Steins Theory: A New Axiomatic System Concerning Identity

1. Introduction

The problem of identity, namely “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, sets the lofty ideal of individual individuation—to Kripke’s (1980) groundbreaking theory of necessary identity based on rigid designators and origin, which provides what seems to be a solid foundation for the stability of reference through its profound grasp of modal intuitions. The excellent efforts of generations of philosophers have collectively constructed the grand intellectual tradition through which we understand the persistence and identification of individuals.

However, a thought-provoking phenomenon is that these theoretical frameworks, which are highly persuasive within their respective domains, often exhibit a regrettable, systematic limitation when dealing with complex boundary cases posed by reality and thought experiments. Leibniz’s strong PII principle encounters fundamental difficulties that its original conception perhaps did not foresee when facing quantum identical particles (French & Redhead, 1988). The inherent rigor of its principle instead leads it into a near-paradoxical impasse when dealing with multiple entities that are completely indistinguishable in all intrinsic properties yet remain numerically distinct (such as two electrons). Similarly, Kripke’s elegant rigid theory, aimed at anchoring reference, becomes rather entangled in its explanatory power when handling 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 attribute identity or Kripke’s historical path focusing on the necessity of origin, each successfully illuminates one wing of the edifice of identity but unfortunately leaves the other side in deeper shadow. Does their common, perhaps unstated ambition—that is, to find a single, absolute criterion for identity—itself constitute an a priori obstacle that prevents us from truly understanding the multi-dimensional nature of the problem?

This paper argues that the key to breaking this impasse is not to make an either-or choice among existing paths or to undertake another round of patching. Instead, 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 attributes 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 difficulties 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, offering a new way out for a series of philosophical anxieties arising from category mistakes. Some may think this theory is merely a re-description of relative identity in different language, but this is not actually the case (see section 2.3 for clarification).

2. Analysis

2.1. Axioms

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: Two (n) with identical content must be the same (see section 2.5).

Note: See examples in 3.1, Chapter 2, and 3.7. Unless explicitly stated otherwise (for example, the question in 3.7 specifies consideration from a cognitive perspective), this paper defaults to the premise that propositions are effective in reality (physics). This default is because if a proposition does not pursue real-world effectiveness, any unexpected errors in its conclusions due to unstated shifts in perspective are naturally unimportant.

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 holds that an object’s identity is not guaranteed by its instantaneous properties at any given moment but must be carried by its spatiotemporal continuity and an uninterrupted historical causal path. The great advantage of this approach lies in its successful capture of our deep intuitions 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 starkly different and metaphysically parsimonious picture (see Heller, 1984; Sider, 2001). This theory, with its thorough clarity, very cleverly avoids many pitfalls of diachronic identity. Four-dimensionalism holds that the Ship of Theseus is not a three-dimensional entity that “exists wholly” 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 possess 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 the troubling question of “identity” into the relatively clear relation of “part-whole.”

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 of it 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, maintained by some similarity and causal continuity. With its conceptual economy, Stage Theory avoids presupposing a cross-temporal identity relation and thus exhibits strong theoretical appeal.

However, despite the ingenuity and often self-consistency of the internal logic 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. 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 checking 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 schemes 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 category of comparison? A reasonable interpretation is that we are concerned with its identity as an “objectively verifiable ship,” i.e., the category (physical ship), with relevant attributes {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), with relevant attributes {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 is implicitly asking 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 of (physical ship) constitutes a category mistake. The true value of the historical path theory lies in revealing “history” as an important category, but it erroneously treats it as the sole decisive category.

We can reconstruct this formally as follows:

1. Initial question: Determine whether the entity “ship” at time points t₁ and t₂ is the same.

2. Correct (n) domain (according to the initial question): Should contain only attributes related to the “ship’s” matter, i.e., physical ship: (all particles of the ship and their arrangement and shape).

3. Category mistake of historical/four-dimensional/stage theories:

· Wiggins effectively adopts history + physical ship (all particles of the ship and their arrangement and shape, historical causal path).

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

· Stage Theory effectively adopts stage + physical ship (all particles of the ship and their arrangement and shape, counterpart relation).

