Subjective Numbers: Two-Tier Semantics, Categorical Provenance, and Measurable Discrepancies

Abstract

A subjective number is a value endowed with an intrinsic perspective. This article presents a two-tier semantics for subjective numbers. Base directed judgments are separated from inferential closure. For each evaluating perspective p, a base kernel R^∘_p records initial perspective-indexed judgments. Closed judgments R^⋆_p are generated as the least simultaneous fixpoint of a monotone one-step operator.

The closure operator enforces reflexivity and transitivity inside each evaluating perspective and adds cross-perspective adoption only when a Boolean guard CPA_p(q,u) permits it. The resulting fixed-perspective relations are preorders. The framework tracks provenance through proof terms and derivation categories, using the empty history ε as the history identity. This keeps categorical provenance separate from Boolean or numeric identity elements.

Under explicit standard Borel hypotheses on both perspectives and values, the closed sections are analytic and universally measurable. This supports a discrepancy functional between perspectives and yields a pseudometric Δ. The article also gives algebraic operations on subjective numbers, an explicit asymmetric model, selective fusion theorems, a restrained comparison with modal and many-valued logic, and finite-universe termination and complexity bounds.

Notation Guide

Symbol	Description
P	Set of perspectives.
V	Value space.
X	Set of subjective numbers, X=P×V.
a(s)	Subjective number with value a and intrinsic perspective s.
R^∘_p	Base kernel of initial positive directed judgments at evaluating perspective p.
R^⋆_p	Closed relation generated by the least simultaneous fixpoint.
a(s) ⊑_p b(t)	Closed judgment R^⋆_p(s,t,a,b)=1.
CPA_p(q,u)	Guard allowing p to compose a p-validated left edge with a q-validated right edge whose target intrinsic perspective is u. It is not bare import of arbitrary q-judgments.
N^∘_p	Optional rejection log used only when examples explicitly record negative information.
Deriv_p	Derivation category for judgments concluding in perspective p.
Hist	History category whose morphisms are typed finite histories.
ε	Empty history and identity history.
H_p	Strict provenance functor from derivations to histories.
δ(p,q;s,t)	Localized discrepancy between perspectives p and q at intrinsic pair (s,t).
Δ(p,q)	Aggregated discrepancy pseudometric between perspectives.
p ⊕ q	Fused perspective, when a fusion operation is specified.

1. Introduction

1.1 Motivation and Central Idea

Many mathematical settings require directional judgments. Preference aggregation, distributed knowledge, and multi-agent coordination often involve claims that one perspective accepts while another does not. A useful formalism must allow local coherence without forcing every judgment into one global equivalence relation.

A subjective number is written a(s), with value a ∈ V and intrinsic perspective s ∈ P. For each evaluating perspective p, a base kernel R^∘_p records initial judgments. The closed relation R^⋆_p is generated by a least fixpoint closure.

This two-tier separation is the main formal move. Base kernels can be sparse and asymmetric. Closure supplies reflexivity, transitivity inside a fixed evaluating perspective, value substitution, and guarded cross-perspective adoption. Cross-perspective adoption is controlled by CPA_p(q,u).

This makes the framework useful for analyzing the liar paradox, where a directed role relation becomes contradictory only after an unindexed collapse.

1.2 Positive Judgments, Absence, and Rejection Logs

The primary system is positive. A judgment is either present in a base kernel, derivable after closure, or absent from the positive relation. Absence is not the same as falsity. Some examples also record explicit rejections through an optional negative log N^∘_p. These rejections are annotations unless a separate negative logic is added. They do not participate in the positive closure operator.

This distinction prevents a common ambiguity. Saying that a(s) ⊑_p b(t) is not derivable means that the positive system has no proof of that judgment. Saying that perspective p rejects the judgment means that the optional log N^∘_p records a negative attitude. The two notions are kept separate.

Two-tier subjective-number semantics separates base kernels from closure

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2. Formal Data

2.1 Basic Notation

P is a nonempty set of perspectives.
V is a value space equipped with equality =.
The set of subjective numbers is X=P×V. The pair (s,a) is written a(s).
For each p ∈ P, the base kernel is R^∘_p⊆X×X, equivalently R^∘_p:P×P×V×V→{0,1}.
The closed relation is R^⋆_p⊆X×X.
The closed directed-judgment notation is a(s) ⊑_p b(t), meaning (a(s),b(t))∈R^⋆_p.

2.2 Value Substitution

Kernels respect equality on V. If a=a' and b=b' in V, then R^∘_p(s,t,a,b)=R^∘_p(s,t,a',b'). When equality on V is literal identity, this condition is automatic. If V is presented with an equivalence relation rather than literal equality, the same rule says that the kernels are invariant under replacement of representatives. One may instead quotient V first and then form X=P×V from equivalence classes.

2.3 Cross-Perspective Composition Guard

For each evaluating perspective p, the cross-perspective composition guard is a Boolean predicate

CPA p :P\timesP\to{0,1}.

