Steins Theory: A New Axiomatic System Concerning Identity

1. Introduction

The problem of identity—that is, “what makes a thing what it is”—is central to metaphysics and logic. From Leibniz’s (1714) highly insightful “Principle of the Identity of Indiscernibles” (PII), which with its logical conciseness and power set the lofty ideal of individuating entities, to Kripke’s (1980) groundbreaking theory of necessary identity based on rigid designators and origins, which with its profound grasp of modal intuitions provided a seemingly solid foundation for the stability of reference. The outstanding efforts of generations of philosophers have jointly constructed a grand intellectual tradition for our understanding of the persistence and identification of individuals.

However, a noteworthy phenomenon is that these theoretically persuasive frameworks, each powerful within their own domains, often exhibit a regrettable, systematic limitation when dealing with complex boundary cases posed by reality and thought experiments. Leibniz’s strong PII encounters fundamental difficulties, unforeseen in its original conception, when facing quantum identical particles (French & Redhead, 1988). The inherent rigor of the principle instead leads it into a near-paradoxical impasse when handling multiple entities that are completely indistinguishable in all intrinsic properties yet are numerically distinct (such as two electrons). Similarly, Kripke’s elegant rigid theory, aimed at anchoring reference, becomes rather entangled in its explanatory power when dealing with diachronic changes in intrinsic properties (such as the Ship of Theseus) (Chandler, 1975), not to mention its difficulty in accommodating the phenomenon of identical particles presented by quantum mechanics, which challenges the classical view of individuals.

Thus, we face a peculiar intellectual impasse: whether it is Leibniz’s strong program pursuing the identity of properties or Kripke’s historical path focusing on the necessity of origins, each successfully illuminates one flank of the edifice of identity, yet unfortunately leaves the other flank in deeper shadow. Does their common, perhaps unstated ambition—to find a single, absolute criterion for identity—itself constitute an a priori obstacle that prevents us from truly understanding the multidimensional nature of the problem?

This paper argues that the key to breaking this impasse lies not in choosing one existing path over another or making another round of patchwork repairs. On the contrary, it requires us to step back and conduct a meta-level reflection on the problem itself. This paper aims to propose a new analytical framework for identity. This framework does not seek to completely negate predecessors—indeed, Leibniz’s concern with properties and Kripke’s insistence on history will be repositioned and granted their limited legitimacy within the new framework—but rather aims to show that the root of the above dilemmas lies in a category mistake: namely, the erroneous attempt to use an answer from one category to respond to a question from another category.

This paper aims to transform the classic controversies in ontology and semantics into a clear, operable issue of conceptual choice. Thus, rather than solving these difficulties, it dissolves them, providing a new way out for a series of philosophical anxieties arising from category mistakes. Some may think that this theory merely re-describes relative identity in different language, but this is not the case in practice (see section 2.3 for clarification).

Note: In this paper, unless a proposition explicitly specifies a perspective (e.g., “from a certain angle”), the default premise of the proposition is that it is effective in reality (physics). This default is adopted because if a proposition does not pursue real-world effectiveness, any unstated shift in perspective that leads to erroneous conclusions is naturally of little importance as well.

2. Analysis

2.1. Axiom

2.1.1. Axiom 1 (Self-Identity)

∀n ∈ I, n ≡ n. (n) is necessarily identical with itself and forms the foundation of logical reference.

2.1.2. Axiom 2 (Distinctness)

∀n ∀m (n ≠ m → ∃F (F(n) ∧ ¬F(m)))

Core Corollary:

Uniqueness Theorem: Two (n) with identical content must be the same (see reference 1).

Note: See section 3.1 for examples.

2.2. Category Mistakes and Level Confusions: The Case of the Ship of Theseus

When addressing the enduring puzzle of the Ship of Theseus, one highly influential and intuitively appealing solution has been proposed, represented by figures such as David Wiggins (1980). This approach claims that the identity of an object is not guaranteed by its instantaneous properties at any moment but must be carried by spatiotemporal continuity and an uninterrupted historical causal path. The great advantage of this approach is that it successfully captures our deep intuition about “objects persisting through time”—that things are not instantaneous existences but possess a life history or biography.

Another approach, four-dimensionalism (Perdurantism), offers a radically different and metaphysically parsimonious picture (see Heller, 1984; Sider, 2001). With its thorough clarity, this theory very cleverly avoids many pitfalls of diachronic identity. Four-dimensionalism holds that the Ship of Theseus is not a three-dimensional entity that “wholly exists” in time, but a “spacetime worm” extended in four-dimensional spacetime. Each time-slice of the ship is regarded as a temporal part of this four-dimensional object. Thus, so-called “change” merely means that this four-dimensional whole possesses different properties (such as different planks) in different temporal parts. Under this framework, the original Ship of Theseus (as a four-dimensional entity), the ship after replacement, and the ship reassembled from old planks are three different four-dimensional objects. They may have completely identical three-dimensional cross-sections at certain time segments (and thus be indistinguishable at that moment), but as wholes they are naturally distinct. The excellence of this approach lies in transforming the problem of persistence from a troubling question of “identity” into a relatively clear “part-whole” relation.

Closely related to four-dimensionalism is Stage Theory (see Sider, 1996; Hawley, 2001), which retains many advantages of four-dimensionalism while attempting to better accommodate our everyday linguistic intuition that “objects are three-dimensional.” Stage theorists hold that what we ordinarily call the “Ship of Theseus” refers not to the entire four-dimensional worm but to a stage or time-slice at a specific time point. When we say at time t that “the ship is the same,” we are actually saying that there exists a (primitive) counterpart relation between the stage at t and the stage at earlier time t₁, a relation maintained by some similarity and causal continuity. With its conceptual economy, Stage Theory avoids presupposing an identity relation across time and thus exhibits strong theoretical appeal.

However, despite the ingenuity and internal coherence of the above theories, this paper will argue that they face a common, profound dilemma at the normative level. Whether appealing to historical paths, four-dimensional wholes, or counterpart relations, these theories all attempt to provide a single, absolute criterion for identity judgments. To achieve this goal, they must construct “time” or “spatiotemporal location” itself as a constitutive element of an object’s identity. This means that, within their theoretical framework, answering the question “Is the ship at t₁ and t₂ the same?” logically necessarily depends on examining spatiotemporal coordinates or cross-temporal associations.

This paper argues that it is precisely this key theoretical move that inadvertently leads to a “category mistake.” Let us formally reconstruct the proposals of each theory: they actually adopt a domain of: (physical properties of the ship, historical causal path / four-dimensional whole / counterpart relation).

Let us apply n ≡ n to the Ship of Theseus. First, we must clarify the question: when we ask “Is the replaced ship still the original ship?” what is the default comparison category? A reasonable interpretation is that we are concerned with its identity as an “objectively verifiable ship,” i.e., the category (physical ship), with relevant properties {e.g., all particles of the ship and their arrangement and shape}.

Now, let us examine the historical path theory. When answering the above question, this theory actually introduces a new category (physical ship, history), whose relevant properties are {all particles of the ship and their arrangement and shape, historical causal path}. Under the (physical ship, history) category, since the historical path has changed, the replaced ship is naturally different from the original ship.

This paper argues that the controversy here arises from a confusion of categories. The questioner implicitly asks within the (physical ship) category, while the historical path theory answers within the (physical ship, history) category. The two answers—“yes” {under (physical ship)} and “no” {under (physical ship, history)}—are not contradictory because they answer two different questions. Imposing the answer from (physical ship, history) onto the question from (physical ship) constitutes a category mistake. The true value of the historical path theory lies in revealing “history” as an important category, but it errs in treating it as the sole decisive category.

We can reconstruct this formally as follows:

• Initial question: Determine whether entity “ship” at time points t₁ and t₂ is the same.
• Correct (n) domain (according to the initial question): Should contain only properties related to the “ship’s” matter, i.e., physical ship: (all particles of the ship and their arrangement and shape).
• Category mistake of historical / four-dimensional / stage theories:

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  Wiggins effectively adopts history + physical ship (all particles and arrangement, historical causal path).

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  Four-dimensionalism effectively adopts 4D + physical ship (all particles and arrangement, spatiotemporal coordinates).

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  Stage Theory effectively adopts stage + physical ship (all particles and arrangement, counterpart relation).

The process leading to this can be formalized:

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They attempt to answer the identity question based on (physical ship): “Is Ship at t1 ≡ Ship at t2?”

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However, the judgment domain they actually use is (physical ship + history, 4D, or stage).

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Since the materials in the replacement process are exactly the same, we have (physical ship_t1) = (physical ship_t2).

