Appendix A The formal system and proofs¶

This appendix serves readers taking the slow route. It gives fully symbolic statements and derivation outlines for the numbered propositions in the chapters, with the corresponding chapter number in parentheses after each entry. Readers taking the fast route can skip every page without losing the conclusions. Two sentences define the division of work. Chapter 8 applies the theorem outside its domain, and its general conditional schema is not formalized again. Section 10 of this appendix is the master record for the premise audit in Chapter 9, Section 1.

1. Notation. E1(x): the quantificational sense, x taking a value in the domain. E2(x): the designated sense, x established through individuation in a network of language and names. E3(x): the sense of real existence, surviving decomposition and analysis or giving rise to cognition as an object, by the opponent's criteria. D2(x): x has intrinsic nature, a conjunction of three conditions, D2(i) not constituted in dependence on other dharmas, D2(ii) not changing with conditions, and D2(iii) not produced by conditions. A D3-type proposition: an assertion of the form ∃x[E3(x) ∧ D2(x)] and its instances. ⊢A: assert A. ⊣A: reject A; ⊣ marks force and is no new truth-functional connective. π(t): the subject t has a referent in the manner of E3 and D2. ≠: numerical difference within a framework. ≠*: independent difference, being distinct entities established without depending on another. Rel(x): x's identity conditions involve difference and relations, with difference at either the level of things or the level of types. P: ∀x Rel(x). See Chapters 1, 3, and 4 for the above.

2. Definitions and premises. D1 establishes the three senses as initial predicates. Either of the opponent's two criteria supplies the substantive content of E3. Chapter 1 records their division of work and the fact that they need not coincide. In D2's threefold conjunction, D2(i) is the root. D2(ii) and D2(iii) manifest it in the directions of change and origin. The derivation uses only D2(i). D3 and the scope clause: the theorem's quantifier domain comprises all D3-type propositions, with all E1 and E2 assertions excluded. The clause delimiting definition: identity and difference are logical constants, given rules but no definitions. Leibniz's law is restricted to extensional contexts. Change := difference across indices. Dependence, constitution, and production serve according to the opponent's usage. Rules of grading: dependence has three levels, causal, contrastive, and that of statement. The dependence D2(i) rejects includes at least the contrastive level. The warrant is the texts both parties use, entered in evidence in Chapter 4. P has the status of analytic truth, established in three steps: content is exclusion, at the granularity of options open so far as we know; identity and difference work as a pair; the test of a singleton domain, with difference allowed to be difference of type.

3. The lemmas. Lemma 1, weak rejection: ⊣A does not entail ⊢¬A, where ¬ is internal negation. Rejection can result from presupposition failure, in which case ¬A inherits the same presupposition. Asserting ¬A instead keeps π alive. If ¬ is external negation, mutual inference is restored, but the result merely registers rejection and adds no commitment (Chapter 3).

Lemma 2, presupposition cancellation: when ⊢¬π(t) holds, F(t), ¬F(t), and their combinations, wherever they have t as subject and inherit π, are all rejected. Their unified authorization comes from ⊢¬π(t) (Chapter 3). There are three criteria for its application, the π criteria. First, resistance to reformulation. The proponent rejects an E2 reformulation of the sentence as changing the question. The opponent's usage establishes π's presence. This extends Chapter 2's discipline of authorization. Second, load. The sentence must carry an inferential load in the opponent's argument that requires E3 strength. Where an E2 reading can carry it, the scope clause preserves it, and the judging machine does not apply. Third, form of expression. Classical texts are sorted by their closing forms of expression. The refutational family enters the judging machine; the family of stages of teaching does not. See the table in Section 12 of this appendix. All three are criteria of type, publicly testable, in a different category from discretionary sentence-by-sentence judgement.

Lemma 3, fixing the scope of application: the gap between ≠ and ≠ is exactly D2(i). Those who abandon independence keep ≠, untouched by the core derivation. For those who retain independence, ≠ is ≠, and the identity-or-difference dilemma closes (Chapter 3). Its outline: on the identity alternative, Leibniz's law transfers properties and the two ends collapse. On the difference alternative, if x ≠* y, x should be available apart from y, yet it is unavailable apart from y. The relational lemma: a relation with intrinsic nature depends on neither end and can relate to neither. It fails to be a relation, so all relations fall within E2 (Chapters 3 and 7).

