist after death , or (10) he both (at the same time) not exist and not not-exist after death 非非. The first 6 are actually 3 pairs of
dualities while the last 4 compose a quadratic structure.
Around 750 years after Shakyamuni, Nagarjuna began his Fundamental Verses on the Middle Way” (Mulamadhyamakakarika) with 8-NOTs “Not born and not annihilated; Not permanent and not continuous; Not single and not various; Not coming and not going…” These 8-NOTs can be viewed as 4 pairs of non-dualities. Significantly, each pair takes up the quadratic structure of the 10th inexpressible does he both (at the same time) not exist and not not-exist”.

During 557-715 AD the 1st through the 3rd Patriarch of the Avatamsaka School classified the Buddhas teachings into 5 designations: Hinayana, Mahayana Beginning, Mahayana Final, Sudden, and Mahayana Complete. The Chán (Zen) School which preaches Sudden Awakening refuted the effectiveness of the Avatamsaka designations. A Chán master openly debated the Avatamsaka School master in the
presence of the 8th Emperor Huizong (宋徽宗1082 – 1135AD) of the
Song Dynasty and his Chief of Staff. The Chán master shouted once and
then began to illustrate how the one shout permeated all 5 Avatamsaka
designations of Buddhist teachings. He used a rhetoric based entirely on
quadratic category logic and accomplished a glorious victory.
Realizing the detrimental mental block when one fails to comprehend the quadratic category logic, the 1950 Nobel Laureate Bertrand Russell told the world to contemplate on the barber paradox”.

It is high time I decipher these mental contests spanning from Shakyamuni to Bertrand Russell by using simple Venn Diagrams. Learning the Buddhist non-duality logic and quadratic logic would help to attain correct mindfulness that pierce into infinite possibilities. Significantly, the Buddhist quadratic category logic is distinct from the category logic of Aristotle (384-322 BC).

1. INTRODUCTION
1.1. The emergence of category logic
What is truth? An operational definition could be an accurate and thorough  description  of reality.  The  metaphor “Blind me
and the elephant” speaks to the common mental inadequacies to describe reality accurately and thoroughly. To seek truth, we must acknowledge as many facts and discover as many propositions as possible, i.e. the blind men must be in touch with as many areas of the elephant as possible. If the areas of an elephant are divided into two categories, front or back, this is dualistic thinking. Duality is how we learn to know our world. We perceive realities in contrasting pairs: yes/no, light/dark, hungry/full, hot/cold, life/death, good/ evil, samsara/nirvana, finite/infinite, etc. Non-duality is to embrace both front and back, yes and no, viewpoints.

Besides the three viewpoints of front, back, both front and back, there are other viewpoints such as left-side, right-side, top, bottom, which maybe collectively designated as neither front nor back. We now have a total of 4 categories of viewpoints: Yes; No; Both yes and no (non-duality); Neither yes nor no. This is the meaning of quadratic category logic, a tool to overcome the common mental limitations in pursuit of the truth. This thinking tool forces the mind to systematically discover the contents in each category and acknowledge the facts that these categories exists.

1.2. Time line on the developments of category logic systems
The incredibly long timeline (Fig 1) for the development of the aforementioned logic systems indicates how slow the progress and how difficult the thought experiments have been. It should not be surprising to find many of our best and brightest predecessors had fallen into logical pitfalls along this rough intellectual journey. This monumental task is finally completed and elucidated in this article. Historic contributions of our predecessors will be presented chronologically according to the mind map in Fig 1.


Fig 1. Time line on the developments of category logic systems

2. DEVELOPMENT OF CATEGORY LOGIC SYSTEMS
2.1. 
Ten Inexpressible in Agama Sutra

The concept of category logic first emerged in early Buddhism but its intellectual value has been largely ignored. A disciple by the name Malunkyaputta (鬘童子) posed ten questions (paraphrasein  the  Abstract)  to  Shakyamuni  (563~480  BC).  He  asked  the Buddha to either provide definite answers or to admit honestly thathe Buddha himself did not know. The Buddha refused to answerWe assume Shakyamuni would be around 30-40 years old when this happened and thus assigned a speculative date of 526 BC. The unanswered questions became known as the Ten Inexpressible1 or undeclared questions (Sanskrit avyākta, Pali: avkata – unfathomable, un-expounded”). Another Pali term is Acinteyya commonly translated as  imponderable  or  incomprehensible. The Theravada tradition recorded the story in Pali in the Agama Sutra Majjhima Nikaya canon 61, Cūla-Mālunkya-sutta (1998). The Buddha responded to Malunkyaputtas 10 questions with the poison arrow metaphor. When a man is injured by a poison arrow, he should seek medical treatment immediately. He should not waste time asking irrelevant question such as who shot the arrow, what is his cast, his skin colour, his height, where is he fromwhich type of bow and bow string… etc. The Buddha considered metaphysical questions unrelated to, and created distractions from, the spiritual


1 http://en.wikipedia.org/wiki/The_unanswered_questions


practices leading to enlightenment.  Although this traditional interpretation is indisputable, the author begs to differ and wishes to disclose hidden treasures in this lesson.

Firstly, it should be obvious that the first 2 questions are deliberately written as a pair of dualities. Conceivably, no languages in the world would ask questions in such clumsy, redundant grammar. It goes without saying for the rest of the 8 questions. Instead of asking Have you had breakfast?Malunkyaputta statedi) You had breakfast. ii) You did not have breakfast.These are position statements. Therefore, question marks “?should be removed at the end of all of Malunkyaputtas 10 statements. Malunkyaputta was asking the Buddha to take a position regarding each statement.

