Metaphysics of Self-as-an-End: Dimensional Sentence-Form Theory
SAE形而上学：维度句式论  /  SAE-Metaphysik: Dimensionale Satzformenlehre

Terminology

Term	German	Sentence-Form	Level
Law of Deduction	Schlussgesetz	A therefore B	1DD–4DD
Instrumental Hypothetical Imperative	instrumentaler hypothetischer Imperativ	Want A, therefore do B	5DD–12DD
Self-aware Hypothetical Imperative	selbstbewusster hypothetischer Imperativ	I want A, therefore I do B	13DD
Teleological Hypothetical Imperative	teleologischer hypothetischer Imperativ	My purpose is A, therefore I do B	14DD
Absolute Categorical Imperative	absoluter kategorischer Imperativ	The other's purpose is A, therefore I cannot not do B	15DD
Cooperative Categorical Imperative	kooperativer kategorischer Imperativ	I for purpose A, the other for purpose B, we cannot not do C	16DD

The Law of Deduction (Schlussgesetz) uses Gesetz rather than Imperativ because an imperative presupposes a subject who can receive a command; at this level there is no subject. Kant's kategorischer Imperativ is split in this framework into three distinct levels: the Teleological Hypothetical Imperative (14DD), the Absolute Categorical Imperative (15DD), and the Cooperative Categorical Imperative (16DD). Levels 15DD and 16DD retain kategorisch because they are genuinely unconditional. Level 14DD, though classified by Kant as kategorisch, remains hypothetisch in this framework: an anchored purpose is not the same as an unconditional one.

I. The Problem: Why 0D Is Needed

There is a class of philosophical disputes that cannot be resolved. Not because evidence is lacking, and not because reasoning is flawed, but because the two sides are using different levels of sentence-form to talk about the same thing. One person says "A causes B." Another says "I chose B for the sake of A." Both sentences appear to address the relation between A and B, but their sentence-forms are entirely different — the first is a causal statement, the second is a purposive choice. Unless the level-difference between these two sentence-forms is identified, the dispute will never terminate.

This paper addresses that problem. Not any particular dispute, but the structural reason why disputes fail. The answer is: each dimensional level has its own sentence-form. Mixing levels produces error. This is not a moral error, not an epistemological error, but a metaphysical error.

To explain why sentence-forms stratify, why they misalign, and why they produce remainders, we need an origin point. That origin is 0D. In this paper, 0D is a semantic zero-point (semantischer Nullpunkt): the state in which no distinction has yet occurred. It is not an entity, not the historical origin of the physical universe; it is the origin of sentence-form generation. 0D is a structural origin, not a temporal one. Just as a coordinate system presupposes an origin, the dimensional sequence presupposes 0D.

A meta-question must be addressed: what sentence-form does this paper use to describe 0D? This paper's own language operates at least at the level of the Self-aware Hypothetical Imperative (13DD) to the Teleological Hypothetical Imperative (14DD). Using these sentence-forms to describe "no sentence-form" is itself a level-crossing. Three responses: First, this level-crossing is unavoidable — this paper is itself a fish-trap (Zhuangzi's metaphor). Second, this paper's description of 0D is not a reproduction of 0D but a retroactive marking of a logical position from a higher level. Third, this paradox is a self-aware boundary of the framework, not a defect.

II. Axiom and Corollaries: The Hundun Principles (Hundun-Prinzipien)

Axiom: Necessity of Distinction (Prinzip der Unterscheidungsnotwendigkeit). Once any repeatable difference appears, "distinction" is irreversibly established. 0D cannot sustain itself; structure necessarily emerges. This is not because some subject "wants" development — the option of not-developing does not structurally exist.

Corollary I: Cost of Fixation (Prinzip der Fixierungskosten). Once a distinction is established, it must be fixed by some sentence-form or rule in order to be transmissible. Fixation necessarily loses information — it compresses continuous, overflowing difference into discrete, repeatable form. Within any level, the boundary is invisible; the remainder is always outside the field of view. This is structural ignorance, not epistemological limitation.