The reading of CPA_p(q,u)=1 is not that p may import every judgment validated by q. It means that p may compose through a q-validated edge when the edge begins where a p-validated edge ends and when the resulting target intrinsic perspective is u.

x ⊑p y,\quad y ⊑q z,\quad CPAp(q,tgt(z))=1 \quad ⇒ \quad x ⊑p z.

The guard is directional. In general, CPA_p(q,u) and CPA_q(p,u) may differ. The target parameter u records where the adopted right edge lands, so two right edges validated by the same q may be treated differently by p.

2.4 Measurable Structure

The discrepancy construction requires additional measurable data. For that section, assume that P and V are standard Borel spaces, that ν is a complete probability measure on P, and that μ_V is a complete probability measure on V. The product measure on V×V is μ_V×V.

Let the global base set and global guard set be

B={(p,s,t,a,b):R \circ p (s,t,a,b)=1}

C={(p,q,u):CPA p (q,u)=1}.

For the measurable results, require B and C to be Borel. This implies that each base section A^∘_p;s,t={(a,b):R^∘_p(s,t,a,b)=1} is Borel.

3. Two-Tier Semantics: Closure and Derivability

3.1 Global Atomic Predicates

For each r∈P, let E_r(s,t,a,b) be an atomic predicate. A base axiom asserts E_r(s,t,a,b) whenever R^∘_r(s,t,a,b)=1.

3.2 Derivation Rules

Base: infer E_p(s,t,a,b) from R^∘_p(s,t,a,b)=1.
Refl: infer E_p(s,s,a,a).
Subst: from E_p(s,t,a,b) and a=a', b=b', infer E_p(s,t,a',b').
Trans: from E_p(s,t,a,b) and E_p(t,u,b,c), infer E_p(s,u,a,c).
CPA composition: from E_p(s,t,a,b), E_q(t,u,b,c), and CPA_p(q,u)=1, infer E_p(s,u,a,c). This rule composes a p-edge with a guarded q-edge; it does not copy all of R^⋆_q into R^⋆_p.

3.3 Least Fixpoint Closure

Let D=X×X and let L=∏_r∈P𝒫(D) be the complete lattice of perspective-indexed relations ordered by componentwise inclusion. For S∈L, define a one-step operator T:S↦T(S). For each p, T_p(S) is the union of:

R^∘_p.
The diagonal {(a(s),a(s)):s∈P,a∈V}.
All value-substitution variants of members of S_p.
All (a(s),c(u)) such that for some b(t), (a(s),b(t))∈S_p and (b(t),c(u))∈S_p.
All (a(s),c(u)) such that for some q∈P and b(t), (a(s),b(t))∈S_p, (b(t),c(u))∈S_q, and CPA_p(q,u)=1. This is guarded cross-perspective composition, not unrestricted import of S_q.

Define S⁰=⊥, the family of empty relations, and Sⁿ⁺¹=T(Sⁿ). The closed relation is

R ⋆ = ⋃ n<ω S n .

a(s) ⊑ p b(t) \Leftrightarrow (a(s),b(t))\inR ⋆ p .

Theorem 3.1 (Existence and Finite-Stage Closure): The operator T is monotone and ω-continuous. Hence R^⋆=⋃_n<ωTⁿ(⊥) is the least simultaneous fixpoint of T.

Proof: If S⊆S' componentwise, then every premise available in S is available in S'. Each clause defining T is therefore inclusion-preserving, so T(S)⊆T(S'). Thus T is monotone.

Let S⁰⊆S¹⊆... be an increasing chain and let S=⋃_n<ωSⁿ. A judgment in T(S) is obtained by one of the defining clauses. Base and reflexive judgments are already in T(S⁰). A substitution judgment depends on one premise, so that premise occurs in some Sⁿ. A transitivity or CPA judgment depends on two premises; if those premises occur in Sⁱ and S^j, then both occur in S^max(i,j) because the chain is increasing. Therefore the one-step conclusion occurs in T(S^max(i,j)). Hence T(⋃Sⁿ)⊆⋃T(Sⁿ). The reverse inclusion follows from monotonicity. Thus T is ω-continuous.

Now let U=⋃_n<ωTⁿ(⊥). By ω-continuity, T(U)=⋃_n<ωT(Tⁿ(⊥))=⋃_n<ωTⁿ⁺¹(⊥)=U. So U is a fixpoint. If Y is any fixpoint, then ⊥⊆Y, and monotonicity gives Tⁿ(⊥)⊆Tⁿ(Y)=Y for every n. Hence U⊆Y. Therefore U is the least fixpoint. □

3.4 Local Preorders

For each fixed evaluating perspective p, the relation ⊑_p is reflexive and transitive after closure. Symmetry is not imposed globally.

Theorem 3.2 (Fixed-Perspective Preorders): For every p∈P, the closed relation R^⋆_p is a preorder on X.

Proof: Reflexivity holds because the diagonal (a(s),a(s)) is added by the Refl clause of T and hence belongs to the least fixpoint. Transitivity holds because if (a(s),b(t))∈R^⋆_p and (b(t),c(u))∈R^⋆_p, then applying T to the fixpoint adds (a(s),c(u)) by the Trans clause. Since T(R^⋆)=R^⋆, the conclusion is already in R^⋆_p. Therefore R^⋆_p is reflexive and transitive. □

3.5 Soundness and Completeness

The global derivation rules are sound and complete for the least-fixpoint semantics.

