Semiotic Elucidation of the Emptiness View

• Hajime Nakamura PDF This is a research paper by Gen Nakamura titled “A Semiotic Clarification of the Ku-kan”. The main contents are as follows:.

1. an attempt to elucidate the empty thought of Buddhism, especially Chukan sect (of Shin Buddhism), using modern semiotic logic rather than conventional traditional logic.
2. by applying symbolic logic, it is shown that there are cases in which the logic of the middle-of-the-road school, traditionally considered illogical, is actually logically consistent.
3. by re-presenting arguments such as Tatsuki’s “Middle Theory” in the form of symbolic logic formulas, we clarify the logical structure of these arguments.
4. he also attempts to interpret the unique logic of Buddhism, such as four-phrase discrimination, in terms of semiotics, but also points out that there are some aspects that cannot be fully explained.
5. raises the issue of whether “hollow” (śūnya) can be equated with the mathematical “zero” and is an issue for future work.
6. this study is a pioneering attempt to link Buddhist thought and modern logic, and suggests the possibility of a dialogue between the two. The paper is an ambitious study that offers a new perspective on the comparative study of Buddhist and Western logic and attempts to shed light on the logical aspects of Buddhist thought.

causality=Buddhist logic. Formal Logic ⇔ Symbolic Logic Describing Buddhist logic in terms of symbolic logic rather than formal logic From the standpoint of traditional logic, Indian logic appears to be asymmetric, but from the standpoint of symbolic logic, it is not asymmetric but rather traditional logic’s adherence to useless rules.

Concepts are always genitive

• This mountain has fire in it
• There is fire here!
  • Fire is “here.”
  • Indian logic
    • Here” is the subject.
  • traditional logic
    • Fire, here and there Inclusion Decision
    • There is fire here. Existence judgment

$-a+b=1$ $a<b$

• The meaning of the former expression is that “everything is either a or b.

  • $\neg A\vee B=\omega$ that $a-b=0$

• There can be nothing that is “a” and not “b”.

• $A\wedge \neg B=\varnothing$. $ab=a$

• a and what is b is a

• $A\wedge B=A$.

These have no principalities.

• I don’t understand the usage of this symbol.
  • Understood! I added it.
  • I’m not sure whether to write it in a set style or logical formula style for a modern writing style.

All a is b: $a\subset b;ab=a;a-b=0.$

• [$ A \subset B; A \cap B = A; A\cap\overline B=\emptyset.
  • So this is the same thing expressed in three different ways.

No a is b: $a\subset -a;a-b=a;ab=0.$

• $a\subset -a$？？typo?
  • Opinion that AI is also a typo
  • $a\subset -b$.
• [$ A \subset \overline B; A \cap \overline B = A; A\cap B=\emptyset.

Some a is b: $a-b\ne a;ab\ne 0.$

• [$ A \cap \overline B \neq A; A\cap B \neq \emptyset.

Some a is not b: $ab\ne a;a-b\ne 0.$

• [$ A \cap B \neq A; A\cap \overline B \neq \emptyset.

(formal) proposition

$(x).\varphi x.\subset (\exists x).\varphi x$

• $\varphi a.\varphi b.\varphi c.\cdots \cdots :\subset \varphi a\vee \varphi b\vee \varphi c\cdots \cdots .$
• Eh,$\subset$がその位置にくるのこの記号の意味の意味は今の$\subset$ ではなかったりする？
• These equations will be expressed and explained in modern TeX.
  • $(\forall x)\varphi (x)⟹(\exists x)\varphi (x)$
  • Meaning: if φ(x) is true for all x, then φ(x) is true for at least one x.
  • Explanation: This is a basic logical rule that expresses the inference from all quantifiers to existential quantifiers. If “all” holds, then “at least one” must also hold.
  • $\varphi (a)\wedge \varphi (b)\wedge \varphi (c)\wedge \cdots ⟹\varphi (a)\vee \varphi (b)\vee \varphi (c)\vee \cdots$
  • Meaning: φ(a), φ(b), φ(c), … are all true, then φ(a) or φ(b) or φ(c) or … is true.
  • Explanation: This shows a logical implication relation. It represents an inference from a conjunction (AND) to an election (OR). If all propositions are true, it means that at least one (in fact all) of them is true.