The process leading to this can be formally expressed as:

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

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

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

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

· Thus, they reach the conclusion Ship_t1 ≠ Ship_t2 to answer 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 t₁ and t₂ for the ship?” 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 comply with Article X of the Criminal Law?” but consulting the Civil Code and issuing a verdict. The conclusion may be coherent within its own system, but it has quietly shifted 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 attributes, as powerful explanatory properties, influence our identity judgments. Its limitation lies in attempting to elevate such explanatory properties to metaphysical necessities, thereby having to expand the criteria for judging identity questions to maintain theoretical completeness. The value of this theory lies in the fact that it does not need to make such difficult expansions but, by clarifying the attribute sets of the questions, 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 surprisingly close affinity with this paper in intuition: both reject the naive notion of “absolute identity” and hold that “whether x and y are identical” cannot be discussed in a vacuum detached 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 noun A in “the same A”—undoubtedly captures the essence of a great deal of real linguistic practice. Moreover, Geach insightfully emphasized that things within the same domain of objects can appear identical or non-identical under different standards of identity (i.e., different “same A” predicates), without the things themselves changing. This insistence on “the things themselves not changing” makes his theory particularly elegant and economical in metaphysics.

On the surface, the theory in this paper is extremely similar to Geach’s relative identity. Taking the Ship of Theseus as an example: Geach would say that under the standard of “same ship” (as a material object), the old ship and the new ship are the same; under the standard of “same ship-with-history” (with a historical trajectory), they are not. This is outwardly identical to this paper’s distinction in section 2.2 between the two categories of (physical ship) and (physical ship, history). For this reason, a careful reader might well think that this paper is merely restating 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 detailed reconstruction of Geach’s thesis than in 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 noun A in “the same A.” Things within the same universe of discourse can be identical or non-identical under different standards of identity (i.e., different “same A” predicates), without the things themselves changing.

That is to say, Geach’s theory relies on a key commitment: when we switch from one standard of identity f1 (e.g., “same ship as material object”) to standard 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 does not undergo any change due to the switch in standards.

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

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

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

When we examine 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 the planks correspond one-to-one identically), so according to the standard of f1, x ≡ y.

When we examine 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 the attributes in A_f2? Clearly not, because x at time t₁ has not yet acquired the historical path from t₁ to t₂—historical path is a relational property added after the fact, and it is not part of x’s intrinsic constitution in isolation. Thus, the “old ship entity” judged under f2 is actually no longer the original x, but a composite x’ = (x, history_path) associated with t₂. 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 (x’, y’). And x’ ≠ x (as attribute sets), y’ ≠ y. Geach asserts that “the things themselves have not changed,” but strictly speaking, when we switch identity standards, we have quietly replaced the entities under discussion—because the applicability of each standard itself presupposes different attribute packages. This is not the same batch of things exhibiting different identity relations under different standards, but different sets of things naturally yielding different conclusions under different judgment frameworks.

Consider, for example, when we discuss whether Superman is a human or a superhuman. When judging whether he is human, we use his human attributes for comparison. At this point, he only needs to satisfy human attributes (at this point he is A); whether he is an ordinary person has no effect, because we are looking at his human attributes. But when we ask if he is a superhuman, we compare his superhuman attributes (at this point B) and not just his human attributes. Here A is not equal to B; A is A, B is B. A and B may overlap, but because their boundaries differ, they are not actually the same thing.

Assume there exists an object O.

When we discuss: Is O a human?

We actually focus only on the “set of human attributes.”

Thus we obtain: A = O as an object under the human boundary.

When we discuss: Is O a superhuman?

People focus on: “the set of superhuman attributes.”

Thus we obtain: B = O as an object under the superhuman boundary.

But A ≠ B

Because:

The constitutive boundary of A is the set of human attributes;

The constitutive boundary of B is the set of superhuman attributes;

Although they largely overlap, they are not the same object.

This has a subtle difference from ordinary relative identity, because Geach’s general idea is: the same object, under different standards, yields different identity judgments.

This paper holds: when the standard changes, the object changes.

Let:

A = {human attributes}

B = {human attributes + superhuman attributes}

Then: A ⊂ B

But: A ≠ B

Therefore: the object defined by A and the object defined by B are not the same object.

This reasoning is itself self-consistent, but most philosophers might question the first step: why is an object defined by its attribute set? Rather than: the object exists first, and attributes attach to it afterward?