Theorem 3.3 (Soundness and Completeness): E_p(s,t,a,b) is derivable from the rules if and only if (a(s),b(t))∈R^⋆_p.

Proof: For soundness, use induction on the height of a derivation. A Base derivation is included in R^⋆ because every base kernel is included in T(R^⋆)=R^⋆. A Refl derivation is included because the diagonal is included. A Subst, Trans, or CPA derivation has premises that belong to R^⋆ by the induction hypothesis, and the corresponding clause of T places its conclusion in T(R^⋆)=R^⋆.

For completeness, use induction on the least finite stage n at which a judgment enters Sⁿ. At stage 1, the judgment is produced by Base or Refl, and hence has an immediate derivation. If a judgment first appears at stage n+1, it is produced by one of the clauses of T from premises that occur at stages at most n. By the induction hypothesis those premises have derivations. Applying the corresponding derivation rule gives a derivation of the stage n+1 judgment. Because R^⋆ is the union of all finite stages, every closed judgment is derivable. □

3.6 Positive Consistency of the Core Language

The core language derives positive judgments only. It does not contain a rule that converts absence into negation or a rule that turns an optional rejection log into a positive contradiction. If a signed or paraconsistent extension is desired, it must be added explicitly.

Theorem 3.4 (Absence Is Not Contradiction): In the core system, the failure to derive a(s) ⊑_p b(t) never entails a derivation of its negation.

Proof: The derivation language contains atomic predicates E_p(s,t,a,b) and the rules Base, Refl, Subst, Trans, and CPA. None of these rules has a conclusion containing a negation symbol, and no rule has the form "from non-derivability infer negation." Therefore a missing derivation supplies no derivation of a negative judgment. Optional rejection logs are not premises of the positive closure rules, so they also cannot create a positive contradiction. □

Categorical provenance records derivation histories for subjective-number judgments

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4. Categorical Provenance

Category theory enters through derivational provenance. The objects are subjective numbers. The morphisms are typed proof terms witnessing closed judgments. This avoids treating raw values such as 0 or true as categorical identities.

4.1 Proof Terms

Proof terms record whether a judgment came from a base kernel, reflexivity, substitution, transitivity, or guarded cross-perspective composition. A term concluding E_p(s,t,a,b) is a derivation from a(s) to b(t) at perspective p. Write src(π) and tgt(π) for the source and target subjective numbers of a proof term π.

The proof-term grammar is generated by the following constructors:

π ::= base p (s,t,a,b) | refl p (x) | subst(π,a=a',b=b') | trans(π,ρ) | cpa p,q,u (π,ρ).

The constructor base_p(s,t,a,b) records the source kernel entry R^∘_p(s,t,a,b)=1. The constructor refl_p(x) records identity at the subjective number x. The constructor subst records the value-identifications used. The constructor trans(π,ρ) records ordinary same-perspective composition. The constructor cpa_p,q,u(π,ρ) records guarded composition of a p-validated left edge with a q-validated right edge landing at intrinsic perspective u.

The quotient on proof terms identifies only identity and associativity rearrangements of trans. It does not identify distinct base entries, substitution events, or CPA-composition events. Provenance labels therefore survive the quotient used to form the derivation category.

4.2 Derivation Category

For each p, the category Deriv_p has objects X. A morphism x→y is an equivalence class of proof terms deriving E_p(x,y). The identity at x is the reflexivity proof refl_p,x. Composition is induced by the Trans rule:

[ρ]\circ[π]=[trans(π,ρ)].

The equivalence relation identifies associativity and identity rearrangements of proof terms.

Theorem 4.1 (Derivation Categories): For each p, Deriv_p is a category.

Proof: The source and target of a proof term determine its hom-set. If π:x→y and ρ:y→z, the Trans rule gives trans(π,ρ):x→z, so composition is typed. Reflexivity gives refl_p,x:x→x for every object. The quotient relation identifies trans(refl_p,x,π) with π, trans(π,refl_p,y) with π, and trans(trans(π,ρ),σ) with trans(π,trans(ρ,σ)). Therefore identities and associativity hold in the quotient. □

4.3 History Category

Let Σ be an alphabet of provenance events, including base events, adopted-perspective events, and value/perspective labels. A typed history from x to y is a triple (x,h,y) with h∈Σ^*. The history category Hist has objects X and morphisms (x,h,y):x→y. The identity at x is (x,ε,x). Composition is concatenation:

(y,k,z)\circ(x,h,y)=(x,hk,z).

Theorem 4.2 (History Category): Hist is a category.

Proof: Composition is defined exactly when the target of the first typed history equals the source of the second. The result has source x and target z. The empty word ε is the identity for word concatenation, so (x,ε,x) acts as a left and right identity. Concatenation of finite words is associative, so typed history composition is associative. □

4.4 Strict Provenance Functor

The functor H_p:Deriv_p→Hist sends a derivation to its recorded history. It sends a reflexivity proof to the empty history and sends transitive composition to concatenation. For a CPA step, it records both the importing perspective and the perspective whose judgment was adopted.