EIO $ab=0.bc\ne 0.a-c\ne 0$

• $A\cap B=\varnothing ,B\cap C\ne \varnothing ,A\cap \overline{C}\ne \varnothing$
• EIO refers to three of the four propositional forms (A, E, I, O) used in traditional logic. - E: Full-symbol negation (No A is B) - I: Singular Affirmative (Some A is B) - O: Special-person negation (Some A is not B)
  • This formula expresses these three propositional forms in set-theoretic terms:
    • $A\cap B=\varnothing$No intersection of A and B (E: No A is B)
    • $B\cap C\ne \varnothing$ There is an intersection of B and C (I: Some B is C)
    • $A\cap \overline{C}\ne \varnothing$: Some A is not C (O: Some A is not C)
  • These formulas are a set-theoretic interpretation of the propositional forms of traditional logic, expressed in modern symbolic logic.

It is not illogical that the idiosyncratic judgment takes the form of an existence judgment.

This is the story of “Japanese omits the subject! Japanese is ambiguous!” I believe that this is a counterargument to the Western view that “Japanese omits the subject!

• That is, that people in languages that do not omit the subject are limited to certain thoughts by the constraints of the language and therefore cannot understand some of the conversations of people in languages that do omit the subject.

$(\exists x).\varphi (x).\equiv .\sim (x).\sim \varphi x$

• x for “people” and phi for “going to see cherry blossoms”.

• What do equiv and sim mean?
  • The one like sim is negative, equiv means equivalence as it is.
• $(\exists x)\varphi (x)\equiv \neg (\forall x)\neg \varphi (x)$
  • Meaning: “some x exists and φ(x) is true” is equivalent to the negation of “φ(x) is false for all x”.
  • Commentary:.
    • $(\exists x)\varphi (x)$ uses the existential quantifier, which states that “some x exists and φ(x) is true”.
    • $\neg (\forall x)\neg \varphi (x)$ uses the negation of the full symmetric quantifier.
      • $(\forall x)\neg \varphi (x)$ means “φ(x) is false for all x”.
      • By negating its entirety, it means “φ(x) is not false for all x”.
    • $\equiv$ denotes logical equivalence.
  • This expression is one of the key theorems for quantifier duality. It expresses the relationship between existential and omnisymmetric quantifiers and shows that one can be defined by the other.

- [[Special name judgment]]: "Some A is B." - [Existence judgment: "A exists which is B." - They logically express the same content.

What has smoke is what has fire Where there is smoke, there is fire.

• Furthermore, the above rewrite becomes “there is no smoke without fire” (=where there is no fire, there is no smoke)

In traditional logic, which was perfected in the Western Middle Ages following Aristotle, only the former was treated. The latter had to be rewritten in the form of a Priestly Affirmative Inclusion (A) like the former, or else in the form of a hypothetical inclusion.

• We will explain this part using the notation of modern symbolic logic.
• All symmetric affirmative decisions (A):$(\forall x)(Sx\to Px)$“All S are P”
• Hypothetical decision:$(\forall x)(Sx\to Px)$“If x is S, then x is P”
• Existence Decision: $(\exists x)(Sx\wedge Px)$ “There exists x such that there is S and it is P.”
• Traditional logic since Aristotle dealt primarily with forms of the whole-person affirmative judgment (A). Existence judgments and judgments expressing special relations were often rewritten as either whole-person affirmative judgments or hypothetical judgments.
• For example, the existence judgment “some S is P” could traditionally be rewritten as
  • As a full name positive judgment:
    • $(\forall x)((Sx\wedge (\exists y)(Sy\wedge Py))\to Px)$
    • “For every x such that there exists an S which is S and which is P, x is P.”
  • As a hypothetical judgment:
    • $(\forall x)(Sx\to (\exists y)(Sy\wedge Py))$
    • “If x is S, then there exists S such that P”
• These rewrites were a way to get around the limitations of traditional logic, which cannot deal directly with existence judgments.
• However, in modern symbolic logic, such complicated rewriting is no longer necessary because existence judgments can be expressed directly as $(\exists x)(Sx\wedge Px)$.

• $\psi \alpha$
• $\varphi \alpha$
• $(x)\varphi x.\supset \psi x.$
• $\varphi \alpha .\supset \psi \alpha .$

$(x).\varphi x.\supset \psi x:\varphi \alpha .\supset \psi \alpha$ (Ratzel’s theory of apparent variables)

• this (something or someone close to the speaker (including the speaker), or ideas expressed by the speaker)$(x)$ってのは今で言うところの$\forall x$なのかな
  • Claude said it was.