When an entity interacts with the world, we call the features left behind its attributes. If an entity interacts with the world without producing any features—that is, without any attributes—then its existence has no explanatory relevance to us, like the dragon in the garage.

Thus, it is not that attributes determine the existence of matter, but that the boundary of observable matter is the boundary of attributes. The “object” here is no longer the thing-in-itself in the Kantian sense, but an object incorporated into the cognitive and explanatory system. Therefore, this paper does not claim that objects must be defined by attributes, but that objects defined by attribute sets have significance for explanation; otherwise, they are redundant in explanation (see section 3.7).

Therefore, this paper holds 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 the existence of a “thing itself” that can freely shuttle between different A’s while remaining unchanged and retaining explanatory significance.

2.4. Conservation

2.4.1. The Problem of the Indistinguishability of Identical Particles in Quantum Mechanics in Philosophy Poses the Most Severe Challenge to Leibniz’s PII, yet it 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 the principle of individuality, while the latter abandons individuality itself.

Saunders’ scheme undoubtedly represents one of the most ingenious and technically rigorous attempts in the revisionist path. Through his subtle definition of “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 itself 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 argumentation (Muller & Saunders, 2008). More centrally, the entire theoretical edifice of the scheme is built upon an unsettling presupposition: that the “individuality” of identical particles must, and can only, be “salvaged” by finding some (even relational) individuating property. This makes its theoretical efforts—no matter how ingenious—essentially a kind of 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 ad-hoc: it no longer seems like 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 chooses 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 notable for its conceptual thoroughness and consistency; it wholeheartedly embraces the most counterintuitive features of quantum mechanics and decisively breaks with our entire classical framework of objects and spatiotemporal localization. This resolute stance is undoubtedly clean and efficient in theory. However, the corresponding cost of this efficiency is that the concept of “non-individual” itself creates considerable explanatory burdens in metaphysics, 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 approaches 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 “In 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, coordinate), where the particle attribute set (mass, charge, spin, etc.).

· 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 in this domain. This explains the root of their indistinguishability.

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

Therefore, the confusion brought by quantum identical particles stems from erroneously substituting the difference in coordinate attributes into the judgment of identity for 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 specific spacetime). Particle annihilation and creation merely represent the decoupling and re-coupling of the e with different coordinates.

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

2.4.3. Formal Derivation of Conservation:

Assume a basic particle state can be expressed as: particle = (particle attributes, coordinate), where:

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

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

Formalization:

1. When the particle’s domain is particle = (particle attributes, coordinate₁), meaning an electron at a certain coordinate.

2. Coordinate decoupling (destruction): (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:

· When two particles’ eigen-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:

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

· Particle creation ⇨ The same e binds to coordinate₂ ⇒ Observed as reappearance. Example: Electron e disappears at coordinate₁ and appears at coordinate₂, which is actually 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, i.e., what logic permits exists, will neither annihilate nor be updated.

2.5. Symmetry

Max Black’s (1952) thought experiment of a symmetric universe poses the most extreme challenge to Leibniz’s strong PII principle. 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 argues from this that this is a real scenario of “two” things, thereby refuting PII—that there is no attribute 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 attribute); 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 difficult to work in Black’s original setup, because each sphere’s relational properties (“X miles from the other sphere”) are still completely identical.

This paper argues that the difficulties of Black’s challenge and 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 the framework of this paper, we must first clarify the domain of (n).

· 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 sought to refute.

· However, Black’s intuition—”there are obviously two spheres here”—is not entirely without basis. This theory explains it as a mental fixation. The reason observers report “seeing two” is that their perspective itself is embedded in this symmetric spatiotemporal coordinate system. This paper argues that a 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 a dilemma between “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).

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

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

However, some may say that the number 2 is a primitive fact.

But consider: suppose there exist two completely indistinguishable things, but as soon as symmetry is broken they are no longer indistinguishable. Any third party that creates asymmetry can directly prove that they are not the same in a global sense, which means distinguishable items have appeared. Of course, if we say global symmetry, then it becomes Black’s symmetric universe. The solution is as in section 2.5; essentially, it is not two. Therefore, the description “there exist two indistinguishable things” is contradictory. Assume there exist two objects A and B. At the same time:

A and B have completely identical attributes;

There are no distinguishable items;

No positional differences;

No relational differences;

No historical differences;

No third-party observer can distinguish them.