Secondly, the first 6 questions were 3 pairs of dualities. Let Arepresent the first statement and “B” the second statement. If the answer to Ais affirmative, then the answer to “Bmust be negative. The propositions Aand “B are mutually exclusive. One has to take a position between the two propositions or categories. Thirdly, Malunkyaputta structured his last 4 questions in a quadratic format; affirmative, negative, both affirmative and negative, neither affirmative nor negative. Malunkyaputta designed 4 positions or categories. Quadratic category logic was born.

Fourthly, Malunkyaputtas statement #9 The Tathāgata both exist and not exist after death.takes the form of affirmative
to both Aand “B” meanwhile Aand “Bare supposed to be mutually exclusive. This is the birth of the non-duality concept of
embracing opposites .
Fig 2. Venn diagram depicting Buddhist quadratic categories
Fifthly, Malunkyaputtas statement #10 takes the form of negative to both Aand “B” meanwhile Aand “Bare supposeto be mutually exclusive. This is the birth of thinking outside the Aand “B” boxes 非是非非.

Malunkyaputtas mind-boggling syntax can be elucidated by a Venn diagram (Fig 2). He structured his first 6 questions as 3 pairs of dualistic categories. His last 4 questions were in the format of quadratic categories logic i.e. 4 viewpoints. In Fig 2, “No” can be substituted by “Not-yes, “Bothsubstituted by “Non-duality. The label “Neither” can be substituted by “Not-yes and Not-no. Its area is outside the Yes” “No” “Both” thinking boxes.

Perhaps the best part of Malunkyaputtas so called questions were his intention to invent category logic as a tool to not only guide the mind to lay out all possibilities but also to force the mind to systemically consider/imagine the content and meanings in each possible category. The metaphysical nature of his questions was probably insignificant. He would be silly to ask the Buddha “(1) You had breakfast. (2) You did not have breakfast. (3) You both had and not had breakfast. (4) You neither had nor not had breakfast. So, Malunkyaputta might just as well ask something meaningful.

Inspired by this novel analysis of Malunkyaputtas true intent, we could speculate on an alternative reading of the Buddhas silence. Perhaps, he realized that the world was not yet ready to listen to true answers due to the lack of logic tools. Also, the views of absolute existence and absolute nonexistence do not correspond to the way things really are. He preferred the “Middle Way. While the story in Sayutta Nikāya (相应部) in the Pali Tipitaka also told of 10 Inexpressible, a slight discrepancy occurred in the Majjhima Nikaya (中阿含经) in that the questioner was Vacchagotta (婆磋 ) and not Malunkyaputta (鬘童子). The geographic location (Jetas Grove in Savathi) was identical in both canons. For surboth these Theravada canons reported 10 questions.
The Sayukta Āgama 杂阿含经 translated from whole Sanskrit texts to Chinese corresponds well to the Pali SayuttNikāya   (相应部).   Mere   fragments   of   the   original   Sanskrit survived.     Peculiarly, the Chinese sutra presented 14 instead of 10 inexpressible. Scroll 16, #408 [http://www.cbeta.org/result/normal/T02/0099_016.htm] described the Buddha overheard a group of bhikkhu discussing the 14 metaphysical questions in the
 

The 14 inexpressible were written in the following format:

Questions concerning the existence of the world in time
Is the world eternal?
...or not?
...or both?
...or neither?
(Pali texts omit "both" and "neither")
Questions concerning the existence of the world in space
Is the world finite?
...or not?
...or both?
...or neither?
(Pali texts omit "both" and "neither")
Questions referring to personal identity
Is the self identical
with the body?
...or is it different from the body?
Questions referring to life
after death
Does
the Tathāgata (Buddha) exist after death?
...or not?
...or both?
...or neither?




 

 

 
documented 10 Inexpressible.


dining hall of Kalanda Bamboo Garden, Rajagaha City and then responded in the same manner as    towards    Malunkyaputta but without the poison arrow metaphor. Elsewhere [http:// www.cbeta.org/result/normal/ T02/0099_034.htm] in Scroll 34,   #962   the   story   involves
Vacchagotta        (婆磋种)        at
Kalanda Bamboo Garden asking
whether the Buddha subscribes
to the 14 questions. The Buddha
denied   he   ever   contemplated
on    these    wrong    views.    He
considered    the    Four    Nobel
Truth as the sole correct view.
Curiously,    the     emphasis of the 14 Inexpressible is on presenting 3 sets of quadratic categories and minimized the duality category down to a single pair (Question 9 & 10).

The chronology of the various Āgama sutra versions are:

~483 BC Pali versions of Sayukta Āgama 《巴利文 杂阿含经》   and    Sayutta-
nikāya   巴利文相應部經》
documented 10 Inexpressible.

~383 BC Pali Majjhima- nikāya   《巴利文中阿含经》
 
~83   BC   Sanskrit   Sayukta  Āgama   《梵文杂阿含经》
documented 14 Inexpressible.
 


~439  AD  Chinese  Sayukta Āgama         《汉文杂阿含经》
documented 14 Inexpressible.
Clearly, the 10 Inexpressible featuring 3 pairs of dualities and 1 set of quadratics was at the time of the Buddha. In contrast, the 14 Inexpressible highlighting 3 sets of quadratics and 1 pair of dualities emerged ~400 years later.