Corollary II: Remainder Generation (Prinzip der Restgliedgenese). The information lost through fixation does not disappear. It persists as a "point of non-closure" — a remainder (Restglied) — and drives the next round of distinction. No closed system exists.

The internal relation among the three is generative, not parallel: the Axiom is the engine, Corollary I is the cost of the engine's operation, Corollary II is the consequence of that cost. One generates two, two generates three. This mirrors Laozi's Dao De Jing, Chapter 42: "The Dao generates one, one generates two, two generates three, three generates the ten thousand things." The Dao is 0D; the Dao generates one is the Axiom; one generates two is Corollary I; two generates three is Corollary II; three generates the ten thousand things is the unlimited unfolding of the dimensional sequence.

III. Core Theorems

Theorem 1: Chisel-Construct Identity (Meißel-Konstrukt-Identität)

Chisel (Meißel — the negating penetration of existing structure) and Construct (Konstrukt — the establishment of new structure) are not two independent operations but two sides of the same process. The deeper the chisel, the higher the construct; the higher the construct, the deeper the chisel.

Derivation. By the Axiom (Necessity of Distinction), structure necessarily emerges — this is construct. By Corollary I (Cost of Fixation), the fixation of existing structure necessarily loses information, and the lost portion constitutes the direction of penetration — this is chisel. The two are inseparable. Chisel is construct.

Criterion: if a putative "chisel" does not give rise to a new transmissible sentence-form, it is not a chisel. If a putative "construct" does not produce a new point of non-closure, it is not a construct.

Theorem 2: Dimensional Sequence Generation (Dimensionssequenzgenese)

Starting from 0D, each chisel-construct cycle generates a new dimensional level. Each level has its own sentence-form, and the differences between sentence-forms are not differences in wording but differences in the source of compulsion.

Law of Deduction (Schlussgesetz), 1DD–4DD: A therefore B. Source of compulsion: causal or structural necessity. No subject, no desire, no choice. 1+1=2 does not require anyone to agree for it to be true. Logic, mathematical proof, and causal inference all operate at this level.

Instrumental Hypothetical Imperative (instrumentaler hypothetischer Imperativ), 5DD–12DD: Want A, therefore do B. Source of compulsion: conditional instrumental rationality. This level has desire, has purposive drive, but no self-awareness of "I." Kant's hypothetical imperative covers this level, but he did not see the eight dimensions of variation within it.

Self-aware Hypothetical Imperative (selbstbewusster hypothetischer Imperativ), 13DD: I want A, therefore I do B. Source of compulsion: self-reference of the subject — "I" becomes the source of choice. The difference between "want A" and "I want A" is not the addition of a pronoun but a qualitative shift in the attribution of choice.

Teleological Hypothetical Imperative (teleologischer hypothetischer Imperativ), 14DD: My purpose is A, therefore I do B. Source of compulsion: purpose-fixation. From "I want A" to "my purpose is A," the purpose no longer drifts. But the purpose is still "mine"; the structure is still hypothetical. Kant called this level the kategorischer Imperativ, mistaking the anchoring of purpose for unconditionality.

Absolute Categorical Imperative (absoluter kategorischer Imperativ), 15DD: The other's purpose is A, therefore I cannot not do B. Source of compulsion: the other's purpose enters my constraint conditions. Two qualitative shifts: first, the purpose is no longer "mine" but "the other's"; second, the modality shifts from "therefore" to "cannot not" — it is not that I choose to do B, but that the other's existence as a subject eliminates the option of not doing B. "Ought" is the voice of 14DD; "cannot not" is the voice of 15DD.

Cooperative Categorical Imperative (kooperativer kategorischer Imperativ), 16DD: I for purpose A, the other for purpose B, we cannot not do C. Source of compulsion: the encounter of multiple subjects' purposes. There are two independent purposes (A and B), two independent subjects, and one joint action (C). C belongs neither to A nor to B; it is forced into existence by the encounter of two different purposes. The two subjects each maintain their independence — they do not fuse into a "we" sharing a single purpose — but each retains their own purpose and, in a situation of "cannot not," produces joint action.