Then: What exactly distinguishes A and B?

If one says: “They are just numerically different.”

Then you have actually introduced a new property:

A possesses the property of “being A rather than B.”

B possesses the property of “being B rather than A.”

Thus: They have already become distinguishable.

Conversely, if there are no distinguishable attributes, then it can only be a symmetric state, but once it enters the symmetric state, it means it has undergone self-involvement with itself.

Therefore, once there are no distinguishable attributes, their numerical distinction loses its existential basis. Thus, the expression that they exist as two requires stronger argumentation, because a number without basis is not a number but an artificial label.

Some may say that numerical difference itself is the basis; the world is just like that.

This leads to a question: If indistinguishable things can be more than one thing, then what is it itself? Is it identical to itself? If it is identical to itself, yet there can exist other things identical to it but not identical to it?

Let: A and B are completely identical.

Then: A = A holds.

Why?

Because of the law of self-identity.

Then why does A ≠ B also hold?

If there is no difference whatsoever between A and B.

Then: Where does the fact of “inequality” come from?

Therefore, this paper holds that numerical multiplicity must have a distinguishable structure as its basis; “two” without basis is merely artificial naming, not additional existence.

3. Examples

3.1. The Replica Paradox

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

· Solution:

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

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

· Conclusion: Replicas are the same information forming sets with different spatiotemporal coordinates, leading to observability.

3.2. Gibbs Paradox

Category mistake:

· The goal should be particle type identity → (n) = (mass, spin, ...)

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

Correction: (n) and (n, label) ⇒ (n) ≡ (n). The entropy increase error stems from the wrong choice of (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 coordinate).

Correct solution:

· Define the goal: internal attribute identity → (n) = quantum attributes.

· The black hole disassembles the set (quantum attributes, coordinate); unpaired coordinate content leads to unobservability, but (quantum attributes/coordinates) as logical concepts do not disappear.

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

3.4. The Chinese Room Thought Experiment

Assume the target entity is the Chinese understanding function, defined as (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. The Twin Earth Paradox

· Traditional contradiction: The chemical formula of “water” on Earth and Twin Earth differs (H₂O vs. XYZ), but is the “water” concept of the inhabitants of the two planets the same?

· Theoretical solution:

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

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

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

3.6. The Grandfather Paradox

· Point of contradiction: If one returns to the past and kills one’s grandfather ⇒ one should not exist ⇒ the assassination cannot be carried out.

· Theoretical dissolution:

· Define the goal: worldline identity (worldline) = event causal historical logical structure.

· The assassination event leads to:

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

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

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

3.7. The Brain in a Vat

Then, when we face a situation where we do not know the rules, who decides which attributes should be included to satisfy the object we have in mind for comparison?

This paper holds that we should look at whether these attributes will affect the object we truly want to compare.

For example: If we are discussing consciousness (a phenomenon produced by neural activity).

Then: Changes in neural material and culture vat changes, if they do not affect consciousness itself, belong to redundant attributes.

Conversely: If a change in some attribute necessarily leads to a change in consciousness, then it should enter the domain.

This is close to scientific modeling:

Retain only explanatory variables.

Delete irrelevant variables.

For example, studying free fall:

The color of the apple is unimportant;

Mass may be important.

So color is excluded.

In general, only attributes that affect the target object should enter the domain.

Therefore, in the current debate about the “brain in a vat,” whether skeptical or realist interpretations, they all implicitly and without examination sneak the attributes of the “external carrier” (biological brain or vat) into the judgment of “cognitive” identity:

· Question: How to prove that one is not a brain in a vat? Perception cannot distinguish reality from simulation.

· Applying the theory’s formula:

· Define (cognition) = perceptual information.

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

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

· According to the axiom (n) ≡ (n), B ≡ B (same cognition).

3.8. Mary’s Room

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

· Theoretical answer:

· Define types of knowledge:

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

· Qualia knowledge: (K_qualia) = subjective red experience.

· ∵ (K_prop) ≠ (K_qualia)

· ∴ The two 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

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

· Theoretical deconstruction:

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

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

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

· Uniqueness theorem adjudication:

· If the goal is unconstrained rational decision-making → Use (n) ⇒ Choose two boxes.

