This document formalizes how objects relate. It is the companion to [[01-objects]], which defines what exists; this document defines the connective tissue. The ontology has one primitive relation — belongs-to — and a single extension that gives it expressive power: the quality. Qualities are themselves objects in the ontology, which means the relational structure describes itself. This reflexive closure is the document's central result.
The document resolves several deferred questions: the representation relation (P.F from [[06-ai-writing-scenario]]), the institution as a compound with jointly high axis-properties (IV.1 from [[04-correspondence_v1-2]]), compounds as operands (IV.2), and the general claim from [[08-relational-ground]] that all ontological properties are relational. The resolution in every case is the same: the property that seemed to require a new primitive is a quality of a belonging.
Four running examples from [[01-objects]]: (a) a student sitting an examination, (b) a doctor diagnosing a patient, (c) a lawyer arguing a case in court, (d) a chef preparing a meal. Two additional examples from the data infrastructure: (e) a file triaged to cloud storage, (f) a person participating in a communication recorded in an archive.
#The Primitive Relation
Definition R1 (Belongs-to). Belongs-to is the single primitive relation. Every assertion in the ontology takes the form: entity A belongs-to entity B. The relation is directed: A belongs-to B does not entail B belongs-to A. The positions are not interchangeable — A is the dependent, B is the context. A file belongs to a paper; the paper does not thereby belong to the file. A participant belongs to a situation; the situation does not thereby belong to the participant.
Belongs-to is the only stored relation. Every other relation — co-participation, authorship, citation, containment, temporal ordering, causal derivation — is either a quality of a belonging (see R2) or a derived view computed over the graph of belongings (see [[09-situation-graph]]).
Terminology. This document is titled "Relations" because it describes how objects relate — the general theory of connection. The primitive it defines is the specific directed relation belongs-to, and its instances are called belongings. "Relation" is the genus; "belonging" is the species. The word "belonging" is a term of art within this ontology: it names any directed edge from a dependent to a context, regardless of whether the ordinary English sense of "belonging" fits the case. A paper's belonging to its author does not mean the paper is the author's possession — it means the paper has a directed dependency on the author, qualified by "authored-by." The quality carries the semantics; "belonging" names only the structural fact that a directed link exists.
Why one relation. Multiple named relations (authored-by, contains, published-in, member-of) encode semantics in the relation label. This creates a growing vocabulary of edge types, each requiring its own interpretation rules. A single relation with a quality field inverts the architecture: the relation is uniform, and the semantics live in the quality, which is itself an entity subject to the same ontological treatment as everything else. The relation does not carry meaning; the quality does. And because the quality is an entity, it can be inspected, classified, and related to other qualities — all within the same framework.
Implementation. In the data model ([[22-lifecycle]]):
belongings (
entity_id TEXT -- the dependent
target TEXT -- the context
quality TEXT -- the nature of the belonging (see R2)
start_date TEXT -- temporal bound
end_date TEXT -- temporal bound
source TEXT -- provenance
)#Quality
Definition R2 (Quality). A quality is a property of a belonging that specifies the nature of the dependence. It answers: in what way does A belong to B?
Without a quality, belongs-to records only that a dependence exists. With a quality, it records what kind of dependence. The student belongs to the examination as a participant. The exam paper belongs to the examination as material. The invigilator belongs to the examination as authority. The examination hall belongs to the examination as venue. The silence rule belongs to the examination as policy. All five are belongs-to relations with the same target. The qualities distinguish them.
Qualities are not labels drawn from a flat vocabulary. They are entities.
Definition R3 (Quality as entity). Every quality is an entity in the ontology. As an entity, it can itself participate in belongs-to relations with other entities, including other qualities. The quality "author" is an entity that belongs to the quality "creation" with quality "instance." The quality "sender" is an entity that belongs to the quality "receiver" with quality "inverse."
This is the reflexive closure. The system that describes the world also describes its own descriptive apparatus. Qualities are part of the world the ontology models — they are the names for ways things depend on each other, and those names have their own dependencies.
The reflexive closure terminates. A quality's quality is always drawn from a small set of meta-qualities (see R5). Meta-qualities do not themselves require further qualification — their meaning is given by their structural role in the quality graph. This is not a stipulation to avoid infinite regress; it is an empirical observation. When you ask "in what way does 'author' belong to 'creation'?", the answer — "as an instance" — does not itself demand further qualification. The recursion bottoms out in structural roles.
#The Axis-Roles
Definition R4 (Axis-role as quality). The three axes of [[01-objects]] — actor, method, domain — and the fourth category, material, are not types of entities. They are qualities of belonging. Nothing is an actor in absolute terms. Things participate as actors in specific situations. The axis-role is a property of the belonging, not of the entity.
This resolves the central tension identified across [[04-correspondencev1-1]], [[04-correspondencev1-2]], and [[06-ai-writing-scenario]]: entities that seem to occupy multiple axes simultaneously.
The student belongs to the examination as actor (they process the exam paper) and as material (the pedagogical method transforms their capabilities). These are two belongings with different qualities, not a contradiction.