Theorem 3: Remainder Persistence (Restgliedpersistenz)

Every level's construction produces a remainder. Remainders do not disappear, cannot be dissolved, and serve as the source of the bridge to the next level.

Derivation. By Corollary I, fixation necessarily loses information; remainders necessarily exist. By Corollary II, remainders cannot be enclosed within the current level; they persist as points of non-closure driving the next round of distinction. In the context of this paper, remainder persistence acquires a specific linguistic meaning: every sentence-form, in fixing the content of its level, omits whatever that sentence-form cannot capture. This omission persists as a sense that "something is not right about this sentence," until a new sentence-form emerges to accommodate it.

Corollary: Termination Structure of Inquiry (Terminierungsstruktur der Rückfrage)

Any hypothetical imperative chain, when interrogated with "why A?", will terminate at a sentence-form of some higher level. The Instrumental Hypothetical Imperative (5DD–12DD) has four possible termination points: 13DD, 14DD, 15DD, 16DD. The Self-aware Hypothetical Imperative (13DD) has three: 14DD, 15DD, 16DD. The Teleological Hypothetical Imperative (14DD) has two: 15DD, 16DD. The Absolute Categorical Imperative (15DD) has one: 16DD. The Cooperative Categorical Imperative (16DD) has no higher termination point — it is itself the terminus.

This yields two consequences. First, the lower the sentence-form, the more termination points, and therefore the more susceptible it is to distortion. Second, the higher the sentence-form, the fewer termination points and the greater the determinacy. At the Cooperative Categorical Imperative there is only one possibility — no retreat, no substitution, no room for distortion.

IV. Subject Condition: Ignorance and Presumption (Unwissenheit und Anmaßung)

The subject's initial condition for entering the dimensional sequence is "ignorance and presumption" — not knowing which level one occupies, yet assuming one's sentence-form applies to all levels. This is not psychological criticism but structural condition: the subject always begins speaking before having located its own level; speaking is fixation, fixation produces a remainder, and chisel thereby begins.

By Corollary I, ignorance is structural — within any level the boundary is invisible. "Presumption" is the subjective expression of the Axiom — "cannot not develop," experienced from the subject's perspective as "I can handle this." The subject does not begin to chisel out of humility. It is precisely out of presumption — assuming its sentence-form is sufficient — that the subject collides with a remainder and is forced into the next level.

V. Four Types of Sentence-Form Misalignment (Vier Typen der Satzformverschiebung)

Type 1: Causalization (Kausalisierung). Purpose or normativity is rendered as causation. "For A" becomes "A causes." This uses the sentence-form of the Law of Deduction to process content belonging to the Teleological Hypothetical Imperative or higher. Example: "He chose this path" is rendered as "The environment caused him to take this path."

Type 2: Instrumentalization (Instrumentalisierung). A non-exchangeable purpose is downgraded to an exchangeable means. The "cannot not" of the Absolute Categorical Imperative is replaced by the "if you want, then" of the Instrumental Hypothetical Imperative. Example: "I cannot not respect him" becomes "If you want to be a good person you should respect others."

Type 3: Self-reference Loss (Selbstreferenzverlust). Subject-choice is disguised as natural necessity. "I choose" is deleted, causing the Self-aware Hypothetical Imperative to regress to the Instrumental Hypothetical Imperative. Example: "I decided not to lie" becomes "One should not lie."

Type 4: Alterity Erasure (Alteritätstilgung). Mutual recognition or conflict is disguised as single-subject coherence. "The other's purpose" is deleted, causing the Absolute Categorical Imperative or Cooperative Categorical Imperative to regress to the Teleological Hypothetical Imperative. Example: "He and I each have our own purpose, and we cannot not negotiate" becomes "My plan is the most reasonable."

