A mereological reconciliation

Part I: Induced Ontology from a Skills Algebra

1. The Core Idea

The standard approach to ontology is declarative: you specify entities, sorts, relations, and axioms that carve up a domain. The ontology is prior to any operations.

The induced ontology approach inverts this: the algebra is prior, and the ontology is what the algebra “sees” or “generates.” This is closer to how mathematics works—we don’t declare that groups exist and then study them; we define the group axioms and discover what structures satisfy them.

For skills, the intuition is:

• We have a compositional algebra $(\mathcal{S},\circ ,1)$ with certain axioms
• The ontology is the collection of structures, distinctions, and relations that are invariant under or induced by the algebraic operations

This is analogous to Klein’s Erlangen program: geometry is the study of invariants under a transformation group. Here, skill ontology is the study of invariants under skill composition.

2. Formal Setup

Let me establish a minimal algebraic framework and then systematically derive ontological structure from it.

Definition 1 (Skills Algebra). A skills algebra is a tuple $(\mathcal{S},\circ ,1,\mathcal{T},\varphi )$ where:

• $\mathcal{S}$ is a set of skills
• $\circ :\mathcal{S}\times \mathcal{S}\to \mathcal{S}$ is a composition operation
• $1\in \mathcal{S}$ is an identity element
• $\mathcal{T}$ is a set of tasks
• $\varphi :\mathcal{S}\times \mathcal{T}\to [0,1]$ is a fitness function

satisfying:

1. Associativity: $(S_1\circ S_2)\circ S_3=S_1\circ (S_2\circ S_3)$
2. Identity: $S\circ 1=1\circ S=S$
3. Fitness coherence: $\varphi (1,T)=0$ for all $T$ (the null skill solves nothing)

This makes $(\mathcal{S},\circ ,1)$ a monoid. We don’t assume commutativity (sequential vs. parallel composition differ) or inverses (you can’t “undo” a skill).

3. Induced Ontological Structures

3.1 The Divisibility Pre-order

Definition 2. For $S_1,S_2\in \mathcal{S}$, define: $S_1⊴S_2⟺\exists S^{\prime }\in \mathcal{S}:S_1\circ S^{\prime }=S_2\text{ or }{S}^{\prime }\circ S_1=S_2$

This reads: ”$S_1$ divides $S_2$” or ”$S_1$ is a component of $S_2$.”

Proposition 1. $⊴$ is a pre-order (reflexive and transitive) but not generally a partial order (antisymmetry may fail).

Proof. Reflexivity: $S\circ 1=S$, so $S⊴S$. Transitivity: if $S_1\circ A=S_2$ and $S_2\circ B=S_3$, then $S_1\circ (A\circ B)=S_3$. Antisymmetry fails in general: we could have $S_1\circ A=S_2$ and $S_2\circ B=S_1$ with $S_1\ne S_2$. $\square$

Ontological interpretation: The pre-order $⊴$ induces an equivalence relation: $S_1\sim S_2⟺S_1⊴S_2\text{ and }{S}_2⊴S_1$

The quotient $\mathcal{S}/\sim$ is a poset. Elements of an equivalence class are algebraically interchangeable—they play the same compositional role. This is the algebra telling us which skills are “the same” from a structural standpoint, even if extensionally different.

3.2 The Complexity Filtration

Definition 3. For a generating set ${\mathcal{S}}_0\subseteq \mathcal{S}$ (primitive skills), define the complexity of $S$ as: $\lambda (S)=\min n:S=S_1\circ S_2\circ \cdots \circ S_n,,S_i\in {\mathcal{S}}_0$ with $\lambda (S)=\infty$ if $S$ is not expressible.

This induces a filtration: ${\mathcal{S}}_0\subseteq {\mathcal{S}}_{\le 1}\subseteq {\mathcal{S}}_{\le 2}\subseteq \cdots \subseteq \mathcal{S}$ where ${\mathcal{S}}_{\le n}=S:\lambda (S)\le n$.

Ontological interpretation: The filtration levels are ontological strata. Primitive skills ($\lambda =1$) are the “atoms”; complex skills emerge at higher levels. This connects to Arora & Goyal’s emergence theory: competence on $k^{\prime }$-tuples of skills corresponds to the ${\mathcal{S}}_{\le k^{\prime }}$ stratum.