Theorem 4.3 (Strict Provenance Functor): With histories defined by recursive recording of proof terms, H_p is a strict functor.

Proof: On objects, H_p is the identity function on X. On morphisms, it maps a proof term π:x→y to the typed history (x,h(π),y). By definition, h(refl_p,x)=ε, so identities are preserved. For composable proof terms π:x→y and ρ:y→z, define h(trans(π,ρ))=h(π)h(ρ). Then H_p([ρ]∘[π])=(x,h(π)h(ρ),z)=H_p([ρ])∘H_p([π]). The quotient on proof terms identifies only associativity and identity rearrangements, and those have the same concatenated history because word concatenation is associative and ε is neutral. Hence H_p is well-defined and strictly preserves identities and composition. □

5. Measurable Discrepancies and a Pseudometric

For each p,s,t∈P, define the closed section

A ⋆ p;s,t ={(a,b)\inV\timesV:R ⋆ p (s,t,a,b)=1}.

5.1 Analyticity and Universal Measurability

Theorem 5.1 (Analytic Closed Sections): Under the standard Borel hypotheses of Section 2.4, every closed section A^⋆_p;s,t is analytic and therefore universally measurable.

Proof: Let Sⁿ be the finite-stage construction from Section 3.3, represented globally as subsets of P×P×P×V×V. The base stage is Borel because the global base set B is Borel, and the reflexive diagonal is Borel in a standard Borel product. Suppose Sⁿ is analytic. The substitution clause preserves analyticity because equality on a standard Borel space is Borel and analytic sets are closed under Borel preimages and projections. The transitivity clause is obtained by taking the analytic set of tuples satisfying two membership conditions with a shared middle subjective number and projecting away the middle variable. Analytic sets are closed under finite intersections with Borel constraints and under projection. The CPA clause is handled the same way, with the additional Borel guard set C. Therefore Sⁿ⁺¹ is analytic. By induction, every finite stage is analytic, and R^⋆=⋃_n<ωSⁿ is analytic because analytic sets are closed under countable unions. A section of an analytic set is analytic. Analytic subsets of standard Borel spaces are universally measurable. Hence each closed section is universally measurable. □

5.2 Localized Discrepancy

δ(p,q;s,t)=\int V\timesV |1 A ⋆ p;s,t (a,b)-1 A ⋆ q;s,t (a,b)| dμ V\timesV (a,b).

This is the measure of the symmetric difference between the closed sections for p and q at intrinsic perspectives s,t:

δ(p,q;s,t)=μ V\timesV (A ⋆ p;s,t △ A ⋆ q;s,t).

5.3 Aggregated Pseudometric

Δ(p,q)=\int P\timesP δ(p,q;s,t)d(ν\timesν)(s,t).

The integral is taken in the completed product measure. The global closed relation R^⋆ is analytic by Theorem 5.1. We use the standard fiber-measure theorem: if A⊆X×Y is analytic in a standard Borel product and μ is a probability measure on Y, then the function x↦μ(A_x) is upper semianalytic and therefore universally measurable.

Apply this theorem to the closed-section families for p and q and to their intersection. Since

μ(A△B)=μ(A)+μ(B)-2μ(A\capB),

the function (s,t)↦δ(p,q;s,t) is universally measurable and bounded by 1. Hence the integral defining Δ is well-defined in the completed product measure.

Theorem 5.2 (Pseudometric): The function Δ satisfies reflexivity, symmetry, and the triangle inequality. It is a pseudometric because distinct perspectives can have distance zero when their closed sections differ only on a ν×ν-null family of intrinsic pairs and a μ_V×V-null set of value pairs.

Proof: The preceding fiber-measure argument makes Δ well-defined. For fixed s,t, δ(p,q;s,t) is the measure of a symmetric difference. Thus δ(p,p;s,t)=0, and symmetry follows from A△B=B△A. For the triangle inequality, use the set inclusion A△C⊆(A△B)∪(B△C). Taking measures gives μ(A△C)≤μ(A△B)+μ(B△C). Hence δ(p,r;s,t)≤δ(p,q;s,t)+δ(q,r;s,t). Integrating this pointwise inequality over P×P gives Δ(p,r)≤Δ(p,q)+Δ(q,r). Distinct perspectives can have equal closed sections almost everywhere, so Δ(p,q)=0 need not imply p=q. Therefore Δ is a pseudometric. □

Perspective discrepancy compares closed sections across evaluators

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6. Finite-Universe Termination and Complexity Bounds

Suppose |P|=m and |V|=n. Then |X|=mn. For each evaluating perspective there are at most |X|²=m²n² possible positive atoms, and across all evaluating perspectives there are at most

M=m|X| 2 =m 3 n 2

possible atoms.

Theorem 6.1 (Finite Termination): In a finite universe, the least-fixpoint saturation terminates after at most M strict atom-insertion stages.