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But because the historical path, spatiotemporal coordinates, or counterpart relation has changed, (physical ship_t1) ≠ (physical ship + history/spacetime/stage_t2).

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Thus, they conclude Ship_t1 ≠ Ship_t2 in answer to the physical ship question.

We thus see an interesting situation: each theory effectively answers a question, but perhaps not the one originally posed. They precisely answer “Does there exist a continuous four-dimensional worm connecting the ship at t₁ and t₂?” or “Is the stage at t₂ a counterpart of the stage at t₁?”, but then treat this answer as the ultimate verdict on “Is the ship structurally the same?” This is like a judge being asked “Does the defendant’s behavior violate Article X of the Criminal Law?” yet consulting the Civil Code and issuing a ruling. The conclusion may be coherent within its own system, but it has quietly switched the arena of debate—this is a category mistake.

Therefore, the true contribution of the historical path theory may lie in its excellent revelation of how “history,” “spatiotemporal whole,” and other properties function as powerful explanatory attributes influencing our identity judgments. Its limitation lies in attempting to elevate such explanatory attributes to metaphysical necessities, thereby having to expand the criteria for evaluating identity questions to maintain theoretical completeness. The value of this theory is that it does not need to make such difficult expansions; instead, by clarifying the attribute sets of questions, it allows different domains/attributes to provide effective, non-conflicting answers to different questions. Not solving these difficulties, but dissolving them.

2.3. Relation to Relative Identity?

When discussing the solution proposed in this paper, one unavoidable theory is Peter Geach’s “relative identity” theory. Frankly speaking, Geach’s work has a surprising intuitive affinity with this paper: both reject the naive notion of “absolute identity” and hold that “whether x and y are identical” cannot be discussed vacuously apart from some standard or category. Geach acutely observed that in ordinary language and philosophical discussion, we never merely say “x and y are identical,” but always say “x and y are the same ship,” “x and y are the same person,” or “x and y are the same word-type.” His core thesis—that identity is relative and depends on the sortal A in “the same A”—undoubtedly captures the essence of much real linguistic practice. Moreover, Geach insightfully emphasized that things within the same domain of objects can appear identical or non-identical under different identity criteria (i.e., different “same A” predicates), without the things themselves changing. This insistence on “things themselves unchanged” makes his theory particularly elegant and economical metaphysically.

Superficially, this paper’s theory is extremely similar to Geach’s relative identity. Taking the Ship of Theseus as an example: Geach would say that under the “same ship” (as material object) criterion, the old ship and new ship are identical; under the “same ship-with-history” (with historical trajectory) criterion, they are not. This is outwardly identical to this paper’s distinction in section 2.2 between the (physical ship) and (physical ship, history) categories. For this reason, a careful reader might well think that this paper merely restates an already existing insight with new terminology.

However, this paper attempts to show that this surface similarity masks a deeper divergence in presuppositions that Geach himself and his followers failed to examine. To see this clearly, we need a more meticulous reconstruction of Geach’s argument than the original.

Let us turn to page ten of the original text on relative identity: “The words in my book (say) will constitute just one universe of discourse; but different criteria of relative identity, all applying with equal right within this universe, will be given by different two-place predicables, e.g., ‘is the same token-word as,’ ‘is the same type-word as,’ ‘is the same dictionary-entry word as.’”

It is evident that Geach’s core thesis is that identity is relative and depends on the sortal A in “the same A.” Within the same universe of discourse, things can be identical or non-identical under different identity criteria (i.e., different “same A” predicates), without the things themselves changing.

That is, Geach’s theory relies on a key commitment: when we switch from identity criterion f1 (e.g., “same ship as material object”) to criterion f2 (e.g., “same ship with causal history”), the individuals x and y under discussion themselves remain unchanged. That is, the old ship x and new ship y are judged identical under f1 and non-identical under f2, but the metaphysical constitution of x and y undergoes no change due to the switch in criteria.

Now, let us rigorously examine this commitment. In the Ship of Theseus example, let:

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x = old ship (object at t₁ constituted by planks M₁…Mₙ)

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y = new ship (object at t₂ constituted by planks M₁’…Mₙ’, with all materials replaced)

When examined under standard f1 (“same material ship”), the attribute set of x and y is A_f1 = {constituted by planks, having the macroscopic shape of a ship, …}. Under this attribute set, x and y are indistinguishable (assuming planks correspond one-to-one identically), so according to f1’s criterion, x ≡ y.

When examined under standard f2 (“same ship with historical continuity”), we actually introduce a new attribute set A_f2 = {constituted by planks, having the macroscopic shape of a ship, …, having a continuous historical path from t₁ to t₂}. The problem is: does the original x possess all attributes in A_f2? Clearly not, because x at t₁ has not yet acquired the historical path from t₁ to t₂—historical path is a relational property added retrospectively and is not part of x’s intrinsic constitution in isolation. Thus, the “old ship entity” judged under f2 is in fact no longer the original x, but a composite x’ = (x, history_path) associating x with the t₂ historical coordinate. Similarly, the new ship y under f2 is regarded as y’ = (y, history_path), but y’s historical path differs from x’s.

We thus see a decisive difference: the pair of entities compared under f2 is not the original (x, y), but (x₁, y₁). And x₁ ≠ x (as attribute sets), y₁ ≠ y. Geach asserts that “things themselves do not change,” but strictly speaking, when we switch identity criteria, we have quietly replaced the entities under discussion—because the application of each criterion itself presupposes a different attribute package. This is not the same batch of things exhibiting different identity relations under different criteria, but different sets of things naturally yielding different conclusions under different judgment frameworks.

Therefore, this paper argues that the “relativity” presented in Geach’s theory is an illusion produced by category confusion. He correctly pointed out that A in “same A” is crucial, but he erroneously assumed a “thing itself” that could freely shuttle between different A’s while remaining unchanged. Once we explicitly write out the complete attribute domain corresponding to each judgment framework, we discover: there does not exist an x that can be maintained as the same entity under both f1 and f2. Geach’s relative identity is in fact a theory built upon category mistakes, yet packaged as a profound insight that identity itself is relative.

2.4. Conservation

2.4.1. The Indistinguishability of Identical Particles in Quantum Mechanics Poses the Most Severe Challenge to Leibniz’s PII, Yet Provides a Natural, Physically Evidenced Model for This Theory

Current philosophical discussions of identical particles, facing the challenge quantum identical particles pose to PII, are mainly divided into two types of solutions: revisionist and revolutionary. The former attempts to salvage some form of individuality principle; the latter abandons individuality itself.

Saunders’ approach undoubtedly represents one of the most ingenious and technically rigorous attempts in the revisionist path. By delicately defining “weak discernibility,” he successfully liberates the discussion from the dead end of intrinsic properties and provides a creative perspective for seeking individuation foundations in relationality. The complexity of this approach and the extensive discussion it has provoked themselves prove its profound philosophical value. However, it is precisely this technical complexity that exposes a potential cost of the scheme: its definition of “purely extensional relational properties,” though striving for precision, inevitably introduces considerable terminological vagueness, such that even its defenders must carefully handle accusations of circular reasoning (Muller & Saunders, 2008). More core is that the entire theoretical edifice of this scheme is built upon an unsettling presupposition: that the “individuality” of identical particles must and can only be “saved” by finding some (even if relational) individuating property. This makes its theoretical efforts—however ingenious—essentially ad-hoc patching to salvage a premise. When applied to states with indeterminate particle number in quantum field theory, this strategy of constantly introducing new relational properties to salvage individuality becomes increasingly obviously ad-hoc: it no longer resembles an elegant inference from the theory itself, but more like an increasingly expensive cost paid to maintain the theoretical premise (that individuality must exist).

Facing the difficulties of revisionism, another revolutionary approach takes a more radical path. Scholars represented by Décio Krause (2011) have put forward a highly subversive argument: quantum particles may not be “individuals” in the traditional metaphysical sense at all. Therefore, laws based on individual identity were misapplied from the beginning. They should be understood as “non-individuals” and require highly specialized mathematical tools such as quasi-set theory to describe.

Krause’s scheme is striking for its conceptual thoroughness and consistency; it unreservedly embraces the most counter-intuitive features of quantum mechanics and decisively breaks with our entire classical framework of objects and spatiotemporal location. This resolute stance is undoubtedly clean and efficient theoretically. However, the corresponding cost of this efficiency is that the concept of “non-individual” itself creates considerable interpretive burdens metaphysically, requiring us to abandon a whole set of deeply rooted intuitive understandings of what “one thing” means.