4. The tetralemma and the decision on S4. Notation for the four statements: S1 is F(t), S2 is ¬F(t), and S3 is F(t) ∧ ¬F(t). S4 has two candidates, left unresolved in Chapter 3 and decided here. Candidate (a), propositional notation: S4 is ¬F(t) ∧ ¬¬F(t), literally corresponding to the two negations in the verse form "neither a certain thing nor not that thing." Candidate (b), stance notation: S4 rejects both F(t) and ¬F(t) while retaining π(t), both rejected, yet the referent preserved. The decision adopts (b), recording (a) as its propositional projection.

Three reasons follow. First, agreement of levels. Nāgārjuna's move is ⊣S1 through ⊣S4. If S4 is already a rejecting stance, the fourth rejection targets not rejection itself but the π hidden beneath it. The "Examination of Nirvāṇa" asks a question of exactly this form: "By what is this, which is neither existent nor nonexistent, distinguished?" Those who maintain the neither-existent-nor-nonexistent account are still distinguishing that nirvāṇa. The subject's E3 presupposition is untouched.¹ Second, the four branches remain distinct. Under candidate (a), S3 and S4 both reduce to contradictions in classical logic, and the four statements lose their fourfold character. Under candidate (b), each branch is a different repair strategy for keeping π alive. Pragmatics supplies their distinctness without requiring nonclassical logic. Third, the text's own evidence. The commentary on the "Examination of Dharmas" poses a question: "Elsewhere the Buddha speaks of leaving behind 'neither existence nor nonexistence.' Why is it said here that 'neither existence nor nonexistence' is what the Buddha taught?" The distinction between leaving behind the "neither existence nor nonexistence" position and stating "neither existence nor nonexistence" is precisely the distinction between ⊣S4, which rejects stance and presupposition together, and S4 as a stance retaining the presupposition.¹

Record one further observation. Qingmu’s commentary on the “Examination of Wrong Views” states explicitly that the four branches are mutually dependent: "there is the unbounded because it is opposed to the bounded"; "because of mutual dependence." Once the third branch is refuted, the fourth cannot be established "because there is no mutual dependence." The tetralemma's grid is itself a network of mutual dependence. Each branch in the grid has already demonstrated P on its own.¹

5. The core derivation, T-strong. These steps correspond one by one to Chapter 4's five steps. First, suppose ⊢ E3(a) ∧ D2(a), the content of a D3-type assertion, with a as witness. Second, as the subject of the assertion, a is already individuated and its identity conditions are present. Third, by P, Rel(a). Fourth, by D2(i), ¬Rel(a). Fifth, steps three and four concern the same object in the same respect: ⊥. Thus D3, the definitions, and P jointly entail contradiction. T-strong holds, its strength depending on P's analytic truth. The theorem sentence is its natural-language equivalent: definition requires identity and difference; intrinsic nature cancels identity and difference; hence contradiction.

6. T-middle and T-weak. T-middle has two premises. A1: assertion must individuate its subject. A2: individuation requires contrast. Derivation: the act of asserting D3 presupposes Rel(a), while the assertion's content includes D2(i), hence ¬Rel(a). The contradiction lies between preparatory conditions and content, a pragmatic self-defeat. The reply about reference: a D3 subject has no causal encounter. Its reference is fixed only by a schema, and every sentence of that schema contrasts (Chapter 4). Denying A1 and A2 is harder than denying P. This is what it means for the middle version to be more secure. T-weak is a programme plus established results: refute all four statements for every D3 candidate t that has been advanced. Authorization comes case by case from ⊢¬π(t), or is supplied at once by T-strong. The archive is the twenty-seven chapters (Chapters 3 and 4).

7. Downgrade branches and exits. P loses analytic truth but retains transcendental truth: T-strong is downgraded to the transcendental, exit two. Only an empirical generalization remains: reductio case by case, exit three. The opponent rejects the definitions: a conditional downgrade yielding schematic conditionals, with whoever satisfies the definitions receiving the conclusion (Chapter 2). The opponent rejects logic: no entry into the arena of argument, with four costs placed on record (Chapter 2). Verdict: exit one is reached; two and three have not been triggered; the channels remain open (Chapters 7 and 9).