Historically, the Sanskrit version appeared after the earlier Pali version of Sayukta Āgama. Although the originals were mostly lost, a Chinese translation from Sanskrit of the complete Sayukta
Āgama 《杂阿含经》 belonging to the Sarvāstivāda (說一切有 ) tradition was done by Guṇabhadra (求那跋陀羅 394 468 AD) during the Liu Song (刘宋) Dynasty of the South Dynasties ( 南朝) Yuanjia Period (元嘉年间 435 443AD) ~900 years later
than the Pali version.
Based on these chronologic and historic observations, we hypothesize that the quadratic category logic is the actual highlight; the inexpressible are mere examples to bring out the problem of immature logic systems.

The following rationale further supports our hypothesis: It is generally agreed that the Sanskrit Sayukta Āgama was the work of Sarvāstivādins who were precursors of Mahayana. Ideological schism between Sarvāstivāda and Vibhajyavāda is a historical fact. We could reasonably speculate that the Sarvāstivādins put a lot of thought into editing and conferring the true meaning of the Tathāgata rather than mere literal translations from Pali to Sanskrit to Chinese. Omission due to forgetfulness is understandable after generations of oral transmission. But insertion of extra text is astounding. Thus, the addition of new content (14 Q with 3 quadratic sets) to the original text (10 Q with 1 quadratic set) in such a logical manner is very likely to be deliberate. The deletion of the poison arrow passage was probably intentional so as to avoid distraction from the logic highlight.

Aristotelian Category Logic
In order to describe reality accurately and thoroughly Aristotle (384~322 BC) invented 10 categories [https://en.wikipedia.org/ wiki/Categories_(Aristotle)] to identify/distinguish all things in
 






the world. This is not to be confused with his category logic below. His ultimate goal is to develop syllogism [https://en.wikipedia.org/ wiki/Syllogism], defined as logical argument that applies deductive reasoning [https://en.wikipedia.org/wiki/Deductive_reasoning] to derive a conclusion based on two or more propositions that are asserted or assumed to be true. For example, when we say all dogs are mammals, we cannot turn it around and say all mammals are dogs. Why? because mammal is a category that contains a subset of dog. Fig 3 illuminates the grave problem of thinking or arguing in language. Using English syntax as stated within brackets, the colour of swan could be argued in six ways, deceptively creating six categories. The language syntax is correct but the logic is erroneous. Using pictures, the six categories collapsed into three, which is also wrong. This exercise demonstrates our common mental inadequacies to describe reality accurately and thoroughly. Errors are plentiful when we use only either our left-brain language or our right-brain picture to think. Venn diagrams uses both words and pictures together forcing the left- and right-brain to work together to arrive at a correct mindset. Venn diagrams accurately describes all the possibilities of the colour of swans pertaining to the colour white into four quarters. The arrow in the second column points to the overlapping area (white swans) which is different from the arrow in the fifth column pointing at the area outside white things (non-white swans).
All A (Swan) are B (All swans are white) Some A are B (Some swans are
white)
No A are B (No swans
are white)
All A are not-B (All
swans are not white)
Some A are not- B (Some swans are not white) No A are not- B (No swans are not white)




 




 




 


 
 

 


 
 

 



 

 
Fig 3. Aristotelian Category Logic (with examples in brackets) depicted by Venn diagrams
 


Comparing Fig 3 Aristotelian categories to Fig 2 Buddhist quadratic categories, obviously the two logic systems are entirely different. Aristotle tames the mind to tidy up ideas whereas Buddhists challenge the mind to explore unconventional ideas and instigate new viewpoints.

Nāgārjuna Mulamadhyamakakarika and Great Treatise on the Perfection of Wisdom
Nāgārjuna    (龙树 150~250      AD)   [https://en.wikipedia.
org/wiki/Nagarjuna]  is  one  of  the  most  influential  Buddhist
philosophers after Shakyamuni. Born a Brahman, he converted to
Sarvāstivāda Buddhism. Chinese Buddhism honoured Nāgārjuna
as the founding father of eight traditions.
Definition of “Middle Way” by Shakyamuni
In the English translation of the Chinese Sayukta Āgama Sutra
《杂阿含经 pertaining to the Middle Way(Choong 2010),
Shakyamuni define the Middle Wayin four passages to consistently
meanavoidingextreme/oppositeviews.Thepassagenumbersandtitles are: 3) the excellent Dharma (Right view, the Middle Way) 胜妙法 (正见中道); 4) Empty of the externalist and annihilationist view
(The Middle Way) 空常见断 (中道); 5) The great discourse on the emptiness of dharmas 大空法经; and 6) Establishing right view 施设正见. Based on the Buddhas explicit words in these four
passages, the scripts did not represent non-duality, which is to embrace two opposite sides. Avoidance is very different from embracing.
Following this doctrine Nāgārjuna founded the Madhyamaka school of Mahayana Buddhism. His written contribution to the advancement of Buddhist quadratics category logic is expounded below.

2.3.2.FundamentalVersesontheMiddleWay,Mūlamadhyamakakārikā
《中论》

This treatise [https://jampasmandala.wordpress. com/2011/07/06/mulamadhyamakakarika-by-nagarjuna/]  is the foundation of the Madhyamaka School, which focuses on the analysis of emptiness, and was thus also known as Śūnyatavāda. Nāgārjuna defined the “Middle Way” as:
 


“… Conditioned origination is emptiness. It is mere designation depending on some-thing, and it is the middle path (24.18). Since nothing has arisen without depending on something, there is nothing that is not empty (24.19) (Bronkhorst, 2009).”