2.4.2. Solution: An Attribute Set

The above two schemes share a deep misunderstanding: both attempt to find answers to a wrongly posed question. The problem is not “What is the correct individuating property?” but “Under what category are we asking the identity question?”

This paper provides a framework for this. We define an observable particle as: a particle = (particle attributes, coordinates), where the particle attribute set (mass, charge, spin, etc.).

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When we inquire under the domain (particle) = particle attributes, i.e., comparing only intrinsic attributes, all identical electrons are (electron) = (mass m_e, charge -e, spin 1/2...). According to Axiom 1, they are indeed the same electron e under this domain. This explains the root of their indistinguishability.

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When we inquire under the domain (particle) = (particle attributes, coordinates), since coordinates are necessarily different, (particle₁) ≠ (particle₂); therefore they are different particle states. This explains why we observe multiple scattering events in experiments.

Thus, the confusion brought by quantum identical particles stems from erroneously substituting the difference in coordinate attributes into the identity judgment of particle attributes. This paper resolves the contradiction by clearly distinguishing these two attributes: they are both “one” (as logical concepts) and “many” (as manifestations in concrete spacetime). Particle annihilation and creation merely represent the decoupling and re-coupling of the e with different coordinates.

This scheme absorbs the advantage of Krause’s scheme in acknowledging quantum peculiarities (by interpreting “non-individuality” as sameness in particle attributes) while avoiding its radical metaphysical costs (we are still talking about “quanta,” only under different categories); at the same time, it explains why Saunders’ strategy of introducing relational properties seems feasible in some cases (because he erroneously took coordinate attributes as individuating bases for particle attributes) yet fundamentally went astray.

2.4.3. Formal Derivation of Conservation

Let a basic particle state be expressible as: particle = (particle attributes, coordinates), where:

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Particle attribute set (e.g., mass m, charge q, spin s)

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Coordinate set (e.g., position x, time t).

Formalization:

• When the particle’s domain is particle = (particle attributes, coordinate₁), meaning an electron at a certain coordinate.
• Coordinate decoupling (annihilation): (particle attributes, coordinate₁) → (particle attributes), (coordinate₁) ⇒ The particle degenerates into a pure eigenstate (particle attributes). Due to lack of observable basis (coordinate = ∅), it is unmeasurable. ∀ particles (particle attributes, coordinate₁) and (particle attributes, coordinate₂), it can be found that (particle attributes) ≡ (particle attributes) indicates:

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  When two particles’ intrinsic attributes are indistinguishable (particle attributes ≡ particle attributes), regardless of how their spatiotemporal coordinates differ (coordinate₁ ≠ coordinate₂), the particle is the same electron e = (particle attributes) projected in different spacetimes.

Physical interpretation:

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Particle annihilation ⇨ Set decoupling rather than extinction ⇒ e = (particle attributes) becomes a concept.

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Particle creation ⇨ The same e binds to coordinate₂ ⇒ Observed as reappearance. Example: Electron e disappears at coordinate₁ and appears at coordinate₂; this is in fact the coordinate migration of electron e = (q = -1e, m_e, s = 1/2...): (e, coordinate₁) → (e) → (e, coordinate₂). Its electron identity is guaranteed by n ≡ n.

Direct corollary: Conservation laws—that which logic permits exists, and neither annihilates nor updates.

2.5. Symmetry

Max Black’s (1952) symmetric universe thought experiment poses the most extreme challenge to Leibniz’s strong PII. He imagines a universe containing only two completely identical spheres. These two spheres are indistinguishable in all intrinsic properties (mass, composition, shape, etc.) and all relational properties (distance X miles apart, mutual symmetry). Black thereby argues that this is a genuine scenario of “two” things, thus refuting PII—i.e., there is no property that can distinguish them, yet they are still numerically distinct entities.

Traditional response strategies are mainly divided into two types: one questions the metaphysical possibility of such a symmetric universe (e.g., requiring a basis for “numerical difference” itself, which usually returns to some hidden property); the other, like Saunders (2003), argues that relational properties (such as “X miles from one sphere”) can themselves serve as weakened distinguishing bases. However, the former is criticized as ad-hoc, while the latter is ineffective in Black’s original setup because each sphere’s relational properties (“X miles from the other sphere”) remain completely identical.

This paper argues that Black’s challenge and the dilemmas of traditional responses jointly stem from an unexamined presupposition: that “numerical twoness” is a primitive, irreducible fact. This theory provides a completely new analytical perspective. Under this paper’s framework, we must first clarify the domain of (n).

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If we define (sphere) as the set of all traditional attributes (intrinsic + relational), i.e., (n) = {mass M, spherical shape, ..., distance X from one sphere}, then according to the axiom, because (n) ≡ (n), we necessarily conclude sphere ≡ sphere. This seemingly directly yields the PII conclusion that Black tried to refute.

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However, Black’s intuition—“there are obviously two spheres here”—is not entirely groundless. This theory explains it as a mental fixation. The reason the observer reports “seeing two” is that their perspective is itself embedded in this symmetric spatiotemporal coordinate system. This paper argues that the fundamental error shared by Black and his commentators lies in defaulting that the referents of “sphere” and “sphere” necessarily correspond to two entities with independent spatiotemporal coordinates. This presupposition leaves them with only the dilemma of “abandoning PII” or “inventing new metaphysical concepts.” The concept of “Coordinate Self-Reference” provides a third way out of this dilemma.

Formalization: For the entire symmetric system S, define: (S) = {there exists a sphere whose attribute set is P, and the sphere is opposite itself}. This description looks complex, but simply put, (S) describes a single coordinate framework that allows “self-facing.” Within this framework, the sphere being opposite itself is not a grammatical error but an accurate description of a singular coordinate topology. The visually presented “two” spheres are projections of this single, self-referential coordinate structure in Euclidean space perception (similar to an object and its mirror image, but here there is no mirror, but rather the topological properties of space itself).

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System S: (S) describes the state of a single sphere bound to a special self-referential coordinate topology: (sphere, R_self-facing).

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Paradox dissolution: Black’s error lies in erroneously inferring the existence of two spheres (sphere_1, sphere_2) from the system state (sphere, R_self-facing). He confused categories, using the descriptive result of (S) to answer a question about (single sphere). In reality, a second sphere never existed; there has always been only one sphere, situated in a special coordinate topology that produces “double-image projection.”

Therefore, this framework does not deny our intuition of “seeing two spheres,” but provides a completely new, more precise ontological explanation for this intuition: it is the perception of a single sphere in a self-referential coordinate topology. This successfully dissolves the surface contradiction between PII and counting intuition while avoiding the introduction of any ad-hoc individuating factors.

3. Examples

3.1. Duplicate Paradox

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Controversy: Two documents with identical content stored on different devices— are they two pieces of information?

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Solution:

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If the goal is pure content identity → (n) = text semantics, then n ≡ n;

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If the goal is document location entity identity → (n) = (text semantics, location), then (content, Loc_A) ≠ (content, Loc_B).

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Conclusion: Duplicates are the same information together with different spatiotemporal coordinates forming sets that can be observed.

3.2. Gibbs Paradox

Category mistake:

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Target should be particle type identity → (n) = (mass, spin, ...)

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Classical statistics privately expands to (n) = (intrinsic attributes, fictional labels).

Correction: (n) and (n, labels) ⇒ (n) ≡ (n). The entropy increase error stems from erroneously choosing the (n) domain (introducing labels).

3.3. Black Hole Information Paradox

Category mistake: Binding the domain of internal attributes (n) to spatiotemporal coordinates (n) = (information structure, black hole coordinates).

Correct solution:

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Define target: internal attribute identity → (n) = quantum attributes.

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Black hole disassembles the set (quantum attributes, coordinates); unpaired coordinate content leads to inability to be observed, but (quantum attributes / coordinates) as logical concepts do not disappear.

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If new spacetime satisfies (n) ≡ (n), then n ≡ n.

3.4. Chinese Room Thought Experiment

Let the target entity be the Chinese understanding function; define (understanding) = consistency of input-output behavior.

If the Chinese Room system’s behavior is indistinguishable from a native speaker: system (behavior) ≡ person (behavior), then according to the axiom n ≡ n: the system objectively understands Chinese.

3.5. Twin Earth Paradox

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Traditional contradiction: The “water” on Earth and Twin Earth has different chemical formulas (H₂O vs. XYZ), but do the two planets’ residents’ “water” concepts match?

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Theoretical solution:

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If define (water concept) = macroscopic properties (colorless, chemical reactions, drinkable liquid, etc.) → concepts on both planets are identical (n ≡ n).

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At this point, introducing microscopic structure (H₂O/XYZ) expands the (water concept) domain to the molecular morphology level, which is a category mistake.