· If the goal is decision-making with predictive causality → Use (n, history) ⇒ Choose one box.

· Paradox dissolution: The two are different levels of decision entities (attribute set 1) ≠ (attribute set 2); the contradiction stems from swapping domains.

3.10. The Raven Paradox

· Core of the paradox: “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?

· Theoretical deconstruction:

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

· Correct definition:

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

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

· Conclusion:

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

· If one claims that “the confirmatory power of the red apple differs from that of the raven,” then it is a category mistake, expanding (p) to the physical category of samples (birds/fruits), violating the initial logical goal.

3.11. The Sorites Paradox (Heap/T Heap Paradox)

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

· Theoretical deconstruction:

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

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

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

· Solution:

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

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

· Root of the paradox: Stealthily swapping (n) domains in the argument (from morphology to quantity).

3.12. The Sleeping Beauty Problem

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

· Theoretical deconstruction:

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

· Attribute set 1 (prior probability): (probability1) = physical state of the coin ⇒ P(heads) = 1/2.

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

· Uniqueness adjudication:

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

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

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

3.13. Modern Contradictions in Pascal’s Wager

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

· Theoretical deconstruction:

· Category mistake: Confusing the domain of (god).

· Attribute set 1: (god) = divine description in a specific religious doctrine.

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

· Adjudication:

· If comparing the reality of specific religious gods → Each (god) differs ⇒ Categories are mutually distinct.

· If asking “Does a supreme entity exist?” → An independent definition of (omnipotent entity) is needed, independent of specific religions.

3.14. The Unexpected Hanging Paradox

· Problem: The judge announces “You will be unexpectedly executed on some day next week.” The prisoner deduces it cannot happen, but execution day still arrives.

· Theoretical deconstruction:

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

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

· 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 (Refer to 3.7)

4.1. The Dilemma of Personal Identity and Existing Theories

The core problem of personal identity is: What makes a person the same person over time? Traditional theories mainly revolve around physical continuity (e.g., continuity of the brain) and psychological continuity (e.g., coherence 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 (such as memory, character, intentions) over time. This theory demonstrates remarkable explanatory power when handling 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 unelucidated premise: namely, on a given time-slice, how do we determine that an entity is a “person,” and how do we perform static, cross-world comparisons between them at different time-slices. In other words, these theories are adept at answering “Why is he still him?” (the dynamic persistence problem) but neglect defining “What exactly is ‘he’ at time t?” (the static identity problem). This static “what” is a prerequisite for any discussion of dynamic “continuity.”

This weakness is exposed in Bernard Williams’ (1970) famous “fission” thought experiment. When a person splits into two psychologically continuous successors, physical continuity theories collapse due to their inability to handle “one dividing into two”; 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 standard for static identity, 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 the attributes of the “carrier” (biological brain) or “history” (causal chain) to intrude into the judgment of “person”’s own identity—this is a category mistake. The debate between Parfit and Williams is essentially a conflict between two different (n) domains (one a flow of psychological attributes, the other a history of physical carriers), but neither side realizes this, thus falling into an insoluble impasse.

An analytical framework based on (n)

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

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

Formally, let:

· Consciousness be a consciousness time-slice.

· We define it as: (consciousness, q)

· q: Represents the carrier and coordinate 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 the two time-slices is the same, they are different instances of the same consciousness, regardless of whether the q between them is continuous.

Thus, we provide a clear analysis for Williams’ fission experiment: the reason the two successors (consciousness, q₁) and (consciousness, q₂) trigger a paradox is that 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 (consciousness, q₁) and (consciousness, q₂) and (consciousness, q₃) 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 in 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 continuity” to proceed on a firm logical basis.

4.1.2. Time

Based on the axiomatic system established earlier, we can reach a thoroughly transformative conclusion about the existence of consciousness in the temporal dimension: Your “present” is (consciousness, q_now). Your “past” is (consciousness, q_past). Your “future” is (consciousness, q_future). They are all 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 experiment at the limit. Assume 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 natural Poincaré recurrence, extreme coincidence of quantum fluctuations, or some cosmic recurrence mechanism we do not yet understand) is instantly assembled and activated, producing an instantaneous (consciousness) 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 a 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 requiring defense, but the iron law of logical identity. The weight of temporal interval is zero in this judgment.