Whatever is dependently co-arisen; that is explained to be emptiness. That, being a dependent designation, is itself the Middle
Way眾因緣生法,我說即是無。亦為是假名,亦是中道 .”          [http://promienie.net/images/dharma/books/nagarjuna_
mulamadhyamakakarika.pdf page 69, Verse 18]
His definition of “Middle Wayadhered to Shakyamunis dependent origination doctrine.

The opening dedicatory verses in the Mūlamadhyamakakārikā are the famous 8-nots: [http://promienie.net/images/dharma/ books/nagarjuna_mulamadhyamakakarika.pdf page 2]

I prostrate to the Perfect Buddha, the best of teachers, who taught that Whatever is dependently arisen is
Unceasing, unborn, 不生亦不灭   =   不生亦不不生 Unannihilated, not permanent, 不常亦不断=   不常亦不不常 Not coming, not going, 不来亦不出   =不来亦不不来
Without distinction,  without identity不一亦不异 = 不一亦 不不一
And free from conceptual construction.
Let us convert the 8-Nots without changing their meanings:
Unborn and unceasing = not born and not not-born 非生亦非 非生

Not permanent and unannihilated = not permanent and not not-permanent 非常亦非非常

Not coming and not going = not coming and not not-coming 来亦非非来

Not dual and not mono = not unison and not not-unison 非一 亦非非一

Employing Venn diagram for correct interpretations, Fig 4 readily displays that Nāgārjuna used the 4th quadrant (designated as
 


the “Neitherarea as per Fig 2) of the Buddhist quadratic categories to express his Śūnyata doctrine of conditioned origination, a new viewpoint of avoiding extremes.


 

 
Fig 4. Śūnyata doctrine of Nāgārjuna depicted by Buddhist quadratic category logic

 
Unborn and unceasing Not permanent and Unannihilated Not coming and not going Without identity and without distinction
不生亦不灭 不常亦不断 不来亦不出 不一亦不异
   

It is critical that  we  must  not  convert  the  double  negatives to a single positive; the meaning will change. For example, Not permanent and Unannihilated cannot be converted to Annihilate and Permanent. Recognizing this conversion rule let us see why the double negative term is a crucial necessity, the only correct way to refer to the 4th quadrant, the area labelled as “Neither” in Fig 2.

Unmistakably, each verse is composed of a duality pair. The first pair refers to genesis, the second to sustainability, the third to migration and the forth to monism; all concerning existentialism. Nāgārjuna no longer shy away from existential questions. He tackled metaphysics head-on using Buddhist quadratic logic to position his Śūnyata theory into the 4th quadrant of double negatives, thereby expressed the Inexpressible. The genius of Nāgārjuna is his recognition of the Buddhist quadratic category logic and used this power tool in his opening verses and throughout his work.
2.3.3. Great Treatise on the Perfection of Wisdom 《大智度论
This treatise [https://en.wikipedia.org/wiki/ Maprajñāpāramitāupadeśa] was work done by Nāgārjuna  in his late period (Shih, Yin-shun 1991). In this literature Nāgārjuna blatantly used quadratic category logic to redefined “Middle Way” as follows:

Non-existent and not non-existent.  Also, again not existent or non-
 



existent. Even this saying too is unacceptable, so this is the name and meaning of the Middle Way. 非有亦非無, 亦复非有無。此語 亦不受,   如是名中道。

[Maprajñāpāramitāupadeśa First Article, Explanations of the Ten Metaphors Number 11, Scroll 6, 《大智度論》大智度初 品中十喻釋論第十一,第6; http://ftp.budaedu.org/ghosa/
C006/T0633/ref/T0633.pdf , page 48, 0105a11]
He finally realized that the “Middle Way” should not be defined as another name to refer to conditioned genesis in general, or the 12-dependent origination in particular. By using quadratic category logic, Nāgārjuna managed to circumvent Sakyamunis definition of “Middle Way” as another name for conditioned genesis. His new definition captured what he meant by Śūnyata.

He then turned around and masterminded that the great wisdom of liberation is derived from a thorough understanding, application and utilization of the quadratic category logic system. In the “First Article, Explanations on the Meaning of Great Loving
Kindness and Compassion, Scroll 42, Number 27 大智度論釋初 品大慈大悲義第四十二卷二十七    [http://ftp.budaedu.org/
ghosa/C006/T0633/ref/T0633.pdf page 198, 0259b29]
So it is for every and all dharma; the so-called the dharma of existence and the dharma of non-existence, of both existence and non-existence, of both not existence and not non-existence; The dharma of emptiness and the dharma of non-emptiness, both empty and non-empty, neither empty nor not-empty; the dharma of birth, the dharma of decease, of both birth and decease; of both unborn and un-decease; the dharma of both non-arising and non-ceasing, the dharma of both not non-arising and not non-ceasing; the dharma of both non-arising and non-ceasing, and not non-arising and not non-ceasing; the dharma including not non-arising and not non-ceasing, not non-non-arising, and not non-non-ceasing; So it is for every and all dharma. The so-called the dharma of existence, the dharma of non-existence, the dharma of both existence and non- existence, the dharma of both not existence and not non-existence are four equanimity statements that any one statement should not be hold onto. The dharma of emptiness, non-emptiness, arising,
 


ceasing, both non-arising and non-ceasing are also five statements to be similarly regarded (with equanimity, without attachment). By this unimpeded wisdom, one knows thoroughly and encompasses all the dharma taken in through a multitude of countless incalculable (Asamkhya) dharma gates. The name of this wisdom is called all pervasive wisdom, all kinds of pervasive wisdom.