Proof: The construction is monotone: atoms are added and never removed. There are only M possible atoms. Every strict stage adds at least one atom not present before. Therefore there can be at most M strict stages. Once a stage adds no atom, the current relation is closed under all rules and is the least fixpoint because the construction started from the empty relation and added only rule consequences. □

Theorem 6.2 (Naive Complexity Bound): A direct saturation algorithm that repeatedly scans all possible rule instances uses O(m³n²) space and has the conservative time bound O(m⁸n⁵).

Proof: Space is bounded by the number of possible atoms, M=m³n². A full scan of Trans instances ranges over an evaluating perspective and three subjective numbers, giving O(m|X|³) checks. A full scan of CPA instances ranges over two perspectives and three subjective numbers, giving O(m²|X|³) checks, which dominates. Since |X|=mn, one scan costs O(m²(mn)³)=O(m⁵n³). By Theorem 6.1, at most M=m³n² strict scans are needed. Multiplying gives O(m⁸n⁵). This is a simple upper bound, not an optimal algorithmic claim. □

7. Operations and Algebraic Structure

Assume V carries a binary operation +. The operation on values is ordinary. The propagation rule determines the intrinsic perspective of the result. The algebraic claims in this section use syntactic equality of the resulting subjective numbers unless a separate closed relation is explicitly invoked.

7.1 Dominant Propagation

a(s)+ D b(t)=(a+b)(s).

The left operand's perspective dominates. The operation is associative when the value operation is associative. It is generally noncommutative because swapping inputs changes the resulting perspective.

Theorem 7.1 (Dominant Propagation): If + on V is associative, then +_D is associative. If results are compared syntactically, +_D is commutative for all inputs only in degenerate cases where the value operation is commutative and the dominant perspective is unchanged by swapping inputs.

Proof: Compute (a(s)+_D b(t))+_Dc(u)=((a+b)+c)(s) and a(s)+_D(b(t)+_Dc(u))=(a+(b+c))(s). If value addition is associative, the values agree and the perspective is s on both sides. For commutativity, a(s)+_D b(t)=(a+b)(s), while b(t)+_Da(s)=(b+a)(t). Even when a+b=b+a, the perspectives differ unless s=t or the perspective component is identified by an additional rule. □

7.2 Fusion Propagation

a(s)+ F b(t)=(a+b)(s\oplust).

Fusion uses a specified operation ⊕ on perspectives. The lifted operation inherits associativity and commutativity from both the value operation and the perspective operation.

Theorem 7.2 (Fusion Propagation): Under syntactic equality of results, +_F is associative if both + on V and ⊕ on P are associative. It is commutative if both operations are commutative. Conversely, if +_F is associative or commutative for all inputs with independent value and perspective components, the corresponding value-level and perspective-level laws hold.

Proof: Associativity compares ((a+b)+c)((s⊕t)⊕u) with (a+(b+c))(s⊕(t⊕u)). These agree for all inputs when the value operation and perspective operation are both associative. Commutativity compares (a+b)(s⊕t) with (b+a)(t⊕s). These agree for all inputs when both component operations are commutative. The converse follows by varying values while holding perspectives fixed to recover the value law, and by varying perspectives while holding values fixed to recover the perspective law. □

7.3 Novel Propagation

a(s)+ N b(t)=(a+b)(new({s,t})).

Novel propagation assigns a fresh perspective to the result. It is commutative at the binary level when + is commutative and new({s,t}) is symmetric. Associativity generally fails because different parenthesizations generate different fresh perspectives.

Theorem 7.3 (Novel Propagation): Binary novel propagation is symmetric when the value operation is commutative and the freshness rule depends only on the unordered pair of input perspectives. Associativity requires an additional coherence law on generated perspectives and otherwise fails in general.

Proof: If new({s,t})=new({t,s}) and a+b=b+a, then swapping the operands gives the same value and the same generated perspective. For associativity, the left parenthesization produces a perspective of the form new({new({s,t}),u}), while the right parenthesization produces new({s,new({t,u})}). These need not be equal. They agree only if the generation rule imposes a coherence law identifying these two fresh perspectives for all s,t,u. □

Propagation rules keep value operations and perspective movement distinct

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8. Asymmetric Models and Perspective Incompleteness

Let P={p,q} and V={0,1}. Define a base kernel by setting R^∘_p(p,q,1,1)=1 and leaving the corresponding reverse judgment absent from R^∘_q. After closure, perspective p validates 1(p) ⊑_p1(q), while q need not validate 1(q) ⊑_q1(p).

This gives a ZFC model of an asymmetric subjective preorder. The closed relations remain reflexive and transitive inside each fixed perspective.

8.1 Perspective Incompleteness

A perspective can fail to derive a judgment directly while deriving it after a guarded adoption step. This is perspective incompleteness. The missing judgment is not a contradiction; it is a judgment whose derivation depends on an explicit path through other perspectives.

Theorem 8.1 (Guarded Incompleteness Model): There are finite systems in which a(s) ⊑_pc(u) is absent from the base kernel but belongs to R^⋆_p only after guarded CPA composition.