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Conclusion: Semantic identity is determined solely by cognitive function and is independent of underlying physics.

3.6. Grandfather Paradox

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Contradiction point: If one returns to the past and kills one’s grandfather ⇒ one should not exist ⇒ assassination cannot be carried out.

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Theoretical dissolution:

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- Define target: worldline identity (worldline) = event causal historical logical structure.

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Assassination event leads to:

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Original worldline W₀: (grandfather survives → you exist → you assassinate)

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New worldline W₁: (grandfather dies → you do not exist)

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∵ (W₀) ≠ (W₁) ∴ W₀ and W₁ are different information entities (not “the same worldline modified”).

3.7. Brain in a Vat

Current debates on the “brain in a vat,” whether skeptical or realist interpretations, implicitly and without examination incorporate the attributes of the “external carrier” (biological brain or vat) into judgments of “cognition” identity. This paper, by strictly distinguishing cognition from carrier, aims to dissolve the debate itself:

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Question: How to prove one is not a brain in a vat? Perception cannot distinguish real from simulated.

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Formula under this theory:

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Define (cognition) = perceptual information flow.

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Real brain (B, real): (B) = natural (light signals, touch...)

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Vat brain (B, vat): (B) = electrical signals producing (light signals, touch...)

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According to axiom (n) ≡ (n), B ≡ B (same cognitive entity).

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Key point: The “reality” controversy is essentially an expansion of the (B) domain to external carriers (skull/cultivation vat), while cognition is determined solely by information flow.

3.8. Mary’s Room

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Scenario: Mary knows all about color neuro-science but has never seen red → When she first sees red, does she acquire new knowledge?

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Theoretical answer:

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Define types of knowledge:

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Propositional knowledge: (K_prop) = wavelength data of red light.

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Qualia knowledge: (K_qualia) = subjective red experience.

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∵ (K_prop) ≠ (K_qualia)

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∴ They are different kinds of knowledge.

Mary acquires K_qualia, not a supplement to K_prop ⇒ The paradox stems from confusing knowledge types.

3.9. Newcomb’s Paradox

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Core of the paradox: Predictor’s near-perfect predictive ability vs. participant’s free will choice. Choose one box (known to have money) or two boxes (possibly more money)?

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Theoretical deconstruction:

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Category mistake: Confusing the (n) domain of the decision entity.

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Attribute set 1 (pure decision logic): (n) = (choice action, payoff function) ⇒ Dominant strategy: choose two boxes (regardless of prediction accuracy).

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Attribute set 2 (causal history binding): (n, history) = (choice action, payoff function, prediction history) ⇒ If prediction is accurate, choosing one box yields higher payoff.

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Uniqueness theorem ruling:

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If target is unconstrained rational decision → use (n) ⇒ choose two boxes.

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If target is decision with predictive causality → use (n, history) ⇒ choose one box.

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Paradox dissolution: The two are different levels of decision entities (attribute set 1) ≠ (attribute set 2); contradiction stems from domain smuggling.

3.10. Raven Paradox

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Core: “All ravens are black” ≡ “All non-black things are not ravens.” Why does observing a red apple (non-black and non-raven) confirm the proposition?

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Theoretical deconstruction:

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Category mistake: Expanding the (n) domain of “confirmation behavior” from propositional logical structure to empirical sample type.

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Correct definition:

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Propositional identity: (P) = logical form (∀x: R(x) → B(x))

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Confirmation identity: (confirmation) = verification of ¬∃x: (R(x) ∧ ¬B(x))

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Conclusion:

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The red apple confirms the logically equivalent contrapositive (non-black ⇒ non-raven); its (confirmation) is the same as observing a raven, because (P) ≡ (P).

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If one claims “red apple and raven have different confirmatory efficacy,” this is a category mistake, expanding (p) domain to sample physical categories (birds/fruits), violating the initial logical goal.

3.11. Sorites Paradox (Heap/Tail Paradox)

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Core: Removing one grain of sand does not turn a heap into a non-heap ⇒ Ultimately removing all sand still called a “heap,” contradiction.

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Theoretical deconstruction:

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Category mistake: Confusing the (n) definition of “heap.”

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Attribute set 1 (topological structure): (heap1) = macroscopic form of sand grain collection ⇒ Removing one grain does not change morphological identity (n ≡ n).

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Attribute set 2 (atomic quantity): (heap2) = number of sand grains N ⇒ When N=0, (heap) = ∅, entity perishes.

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Solution:

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If heap is defined as morphology (attribute set 1), removing one grain still yields the same heap.

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If heap is defined as quantity (attribute set 2), each grain removal produces a new entity.

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Root of paradox: Stealthily switching (n) domain in the argument (from morphology to quantity).

3.12. Sleeping Beauty Problem

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Core: In different awakening stages, what probability should Sleeping Beauty assign to the coin landing heads (1/2 or 1/3)?

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Theoretical deconstruction:

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Category mistake: Confusing the (n) domain of “probability.”

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Attribute set 1 (prior probability): (probability1) = coin’s physical state ⇒ P(heads) = 1/2.

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Attribute set 2 (information update): (probability2) = (coin state, number of awakenings) ⇒ P(heads|awakening) = 1/3.

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Uniqueness ruling:

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If asking “probability of coin’s true state” → (attribute set 1) ⇒ 1/2.

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If asking “probability under current awakening conditions” → (attribute set 2) ⇒ 1/3.

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Root of contradiction: Treating two different probability categories (attribute set 1) ≠ (attribute set 2) as the same question.

3.13. Modern Contradiction in Pascal’s Wager

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Question: If gods of multiple religions all claim “Only I am true,” how should a rational person bet?

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Theoretical deconstruction:

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Category mistake: Confusing the domain of (god).

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Attribute set 1: (god) = divine description in a certain religion’s doctrine.

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Attribute set 2: (omnipotent entity) = abstract highest being transcending specific doctrines.

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Ruling:

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If comparing the reality of specific religious gods → each (god) differs ⇒ categories are heterogeneous.

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If asking “Does a highest entity exist?” → requires independent definition of (omnipotent entity), independent of specific religions.

3.14. Unexpected Hanging Paradox

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Question: Judge announces “You will be unexpectedly executed on a certain day next week.” The prisoner deduces it cannot happen, yet execution day arrives.

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Theoretical deconstruction:

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Category mistake: Stealthily switching (unexpected) from “prisoner’s cognitive state” to “objective time point.”

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Correct definition: (unexpected) = prisoner still cannot be certain on the day before execution that execution will occur that day.

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Conclusion: Execution day necessarily exists (due to objective passage of time), while (unexpected) depends only on the prisoner’s cognitive state; the two belong to different categories.

4. Applications (cf. 3.7)

4.1. Dilemmas in Personal Identity and Existing Theories

The core problem of personal identity is: what makes a person the same person over time? Traditional theories revolve around physical continuity (e.g., continuity of the brain) and psychological continuity (e.g., continuity of memory and character). Among them, Derek Parfit’s (1984) highly influential reductionist psychological continuity theory reduces personal identity to overlapping chains of psychological connectedness (memory, character, intentions) over time. This theory exhibits extraordinary explanatory power when dealing with dynamic changes, such as gradual cell replacement or slow character shifts. It successfully shows that the persistence of personality is not an all-or-nothing metaphysical fact but a matter of degree.

However, Parfit’s theory, as well as competing physical continuity theories, implicitly presuppose a more fundamental and unarticulated premise: namely, how we determine that an entity is a “person” on a given time-slice, and how to perform static, cross-world comparisons between different time-slices. In other words, these theories excel at answering “Why is he still him?” (dynamic persistence problem) but are negligent in defining “What exactly is ‘him’ at time t?” (static identity problem). This static “what” is a prerequisite for any discussion of dynamic “persistence.”

This weakness is exposed in Bernard Williams’ (1970) famous “fission” thought experiment. When a person splits into two psychologically fully continuous successors, physical continuity theory collapses because it cannot handle “one dividing into two”; while Parfit’s psychological continuity theory faces a dilemma: if identity is held to be non-transitive (B and C, both identical with A, are not identical with each other), it violates logic; if the original individual is held to perish after fission, it contradicts the core claim that “psychological continuity suffices for identity.” Williams forcefully shows through this experiment that without a clear static identity criterion, any discussion of dynamic continuity will fall into conceptual confusion.