In todays language, Nāgārjuna could simply be paraphrased as “Do not hold on to any one of the four possible views as per Fig
Take on all four (five) kinds of views, understand each and all viewpoints. Only then will you attain the great liberating wisdom of know-it-all, no matter what the subject matter is.Unfortunately, Nāgārjuna did not have the convenience of Venn diagrams to simplify his presentations. Consequently, he was stuck with spelling out each category relentlessly at length making his composition almost incomprehensible. At this point the author is only a step away from saying that Nāgārjuna considered the quadratic category logic system is the great wisdom that liberates. Throughout this Treatise (which I prefer to translate as Treatise on Great Wisdom Liberates”) Nāgārjuna was fluent in writing in the quadratic format. The above is but a small sample of Nāgārjunas immense contribution to Buddhist quadratic category logic.

3. NON-DUALITY TAUGHT BY THE AVATAMSAKA SUTRA MODULE
The logic that validates the concept of non-duality as per Fig 2 was elucidated for the first time in the Avatamsaka Sutra. Due to space limitations the readers are referred to the article “How to Transcend Duality?” from our Academy previously published at the 2014 UNDV conference (Cheng 2014). Great significant is that the Avatamsaka Module teaches embracing opposite viewpoints in contrast to Shakyamunis definition of “Middle Wayto avoid opposing viewpoints. The teaching is found in the Avatamsaka
Sutra [大方廣佛華嚴經], “Enter the dharma realm [入法界 ]” Section 12 [善財童子第十二參]. The Juvenile Master of Self-reliance [自在主童子] taught the Juvenile Master of Value [Sudhana  善財童子]  to  count  increasingly  large  numbers.  He

started from one koti unit (10 million = 10,000,000 = 8 digits) to unspeakabe2  unit [不可說轉] (18609191940988822220653298
 


843924824065 digits) over 123 operations of squaring. At every squaring step, each infinitesimal number was quantified by the provision of a unit of measurement for that product. This exercise accomplishes the following learning objectives:

Close mindedness can be quantified by discovering ones thinking box at the point when the huge number is regarded as infinity.

Recognize that infinity is a concept created by the close mind or limited thinking box.

The quality/quantity  duality  is  transcended  as the abstract term unspeakable” becomes a concrete speakable” unit.

Finite/infinite can co-exist and not mutually exclusive; yet we can remain comfortable with the traditional definition of finite versus infinite.

The non-duality logic can be visualized as the overlapping area labelled as “Both” in Fig.2. This logical category is so important that for the first time a teaching is repeated in the same sutra, Scroll #45,
Asamkhya Article, Chapter 30 (卷四十五,阿僧祇品,第三十 ) where the Tathāgata instructed the Bodhisattva Ruler of Heart (心王菩薩). In fact, the early foundation of this teaching can be

located in the Lankavatara Sutra Scroll #1, Chapter 1, Section 1, Speaking about the Heart by All Buddha” (楞伽阿跋多羅寶 經卷第一,      一切佛語心品第一之一)   in   the   conversation
between the Bodhisattva Great Awareness (Mahamati 大慧) and
the Tathāgata.

SCHOOL OF THE THREE TREATISES
Developmental history
Kumarajiva (鸠摩罗什) transmitted the Madhyamaka School into China during the Northern and Southern Dynasties (南北朝

420-589 AD). As a result, two major lineages emerged; the Tiantai sect (天台宗) and the School of the Three Treatises (三论宗)
[https://en.wikipedia.org/wiki/East_Asian_Mādhyamaka]. The Three Treatises are Nāgārjunas Fundamental Verses on the Middle
 


Way (Mūlamadhyamakakārikā 《中论》; The Treatise on the Twelve Gates 《十二门论》; and Śataśāstra or The Hundred Verse Treatise 《百论》 [https://en.wikipedia.org/wiki/ Śataśāstra] a work by Nāgārjunas disciple Āryadeva ( 3rd
Century AD). Traditionally, Buddhist schools are established on sutras. The establishment of a school based entirely on three treatises is unprecedented and signifies a major paradigm shift in  ideology.  Kumarajiva  translated  and  promoted  the  three
treatises  during  the  Later  Qin  Dynasty  (  384-417  AD)
thereby setting the ideological foundation of the school. The

doctrines progressively matured after a few generations. By the Sui Dynasty (隋朝) Ven. Jizang ( 546-623 AD) [https://
en.wikipedia.org/wiki/Jizang] amalgamated  the  principles  in the three treatises into his own writings thereby completed the ideological system of the school.

Central ideology
Jizang took the 8-nots from the “Middle Way” of Nāgārjuna
further to construct his own treatise of quadratic category of twofold truth (四重二谛论) which became the central doctrine of  the  School.  Buddhism  subscribe  to  two  kinds  of  truth  
; conventional truth 俗谛 and ultimate truth 真谛 [https:// en.wikipedia.org/wiki/Two_truths_doctrine]. Sengquan 僧诠 and  Falang  法朗 [http://chinabuddhismencyclopedia.com/en/
index.php?title=Fa-lang] advocated ultimate truth to dismantle the view of existence but espoused conventional truth to dispel the view of non-existence. They regarded the twofold truth as skilful means to establish the “Middle Way” doctrine.  Later on, Jizang expanded
this concept in Treatise on the Mystery of the Mahayana 《大乘玄
论》    [http://buddhism.lib.ntu.edu.tw/BDLM/sutra/chi_pdf/
sutra19/T45n1853.pdf]  and  Commentary  on  the  Madhyamika
śhāstra        《中观论疏      [http://buddhism.lib.ntu.edu.tw/
BDLM/sutra/chi_pdf/sutra18/T42n1824.pdf].  He  constructed
a 4-level cascade for the twofold truth (TABLE 1) which became
the central ideology of the School. We can see from TABLE 2 that
the operation to create a level is by combining the two truths from
the preceding level into Conventional truth. Then fill in the content
under ultimate truth.
 