The large language model belongs to the writing session as actor (it transforms the prompt into a completion), as method (it is the repeatable procedure the person invokes), and as material (it is processed by the infrastructure that runs it). Three belongings, three qualities, one entity. This is the "instrument" of Definition P.D — not a new ontological kind, but an entity with jointly high axis-role qualities across multiple belongings.
The institution — Harvard, FIFA, Champalimaud — belongs to the situations it governs as domain (it is the context) and belongs to the situations it acts upon as actor (it changes things). Two belongings, two qualities. The "institution problem" (Gap IV.1) dissolves: there is no contradiction because the axis-roles are qualities of different belongings, not intrinsic properties of the entity.
The relational scalars revisited. [[04-correspondence_v1-2]] introduced three scalar properties: power (actor-property), scope (domain-property), malleability (material-property). Under the quality framework, these are derivative. An entity's power is the count and character of its belongings where the quality is "actor." An entity's scope is the count and character of its belongings where the quality is "domain." Malleability is the count and character of situations where the entity participates with quality "material." The scalars are not stored; they are computed from the quality graph. This is the same principle as [[09-situation-graph]]: relationships are derived from situations, never stored directly.
#Meta-Qualities
Definition R5 (Meta-quality). A meta-quality is a quality that describes a relationship between qualities. Meta-qualities are the grammar of the quality language; all other qualities are vocabulary. The meta-qualities form a small closed set.
R5.1 (Instance). Quality A belongs-to quality B with quality "instance" means A is a specific case of B. "Author" is an instance of "creation." "Python" is an instance of "code." "Email" is an instance of "communication." Instance is the classification relation — it builds the taxonomy of qualities.
R5.2 (Inverse). Quality A belongs-to quality B with quality "inverse" means that if entity X belongs-to entity Y with quality A, then Y belongs-to X with quality B, without storing both. "Sender" is the inverse of "receiver." "Author" is the inverse of "authored-by." "Part-of" is the inverse of "contains." Inverses enable bidirectional graph traversal from unidirectional storage.
R5.3 (Implies). Quality A belongs-to quality B with quality "implies" means that any belonging with quality A entails a belonging with quality B between the same entities, without storing both. "Principal-investigator" implies "member." "Owner" implies "authority." Implication enables inference: if Mainen belongs to Champalimaud as principal-investigator, the system can derive that he belongs as member without an explicit record.
R5.4 (Transitive). Quality A belongs-to "transitive" with quality "tagged" means that if X belongs-to Y with quality A, and Y belongs-to Z with quality A, then X belongs-to Z with quality A. "Part-of" is transitive: if a chapter is part of a book, and the book is part of a series, the chapter is part of the series. "Instance" is transitive: if "python" is an instance of "code" and "code" is an instance of "language," then "python" is an instance of "language." "Author" is NOT transitive: if Mainen authored a paper, and the paper is part of a project, Mainen did not author the project. Transitivity is a property of specific qualities, not of belongs-to in general.
R5.5 (Applies-to). Quality A belongs-to entity-type B with quality "applies-to" means A is only a valid quality for belongings whose target is of type B. "Sender" applies to situations. "Track-on" applies to albums. "Member" applies to organizations. Domain constraints prevent nonsensical belongings: a file cannot belong to a bucket as "sender."
#Quality Families
Qualities cluster into families that reflect distinct modes of dependence. The families are not primitive — they are emergent structure in the quality graph, visible when you trace the instance chains upward. They are listed here for orientation, not as axioms.
Classification. Instance, tagged. X is a kind of Y. The taxonomic backbone of the quality graph. Every tag chain (python → code → language) is built from classification qualities.
Composition. Part-of, contains, member-of, track-on. X is inside Y, or Y is built from X. The mereological family. Part-of and contains are inverses; both are transitive.
Creation. Author, creator, producer, builder. X made Y, or Y was made by X. The causal-generative family. All creation qualities imply participation (you cannot create something without participating in the situation of its creation).
Participation. Actor, participant, sender, receiver, witness, collaborator. X played a role in Y. The situational family. These qualities apply primarily to belongings whose target is a situation. The axis-roles (actor, method, material, domain) are participation qualities at the most general level.
Provenance. Source, origin, sourced-from, materialization, evidence. X came from Y, or X is a trace of Y. The evidential family. Every belonging in the graph has a source field (R1), which itself records provenance. But provenance can also be a quality: a file belongs to a paper as "materialization" — the file is a physical trace of the intellectual object.
Location. Stored-in, published-in, venue, habitat. X is housed in Y. The spatial-institutional family. Storage buckets, journals, courtrooms, kitchens.
Temporal. Predecessor, successor, occurred, concurrent. X is before Y, or X happened at time Y. The temporal family. The startdate and enddate fields on belongings carry most temporal information; temporal qualities are needed only when time itself is the semantic content of the relation.
#Representation
Definition R6 (Representation as quality). A representation is a belonging with a quality from the provenance family — specifically, "materialization," "evidence," "transcript," "recording," or "encoding." The representation relation deferred from [[05-perception-memory]] (P.F) and [[04-correspondence_v1-2]] (Open IV.R) does not require a new primitive. It is a belongs-to with a provenance quality.
