FPF Reference

Advanced Mereology: Components, Portions, Aspects & Phases

Pattern A.14 · Stable Part A - Kernel Architecture Cluster

FPF’s holonic modelling relies on part–whole relations to build structural and conceptual holarchies (systems and epistemes). But U.Holon is not synonymous with “mereological whole”: some holons (notably Roles and Methods) are bounded identity‑bearing objects whose primary composition is handled by other algebras (A.2 role algebra; A.15 method/description graphs), not by partOf. Early drafts distinguished structural vs. conceptual parthood (e.g., ComponentOf, ConstituentOf) but practical modelling kept hitting two recurrent gaps:

Keywords

  • mereology
  • part-of
  • ComponentOf
  • PortionOf
  • PhaseOf
  • composition.

Relations

A.14refinesHolonic Foundation: Entity → Holon
A.14prerequisite forDependency Graph & Proofs
A.14explicit referenceRole Taxonomy
A.14explicit referenceRole–Method–Work Alignment (Contextual Enactment)
A.14explicit referenceCompose‑CAL — Constructional Mereology
A.14explicit referenceHuman-Centric Working-Model
A.14explicit referenceCT2R-LOG — Working-Model Relations & Grounding
A.14explicit referenceU.Work: The Record of Occurrence
A.14explicit referenceSystem-specific Aggregation Γ_sys
A.14explicit referenceContextual & Temporal Aggregation (Γctx & Γtime)
A.14explicit referenceΓ_method — Order-Sensitive Method Composition & Work Enactment
A.14explicit referenceMeta-Holon Transition (MHT): Recognizing Emergence and Re-identifying Wholes
A.14explicit referenceHolonic Foundation: Entity → Holon
A.14explicit referenceΓ_epist — Knowledge-Specific Aggregation
A.14explicit referenceΓ_work — Work as Spent Resource

Content

Context — why an advanced mereology?

FPF’s holonic modelling relies on part–whole relations to build structural and conceptual holarchies (systems and epistemes). But U.Holon is not synonymous with “mereological whole”: some holons (notably Roles and Methods) are bounded identity‑bearing objects whose primary composition is handled by other algebras (A.2 role algebra; A.15 method/description graphs), not by partOf. Early drafts distinguished structural vs. conceptual parthood (e.g., ComponentOf, ConstituentOf) but practical modelling kept hitting two recurrent gaps:

  1. Quantities vs. parts. Engineers routinely need “some of the fuel”, “the first 10 pages”, “a 30% subset of data”. This is not a component; it is a portion of a stuff‑like whole, governed by measures and conservation.

  2. Change vs. replacement. Authors need to say “the prototype before calibration”, “v2 of the spec”, “shift 1 vs. shift 2 of the same run”. That is not a new whole; it is a phase of the same carrier across time.

This section introduces two normative sub‑relations of partOf that close those gaps and lock them to the rest of the kernel:

  • PortionOf — metrical, measure‑preserving parthood of stuffs and other measurables.
  • PhaseOf — temporal parthood of the same carrier across an interval.

It also restates guard‑rails that keep Roles and Methods (as intensional masks/ways‑of‑doing) outside mereology (A.15), while allowing their describing epistemes (e.g., U.MethodDescription, U.WorkPlan) to use ordinary episteme parthood and versioning like any other U.Episteme. It also clarifies how MemberOf fits (preview: collections are grounded constructively in C.13 Compose‑CAL via Γ_m.set; collective agency/composition is handled outside A.14 via B.1.7 Γ_collective and A.15, not via partOf).

Publication note (Working‑Model first). Read A.14 together with E.14 Human‑Centric Working‑Model and B.3.5 CT2R‑LOG: publish relations on the Working‑Model surface; when assurance is sought, ground downward. For structural claims that require extensional identity, use the Constructive shoulder via Compose‑CAL Γ_m (sum | set | slice); order/time stay outside mereology (Γ_time / Γ_method).

Problem — what breaks without these distinctions?

If we only have “generic partOf” (plus Component/Constituent), four classes of errors appear:

  1. Conservation errors. Treating “20 L of fuel from Tank A” as a component leads to nonsense: adding and removing such “components” does not respect quantities; Γ_sys proofs violate Σ‑balance.

  2. Temporal smearing. Flattening “before/after”, or “v1/v2” into one timeless whole collapses history; Γ_time and Γ_method cannot justify order‑sensitive properties; audits cannot reproduce conditions.

  3. Identity confusion. Modelling “new version” as “new component” either breaks identity (it is still the same holon evolving) or hides a Meta‑Holon Transition when identity really changes.

  4. Role leakage. Functional/organisational roles sneak into part trees (“the PumpRole is part of the plant”), violating A.15 and making structural reasoning brittle.