TABLE 1. Jizangs 4-levels of twofold truth 吉藏的四重二谛
 
Level Conventional truth 俗谛 Ultimate truth 真谛
1第一重 Existence Śūnyata/emptiness
2第二重 Existence and emptiness 有、空
Non-existence and non-empti- ness 非有非空
3第三重 Existence and emptiness are dual- ity; Non-existence and non-emp-
tiness are non-duality空、有是 二,非空非有是不二
Not Existence and not empti- ness; Not non-existence and not non-emptiness; Not duality and
not non-duality非二非不二
4第四重
All of the above are merely skilful means for teaching前三重的二 谛都是教门

Ultimate reality is unthinkable, unspeakable 忘虑绝才是 真谛
Formal analysis of Jizangs 4-levels of twofold truth
Subjecting TABLE 1 to quadratic category logic analysis, we can immediately pin point what went wrong (TABLE 2). Level-1 and -2 are correct because the 4 quadrants abide by the definitions in Fig 2. The content in each of the 4 quadrants are knowable epistemologically  speaking  and  will  be  exemplified  in  Section
Levels-3 and -4 committed the informal logical fallacy called red herring. When a pack of hunting dogs are chasing a fox by following its smell, a red herring, which has a very pungent fishy smell, is dragged across the foxs path in a different direction. The dogs are misled to pursue a different subject that has nothing to do with the original pursuit. Likewise, Level-3 and -4 has nothing to do with the issue of existentialism in Level-1 and -2. Remember a fallacy is an invalid argument which appears logical and convincing because it assumes the form of formal logic.

TABLE 2. Derived by subjecting TABLE 1 to analysis by category logic as per Fig 2
 



Level Conventional truth Ultimate truth

 
 

1Existence

Yes    No
 


 
 

2Existence

Yes    No


Outside the 2 circles
 


3→ DualitLevel-1 is duality, Level-2 is non-duality      Not duality and Not non-duality
 
 
4 EpistemologThe 3 levels above are merely skilful
means for teaching purposes

Unknowable, unthinkable, unspeakable
 

To play the devils advocate, suppose I expand Jizangs system one step further and claim that truth known to Shakyamuni (or Jesus Christ or Krishna for that matter) is unknown to the rest of humanity. Then I could add Level-5→Relativism to the bottom of Jizangs two-fold system (TABLE 3). I am totally justified to do so based on the words of 2 sages, namely Shakyamuni and the 6th
Patriarch Hui Neng 六祖惠能  (638-713 AD) of the Chinese
Zen tradition. In the SisaSutta [https://www.dhammatalks.
org/suttas/SN/SN56_31.html]    the     Tathāgata      disclosed      that
what he knows is akin to all the leaves in the forest but what he
taught amounts to the handful he is grasping. In the last will to his
disciples recorded in the Platform Sutra Article 10 (Hua 2011) the
6th Patriarch emphasize that advocating unspeakable is a blasphemy.
TABLE 3. Extension of artificial levels to the twofold truth system of Jizang

 
Level Conventional truth 俗谛 Ultimate truth 真谛
Level-5→ Rela- tivism Ultimate truth is un- thinkable, unspeakable to us Thinkable, speakable by enlightened sages
Level-6→ An- thropocentrism All beings on earths don’t know Extraterrestrial advanced civilizations know
Level-7→ Cos- mocentrism Every photon poten- tially carries informa- tion but itself & even extraterrestrials don’t know that Entanglement with an observer decohere the photons information which becomes known to us
 


 
Level-8→ Mo- nism Everything said above are constructed by human consciousness The entire universe is conscious
Level-9→ Simula- tion hypothesis Our universe is a vir- tual reality projected by programming The projector / program- mer outside our universe God only knows
Now I can perform the same operation and add Level-6 to TABLE 3. Why stop there! Taking all the beings in the entire cosmos, I can argue for Level-7…Level-8…and go outside our universe to insert Level-9. See how simple it is to take the bottom-right cell of ultimate truth, convert it into conventional truth and add a level of two-fold truth? There is no end to adding levels upon levels; something is very wrong with this seemingly logical argument! The reader can easily see I am making a mistake but may not be so astute when it comes from an authoritative sage! To rectify Jizangs error, the correct placement of his Level-3 should be in its own quadratic category by its own right (TABLE 4).

TABLE 4. Quadratic category for Duality

 
Level-3 Yes No

Duality

Yes

No


Non-duality

Yes    No


Outside the 2 circles

The logical pitfall of Jizangs Level-4 is most severe. It concerns epistemology, not skilful means of teaching. Worse, it subscribes to the misnomer that ultimate truth is unknowable, which is a mistake that the Tathāgata aimed to correct twice in the Avatamsaka Modules.

In brief, Jizang had a glimpse of the quadratic category logic system and did utilize it correctly to establish Level-1 and -2. However, he committed the red herring” fallacy and piled upon layers of unrelated issues until he realized that this kind of piling can occur to no end. Thus, he put an artificial end to it at Level-4 and unfortunately come to the dead-end erroneous conclusion that
 


ultimate truth is unthinkable. Nonetheless, Jizang deserves to be credited for finding out that in tackling the truth, one has no choice but to unsuspectingly utilize quadratic category logic. The detriment of inadequate training in handling this power tool is illustrated.