Proof: Take perspectives p,q, subjective numbers a(s),b(t),c(u), base judgments a(s) ⊑_p b(t) and b(t) ⊑_qc(u), and guard CPA_p(q,u)=1. Do not include a(s) ⊑_pc(u) in R^∘_p. The CPA rule composes the p-validated left edge with the guarded q-validated right edge and derives a(s) ⊑_pc(u) in closure. Thus the judgment is not base-accessible and is not imported as a standalone q-judgment; it is derivable through a guarded compositional path. □

9. Connections to Established Mathematics

9.1 Relation to Modal and Preorder Semantics

A fixed evaluating perspective p can be compared with preorder semantics. Since ⊑_p is reflexive and transitive after closure, the single-perspective fragment has the same structural profile as an S4-style preorder frame.

The comparison is limited. A modal translation that sends a(s) ⊑_p b(t) to a formula such as □_p(a=b) loses the target perspective and the guarded adoption structure. A faithful translation must represent the indexed relation itself, for example through primitive predicates E_p(s,t,a,b) and separate guard predicates CPA_p(q,u).

9.2 Multi-Valued Logics

Multi-valued logics vary the value space. Subjective numbers vary the evaluating perspective, the intrinsic perspectives of the values being compared, and the guarded rules for importing judgments. These approaches can be combined when V has more than two truth values.

9.3 Relation to Enriched and Indexed Structures

The framework can be viewed as an indexed family of preorders coupled by controlled adoption rules. It resembles enriched or fibrational thinking in the sense that judgments live over indices, but the formal system here is deliberately elementary: a family of relations, a monotone closure operator, and explicit proof terms.

Asymmetry and incompleteness depend on guarded paths between perspectives

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10. Selective Fusion of Perspectives

Fusion creates a perspective p⊕q from two existing perspectives. Either p⊕q∈P is already specified, or it is adjoined to P together with its base kernel and adoption guard. Adopted perspectives may themselves be fused only when their kernels and guards have been specified.

Source systems and the fused system share the same external kernels for perspectives other than p, q, and p⊕q. This keeps adoption from changing the imported judgment while fusion changes only the local base and guard at the fused perspective.

R \circ p\oplusq (s,t,a,b)=f(R \circ p (s,t,a,b),R \circ q (s,t,a,b))

CPA p\oplusq (x,u)=g(CPA p (x,u),CPA q (x,u)).

When f and g are monotone, fusion respects inclusion of base information and guards.

Theorem 10.1 (Conjunctive Fusion Conservativity): Suppose p⊕q is specified as above, the source systems and fused system share the same external kernels for adopted perspectives, and f=g=∧. Then every judgment in the fused closure R^⋆_p⊕q also belongs to both source closures R^⋆_p and R^⋆_q.

Proof: With f=∧, every base judgment of p⊕q is a base judgment of both p and q. With g=∧, every guard available to p⊕q is available to both p and q. Prove by induction on the finite stage at which a fused judgment appears. Base and reflexive judgments belong to both source closures. Substitution preserves membership in both. If a fused judgment is obtained by transitivity from two fused premises, the induction hypothesis places both premises in each source closure, and source transitivity gives the conclusion in each source closure. If it is obtained by CPA, the fused left premise belongs to both source closures by induction, the adopted right premise is the same external judgment used by all systems, and the fused guard implies both source guards. Therefore each source closure can make the same CPA inference. Hence every fused judgment belongs to both source closures. □

Disjunctive fusion f=g=∨ has the opposite behavior: it can validate paths unavailable to either perspective alone after transitive composition. This is useful for modeling collective perspectives, but it must be marked as an expansive operation.

Selective fusion imports judgments under specified compatibility rules

Universe 00110000

11. Conclusion and Future Directions

Subjective numbers provide a formal language for values with intrinsic perspectives. The two-tier semantics separates base kernels from closed judgments. Closure is generated by a least simultaneous fixpoint and adds only the structural rules specified in the system.

The framework preserves local preorder structure while allowing global asymmetry. Cross-perspective reasoning is guarded by CPA. Categorical provenance records the derivation path through proof terms and typed histories, with the empty history ε as identity.

The measure-theoretic construction yields a pseudometric between perspectives rather than a metric. This distinction is important: two perspectives can differ only on null sets and still have zero discrepancy.

11.1 Future Directions

Develop richer examples of guarded adoption in multi-agent reasoning.
Study non-well-founded provenance for infinite dependency chains.
Improve finite-universe algorithms for restricted classes of CPA graphs.
Combine subjective numbers with many-valued value spaces and signed rejection systems.

References

Aczel, Peter. Non-Well-Founded Sets. CSLI Lecture Notes, 1988.
Awodey, Steve. Category Theory. Oxford University Press, 2010.
Blackwell, David. "The Existence of Certain Invariant Measures." Annals of Mathematics 51, no. 1, 1950.
Davey, B. A., and Hilary Priestley. Introduction to Lattices and Order. Cambridge University Press, 2002.
Desharnais, Josée, Vineet Gupta, Radha Jagadeesan, and Prakash Panangaden. "Metrics for Labelled Markov Processes." Theoretical Computer Science 318, 2004.
Kechris, Alexander S. Classical Descriptive Set Theory. Springer, 1995.
Mac Lane, Saunders. Categories for the Working Mathematician. Springer, 1998.
Tarski, Alfred. "A Lattice-Theoretical Fixpoint Theorem and Its Applications." Pacific Journal of Mathematics 5, 1955.