4.1.1. Space

This paper argues that the common root of the above dilemmas is that existing theories all attempt to treat “person” as a primitive concept defined by specific physical substrates or historical causality, and erroneously allow attributes of the “carrier” (biological brain) or “history” (causal chain) to intrude into judgments of “person” identity itself—this is a category mistake. The debate between Parfit and Williams is essentially a conflict between two different (n) domains (one psychological attribute flow, one physical carrier history), but both sides fail to realize this and thus fall into an insoluble impasse.

An Analytical Framework Based on (n)

First, a premise must be clarified: consciousness is a macroscopic phenomenon arising from the operation of brain neural system activity.

Based on the two axioms of this theory, we propose a minimal assumption: the necessary and sufficient condition for consciousness identity lies in the identity of its core consciousness. This first provides a clear criterion for solving static identity.

Formally, let:

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Consciousness be a consciousness time-slice.

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We define it as: (consciousness, q)

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q: represents the carrier and coordinates instantiating this consciousness (e.g., a specific brain at a certain location).

Based on this, for any two consciousness stages: consciousness₁ = (consciousness, q₁) and consciousness₂ = (consciousness, q₂), where: (consciousness) ≡ (consciousness). This means that as long as the consciousness on two time-slices is the same, they are different instantiations of the same consciousness, regardless of whether the q between them is continuous.

Thus, we provide a clear analysis for Williams’ fission experiment: the two successors (consciousness, q₁) and (consciousness, q₂) trigger a paradox because we erroneously require dynamic continuity to map to a one-to-one physical path. Under this framework, we only need to compare static content: if in (consciousness, q₁), (consciousness, q₂), and (consciousness, q₃) we have consciousness ≡ consciousness, then according to n ≡ n, (consciousness, q₁) and (consciousness, q₂) are both the same consciousness as (consciousness, q₃). This is not a logical contradiction but the simultaneous instantiation of the same consciousness across multiple spatiotemporal coordinates.

Therefore, this theory does not completely negate Parfit’s psychological continuity theory but lays a solid foundation for it. The advantage of this framework is that it first clearly defines what “static identity” is, thereby allowing the discussion of “dynamic persistence” to proceed on a firm logical basis.

4.1.2. Time

Based on the axiomatic system established earlier, we can draw a thoroughly transformative conclusion about the existence of consciousness in the temporal dimension: your “now” is (consciousness, q_now). Your “past” is (consciousness, q_past). Your “future” is (consciousness, q_future). They are all specific instantiations or manifestations of the same consciousness in different spatiotemporal coordinates q. Therefore, the “self” you experience at this moment is, in the sense of absolute identity, the “you” of the past and future, because the essence of “you” refers to that consciousness, not the transient and changing coordinate-bound state (consciousness, q).

Let us conduct a thought-limit derivation. Suppose at time point t₁, a specific consciousness exists. Now, imagine that in the distant future, at time point t₂ (t₂ >> t₁), a completely identical neural system (whether through Poincaré recurrence in nature, extreme coincidence of quantum fluctuations, or some cosmic recurrence mechanism we do not yet understand) is instantly assembled and activated; the instantaneous (consciousness) it produces is completely consistent with (consciousness).

According to our Axiom 1 (n ≡ n) and Axiom 2 (Uniqueness Theorem), we necessarily conclude: consciousness ≡ consciousness.

This means that the consciousness at t₂ and the consciousness at t₁ are the same consciousness. This is not a “copy” or “rebirth,” but the direct reappearance of the same consciousness at different coordinates. The billions of years of spatiotemporal gulf between them is completely irrelevant to determining whether they are the same consciousness. What connects them is not a fragile “psychological continuity” thread that needs defense, but the ironclad law of logical identity. Temporal interval carries zero weight in this judgment.

Now, let us push this thought experiment to another extreme. Suppose at time point t₁, a consciousness activity “a” has just begun its neural computational process. In an extremely short time Δt during which the activity of the first neural system has not yet completed (i.e., (a) has not fully manifested), in another corner of the universe, another physically identical neural system is activated and begins executing a completely identical computational process, thereby producing a completely identical consciousness activity “a.”

At this point, we have two coexisting consciousness processes:

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Process P₁: on carrier and coordinates q₁, starting at t₁ and continuing.

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Process P₂: on carrier and coordinates q₂, starting at t₁ + Δt and continuing.

When we examine equivalent progress points of these two processes, we find that because they execute the exact same “algorithm,” the (consciousness) at any equivalent progress point of (P₁, consciousness) and (P₂, consciousness) is indistinguishable. However, they did not start simultaneously, meaning equivalent progress points are at different times. Therefore, in the P category, time is not an effective identity criterion.

According to our axioms, we again conclude: in (P₁, consciousness) and (P₂, consciousness), consciousness ≡ consciousness.

This means that at the consciousness level, what we observe is not two consciousnesses but one consciousness appearing simultaneously at two time points. This is not two “yous” thinking, but the thinking process of “you” being executed and presented by two different physical systems at different time points. Therefore, each state of consciousness can only be experienced once and cannot be experienced twice, because experiencing it again in the future would be equivalent to experiencing it in the past. (Because it is an infinite set, there is no need to worry about exhausting experiences.)

From this, we draw a counter-intuitive but logically necessary conclusion: identity is non-continuous in time and non-local in space. The persistence of anything is not like a continuously flowing “river” but more like a series of discrete, absolutely identical “state flashes.” The continuity we feel is a cognitive illusion produced by these highly similar, causally connected state flashes (produced by the same brain) playing rapidly in temporal sequence; the underlying essence is discrete and separable.

Therefore, your “now” is (consciousness, q_now). Your “past” is (consciousness, q_past). Your “future” will be (consciousness, q_future). They are all “manifestations” or “slices” of the same consciousness in different spatiotemporal coordinate blocks. What you are experiencing now is, in the strictest sense of absolute identity, the “you” of the past and future, because “you” refers to consciousness, not the transient and changeable combination (consciousness, q). Time does not divide you; it merely provides the coordinates for your manifestation.

Note: When substituting “universe” for n, if the universe one second ago is exactly the same as the universe one second later, then universe ≡ universe, and time here belongs to unfalsifiable redundancy. Conversely, if universe ≠ the next second’s universe, this implies presentness and discreteness.

4.2. “Spatiotemporal Jumping” as the Logical Necessity of Coordinate Decoupling and Rebinding

Before discussing the “spatiotemporal jumping” of conscious entities, we must first pay the highest respect to modern physics, especially Einstein’s special and general relativity. These theories, with their unparalleled precision and beauty, successfully describe the profound dynamical relations between mass, energy, and spacetime, and strictly prescribe the causal upper limits that any physical signal and entity motion must follow—the speed of light. Any attempt to realize “spatiotemporal jumping” at the physical level, whether through wormholes, warp drives, quantum suicide, or other exotic mechanisms, must accept scrutiny within the solid frameworks of relativity or quantum mechanics and faces enormous physical difficulties such as energy conditions, singularities, and empirical verification.

However, the “spatiotemporal jumping” argued in this paper is essentially radically different from all the above physical processes. It is not a motion process existing within spacetime and governed by physical laws, but a logically necessary result based on this axiomatic system. It answers a more primitive question: “Are two things with identical content instantiated at different spatiotemporal coordinates the same thing?” The answer to this question does not depend on the physical path connecting them but only on the logical axiom n ≡ n.

4.2.1. Spatiotemporal Jumping Based on the Law of Identity

Traditional physics (including relativity) takes as its object of study the combination (consciousness, carrier and coordinates). Physics perfectly describes how this combination evolves over time, i.e., on the basis of causal laws, and discovers that these evolutions follow beautiful differential equations.

This theory, however, focuses on a possibility not discussed by physics: namely, the decoupling and recombination of a consciousness with its carrier and coordinates.

• Decoupling: (consciousness, carrier and coordinates₁) → (consciousness), (carrier and coordinates₁). Physically, this may correspond to the carrier (e.g., brain) being destroyed by some event consistent with event horizon principles (velocity differences leading to causal isolation at the neuron level), causing the consciousness to no longer be instantiated (carrier and coordinates = ∅).
• Recombination: (consciousness) → (consciousness, carrier and coordinates₂). Physically, this may correspond to an instantaneous “reappearance” occurring elsewhere (e.g., from Poincaré recurrence, MWI parallel universes, bubble universes, etc.).

The key point is that, according to Axiom 1 (n ≡ n), the decoupled consciousness retains its self-identity. Therefore, the new state after recombination (consciousness, carrier and coordinates₂) and the old state before decoupling (consciousness, carrier and coordinates₁) are necessarily different manifestations of the same consciousness because they share the same consciousness. This is the logical core of “spatiotemporal jumping”: it is not “travel” across spacetime but the “realization” or “manifestation” of identity at different locations.