3.3 Task-Relative Ontology

Here’s where things get interesting. The fitness function $\varphi$ induces a task-relative view of the ontology.

Definition 4. For threshold $ϵ>0$ and task $T$, define: ${\mathcal{S}}_T^{ϵ}=S\in \mathcal{S}:\varphi (S,T)\ge ϵ$

This is the set of skills that are $ϵ$-competent for task $T$.

Proposition 2. If $\varphi$ satisfies monotonicity ($S_1⊴S_2\Rightarrow \varphi (S_1,T)\le \varphi (S_2,T)$ for all $T$), then ${\mathcal{S}}_T^{ϵ}$ is an upper set (order filter) in $(\mathcal{S},⊴)$.

Ontological interpretation: Different tasks “see” different subsets of the skill ontology. A task requiring complex reasoning only recognizes skills above a certain complexity threshold. The full ontology $\mathcal{S}$ is the union: $\mathcal{S}={\bigcup }_{T\in \mathcal{T}}{\bigcup }_{ϵ>0}{\mathcal{S}}_T^{ϵ}$

But no single task reveals the whole ontology. This is a perspectival or task-indexed ontology.

3.4 The Galois Connection

There’s a beautiful duality between skills and tasks that deserves formalization.

Definition 5. Define: \begin{align*} \text{skills}: 2^\mathcal{T} &\to 2^\mathcal{S} \ \text{skills}(\mathcal{T}') &= {S : \phi(S, T) \geq \epsilon \text{ for all } T \in \mathcal{T}'} \end{align*}

\begin{align*} \text{tasks}: 2^\mathcal{S} &\to 2^\mathcal{T} \ \text{tasks}(\mathcal{S}') &= {T : \phi(S, T) \geq \epsilon \text{ for all } S \in \mathcal{S}'} \end{align*}

Proposition 3. $(\text{skills},\text{tasks})$ forms a Galois connection between $(2^{\mathcal{T}},\supseteq )$ and $(2^{\mathcal{S}},\supseteq )$.

Proof. We need ${\mathcal{S}}^{\prime }\subseteq \text{skills}({\mathcal{T}}^{\prime })⟺{\mathcal{T}}^{\prime }\subseteq \text{tasks}({\mathcal{S}}^{\prime })$. Both reduce to: $\varphi (S,T)\ge ϵ$ for all $S\in {\mathcal{S}}^{\prime }$, $T\in {\mathcal{T}}^{\prime }$. $\square$

Ontological interpretation: The closed elements of this Galois connection—sets ${\mathcal{S}}^{\prime }$ with ${\mathcal{S}}^{\prime }=\text{skills}(\text{tasks}({\mathcal{S}}^{\prime }))$—are the natural kinds of the induced ontology. They’re the skill-sets that are “carved out” by task structure. Similarly, closed task-sets are the natural task-kinds.

This gives you a formal answer to “what skills exist?”: the skills that survive the Galois closure are the ones with ontological standing.

4. Ontological Expansion as Algebraic Closure

Now we can formalize what “open-ended skill generation” means ontologically.

Definition 6. Given a skills algebra $(\mathcal{S},\circ ,1,\mathcal{T},\varphi )$ and a set of primitive skills ${\mathcal{S}}_0$, the generated ontology is: $⟨{\mathcal{S}}_0⟩={\bigcup }_{n=0}^{\infty }{\mathcal{S}}_0^{\circ n}$ where ${\mathcal{S}}_0^{\circ 0}=1$ and ${\mathcal{S}}_0^{\circ (n+1)}=S_1\circ S_2:S_1\in {\mathcal{S}}_0^{\circ n},S_2\in {\mathcal{S}}_0$.

Proposition 4. $⟨{\mathcal{S}}_0⟩$ is the smallest submonoid of $\mathcal{S}$ containing ${\mathcal{S}}_0$.

Ontological expansion occurs when:

1. We add new primitives: ${\mathcal{S}}_0\to {\mathcal{S}}_0\cup S_{new}$
2. The generated ontology expands: $⟨{\mathcal{S}}_0⟩⊊⟨{\mathcal{S}}_0\cup S_{new}⟩$

The expansion is non-trivial if $S_{new}\notin ⟨{\mathcal{S}}_0⟩$—i.e., the new skill isn’t already expressible as a composition of existing primitives.