Appendix: Mathematical Examples for Subjective Numbers

This appendix provides detailed examples that illustrate key concepts of the subjective numbers framework, helping to build intuition for this mathematical structure. In the examples, R^∘ denotes base information and R^⋆ denotes the closure generated by the rules above. The displayed consequences are selected consequences, not exhaustive listings of every closed judgment. Once reflexivity, transitivity, substitution, and guarded CPA composition are applied, R^⋆ may contain additional judgments generated by longer paths.

Example A1: Illustrative Model of Subjective Numbers with Explicit Relation Functions

Let us construct a concrete model of subjective numbers with three perspectives P = {p, q, r} and values in ℝ. We explicitly define base relation functions that demonstrate key properties of the framework.

Perspectives:

p, q, and r are distinct perspectives.

Base Relation Functions: For perspective p, define R^∘_p as follows:

R^∘_p(p, p, a, a) = true for all a ∈ ℝ.
R^∘_p(p, q, a, b) = true if and only if a = b.
R^∘_p(p, r, a, b) = true if and only if a ≤ b.

For perspective q, define R^∘_q as follows:

R^∘_q(q, q, a, a) = true for all a ∈ ℝ.
R^∘_q(q, p, a, b) = true if and only if a ≥ b.
R^∘_q(q, r, a, b) = true if and only if a = 2b.

For perspective r, define R^∘_r as follows:

R^∘_r(r, r, a, a) = true for all a ∈ ℝ.
R^∘_r(r, p, a, b) = true if and only if a = b + 1.
R^∘_r(r, q, a, b) = true if and only if a = b/2.

Verification: Reflexivity is present in each base kernel for same-perspective comparisons and is also added by closure for every subjective number. Non-symmetry is visible because R^∘_p(p,r,3,5)=true since 3≤5, while R^∘_r(r,p,5,3) is absent because 5≠3+1. Transitivity is guaranteed in the closed relations R^⋆_p, R^⋆_q, and R^⋆_r by Theorem 3.2.

Specific Evaluations: From perspective p:

5(p) ⊑_p 5(q) by value equality with q.
3(p) ⊑_p 7(r) since 3 ≤ 7.
7(p) ⊑_p 3(r) is absent from the specified base kernel because 7 ≰ 3.

From perspective q:

8(q) ⊑_q 4(p) since 8 ≥ 4.
10(q) ⊑_q 5(r) since 10 = 2 × 5.
3(q) ⊑_q 7(p) is absent from the specified base kernel because 3 ≱ 7.

From perspective r:

6(r) ⊑_r 5(p) since 6 = 5 + 1.
4(r) ⊑_r 8(q) since 4 = 8/2.
3(r) ⊑_r 9(p) is absent from the specified base kernel because 3 ≠ 9 + 1.

This model demonstrates perspective-dependent evaluation, non-symmetric positive relations, and distinctive relation patterns for each perspective.

Example A2: Perspective Propagation in Operations

Let us examine perspective propagation through operations in detail using the three fundamental rules.

Consider the perspectives P = {a, b, c} and subjective numbers 3(a), 4(b), and 5(c). Define base relations such that:

R^∘_a(a, b, 3, 4) = true, while the reverse is absent from R^∘_b.
R^∘_a(a, c, 3, 5) is absent, while R^∘_c(c, a, 5, 3) = true.
R^∘_b(b, c, 4, 5) = true and R^∘_c(c, b, 5, 4) = true.

Dominant Perspective Rule:

3(a) +_D 4(b) = 7(a).
4(b) +_D 3(a) = 7(b).
5(c) +_D 3(a) = 8(c).

Despite having the same numerical value, 7(a) and 7(b) are distinct subjective numbers unless a closed relation identifies them from a specified evaluating perspective.

Perspective Fusion Rule: For perspective fusion, define the composite perspective relation function R^∘_a⊕b by a disjunctive compatibility rule:

R^∘_a⊕b(a⊕b, x, m, n)=R^∘_a(a,x,m,n)∨R^∘_b(b,x,m,n) for x∈{a,b,c}.

Then

3(a)+_F4(b)=7(a⊕b).
4(b)+_F5(c)=9(b⊕c).

The disjunctive fused base may validate relations not present in either component's direct self-view of the result. Such fusion is expansive and must be distinguished from conjunctive conservative fusion.

Novel Perspective Rule: For the novel perspective rule, generate entirely new perspectives:

3(a)+_N4(b)=7(z), where z=new({a,b}).
4(b)+_N5(c)=9(w), where w=new({b,c}).

The new perspective may record component-recognition judgments such as R^∘_z(z,a,7,3)=true and R^∘_z(z,b,7,4)=true, while additional relations are supplied by a generation function. This illustrates how each propagation rule creates distinct patterns of perspective inheritance.