AVATAMSAKA CATEGORIZATION OF BUDDHIST TEACHINGS
Doctrinal classification schemes [https://en.wikipedia.org/ wiki/Classification_of_Buddhas_teaching] proposed by several Mahayana schools was a highly significant development because it arose from divergence and disputes among Buddhist schools themselves. Buddhism is the only religion in the world that allows itself to evolve with time and circumstances, which necessitates the accommodation of new philosophy and diverge ideologies. Any religion or school which claims orthodoxy and monopoly on ultimate truth for all times would be religious fundamentalism, a  major  devastation  to  peace  and  progress  of  civilization.  The
Avatamsaka    School    (Huayan    華嚴)    [https://en.wikipedia.
org/wiki/Huayan#Classification_of_Buddhist_teachings]
developed a 5-fold classification [https://en.wikipedia.org/wiki/
Huayan#CITEREFBuswell1993] listed in TABLE 5.
TABLE 5. Avatamsaka categorization of Buddhist teachings

 
Existence Category Doctrine
1Yes (缘起) Śrāvakas Small vehicle, Hinayana of Sarvāstivāda, Hearer of dharma, the self and phenom- ena are impermanent, pursue nirvana, become an arahat

2No, non-exis- tence
Elementary Mahayana Yogacara, Madhyamaka Emptiness, Sunyata,
3Both Yes & No
亦有亦空
Final Mahaya- na Non-duality between the above two, Tathāgatagarbha-teachings, the Awaken- ing of Faith
4— Both Not yes
& Not No 非有亦
非空
Sudden abrupt Mahayana Immediate awakening followed by grad- ual cultivation. Practices and teachings cannot create what is already there, our Buddha natureBuddhahood is seen as revelation rather than verbalization.
5All of the above together One-vehicle Buddhahood The complete, or perfect, teachings of the Avatamsaka-sutra and the Huayan School, Samantabhadrayana
 





Fig 5. Venn Diagram for Avatamsaka Categorization of Buddhism 华严五   维恩图

In Mahayana Buddhism, śrāvakas (Hinayana) [https:// en.wikipedia.org/wiki/Śrāvaka] are sometimes contrasted negatively with bodhisattvas; its teaching was categorized as beginners or Level-1. Subjecting TABLE 5 to the scrutiny of Venn diagram (Fig 5) two logical flaws are revealed. Firstly, the Sudden/Abrupt Enlightenment in Category-4 leans toward the unthinkable, unspeakable” misnomer. Secondly, the so-called Level-5, which is supposed to be a teaching level all-encompassing the 4 levels prior, is a figment of imagination because it is actually referring to the entire Venn diagram itself. There is no other way to view an elephant over and above the 4 viewpoints. Similarly, there is no one-vehicleteaching method outside of learning all the 4 categories. Nonetheless, the Avatamsaka School deserved to be credited for employing quadratic category logic unintentionally and subconsciously, like any thoughtful Buddhists in history would find themselves doing. According to Fig 5, the School did manage to place the 4 categories correctly, which is a marvellous achievement and promotion of Buddhist logic.


ZEN    SCHOOLS    RHETORICAL    ONE-SHOUT    PERMEATES AVATAMSAKAS FIVE CATEGORIES (一喝透五教)

During the Huizong Period of the Song Dynasty in China, the royal military commander Chen hosted an open symposium that anyone regardless of personal status was free to attend and debate Buddhist dharma at liberty. Numerous famous masters attended.
 


Accompanied by his prime minister, Emperor Huizong was present incognito [http://tripitaka.cbeta.org/X80n1565_012]. An expert Avatamsaka elder charged The Buddha taught us curricula from Sravakas through one-vehicle before we can attain Buddhahood. I often heard the Chán tradition claims that a single shout can bring forth enlightenment. That is contrary to the sutras unless you can show us.Head of the Chán clan called out his junior monk Jing Yin who happily obliged This simple problem does not deserve a response from our elders. He proceeded to first define the 5 Avatamsaka classes (TABLE 6, first column), then shouted loudly and gave his explanation (second column). The Avatamsaka expert succumbed. Chán Master Jing Yin won due to impeccable rhetoric depicted in Fig 6, which is superimposable over Fig 5.

TABLE 6. Zen Master Jing Yins one shout permeated all 5 Avatamsaka classes of teachings

 
Classes Permeation of Avatamsaka Classes
Sravakas 小乘 Existence Yes My shout exists when you hear it. This is an example of the teaching of Sravakas for the dharma hearer.

Elementary teaching of Mahaya- na 大乘始教Existence No
Later, the sound subsided. Since you can’t hear the sound which existed a moment ago, that sound is empty, it is non-existence.
Final teaching of Mahayana 乘終教  Existence both yes and no 亦是亦非 When I shouted, existence emerge from non- existence. When the sound subsided, non- existence comes from existence. So, both existence and non-existence are relatives and complementary.

The sudden teaching of Mahaya- na 大乘頓教  Existence both not yes and not no 非是亦非非
When I shouted, you said yes there is sound. Later you said nowhich emerged from yes. So, your no is based on both not existence and not non-existence.
The all-round complete teaching My  one  shout  is  beyond  existence  and  non-
of one vehicle 一乘圓教  All of existence, analysis and synthesis. When I say
the above. 包含以上四个范 existence, there is nothing. When I say non-
:, , 亦有亦空, existence, there is everything. That is to say, this
有亦非空 shout penetrates hundreds of trillions of shouts,
and vice versa.
 