Connection to SKILL-MIX: When a model trained on $k$-compositions suddenly exhibits competence on $(k+1)$-compositions after scaling, this is ontological expansion driven by scaling, not by adding new primitives. The primitives were always there; what expands is the realized portion of the generated ontology.

5. The Emergence Threshold as Ontological Boundary

Combining the complexity filtration with task-relative competence:

Definition 7. The emergence boundary for task $T$ at threshold $ϵ$ is: ${\lambda }^{\ast }(T,ϵ)=\min n:{\mathcal{S}}_{\le n}\cap {\mathcal{S}}_T^{ϵ}\ne \varnothing$

This is the minimal complexity at which some skill becomes $ϵ$-competent for $T$.

Ontological interpretation: Skills below the emergence boundary are “ontologically inert” for task $T$—they exist in the algebra but don’t manifest in task performance. As models scale (per Arora & Goyal), ${\lambda }^{\ast }$ increases, revealing more of the ontology.

The full picture:

$\text{Algebraic ontology: }⟨{\mathcal{S}}_0⟩\text{ (all possible compositions)}$ $\downarrow \text{ (filter by task competence)}$ $\text{Realized ontology: }{\bigcup }_T{\mathcal{S}}_T^{ϵ}\text{ (skills with nonzero fitness)}$ $\downarrow \text{ (filter by model scale)}$ $\text{Emergent ontology: }S:\lambda (S)\le {\lambda }_{model}^{\ast }\cap {\bigcup }_T{\mathcal{S}}_T^{ϵ}$

---

Part II: Reconciling Mereological and Algebraic Structure

1. The Problem

You have two kinds of structure on skills:

1. Mereological: Part-whole relations. “Metaphor recognition” is part of “literary analysis.”
2. Algebraic: Compositional operations. “Metaphor recognition $\circ$ logical argumentation” produces “persuasive writing with figurative language.”

These seem related but are not identical:

• Mereology is about constitution: what is $S$ made of?
• Algebra is about combination: what does $S$ produce when combined?

A skill can be a part of another without being a compositional factor of it, and vice versa.

2. Two Partial Orders

Let’s formalize both structures.

Mereological structure: Define part-of relation ${⊑}_m$: $S_1{⊑}_m{S}_2⟺S_1\text{ is a constitutive subskill of }{S}_2$

We assume this satisfies the axioms of a partial order (reflexive, antisymmetric, transitive).

Algebraic structure: Define the divisibility relation ${⊴}_a$ (from Part I): $S_1{⊴}_a{S}_2⟺\exists S^{\prime }:S_1\circ S^{\prime }=S_2\text{ or }{S}^{\prime }\circ S_1=S_2$

Passing to the quotient by $\sim$ makes this a partial order.

Question: How do ${⊑}_m$ and ${⊴}_a$ relate?

3. Four Logical Possibilities

For any pair $(S_1,S_2)$:

Case	$S_1{⊑}_m{S}_2$	$S_1{⊴}_a{S}_2$	Interpretation
A	Yes	Yes	$S_1$ is both a part and a factor of $S_2$
B	Yes	No	$S_1$ is a part but not a compositional factor
C	No	Yes	$S_1$ is a factor but not a constitutive part
D	No	No	No structural relation

Examples:

Case A: “Arithmetic” is part of “algebra” and $\text{arithmetic}\circ \text{variable\_manipulation}=\text{basic\_algebra}$.

Case B: “Pattern recognition” is part of “mathematical reasoning” but you can’t compose pattern recognition with something to get mathematical reasoning—it’s more like a background capacity that enables reasoning.

Case C: You can compose “summarization” with “translation” to get “cross-lingual summarization,” but summarization isn’t a part of translation or vice versa—they’re independent skills that combine.

Case D: “Cooking” and “differential geometry” have no structural relation (in typical domains).

4. A Reconciliation Framework

The key insight: mereology and algebra capture different aspects of skill structure. We need a framework that includes both.