Diagnostic Case: Free Will versus Determinism

The determinist says: "Your action is the result of neural firing in your brain. Given the same initial conditions, the same output follows." This is the sentence-form of the Law of Deduction (1DD–4DD): A therefore B. No subject, no choice, no purpose.

The libertarian says: "I chose to do this. I could have done otherwise." This is at minimum the sentence-form of the Self-aware Hypothetical Imperative (13DD): I want A, therefore I do B. A subject is present and claims itself as the source of choice.

The dispute is irresolvable not because one side is right and the other wrong, but because they are operating at different sentence-form levels. All four types of misalignment are visible: Type 1 (Causalization) — the determinist renders purposive choice as causal chain; Type 3 (Self-reference Loss) — the determinist deletes the "I" from "I chose"; Type 2 (Instrumentalization) — the compatibilist attempt downgrades "I chose" to a conditional; Type 4 (Alterity Erasure) — framing as "is the universe deterministic, yes or no?" erases the subject entirely.

The prediction of this paper: the dispute will remain irresolvable as long as participants do not identify the sentence-form levels at which they are respectively operating. Once the levels are identified, the dispute ceases to be a single argument and becomes a set of distinct, level-specific questions.

VI. Rays: Extensions in Multiple Directions

Ray 1: Kant's Imperatives

Kant's hypothetical imperative corresponds to the Instrumental Hypothetical Imperative. His categorical imperative actually corresponds to the Teleological Hypothetical Imperative — the subject has established its own purpose, the purpose is anchored, but it is still "my" purpose. Kant mistook purpose-anchoring for unconditionality.

Kant's three formulas all use the sentence-form "Handle so, daß..." (act such that...) — the imperative structure of the Teleological Hypothetical Imperative. His second formula attempts to express the content of the Absolute Categorical Imperative (15DD), and his third formula attempts to express the Cooperative Categorical Imperative (16DD), but both are written in a 14DD sentence-form. This is a superposition of Type 2 (Instrumentalization) and Type 4 (Alterity Erasure): he wrote "cannot not" as "ought" (instrumentalization) and wrote a two-subject structure as a single-subject command (alterity erasure). Kant saw the right thing — persons are ends, not merely means. But the tools he could write with were limited to 14DD sentence-forms. The pearl is at 15DD–16DD; the casket is at 14DD.

Ray 2: Socrates

Socrates had no theoretical framework for sentence-form levels, but what he did in practice was force interlocutors to expose their sentence-form errors. A typical Socratic dialogue: the interlocutor uses a concrete description from the Instrumental Hypothetical Imperative to serve as a definition belonging to the Teleological Hypothetical Imperative ("Courage is not retreating on the battlefield"). Socrates presses ("Does perseverance in business count as courage?"), and the mismatch is exposed. Socrates' "I know that I do not know" is a naive expression of Corollary I. He encountered the remainder, but he had no theory of remainders.

Ray 3: Analytic Philosophy

The three most important contributions of analytic philosophy to sentence-form diagnosis are Russell's type theory, Ryle's category mistake, and Wittgenstein's language-games. All three diagnose sentence-form errors, but all three work horizontally — distinguishing categories on the same plane, without vertical dimensional levels, without generative logic. Dimensional Sentence-Form Theory explains a more fundamental question: why are type constraints repeatedly violated? The answer is the subject condition — the subject always begins speaking before having located its own level, using old sentence-forms to forcibly fix new-level content.

Ray 4: Eastern Philosophy

Eastern philosophy did not make horizontal category distinctions at the sentence-form level, but it did something else: it directly identified the fundamental limitation of sentence-forms as such. Zhuangzi's "catch the fish, forget the trap" — the trap is the sentence-form, the fish is the level you are trying to reach, and once you arrive you should discard the trap. Chan Buddhism went further: "not established on words" — it canceled the tool of sentence-forms entirely. Wang Yangming's "unity of knowing and acting" pointed out that the sentence-form of knowing cannot replace the practice of acting.