A PDF belongs to a paper as "materialization." Match footage belongs to a past situation as "recording." A student transcript belongs to an engagement as "evidence." The prompt text belongs to the person's idea as "expression." In every case, the representation is an entity that belongs to the thing it represents, with a quality that names the representational mode.
Transitive representation. The chain idea → [expression] → prompt → [generation] → completion (Tension P.F from [[06-ai-writing-scenario]]) is a sequence of belongings:
prompt belongs-to idea quality: "expression"
completion belongs-to prompt quality: "extension"
completion belongs-to idea quality: "derived-expression"The third belonging is derivable from the first two via transitivity of provenance qualities, but only if the intervening method (generation) preserved enough structure for the derivation to be warranted. Whether the completion actually represents the idea — not just the prompt — is an empirical question about structural correspondence, not a logical consequence. The ontology records the chain; the judgment of representational adequacy is a separate act.
#The Quality Graph
The qualities, related to each other by belongs-to with meta-qualities, form a directed graph. This graph is the ontology's relational structure made explicit. It is small (hundreds of quality entities, not millions), stable (new qualities are added rarely), and traversable (meta-qualities enable inference).
The quality graph serves three functions:
- Vocabulary. It defines the controlled vocabulary of qualities available for belongings. New qualities are added by creating an entity and relating it to existing qualities via instance, inverse, and implies.
- Inference. Inverse and implies edges enable the system to derive belongings that were not explicitly stored. If "principal-investigator" implies "member," and Mainen belongs to Champalimaud as PI, then a query for members returns Mainen without an explicit membership record. The inference is principled — it follows the quality graph — and auditable — the implication chain is traceable.
- Navigation. Instance edges build the taxonomy. A query for "all belongings involving creation" traverses from "creation" to its instances (author, creator, producer, builder) and returns all belongings with any of those qualities. The taxonomy enables queries at any level of specificity.
#Withdrawal Revisited
[[08-relational-ground]] argued that withdrawal — the fact that things always exceed any particular encounter — follows from infinite relational connectivity. The quality framework makes this concrete.
An entity has many belongings, each with a quality. Each situation it participates in reveals only the qualities active in that situation. Dr. Chen in the ward round participates as actor (diagnostician), as authority (senior physician), as teacher (the students are observing). In the tenure review, the same person participates as material (their performance is evaluated). In the family, as member, caregiver, dependent. No single situation activates all qualities. Each situation is a partial view — a projection of the entity's full relational character onto the qualities relevant to that situation.
This is withdrawal without essence. The entity is not hiding a secret core. It has more belongings-with-qualities than any situation reveals. The surplus is structural, not metaphysical. It is computable: given the full graph, you can enumerate which qualities a given situation does and does not activate. The "hidden" qualities are not hidden from the ontology — they are hidden from the situation, because the situation's scope does not extend to them.
Harman's object-oriented ontology posits a real object behind every sensual presentation. This ontology replaces the real object with the quality graph: the full set of an entity's belongings, most of which are not visible in any given encounter. The explanatory work is the same — things exceed appearances — but the mechanism is relational rather than essentialist. No hidden natures, no inaccessible cores. Only a graph too large for any single situation to traverse.
#Visualization
The quality graph has a natural visual representation. Entities are nodes. Belongings are edges. The quality of each belonging determines the edge's visual properties — color, arrowhead, dash pattern, thickness. Classification edges might be thin gray arrows; creation edges bold directed lines; participation edges colored by axis-role (actor in one color, material in another, domain in a third). The meta-qualities determine edge style at the quality level: inverse pairs might be drawn as bidirectional arrows; implies edges as dotted lines; transitive edges as double-stroked.
This is not decoration. The visual grammar externalizes the ontological grammar. When you look at a network and see that a person has four red edges (participation as actor) and one blue edge (participation as material) converging on a situation node, you see the person's asymmetric role without reading any labels. When you see a cluster of green edges (creation qualities) fanning out from a person to many work nodes, you see productivity. When you see a dotted inverse pair between two quality nodes, you see that traversal works in both directions.
The graph is browsable at multiple scales. Zoom out: entities are dots, edges are colored lines, clusters of shared-quality edges reveal structural communities. Zoom in: individual belongings are labeled, temporal bounds are visible, provenance is traceable. The quality families provide natural color palettes — classification in gray, composition in blue, creation in green, participation in red, provenance in amber. The meta-qualities provide edge-style conventions. The result is a visual language as expressive as the ontological language it represents.
#Scope and Continuation
This document establishes the theoretical foundation: the belongs-to relation, the quality mechanism, meta-qualities, axis-roles as qualities, and reflexive closure. It is the ontological ground.
The practical consequences — how this structure maps to the data model, how it unifies the domain-specific databases (files, music, papers, books, contacts, public archives), how the quality graph is seeded and maintained, and how the triage and indexing systems feed into it — are developed in a companion document: [[10-relations-applied]] (planned).
ontology · 02 · relations · 2026-03-10 · zach + claude
Ontology 02 — Relations — 2026 — Zachary F. Mainen / HAAK