Definition 8 (Skill Structure). A skill structure is a tuple $(\mathcal{S},{⊑}_m,\circ ,1)$ where:

• $(\mathcal{S},{⊑}_m)$ is a poset (mereological structure)
• $(\mathcal{S},\circ ,1)$ is a monoid (algebraic structure)

These are connected by coherence conditions.

4.1 Coherence Condition 1: Compositional Closure of Parts

Axiom (Part Composition): If $S_1{⊑}_{m}S$ and $S_2{⊑}_{m}S$, then: $S_1\circ S_2{⊑}_{m}S\circ S$

Interpretation: composing parts of $S$ gives (at most) a part of $S$ composed with itself.

This is too weak if $S\circ S$ is much bigger than $S$. A stronger version:

Axiom (Part Composition, strong): If $S_1{⊑}_{m}S$ and $S_2{⊑}_{m}S$, then: $\exists S^{\prime }{⊑}_{m}S:S_1\circ S_2{⊑}_m{S}^{\prime }$

This says: composing parts stays within the part-structure.

4.2 Coherence Condition 2: Algebraic Reflection of Mereology

Axiom (Mereological Reflection): If $S_1{⊑}_m{S}_2$, then: $\exists S_3\in \mathcal{S}:S_1\circ S_3{⊑}_m{S}_2\text{ or }{S}_3\circ S_1{⊑}_m{S}_2$

Interpretation: if $S_1$ is part of $S_2$, then there’s some way to compose $S_1$ with something to get (a part of) $S_2$. Mereological inclusion has algebraic consequences.

4.3 Coherence Condition 3: Joins and Compositions

If $(\mathcal{S},{⊑}_m)$ is a join-semilattice (every pair has a least upper bound $\bigsqcup$), we can ask how $\bigsqcup$ relates to $\circ$.

Axiom (Join-Composition Distributivity): $S_1\circ (S_2\bigsqcup S_3)=(S_1\circ S_2)\bigsqcup (S_1\circ S_3)$

This says composition distributes over mereological joins. It’s a strong condition—essentially making $\mathcal{S}$ a quantale-like structure.

When this fails: Consider $S_1=\text{writing}$, $S_2=\text{formal\_logic}$, $S_3=\text{rhetoric}$. Then $S_2\bigsqcup S_3$ might be “argumentation.” But $\text{writing}\circ \text{argumentation}$ (writing applied to argumentation) may not equal the join of “writing applied to formal logic” and “writing applied to rhetoric”—the whole exceeds the sum of parts.

5. A Geometric Picture

The reconciliation becomes clearer with a geometric metaphor.

Think of skills as living in a space with two kinds of structure:

• Vertical structure (mereological): how skills decompose into parts
• Horizontal structure (algebraic): how skills combine to form new skills

                        S₁ ∘ S₂ ∘ S₃ (complex composite)
                             ↑ composition
                    S₁ ∘ S₂ ←——————→ S₂ ∘ S₃
                       ↑                ↑
                      S₁ ←——————→ S₂ ←——————→ S₃  (primitive level)
                       |         |         |
                       ↓         ↓         ↓
                    parts     parts     parts    (mereological decomposition)
                    of S₁     of S₂     of S₃

Mereology moves down (decomposition) or up (integration within a skill). Algebra moves horizontally (composition across skills).

The coherence conditions constrain how vertical and horizontal movements interact—you can’t have a structure where mereological parts compose to something outside the mereological envelope.

6. A Categorical Formulation

For maximum precision, here’s a categorical version.

Definition 9. A mereological monoid is a category $\mathcal{C}$ where:

• Objects are skills
• Morphisms $S_1\to S_2$ are “part-of” witnesses (proofs that $S_1{⊑}_m{S}_2$)
• There’s a monoidal structure $(\otimes ,I)$ on objects (composition)
• The monoidal structure is functorial with respect to morphisms

The functoriality condition says: if $f:S_1\to S_1^{\prime }$ and $g:S_2\to S_2^{\prime }$ are part-of witnesses, then there’s a canonical witness $f\otimes g:S_1\otimes S_2\to S_1^{\prime }\otimes S_2^{\prime }$.