Traditional logic used the three-stage argument and considered this one of the five items to be redundant

• However, in India, from these five items, instead of a three-stage argument, we proceeded to Russell’s THEORY OF APPARENT VARIABLES
  • Principia Mathematica - Wikipedia
    • Here was also the usage at the time of $\sim$ and $\supset$ and so on In the first place, is the whole nominal quantifier itself a relatively new concept around 1900?
• Full name symbol - Wikipedia
• Theory of apparent variables:.
• This is a concept in logic proposed by Bertrand Russell, a theory that deals with variables bounded by quantifiers (∀ and ∃). The term “apparent variable” refers to a variable bounded by quantifiers.

Each equation in this text is written alongside the modern notation TeX to explain its meaning, and Yes, we will gladly provide an explanation. I will explain the notation of symbolic logic used in this paper, along with the modern TeX notation and its meaning. Aristotle’s Deduction $SaM.MaP.\supset SaP$.

• Modern TeX notation: $(\forall x)(Sx\to Mx),(\forall x)(Mx\to Px)⊢(\forall x)(Sx\to Px)$
• Meaning: All S is M. All M is P. Therefore, all S is P. (Trinitarianism)
• I see what you mean.

Indian logic $(x).\varphi x\supset \psi x:\varphi \alpha :\supset \psi \alpha$ Modern TeX notation:

• $(\forall x)(\varphi (x)\to \psi (x)):\varphi (\alpha )\to \psi (\alpha )$
• This expression can be interpreted as follows
  • $(\forall x)(\varphi (x)\to \psi (x))$: For all x, if φ(x) then ψ(x)
  • $\varphi (\alpha )$: φ(α) holds for some particular α
  • $\to \psi (\alpha )$: Then ψ(α) holds for that same α
• In other words, this entire equation represents
  • There is a general rule that says that if φ(x) for all x, then ψ(x) holds, and if φ(α) holds for a particular α, then ψ(α) also holds for that α.”
  • This formula expresses the application of a general rule (a totally symmetrically quantified proposition) to a specific case, and is an application of Russell’s Theory of Apparent Variables. It shows the relationship between a totally symmetrically quantified x and a particular α. It illustrates the use of variables (x) and free variables (α) bounded by quantifiers.
• Huh, I see. So you’re saying that by using “apparent variables” we can express the three-stage argument this way.
  • The idea of using “apparent variables” that do not correspond to concrete things, which I used to use in programming and accepted without any sense of discomfort, was a “new mathematics” for Hajime Nakamura, which was created 40 years before this paper was written.

Chapter 2 - dragon tree (Dracaena draco) - “Two mutually related things are unparalleled in history (unparallelled).” - Theory of auspiciousness: $\forall (x,y)xRy\to \neg (x=y).\neg (x\ne y)$ - first and fifteenth days of the ninth lunar month (corresponding to new moon and full moon)

“Nothing can be x and non-x.

• Pool: $x(1-x)=x-x^2=x-x=0$
• Schröder: $a(-a)=0=-aa$
• Logic of the time: $\sim (p.\sim p)$
  • Now expression: $\neg (p\wedge \neg p)$

According to Schröder’s Logical Algebra, if a is a and nothing, then a is nothing.

• $a\times 0=0$
• Oh? What’s that funny thing you just said?

Apart from what has already gone and what is yet to go, we do not recognize any other third kind of world path, namely, “what is now passing away.

• $a+-a=1$
• No, I don’t know about that.

I’m talking about how from this point on, the Dragon Tree Empty Argument is an Aristotelian method of reasoning that is NG, but from logical algebra, it’s OK.

• 

• As for the logic of four-phrase discrimination, it’s not even logical algebra.

• That is to say, because it does not assume law of contradiction (logic) or law of the excluded middle.

• While the Saṅghya school adheres to the principle of the law of contradiction or the law of exhaustion, Buddhism tolerates a different kind of thought.

After criticizing you for equating sky with 0 in logical algebra because the original word for sky means zero, which is groundless, I give my reasons why I think sky is a good place to start. But even then, not all of the empty arguments can be expressed by logic algebra, so I’m done talking about how logic may need to be extended - I’m not sure.

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