Fig 6. Chán One Shout (Sound) 一喝透五教图
The friendly fire between the two schools ignited a glorious firework display of advance Buddhist ideologies. Resolving existentialism became the central doctrine; the poison arrow metaphor is a fundamentalism dogma no more. Momentously, both schools employed quadratic category logic  as  foundation for their doctrines. Both dispelled the misnomer of unthinkable and unspeakable truth. The Chán Master display a thorough knowledge of Buddhism not limited to his own tradition; Chán is not meditation in the void. In the end Buddhism must be preached through valid logical argument, not mysticism.

BERTRAND RUSSELLS BARBER PARADOX
Conditioned by Aristotelians restrictive category logic, philosophers have notice inconsistencies and have no choice but to label them as paradox. Such label by our mindset is because there are no known logical systems to accommodate the inconsistence. Amazingly, paradox can be comfortably placed in the “Both yes and no” category in the Buddhist quadratic logic system.


 
The    famous         Bertrand
 
Russell   ‘Barber   Paradox’   is
funny.   You   can   define   the
village   barber   as   one   who
shaves all those, and those only,
who do not shave themselves.

Yes = Men who are shaved by the barber


No = Men who are not shaved by the barber
 


The question is, does the barber shave himself ? Answering this question results in a contradiction. The barber cannot shave himself as he only shaves those who do not shave themselves. If he shaves himself, he ceases to be the barber. If the barber does not shave himself then he fits into the group of people who would be shaved by the barber. This is an excellent example of the breakdown of internal consistence of logic when we think only in terms of duality. This paradox is quickly resolved by a Venn diagram using Buddhist quadratic category logic. The barber (a non-duality entity because he fits into both opposites) should be placed in the overlap of the Yes and No categories. Women and children are outside the two circles. The entire population of the village is confined within the rectangle. See how conflicts between 2 camps can be transcended by considering 2 more other possibilities.

ELUCIDATION OF LOGIC SYSTEMS
Infinity is arguably the largest concept conceivable by the human mind. If the mutually exclusive state of being finite and infinite can be simultaneously  accepted,  opposite  dissolves and non-duality achieved, then conflicts smaller than infinity should be solvable. Let us take the quadratic questions 5, 6, 7 & 8 from the 14 Inexpressible and express the answers completely. We can accomplish this task neatly by employing Buddhist quadratic category logic (TABLE 7).

TABLE 7. Buddhist quadratic category logic inspires answers to perplexing questions

 
Question Category Knowledge Context
1The universe is finite Yes The universe is 13.77 billion years old since the Big Bang. Since space-time is a single thing, finite age means finite space; thus, the universe is finite by definition.
2The universe is finite No Expansion of the universe is accelerating, get- ting bigger every split second with no end in sight; that fits the definition of infinite.
3The universe is both finite and infinite Both Yes
& No
Since the above two categories are both true, logical consistency demands both the Yes and No answers are correct. Non-duality em- braces, not avoid, both extremes.
 


 
4The universe is both not finite and not infinite Both Not yes & Not No The content in here must be distinct from the above 3 categories. Not infinite because cosmological knowledge is finite due to the speed limit of light. The radius of the observ- able universe is 46 billion light years (~1027 m). The radius reachable to us is 14.5 billion light years. Not finite because when we look outwards into every direction we are looking back in time. When we look back far enough, we think we should see the edge of our uni- verse but we see the baby picture of the Big Bang (the 3oK Cosmic Microwave Back- ground Radiation) pervading everywhere, even on our radio and TV screen in between channels. The human concept of both finite and infinite breaks down.
In the Prajna Paramita Heart Sutra …form is not different from emptiness, emptiness is not different from form, form is emptiness and emptiness is form” fits seamlessly into Category-3 embracing opposites; a perfect example of non-duality logic. But that is not enough. We must pursue how conflicts can be totally resolved by exploring the possible solutions in Category-4.

Try answering other  difficult questionsIs an electron (or a photon) a real thing, a particle? Perhaps then we could appreciate the power of Buddhist quadratic category logic,

All truths are describable if you know them. Beware that unthinkable and unspeakable are derived from ignorance of contents in Category-4. Therefore, epistemology should never be in Category-4 which already contains all possibilities of knowledge limits. Otherwise, epistemology can be inserted into Category-4 of any issue, a mistake Jizang made.

In conclusion, classical Aristotelian category logic is divisive and restrictive but Buddhist category logic is inclusive and inspiring. Buried deep in our history for over 2,500 years, this ancient Buddhist wisdom should be offered as contemporary category logic. Quantum physicists and cosmologists are still wrestling with fuzzy logic because they lack an alternative logic system. Anyone in pursuit
 


of the truth, especially peace-makers, world leaders and problem- solvers must learn to overcome the Aristotelian divisiveness. This article has house-cleaned Buddhist ideologies throughout our history and unearthed this treasure of wisdom. Thus, promoting the education of Buddhist quadratic category logic is the first priority for directing civilization onwards. This logic has nothing to do with the ethics of tolerance or inclusiveness even though they maybe by- products. It is a complete, self-consistent logic system, a power tool that enforces an accurate and thorough description of reality. Venn diagrams are indispensable for accurately understanding the logic.

ACKNOWLEDGEMENT
I am indebted to Professor Dr. Ching Lo, principle of the Academy of Wisdom & Enlightenment (AWE) for his mentorship and guidance without which this work is impossible. Many thanks to Tammy Cheng and Jenny Chang for editing, artwork and formatting this article.


***
 

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