Now, let us push this thought experiment to another extreme. Assume at time point t₁, a consciousness activity “a” has just begun its neural computation process. Within an extremely short time Δt before the activity of the first neural system is completed (i.e., (a) has not yet 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:

· Process P₁: On carrier and coordinate q₁, starting at time t₁ and continuing.

· Process P₂: On carrier and coordinate q₂, starting at time t₁ + Δt and continuing.

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

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

This means that at the level of consciousness, what we observe is not two consciousnesses but one consciousness appearing simultaneously at two time points. It is not that there are two “yous” thinking, but that “your” thinking process is executed and presented by two 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 finishing the experiences).

From this, we reach a counterintuitive but logically necessary conclusion: Identity is non-continuous in time and non-local in space. The way anything persists is not like a continuous “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 “present” 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” into n, if the universe of the previous second is completely identical to that of the next second, universe ≡ universe, then time here belongs to unfalsifiable redundancy. Conversely, if universe ≠ the next second’s universe, it means presentness and discreteness.

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

Before discussing the “spatiotemporal leap” 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 law upper limits that any physical signal and entity motion must follow—the speed of light. Any attempt to realize “spatiotemporal leap” 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 evidence.

However, the “spatiotemporal leap” argued in this paper is essentially 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?” 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 Leap Based on the Law of Identity

Traditional physics (including relativity) studies the combination (consciousness, carrier and coordinate) as the object of (spatiotemporal leap). 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, on the other hand, focuses on a possibility not discussed in physics: namely, the decoupling and recombination of a consciousness with its carrier and coordinate.

1. Decoupling: (consciousness, carrier and coordinate₁) → (consciousness), (carrier and coordinate₁). This may physically correspond to the destruction of the carrier (such as the brain) by some event consistent with the event horizon principle (velocity differences leading to causal isolation at the neuronal level), causing the consciousness to no longer be instantiated (carrier and coordinate = ∅).

2. Recombination: (consciousness) → (consciousness, carrier and coordinate₂). This may physically 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 coordinate₂) and the old state before decoupling (consciousness, carrier and coordinate₁), sharing the same consciousness, are necessarily different manifestations of the same consciousness. This is the logical core of “spatiotemporal leap”: 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:

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

· 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 exceed light speed. The reason “spatiotemporal leap” seems “unbelievable” or even “violates physics” is precisely because we erroneously use physical laws describing the motion of combinations to judge a logical theorem about identity. This is also a category mistake.

Conclusion: The “spatiotemporal leap” proposed by this framework is not a physical conjecture awaiting realization but an already established logical 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 logically inescapable yet intuitively highly impactful conclusion: From a strict first-person perspective, any “death” event that can lead to the termination of consciousness in a state of unconsciousness 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.

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

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

1. Foundation of consciousness identity: Consciousness identity is determined by consciousness itself, not by (consciousness and specific carrier coordinate).

2. Decoupling and recombination: When the carrier is destroyed (e.g., plane crash, surgical failure), what occurs is (consciousness, carrier and coordinate₁) → (consciousness), (carrier and coordinate₁). Consciousness continues due to its logical identity (n ≡ n).

3. Survivor effect and observational necessity: Consciousness can be instantiated by “instantiable” carriers and coordinates. It will (necessarily in logic) re-bind with a (carrier and coordinate₂) that can continue to exist, i.e., (consciousness) → (consciousness, carrier and coordinate₂).

4. Continuity of first-person experience: For consciousness, the transition from (consciousness, carrier and coordinate₁) to (consciousness, carrier and coordinate₂) is seamless in experience. Consciousness itself will not experience “death” or “nothingness,” because that would mean no observer. It will only experience a “leap” from one survivable state to another survivable state.

Therefore, under anesthesia:

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

· From the 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 unobservable and non-existent for consciousness itself.

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

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:

· 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.

· The essence of risk is subverted: For awake, continuous consciousness, risk is real (e.g., a cut hurts, jumping off a building involves experiencing falling and impact). But for consciousness crossing risk through anesthesia “leap,” risk is completely eliminated. Risk only exists in those “other” worldlines that will never be experienced.