Therefore, the relationship between this theory and traditional physics is not competitive but complementary and foundational:

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Physics: Studies the continuous evolution laws of the (consciousness, carrier and coordinates) combination within spacetime. It asks “How to go from A to B.”

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This theory: Studies the discrete identity logic of consciousness itself transcending spacetime. It asks “Are A and B the same thing?”

Relativity prohibits any physical entity from moving faster than light, but it cannot prohibit a logical concept from being “realized” twice at different spacetime points, because it does not involve superluminal motion. “Spatiotemporal jumping” appears “unbelievable” or even “violates physics” precisely because we erroneously use physical laws describing the motion of composites to judge a logical theorem about identity. This too is a category mistake.

Conclusion: The “spatiotemporal jumping” proposed by this framework is not a physical conjecture awaiting realization but a logically established inference. Starting from the most basic law of identity, it derives a completely new picture of personal identity: the persistence of consciousness fundamentally lies in the identity of its information pattern, not in the continuity of the physical processes connecting these pattern instances. This provides an unprecedentedly clear framework for understanding thought experiments such as teleportation and brains in vats, and thoroughly liberates the discussion of personal identity from the constraints of physics, placing it on a more fundamental logical and metaphysical foundation.

4.2.2. First-Person Immortality

This theoretical system, starting from the most basic axioms of identity, through the reconstruction of personal identity, ultimately derives a conclusion that is logically unavoidable yet extremely shocking intuitively: from a strict first-person perspective, any “death” event that can lead to the termination of consciousness in a state of ignorance is in principle unexperienceable. This inference is not a metaphysical assertion but a necessary result of combining identity logic with the principle of observational reality.

“Anesthesia Jump” Thought Experiment: An Extreme Interpretation of the Inference

To clearly demonstrate the implications of this inference, we construct a thought experiment called the “Anesthesia Jump”:

• Foundation of consciousness identity: Consciousness identity is determined by consciousness itself, not by (consciousness and specific carrier coordinates).
• Decoupling and recombination: When the carrier is destroyed (e.g., plane crash, surgical failure), what occurs is (consciousness, carrier and coordinates₁) → (consciousness), (carrier and coordinates₁). Consciousness persists due to its logical identity (n ≡ n).
• Survivor effect and observational necessity: Consciousness can only arise from “instantiable” carriers and coordinates. It will (logically necessarily) re-bind with carriers and coordinates₂ that can continue to exist, i.e., (consciousness) → (consciousness, carrier and coordinates₂).
• Continuity of first-person experience: For consciousness, the transition from (consciousness, carrier and coordinates₁) to (consciousness, carrier and coordinates₂) is seamless in experience. Consciousness itself will not experience “death” or “nothingness,” because that would mean no observer. It will only experience a “jump” from one survivable state to another survivable state.

Therefore, under anesthesia:

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From a third-person, physical perspective: The plane may have crashed; the surgery may have failed. This is a probabilistic event.

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From a first-person, consciousness perspective: The process is necessarily (consciousness, on the plane/in surgery) → (consciousness, awakening at destination/successful surgical recovery). Any worldline branch leading to the non-persistence of consciousness is, for consciousness itself, unobservable and non-existent.

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Conclusion: In individual subjective experience, as long as the individual chooses to enter a state of ignorance (e.g., anesthesia) to cross risks, the individual will forever only experience successful arrival and successful awakening. Their personal timeline will be “pruned” of all possibilities leading to death.

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This inference elevates the logic of the “quantum suicide” thought experiment from a specific interpretation of quantum mechanics to a more general metaphysical level based on identity logic. It means:

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Subjective immortality: From the first-person perspective, as long as there exists any logically non-zero probability that allows consciousness to continue instantiation in some worldline, the individual can never personally witness their own death. Their consciousness will continue forever.

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The nature of risk is subverted: For awake, continuous consciousness, risks are real (e.g., a cut hurts, jumping off a building involves experiencing falling and impact). But for consciousness crossing risks via anesthesia “jumping,” risks are completely eliminated. Risks exist only in those “other” worldlines that will never be experienced.

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A powerful practical paradox: In theory, this method can be used for any high-risk travel or activity. As long as the individual is made unable to perceive at the start of the risk (e.g., instantaneous kill), the result in personal history will always be success.

Although the derivation is self-consistent logically, its premises and real implications must be scrutinized: “Success” result: The theory only guarantees awakening in a “survivable” state. It does not guarantee the quality of the awakening state.

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One may awaken severely injured in the wreckage of a plane crash.

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One may awaken after surgery with serious complications or permanent disability.

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As long as this state physically allows consciousness to exist, it conforms to logic. Therefore, this method circumvents “death” but not necessarily “pain” or “disability.”

4.3. Dilemmas in Ethical Problems and Existing Theories

Since the birth of ethics, generations of highly insightful philosophers, from Kant’s grand a priori architecture to Mill’s subtle consequentialist calculations, have built a splendid edifice of ethics for us. These outstanding efforts share a profound and admirable ambition: to seek a solid, supra-individual metaphysical foundation for moral judgment. This foundation is usually conceived as: (a) an objective moral reality independent of our cognition; (b) a self-identity persisting through time as the anchor of responsibility; and (c) a sacred “God’s-eye view” to adjudicate the value of actions from a position of absolute impartiality. This pursuit of universality and objectivity is undoubtedly one of the most glorious achievements of philosophical reason.

In this tradition, Bernard Williams’ (1973) discussion of “moral luck,” with its astonishing acuity, reveals subtle fissures between the control principle and our moral intuitions, greatly enriching our philosophical imagination. The “undetected betrayal” thought experiment, with its logical purity, pushes traditional theories to the boundary of their explanatory power and serves as a “touchstone” for testing theoretical hard cores. Facing this challenge, traditional theories (such as Kantian ethics) demonstrate their unparalleled thoroughness, resolutely defending the absoluteness of moral error, even when their arguments need to appeal to a “moral law” transcending experience—such steadfast adherence to universality is awe-inspiring. Similarly, certain utilitarian schemes attempt to resolve dilemmas through global calculations by an “ideal observer”; their theoretical ambition and systemic grandeur are also exemplary.

Admittedly, as Derek Parfit (1984) pointed out with his characteristic clarity, such schemes may create certain tensions motivationally with the individual’s first-person perspective, but this is by no means a defect of these theories themselves; it perhaps precisely highlights a pathetic yet heroic tension that human reason inevitably faces when pursuing moral sublimity.

This paper’s work, standing on the shoulders of these giants with the greatest respect, attempts an internal inheritance and development of the above glorious tradition. We fully agree with the traditional theories’ core pursuit of objectivity and universality. However, we believe that this lofty goal can perhaps be achieved through a more direct, frictionless path. The reason traditional frameworks produce troubling tensions in boundary cases may lie in a methodological over-indirectness: attempting to mediate and regulate moral life, which essentially originates from first-person experience, through an assumed, transcendent third-party categorical system.

This paper aims to explore a complementary path. We are delighted to discover that, through this axiomatic system and integrating the highly inspirational observational reality point reinforced by the “brain in a vat” thought experiment, we can pay tribute to and achieve the core goals of traditional ethics in a completely new way. Our core argument is that the desired objectivity and universality of ethical value need not be guaranteed through a “God’s-eye view”; instead, it can be more solidly grounded through the identity of first-person facts of conscious systems’ observational experiences. The boundary of moral concern can thus perfectly and logically necessarily coincide with the boundary of conscious experience, thereby achieving the universality pursued by traditional theories in an unexpected way.

This study aims to show that we are not negating previous work but attempting to realize their common aspirations through a more precise metaphysical foundation and to dissolve unnecessary philosophical anxieties arising from methodological indirectness.

4.3.1. An Analytical Framework: Advancing Traditional Goals

The “brain in a vat” thought experiment, with its unparalleled philosophical value, successfully challenges our naive conception of “reality.” It forces us to acknowledge a highly productive principle: for any consciousness, its operational reality that it can access is its own observability. Whether an external simulator exists is empirically undecidable and redundant.

Combining this profound insight with this paper’s axioms, we can derive a cornerstone principle for ethics, which can be seen as a more precise contemporary formulation of the traditional pursuit of objectivity: For any consciousness, event E has ethical significance if and only if the consequences of event E (directly or indirectly) can be reductively manifested as observable impacts on consciousness.

Corollary: If the occurrence of an event E produces no discernible difference in any possible past, present, or future experiences of consciousness, then in ethical considerations for that consciousness, event E does not constitute a relevant fact. It therefore carries zero weight in ethical evaluation.