8. Level clauses and the check for self-application. Clause (a), domain: T's quantifiers range only over object-level D3-type propositions, and ⊣ likewise applies only at the object level. Clause (b), inventory of resources: asserting T uses definitions, logic, and P, without a single E3 item. Clause (c), the exit for self-application: if emptiness is posited as an E3 item, that proposition itself is D3-type and is rejected by the rule. "Emptiness too is empty" is precisely the content of (c) (Chapter 5). The stopping point for the regress of means of knowledge: the denial of P has no content. What has no content has no situation in which it can hold and no functioning role within a framework. The stopping point is warranted by the denial's lack of content, not by a higher means of knowledge (Chapter 5). The check of the scope of application uses the same door: T enters and leaves with long and short, this and that, without a tailored exemption (Chapter 5).

9. The preservation theorem. Part (a), no harm: for every E1 or E2 assertion A, T and the rejections it authorizes leave A's truth conditions unchanged. Two steps: the scope clause supplies the type step; the substantive step is that individuation has never used an E3 foundation. Its universal warrant is conceptual. Grounding is an activity. By definition, a bearer of D2 neither enters into causation nor serves as a condition. As a matter of concepts, it cannot be called upon to do the job (Chapter 6). Part (b), only thus: D2(a) entails that a neither arises, ceases, changes, nor relates, and is wholly inert. The contrapositive gives that whatever functions lacks intrinsic nature (Chapter 6). Taken together: "Because emptiness holds, all dharmas can be established." Formal places of the two truths: conventional truth is the legitimacy of E1 and E2 assertions; ultimate truth is T itself. They distinguish levels of assertion, not worlds (Chapter 6). Corollary, a verdict does not transmit identity: from ⊢¬π(a) and ⊢¬π(b) one obtains ⊣(a ≠ b) and, by the identity horn, the rejection of identity read in terms of intrinsic nature. For any relation R at the level of designation, including = and ≠, the truth conditions of R(a, b) are unchanged; hence the two verdicts yield neither ⊢ a = b nor ⊢ ¬(a ≠ b). The proof has three steps: Lemma 1, ⊣(a ≠ b) does not turn into ⊢¬(a ≠ b); Lemma 2 with the identity-or-difference dilemma, both horns rejected; preservation (a), relations at the level of designation are not altered by the rejections. Passing from joint rejection to nondifference at the level of designation requires two substitutions, writing ⊣ as internal negation and reading ≠ as ≠, and neither belongs to the derivation (Chapter 6).

10. Premise inventory. Lemmas 1, 2, and 3, the relational lemma, and the corollary that a verdict does not transmit identity are derived items, not premises.

11. Dependencies in the assembly. Dependence is one-way. First establish the layer of definitions: D1, D2, D3, and the boundary clause. P is proved independently upon that layer. The core derivation takes P and D2(i) to yield T-strong. T-strong supplies the domain-wide authorization ⊢¬π. Lemma 2 takes ⊢¬π as input and applies to the refutation of all four statements. The results in the chapters fulfil T-weak. Lemma 3 fixes the scope of application at D2(i). The level clauses and criteria put T itself outside the scope of application. Preservation parts (a) and (b) secure the other half through the scope clause and consequences of D2's definition. Two declarations exclude loops. First, the chapters' results do not enter T-strong's premises, as Chapter 7's mode three judged. Second, T supplies ⊢¬π, which then authorizes the treatment of the four statements. The direction is one-way. Refuting the four statements does not feed back into support for T's proof.

The “three-part test” is the set of three π criteria for applying presupposition cancellation. The preservation box names both results (a) and (b), which Section 9 of this appendix sets out separately.

12. Table sorting stages of teaching from refutation. Scope: all places in the verses of T1564 where four branches appear together, plus one quasi-tetralemma, checked throughout the entire work.² Criterion: the closing form of expression, established in Chapter 3. The Mahāvibhāṣā's "A tetralemma should be made" belongs to the family of filling a grid and does not enter this table. See Chapter 3 for comparison.

References¶