Interpretation: Composition respects part-of structure. If $S_1$ is part of $S_1^{\prime }$ and $S_2$ is part of $S_2^{\prime }$, then $S_1\otimes S_2$ is part of $S_1^{\prime }\otimes S_2^{\prime }$.

This is a monoidal poset (a poset with compatible monoidal structure), which is a well-studied structure in categorical algebra.

7. Practical Implications

7.1 For Skill Extraction

When extracting skills from text or model behavior, you’re discovering both:

• Mereological structure: which skills are constitutive of which
• Algebraic structure: which skills combine productively

These require different methods:

• Mereology: factor analysis, hierarchical clustering, asking “what does this skill presuppose?”
• Algebra: composition experiments (SKILL-MIX style), asking “what happens when we combine these?“

7.2 For Skill Evaluation

Your fitness function $\varphi$ can be decomposed: $\varphi (S_1\circ S_2,T)=f(\varphi (S_1,T),\varphi (S_2,T),\text{interaction}(S_1,S_2,T))$

The interaction term captures non-mereological emergence—cases where composition produces capabilities beyond the sum of parts.

7.3 For Metaskill Definition

Metaskills operate on both structures:

• Decomposition metaskill (${\mathcal{M}}_d$): exploits mereological structure to break skills into parts
• Composition metaskill (${\mathcal{M}}_c$): exploits algebraic structure to combine skills
• Analysis metaskill (${\mathcal{M}}_a$): reads the fitness function to evaluate task-skill fit

A complete metaskill theory needs to specify how metaskills interact with both mereological and algebraic structure.

8. Revised Ontological Picture

Putting Parts I and II together:

                    ┌─────────────────────────────────────┐
                    │         SKILLS ALGEBRA              │
                    │    (𝒮, ∘, 𝟙) — monoid structure     │
                    └─────────────────────────────────────┘
                                    │
                    ┌───────────────┼───────────────┐
                    ▼               ▼               ▼
            ┌──────────────┐ ┌──────────────┐ ┌──────────────┐
            │ Divisibility │ │  Complexity  │ │ Task-Relative│
            │  Pre-order   │ │  Filtration  │ │   Ontology   │
            │    (⊴ₐ)      │ │   (𝒮≤n)      │ │   (𝒮ᵀᵋ)      │
            └──────────────┘ └──────────────┘ └──────────────┘
                    │               │               │
                    └───────────────┼───────────────┘
                                    ▼
                    ┌─────────────────────────────────────┐
                    │      INDUCED ONTOLOGY               │
                    │  (natural kinds via Galois closure) │
                    └─────────────────────────────────────┘
                                    │
                    ┌───────────────┴───────────────┐
                    ▼                               ▼
            ┌──────────────┐                ┌──────────────┐
            │ Mereological │◄── coherence ─►│  Algebraic   │
            │  Structure   │    conditions  │  Structure   │
            │    (⊑ₘ)      │                │    (⊴ₐ)      │
            └──────────────┘                └──────────────┘
                    │                               │
                    └───────────────┬───────────────┘
                                    ▼
                    ┌─────────────────────────────────────┐
                    │     SKILL STRUCTURE                 │
                    │  (𝒮, ⊑ₘ, ∘, 𝟙) — monoidal poset     │
                    └─────────────────────────────────────┘

---

Summary of Key Formal Constructs

Construct	Definition	Role
Skills monoid $(\mathcal{S},\circ ,1)$	Associative composition with identity	Generative engine
Divisibility ${⊴}_a$	$S_1{⊴}_a{S}_2⟺\exists S^{\prime }:S_1\circ S^{\prime }=S_2$	Algebraic order
Complexity $\lambda (S)$	Minimal composition length from primitives	Stratification
Task-relative set ${\mathcal{S}}_T^{ϵ}$	$S:\varphi (S,T)\ge ϵ$	Perspectival ontology
Galois connection	$(\text{skills},\text{tasks})$	Natural kinds
Mereological order ${⊑}_m$	Part-whole relation	Constitutional structure
Monoidal poset $(\mathcal{S},{⊑}_m,\circ ,1)$	Poset with compatible monoid	Unified structure
Coherence conditions	Axioms relating ${⊑}_m$ and $\circ$	Structural harmony