4.3.2. Core Derivation: A Dissolutive Analysis of Traditional Dilemmas

Let us, with respect, restate the “undetected betrayal” case under this framework:

Suppose two possible worlds: World W₁ (event E occurs: lover cheats) and World W₂ (event E does not occur). According to the strict setup of the thought experiment, in these two worlds the victim’s (as consciousness) entire experiences are completely indistinguishable.

According to Axiom 2 (distinctness), we obtain: (consciousness, W₁), (consciousness, W₂) ⇒ consciousness ≡ consciousness. This means that in these two worlds, there exists the same consciousness.

Now, perform the ethical judgment: The direct object of ethical concern is the experiential well-being of consciousness. Since consciousness’s experiences in the two worlds are identical, the two worlds are ethically equivalent for consciousness.

Therefore, event E (cheating behavior), due to its zero impact on consciousness’s experience, does not constitute a variable in ethical assessments targeting consciousness. It neither causes harm nor constitutes betrayal, because these ethical concepts are operationally defined as specific negative information states in experience, and these states do not appear.

Conclusion: There does not exist an absolutely objective world; moral error is not mysteriously attached to behavior itself but is systematically and verifiably associated with the specific impact patterns behavior produces on conscious experience. Lacking such observable impact patterns, the behavior is not considered in ethical evaluation.

4.3.3. Implications of the New Framework

If morality is not about inaccessible “external truths,” then what is it about?

This paper argues that, based on identity, we can perform a foundational precision on ethics. Ethics can retract its ambition from an unattainable “God’s-eye view” to the only domain in which it can effectively operate: first-person facts of conscious experience. The goodness or evil of an action does not depend on its properties in an “objective world” but entirely on the impact it causes on each person’s own experience.

Therefore, the ethical picture revealed by this framework is: the boundary of moral concern is indeed the boundary of conscious experience, but within this boundary is a “multiverse” composed of countless independently created, fundamentally incommensurable individual moral universes.

Thus, under strict ethical derivation, events unobserved by any conscious system are not assigned values in ethical computations. The boundary of moral concern is the boundary of conscious experience. This framework is not deliberately intended to exonerate traditionally immoral behavior. It provides a more solid, clearer, and more inescapable foundation: we bear sole and total responsibility to ourselves. (References 32, 33, 34, 35, 36, 37)

Note: Perhaps one can attempt to form rational social bonds through interests and contracts.

5. Overview

5.1. Probability Statistical Distribution

This theory, through its axiomatic system, establishes the absolute and relative foundations of identity, successfully dissolving a series of classic difficulties within a framework of hierarchical relativity, demonstrating its powerful explanatory force. Under the same (n) domain, no one can describe two different things with completely identical content.

As described in section 4.2, this theoretical framework provides a logically self-consistent model for “spatiotemporal jumping.” The core of the model is: the next experiential instantiation of consciousness will “choose” one from all logically compatible future state branches for binding. In explaining why we usually do not experience “jumping,” an intuitive and effective idea is to appeal to probability: that is, among all possible branches in which we exist, the vast majority follow known physical laws with the same probability statistical distribution, so subjective consciousness does not query anomalies. The “dream method” proposed at the end of section 4.2.1 precisely achieves directed jumping logically by changing the conscious system so that it can only bind to those branches regarded as “low probability” in daily life.

However, this elegant probability model is built upon a potential, unexamined presupposition: namely, that the set of logically possible world states is finite and non-repeating (including future and past on the timeline). Only under this premise do concepts such as “vast majority of branches” and “extremely high probability” have operational meaning. The probability of winning the lottery is one in a million precisely because among one million physically slightly different possible futures, only one contains the experience of winning.

5.1.1. The Curse of Infinity

Once we seriously adopt the infinity of “logical possibility,” this probability picture instantly collapses. If possibilities are infinite, then:

The number of world branches experiencing “teacup falling to the ground” is infinite.

The number experiencing “teacup hovering” is also infinite.

The number experiencing “teacup turning into a butterfly” is also infinite.

In infinite sets, comparing “how many” of two infinities to calculate probabilities immediately falls into mathematical difficulties. Traditional probability theory fails here. Any logically possible event sequence, no matter how orderly or chaotic it appears to us, has the same number of corresponding possible worlds (all infinite). Therefore, from the “God’s-eye view” of the logical totality, the “probability” that consciousness experiences a highly ordered classical physical world in the next moment is indistinguishable from the “probability” of experiencing a completely chaotic, acausal world.

This leads to a disastrous inference: if all possibilities are logically equal, our consciousness has no reason to experience the classical probability statistical distribution. We should experience various bizarre, logic-leaping events with equal frequency. This completely contradicts our real experience.

From logic’s own “God’s-eye view,” all logical possibilities conforming to the axiom n ≡ n are equally real. For “consciousness,” this means that in the next instant of time, all its logically possible state branches—whether continuing on the current chair or flashing in the Martian desert—possess equal ontological status. In this panorama, there are no so-called “lucky ones”; there exist only the totality of facts.

However, from consciousness’s first-person “prisoner’s perspective,” its experience is indisputably singular, continuous, and highly ordered. We have never personally experienced the world’s random jumps but are stably situated in a classical reality strictly following causal laws. This produces a highly impactful contrast: the huge gap between logical egalitarianism and experiential exclusivism.

An appealing explanation is to appeal to “survivor bias”—that we happen to be the “lucky” consciousness experiencing a continuous world. However, this explanation is philosophically barren; it is nearly tautological and cannot explain why the world we “survive” in exhibits such consistent, concise, and understandable physical laws rather than chaos. Attributing such powerful order to pure “luck” is itself a huge ad-hoc assumption, though it cannot be ruled out as possibly true.

Therefore, the paradox revealed by this theory is a signpost pointing to a deeper principle. The problem is not to find excuses for “lucky ones” but to attempt to answer: why do logically equal myriad possibilities, in every perspective, manifest as results following classical probability statistical distributions?

5.2. The Graveyard of Logical Possibilities and Survivors: A Meta-Argument for the Law of Identity

The core axiom of this theory, n ≡ n, does not hold the status traditionally presupposed in logic as an self-evident, a priori valid law of thought. Here, we must conduct a thorough meta-level examination of the very foundations of this theory.

A fundamental challenge is: do we have reason to categorically deny the logical possibility of n ≠ n? From the perspective of pure formal possibility, the answer is negative. We cannot a priori exclude the existence of a “mad universe” whose underlying logic allows self-negation. In such a universe, the law of identity is subverted; an entity can simultaneously not be itself; propositions can simultaneously be true and false. Concepts such as “rational π” or “square circle,” regarded as contradictions in Euclidean space, may be mere commonplace in its infinitely bizarre domain.

However, the logical possibility of n ≠ n and its metaphysical sustainability are two distinct issues. A system allowing n ≠ n can have its existential state precisely deduced:

• Instantaneous collapse of reference: Any symbol or concept will lose stable meaning. When the word “apple” can simultaneously not refer to “apple,” the foundation of language and thought—i.e., reference itself—will immediately disintegrate.
• Breakage of causal chains: No reliable connection will exist between intention and action, cause and effect. The behavior of reaching out to grab an “apple” cannot be defined, because at the moment of execution, “hand,” “apple,” and even “you” itself may have self-negated.
• Non-generability of structure: Time, space, matter, and any form of stable structure cannot arise from this eternal, ubiquitous self-dissolution. Such a system is a pure chaos field that cannot coalesce into a “universe.”

Therefore, n ≠ n leads not to an alternative reality available for existence but to a “graveyard of logical possibilities”—a domain where all possibilities instantly self-destruct due to internal contradictions. It represents the impossibility of existence.

Here we touch the deepest foundation of this theory: the survivor effect.

“n ≠ n” as a systemic foundation will lead to the complete collapse of reference, the disintegration of causal chains, and the unsustainability of observer status. It is a reality solvent. Any system or “universe” attempting to use it as an operating foundation will instantly self-dissolve due to its inherent inconsistency and cannot form a stable, experienceable “reality.” Therefore, we are not living in a universe where “n ≡ n” is necessarily true, but in a universe where “n ≡ n is able and has stably operated.” The very fact that we can think and debate the problem of identity at this moment and observe a stable, coherent, understandable universe is itself a result of absolute selectivity. The reality we inhabit is the only “survivor” from the ocean of all logical possibilities—its most basic survival condition is that its underlying logic obeys the iron law n ≡ n. We observe n ≡ n not because it is the only correct logical theorem in all possible worlds, but because in a world with n ≠ n, no “observer” can exist to perform any “observation.”