· 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 outcome in personal history will always be successful.

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

· One may wake up severely injured in the wreckage of a plane crash.

· One may wake up after surgery with serious complications or permanent disability.

· 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. The Dilemma of Ethical Problems and Existing Theories

Since the birth of ethics, generation after generation of highly insightful philosophers, from Kant’s grand a priori architecture to Mill’s subtle consequentialist calculations, have constructed a splendid ethical edifice for us. These excellent efforts share a profound and admirable ambition: to seek a solid metaphysical foundation for moral judgment that transcends the individual perspective. 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 kind 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 the subtle rift between the principle of control 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 can be called 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 the dilemma through global calculations of 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 with the individual’s first-person perspective at the motivational level, but this is by no means a flaw in these theories themselves, but perhaps precisely highlights a pathetic and even heroic tension that human reason inevitably faces when pursuing moral nobility.

The work of this paper, 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 may be achieved through a more direct and 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 realize and pay tribute to the core goals of traditional ethics in a brand-new way. Our core argument is: 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 the axioms of this paper, we can derive a foundational principle of 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) reducibly manifest as observable influences on consciousness.

Corollary: If the occurrence of an event E produces no discernible difference in any possible experiences of consciousness in the past, present, or future, 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:

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

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

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

Therefore, event E (the act of cheating), due to its zero impact on consciousness’s experience, does not constitute a variable in ethical assessment directed at 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 did not appear.

Conclusion: There is no absolute objective world; moral error is not mysteriously attached to the behavior itself but is systematically and verifiably associated with the specific impact patterns the behavior produces on conscious experience. Lacking such observable impact patterns, the behavior is not considered in ethical assessment.

4.3.3. Implications of the New Framework

If morality is not about inaccessible “external truth,” 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 its only effectively operable domain: first-person facts of conscious experience. The good or evil of an action does not depend on its attributes 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 not observed by any conscious system are not assigned values in ethical computation. The boundary of moral concern is the boundary of conscious experience. This framework is not deliberately intended to exonerate traditional immoral behaviors. It provides a more solid, clearer, and inescapable foundation: We bear sole and complete responsibility for 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 leap.” The core of the model is: The next experiential instance of consciousness will “choose” one from all logically compatible future state branches for binding. When explaining why we usually do not experience “leaps,” 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 realizes directional leaps in logic 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: that the set of logically possible world states is finite and non-repeatable (including future and past on the timeline). Only under this premise do concepts such as “vast majority of branches” and “very high probability” have operational significance. The probability of winning the lottery is one in a million precisely because, among one million physically subtly 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 of world branches experiencing “teacup suspended in air” is also infinite.

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

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

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

From the “God’s-eye view” of logic itself, 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—have equal ontological status. In this panorama, there are no so-called “lucky ones”; only the totality of facts exists.

However, from the first-person “prisoner perspective” of consciousness, its experience is indisputably single, continuous, and highly orderly. We have never personally experienced random leaps of the world but are firmly situated in a classical reality that strictly follows 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 impoverished, 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 valid.

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

5.2. The Graveyard of Logical Possibilities and the Survivor: 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 foundation 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 “crazy universe” whose underlying logic allows self-negation. In such a universe, the law of identity is subverted; an entity can simultaneously not be itself, and propositions can simultaneously be true and false. Concepts such as “rational π” or “square circle,” regarded as contradictions in Euclidean space, may be commonplace manifestations of its infinite weirdness in that 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:

1. 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 instantly disintegrate.

2. Rupture of causal chains: No reliable connection will exist between intention and action, cause and effect. The act of reaching out to grab an “apple” cannot be defined, because at the moment of execution, the “hand,” “apple,” and even “you” itself may have already self-negated.

3. 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 instantaneously self-destruct due to their intrinsic contradictions. It represents the impossibility of existence.

Here we touch upon 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 instantaneously self-dissolve due to its intrinsic 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 can and has stably operated.” The fact that we can think and debate the problem of identity at this moment and observe a stable, coherent, understandable universe is itself an absolutely selective result. 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 of n ≡ n. We observe n ≡ n not because it is the only correct logical theorem among all possible worlds, but because in a world of n ≠ n, no “observer” can exist to perform any “observation.”