Example A3: Path-Dependent Derivability

This example demonstrates how positive derivability can be path-dependent. Different guarded paths can make different judgments available to an evaluating perspective.

Consider perspectives P = {α, β, γ} with the following base information:

For perspective α:
- R^∘_α(α,β,5,5)=true.
- R^∘_α(α,α,5,3) is absent from the base kernel.
- No direct base relation with γ is specified.
For perspective β:
- R^∘_β(β,γ,5,5)=true.
- No direct base relation with α is specified.
For perspective γ:
- R^∘_γ(γ,α,5,3)=true.
- No direct base relation with β is specified.

Define the cross-perspective adoption relation:

CPA_α(β,γ)=true.
CPA_α(γ,α)=true.

Direct base access from perspective α: The judgment 5(α) ⊑_α3(α) is not in R^∘_α. This is absence from the base kernel, not a negative theorem.

Evaluation via perspective chain α → β → γ → α:

Base: 5(α) ⊑_α5(β).
Base at β: 5(β) ⊑_β5(γ).
By CPA_α(β,γ), perspective α adopts the second edge and obtains 5(α) ⊑_α5(γ).
Base at γ: 5(γ) ⊑_γ3(α).
By CPA_α(γ,α), perspective α obtains 5(α) ⊑_α3(α).

Thus, 5(α) ⊑_α3(α) is not directly present in α's base kernel but is derivable in R^⋆_α through guarded adoption. This example demonstrates path-dependent derivability without confusing absence with falsity.

Example A4: Perspective Incompleteness

This example demonstrates perspective incompleteness with explicit relation functions. Certain positive judgments are inaccessible from individual base kernels but become available through perspective fusion.

Consider a system with three perspectives P = {α, β, γ} focused on the subjective numbers 7(α), 4(β), and 10(γ).

Define positive base information:

For perspective α:
- R^∘_α(α,β,7,4)=true.
- R^∘_α(α,γ,7,10) is absent.
For perspective β:
- R^∘_β(β,γ,4,10)=true.
- R^∘_β(β,α,4,7) is absent.
For perspective γ:
- R^∘_γ(γ,β,10,4) is absent.
- The optional rejection log records N^∘_γ(γ,α,10,7)=true. This rejection is not a positive directed judgment and does not enter the closure operator.

Now create the fused perspective α⊕β with a disjunctive fused base kernel:

R \circ α\oplusβ (s,t,x,y)=R \circ α (s,t,x,y)\lorR \circ β (s,t,x,y).

From this fused perspective:

7(α) ⊑_α⊕β4(β) is inherited from α's base information.
4(β) ⊑_α⊕β10(γ) is inherited from β's base information.
By transitivity, 7(α) ⊑_α⊕β10(γ).

This demonstrates perspective incompleteness: the relation between 7(α) and 10(γ) is not directly present in the individual base kernels of α or β, yet becomes accessible through fusion. The rejection log at γ records a conflicting negative attitude, but because it is not part of the positive closure relation, it does not create a formal contradiction in the core system.

Example A5: Constructing a Penrose Stairs Analogue with Subjective Numbers

To further solidify the intuition behind subjective numbers and their capacity to model paradoxical, viewpoint-dependent structures, we present a concrete construction that mirrors the Penrose stairs illusion. This example demonstrates how subjective numbers can create a mathematical analogue of an impossible object, exhibiting local consistency while forming a globally cyclic loop due to perspective-dependent relations.

Construction of the Penrose Stairs Analogue:

Perspectives as Steps: Define a set of four perspectives, P = {p₁, p₂, p₃, p₄}, each representing a step in the subjective staircase.
Subjective Numbers as Levels: Consider subjective numbers representing levels of ascent:
- level₀(p₁): value 0 at perspective p₁.
- level₁(p₂): value 1 at perspective p₂.
- level₂(p₃): value 2 at perspective p₃.
- level₃(p₄): value 3 at perspective p₄.
Asymmetric Directed Judgments for Ascent: Add the following base judgments:
- R^∘_p₁(p₁,p₂,0,1)=true.
- R^∘_p₂(p₂,p₃,1,2)=true.
- R^∘_p₃(p₃,p₄,2,3)=true.
- R^∘_p₄(p₄,p₁,3,0)=true.
- Direct same-perspective comparisons of different levels remain absent from the positive base kernel, while reflexivity still supplies n(p_i) ⊑_{p_i}n(p_i).
The Penrose Stairs Effect: Tracing the locally validated edges gives level₀(p₁) → level₁(p₂) → level₂(p₃) → level₃(p₄) → level₀(p₁) . Each step appears to ascend locally from one perspective to the next, yet the chain returns to its starting level. The path is a cyclic sequence of indexed judgments, not an unguarded proof that all numeric levels are equal inside one perspective.

If an observing perspective is given CPA guards for all four transitions, it can import the entire cycle as a path-dependent history. Without those guards, the construction remains a directed graph of local judgments. This is the mathematical analogue of the visual illusion: local transitions are coherent, while global flattening of all perspectives into one unindexed comparison creates the apparent impossibility.

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