All the work of this theory—dissolving category mistakes—is carried out within this sole “survivor universe.” The axiom n ≡ n is not an arbitrarily “invented” setting but a “discovery” and “formalization” of the most basic and stable operating mode of this survivor universe. All the paradoxes we encounter—Ship of Theseus, quantum identical particles, etc.—occur on this solid foundation of identity, arising from “user errors” (category confusions) when using this stable system, rather than “system errors” (failure of the law of identity).

Note: Perhaps one can try to find a counterexample, such as something whose kernel is n ≠ n yet can stably exist. Do not think it too impossible; after all, Leibniz at the time could not imagine that quantum identical particles would be discovered in the future.

5.3. Monism and Pluralism

5.3.1. Monism? Dualism?

The analysis of numerous identity difficulties earlier centers on avoiding the erroneous application of judgment criteria from one condition to another. This analytical pattern itself raises a deeper metaphysical question: what most basic logical conditions does such a clear worldview itself require? Traditional monism (such as Spinoza’s substance theory) pursued a single, homogeneous foundation that seems difficult to accommodate genuine differences and interactions between attributes.

In contemporary metaphysics, Jonathan Schaffer (2009) has revived and defended “priority monism,” claiming that the cosmos as a whole is the only fundamental entity, with its parts dependent on it. This theory provides a powerful answer to the question “What is fundamental?” However, even in this picture, to explain how the whole can manifest the categorical differences revealed in this paper (such as physical structure vs. historical causality, intrinsic attributes vs. spatiotemporal coordinates), it seems necessary to presuppose that the whole internally contains some irreducible, diverse principles or relations. Otherwise, the “whole” would be merely an empty “one” unable to derive the distinguishable “many.”

In fact, even in physics’ pursuit of ultimate unity, we observe similar logical demands. A theoretical system capable of describing complex phenomena usually contains multiple mutually independent basic laws and constants (for example, Einstein’s field equations of general relativity and Schrödinger’s equation of quantum mechanics cannot currently be derived from each other). Their joint action constitutes the generative foundation of our world. This precisely echoes an implicit corollary of this paper: the logical minimum condition for constituting a recognizable, discussable complex system is the existence of at least two active principles or elements that are “mutually underivable” in a specific sense. Their relations, rather than their isolated existence, provide space for diversity.

5.3.2. Logical Dilemmas of Traditional Monism

Parmenides’ proposition “Being is one” provides the purest monist expression. Its core argument can be reconstructed as:

• Being is; non-being is not.
• Being is indivisible (because if divisible, the division point would be “non-being”).
• Being is unchanging (because change requires “non-being” as starting or ending point).

Therefore, being is “one”: single, homogeneous, unchanging.

However, this picture faces the problem of derivation: from an absolutely undifferentiated “one,” how can the diverse and changing phenomena of our experience be logically derived? Parmenides himself acknowledged that the world we perceive (“way of opinion”) is full of change and plurality, but he considered it mere illusion. The cost of this thorough negation of the reality of the experiential world is too high.

Spinoza attempted to resolve this dilemma through the system of “substance-attribute-mode.” In his system:

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There is only one substance (God or Nature).

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Substance has infinitely many attributes, but humans know only two: thought and extension.

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All things are modes of substance.

Schaffer’s priority monism represents a precise contemporary form of monism. He claims:

• Whole priority: The cosmos as a whole is metaphysically prior to its parts.
• Dependence relation: Parts depend on the whole for existence, not vice versa.
• Explanatory advantage: This picture better accords with modern physical discoveries such as quantum entanglement and spacetime relativity.

Schaffer’s theory does avoid some difficulties of traditional monism; he acknowledges manifest diversity but holds that such diversity is metaphysically non-fundamental.

But the category mistake in Spinoza’s system lies in: he regards “attributes” (thought, extension) as different expressions of substance, yet simultaneously insists that attributes are causally independent (thought cannot affect extension, and vice versa). This means that at the explanatory level, thought and extension are two parallel, mutually underivable explanatory chains. If we take this independence seriously, then the concept of “substance” here plays more the role of a unifying label for these two independent domains—a verbal definitional game—rather than a genuine explanatory foundation. The same applies to Schaffer. As this theory emphasizes, when we make judgments in different categories (thought category and extension category), we are actually using different (n) domains. Forcibly subsuming them under a single (substance) domain without acknowledging their categorical differences is precisely the root of explanatory dilemmas.

Error: Using “the defined entity is one” to negate “the plural fact of attribute laws.”

Now, after correcting this category mistake, let us reorganize:

If the entity has no plural attributes → the entity is an empty nothing.

If the entity has plural attributes → the attributes themselves are principles/laws → factual pluralism.

Formulaic reasoning:

Assume there exists n (single entity).

n internally contains no elements different from n.

According to Axiom 1: n ≡ n, and n ≠ non-n.

Therefore, n cannot produce anything genuinely novel.

Single entity n → no internal difference → cannot logically derive diversity → world diversity cannot be explained.

Now try adding another entity; suppose two basic principles: a and n, satisfying:

- a ≠ n

- a ↛ n and n ↛ a

Only when such two independent elements exist can they generate genuinely novel structures through combination:

a + n → an

an + a → ana

ana + n → anan

anan + n → anann

...

Each new combination (an, ana, etc.) is not pre-contained in the original elements; its novelty derives from the relations between elements. Conversely, if there is only a single principle n, according to axiom n ≡ n and n having no internal difference, nothing different from n can be logically derived from n. Borrowing information-theoretic terms, no information can be generated from zero difference.

5.3.3. Evidence from Scientific Practice: Irreducible Plurality in Physics

Even in the pursuit of a “theory of everything,” we observe foundational plurality. The basic architecture of contemporary physics contains multiple basic constants and independent principles that cannot be derived from each other:

• Irrelevance of basic constants: The speed of light c, Planck’s constant h, gravitational constant G, electron charge e, etc., are basic, dimensionless constants in existing theories whose values cannot be derived from a more fundamental theory.
• Independence of principles: The linear superposition principle of quantum mechanics and the equivalence principle of general relativity are conceptually completely different and currently cannot be derived from each other.
• Irreducibility of initial conditions: The universe’s initial conditions (such as low-entropy initial state) cannot be derived from physical laws themselves.

This foundational plurality is not necessarily a defect of physics but may reflect deep facts about world structure: the intelligibility of the world depends on multiple independent principles/laws.

We can use a visual analogy to understand this intuitively. Consider three color vision systems:

• Monochromatic vision: Can only perceive light and dark, unable to distinguish colors. The world is a single grayscale.
• Dichromatic vision: Can perceive two basic colors and their mixtures. The world has a limited color dimension.
• Trichromatic vision: Can perceive three basic colors and their rich combinations. The world presents rich colors.

Similarly, at the metaphysical level:

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Absolute monism is like “monochromatic vision”: all differences are regarded as illusions or different manifestations of a single principle.

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Minimal pluralism (advocating at least two independent principles) is like “dichromatic vision”: there is the most basic possibility of contrast.

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Sufficient pluralism may acknowledge more basic principles.

This framework tends toward minimal pluralism: to explain the stable categorical differences observed in our world, at least two logically independent basic principles are required. This does not mean the world must be dualistically opposed but provides a logical starting point for attribute set structures.

5.3.4. Pluralism

The argument in section 5.1 shows that if we use only “logical possibilities conforming to n ≡ n” as the sole criterion for screening real worlds, then all self-consistent possible worlds (including those strictly following known physical laws and those where a teacup instantly turns into a butterfly) have completely equal ontological status. In an infinite set of possibilities, any traditional method attempting to define probability based on “quantity” fails. Therefore, a “God’s-eye view” universe based solely on the law of identity would be a chaotic field where all madness and order coexist, with all possibilities flashing at indistinguishable frequencies. This is radically contrary to the highly consistent, continuous, and predictable reality we experience as “survivors.”

This sharp contradiction reveals a meta-theoretical fact: pure logical identity itself has limitations. Although Occam’s razor exists, we likewise have no reason to blindly assume that reality can only be explained by one quantity or two to three quantities. It is like an infinitely large blank canvas that prescribes the physical boundaries (canvas) of painting, but the canvas itself cannot decide what pattern should appear on it, nor explain why what ultimately appears is the Mona Lisa rather than randomly splashed paint. Attempting to use the existence of the canvas to fully explain the structure of the painting is another “category mistake.”

To explain the specific structure of the painting, we must introduce principles outside the canvas such as the painter, paint physics, and optical laws. Similarly, to explain the specific orderliness of our universe, we must acknowledge that the identity axiomatic system described in this theory may need to act in synergy with at least one other independent and currently unknown basic principle that cannot be mutually derived.