Eastern philosophy saw the ultimate consequence of Corollary I at the sentence-form level: every sentence-form necessarily has a remainder, and therefore no sentence-form is final. But the Eastern solution is "transcend sentence-forms" without providing the level-structure or generative logic. Dimensional Sentence-Form Theory's position: acknowledge the limitation of sentence-forms (consistent with the East) while providing the level-structure of sentence-forms (consistent with the West).

VII. Non-Trivial Predictions

Prediction 1: There Is Always a Remainder (Restglied-Persistenz)

No level can close itself off. There is always overflow. Falsification condition: find a completely self-consistent system with no remainder of any kind. Operationalization: take any theoretical system considered "complete" and examine whether its axiom system contains undecidable propositions or external dependencies. Historical case: Gödel's incompleteness theorems demonstrated the existence of remainders in arithmetic systems.

Prediction 2: High-DD Content Tends to Misuse Low-DD Sentence-Forms (Abwärts-Satzformverschiebung)

Content at higher dimensional levels, when expressed, tends to slide into lower-level sentence-forms. Falsification condition: produce a case in which high-DD content in public discourse stably maintains a high-DD sentence-form over the long term. Operationalization: take canonical philosophical texts, annotate the sentence-form level of the content the author is attempting to express, and compare it with the sentence-form level actually used. This paper has performed this on Kant's second formula: content at 15DD, sentence-form at 14DD.

Prediction 3: Low-DD Subjects Cannot Comprehend High-DD Sentence-Forms (Aufwärts-Satzforminkompatibilität)

A subject at a lower dimensional level cannot fully comprehend the content of a higher-level sentence-form. Falsification condition: without high-DD sentence-form training, produce stable reproduction of high-DD judgments through information input alone. Case: post-Kantian readings of the categorical imperative that instrumentalize it — downgrading "cannot not" to "ought," then further to "if you want to be a good person, you should." This degradation chain (15DD → 14DD → 5DD–12DD) is a concrete instance.

Prediction 4: Contradiction Always Occurs Across Levels (Interlevel-Widerspruchsprinzip)

When participants share the same level of sentence-form, disagreements can be compressed to different values within the same rule; irresolvable disputes are more commonly generated by cross-level sentence-form substitution. Falsification condition: find an irresolvable pragmatic dispute arising within a single level, involving no sentence-form elements from any other level. Case: Ryle's diagnosis of Cartesian mind-body dualism is a cross-type case; Dimensional Sentence-Form Theory specifies it further as cross-level — Descartes used the causal sentence-form of the Law of Deduction to process the subject-choice content of the Self-aware Hypothetical Imperative.

VIII. Conclusion

This paper begins from a simple observation: philosophical disputes are often irresolvable not because of insufficient evidence or faulty reasoning, but because the disputants are using different levels of sentence-form to talk about the same thing. Sentence-form misalignment is not a moral error, not an epistemological error, but a metaphysical error.

The contributions of this paper are three: (1) Sentence-form / level isomorphism (Satzform-Dimensionsisomorphie): different levels have different sources of compulsion and different sentence-forms, constituting a complete sequence from the Law of Deduction to the Cooperative Categorical Imperative. (2) Generative skeleton: 0D → distinction-fixation → remainder → next-level sentence-form, extending the horizontal category distinctions of Russell, Ryle, and Wittgenstein into a vertical dimensional-level system. (3) Diagnostic tools: four types of sentence-form misalignment applicable to concrete sentence-form diagnosis of philosophical texts, together with four falsifiable structural predictions.

Questions left for subsequent treatment: the status of the Hundun Principles themselves (do they occupy a dimensional level, or are they prior to all levels?); the precise structure of bridges between dimensions; whether the Eastern path of "transcending sentence-forms" and the SAE path of "specifying sentence-forms" ultimately converge; and how Dimensional Sentence-Form Theory accounts for the sentence-form generation of a posteriori consciousness.

This paper is part of the Self-as-an-End (SAE) framework series.