All the work of this theory—dissolving category mistakes—is carried out within this unique “survivor universe.” The axiom n ≡ n is not an arbitrarily “invented” setting but a “discovery” and “formalization” of the most basic and solid operating mode of this survivor universe. All the paradoxes we encounter, such as the Ship of Theseus and quantum identical particles, occur on this solid foundation of identity and arise from “user errors” (category confusions) when using this stable system, not from “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. There is no need to feel it is unlikely; 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 in the preceding text centers on avoiding the erroneous application of a criterion from one condition to another. This analytical pattern itself raises a deeper metaphysical question: What kind of most basic logical conditions does such a clear worldview itself require? Traditional monism (such as Spinoza’s substance theory), which pursues a single, homogeneous foundation, seems unable to accommodate genuine differences and interactions between attributes.

In contemporary metaphysics, Jonathan Schaffer (2009) revives and defends “priority monism,” asserting that the cosmos as a whole is the only fundamental entity, with its parts depending 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 different attributes with 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” incapable of deriving the “many” we can distinguish.

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, the Einstein field equations of general relativity and the Schrödinger 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 inference 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 “not mutually derivable” in a specific sense. It is their relation, rather than their isolated existence, that provides space for diversity.

5.3.2. The Logical Dilemma of Traditional Monism

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

1. Being is; non-being is not.

2. Being is indivisible (because if divisible, the division point would be “non-being”).

3. Being is unchanging (because change requires “non-being” as a starting or ending point).

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

However, this picture faces the problem of derivation: How can diversity and change as we experience them be logically derived from an absolutely undifferentiated “one”? Parmenides himself acknowledged that the world we perceive (“the way of opinion”) is full of change and plurality, but he held that this is merely illusion. The cost of this thorough denial of the reality of the experiential world is too high.

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

· There is only one substance (God or Nature).

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

· All things are modes of the substance.

Schaffer’s priority monism represents the precise form of contemporary monism. He asserts:

1. The whole is prior: The cosmos as a whole is metaphysically prior to its parts.

2. Dependence relation: Parts depend on the whole for existence, not vice versa.

3. Explanatory advantage: This picture better fits 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 the substance, yet simultaneously insists on the causal independence of attributes (thought cannot affect extension, and vice versa). This means that at the explanatory level, thought and extension are two parallel, non-mutually derivable explanatory chains. If we take this independence seriously, then the concept of “substance” here plays more like 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 the explanatory dilemma.

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 has 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 differences → cannot logically derive diversity → world diversity cannot be explained.

Now attempt to add another entity, let there be 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.) does not pre-exist in the original elements; its novelty stems from the relations between elements. Conversely, if there is only a single principle n, according to the axiom n ≡ n and n has no internal differences, then 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: The Irreducible Plurality of 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:

1. 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.

2. Independence of principles: The linear superposition principle of quantum mechanics and the equivalence principle of general relativity are conceptually completely different and cannot currently be derived from each other.

3. Irreducibility of initial conditions: The initial conditions of the universe (such as the low-entropy initial state) cannot be derived from the physical laws themselves.

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

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

1. Monochromatic vision: Can only perceive light and dark, unable to distinguish colors. The world is a single grayscale.

2. Dichromatic vision: Can perceive two basic colors and their mixtures. The world has a limited color dimension.

3. Trichromatic vision: Can perceive three basic colors and their rich combinations. The world presents rich colors.

Similarly, at the metaphysical level:

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

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

· Sufficient pluralism may acknowledge more basic principles.

This framework tends toward minimal pluralism: To explain the stable categorical differences we observe in our world, at least two logically independent basic principles are needed. 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 worlds strictly following known physical laws and worlds 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 starkly contrary to the highly consistent, continuous, and predictable classical 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 similarly have no reason to blindly assert that reality can only be explained by one quantity or two to three quantities. It is like an infinitely large blank canvas that stipulates the physical boundaries (the canvas) for painting, but the canvas itself cannot determine what patterns should appear on it, nor can it explain why what is ultimately presented is the “Mona Lisa” rather than randomly splashed paint. Attempting to fully explain the structure of the painting using the existence of the canvas 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 concert with at least one other independent and currently unknown basic principle that is not mutually derivable.