Forces

ForceTension
Expressiveness vs. ParsimonyWe need new relations (Portion, Phase) ↔ we must keep the catalogue minimal and orthogonal.
Universality vs. Domain nuanceOne set of rules must serve physical systems and epistemes ↔ measurement and time behave differently by domain.
Identity vs. ChangePreserve “the same carrier through change” ↔ allow explicit re‑identification when invariants fail.
Static structure vs. HistoriesPart trees should be simple ↔ real work requires phased histories and measured slices.

Solution — extend the mereology catalogue, keep it clean

A.14 defines two additional sub‑relations of partOf and re‑affirms the firewall between mereology and the role/recipe layer:

  1. PortionOf — for measured parts of a whole (stuffs and other extensives).
  2. PhaseOf — for temporal parts of the same carrier.
  3. No Roles/Methods in mereology. U.Role and U.Method are never parts (A.15). A U.MethodDescription is an Episteme and may be versioned/structured; that does not make the described U.Method a part.
  4. MemberOf stays, but constructive aggregation and agency live elsewhere. MemberOf remains available to state collections and collectives; a collection‑as‑whole may be constructed via Γ_m.set (Compose‑CAL, C.13), while collective agency/composition is handled in B.1.7 Γ_collective and A.15 (not in A.14).

The classical pair ComponentOf (structural, discrete) and ConstituentOf (conceptual, logical/epistemic) remain as in the kernel; A.14 only clarifies how to tell them apart from Portion/Phase (§ 6).

Formal cores (normative semantics)

PortionOf — metrical part of a measurable whole

Intent. Capture “some of the same stuff/extent”, governed by a measure that adds up.

Applicability. Any U.Holon that carries an extensive measure μ on the chosen scope (examples: mass, volume, length‑of‑text, byte size, wall‑time budget).

Primitive. PortionOf(x, y) means: x is the same kind of stuff/content as y, but less.

Axioms (A14‑POR‑*)

  • POR‑1 (Partial order). PortionOf is reflexive, antisymmetric, transitive on its domain.
  • POR‑2 (Metrical dominance). If x ProperPortionOf y then 0 < μ(x) < μ(y) for the agreed μ.
  • POR‑3 (Additivity on disjoint portions). If x ⟂ y (no overlap) and both PortionOf y, then μ(x ⊔ y) = μ(x)+μ(y) and x ⊔ y PortionOf y.
  • POR‑4 (Kind integrity). x and y must share the same measure kind and unit (or a declared conversion).
  • POR‑5 (Boundary compatibility). For physical wholes, the whole’s boundary encloses the union of its portions; cross‑boundary “leaks” are interactions, not portions.

Didactic tests. ✔ “5 kg from a 20 kg billet” — PortionOf. ✔ “Pages 1–10 of the report” — PortionOf (μ = page or token count). ✘ “The pump module of the plant” — ComponentOf, not PortionOf. ✘ “The Methods section of the paper” — ConstituentOf, not PortionOf.

PhaseOf — temporal part of the same carrier

Intent. Capture “the same holon during a sub‑interval”, preserving identity through change.

Applicability. Any U.Holon that persists across time with a recognised carrier identity.

Primitive. PhaseOf(x, y) means: x is y restricted to a proper time interval.

Axioms (A14‑PHA‑*)

  • PHA‑1 (Partial order). PhaseOf is reflexive, antisymmetric, transitive (on the same carrier).
  • PHA‑2 (Coverage). The whole is the union of its maximal, non‑overlapping phases over its lifetime interval.
  • PHA‑3 (No paradoxical overlap). Phases of the same carrier do not overlap in time; overlapping variants require PhaseOf on aspects or different carriers.
  • PHA‑4 (Identity through change). Properties may vary between phases, but the carrier’s identity criteria hold continuously (e.g., same serial number, same legal identity, same theorem statement).
  • PHA‑5 (Escalation to MHT). If identity criteria break (e.g., metamorphosis with new objectives), declare a Meta‑Holon Transition (B.2) rather than a PhaseOf.

Didactic tests. ✔ “PumpUnit#3 before calibration” — PhaseOf(Pump#3_pre, Pump#3). ✔ “Spec v2” — PhaseOf(Spec_v2, Spec), on the MethodDescription artefact. ✔ “Shift 1 of the same batch run” — PhaseOf(Work_shift1, Work). ✘ “Prototype vs. production unit” — likely different carriers; use ComponentOf/ConstituentOf or MHT per criteria.

CT2R‑LOG & Compose‑CAL handshake (normative link)

  • Structural claims published on the Working-Model surface SHALL be justified, when assurance is required, by a Constructive grounding narrative using Γ_m.sum | Γ_m.set | Γ_m.slice and linked with tv:groundedBy (see B.3.5; C.13).
  • PhaseOf is temporal parthood; it SHALL NOT be grounded via Γ_m. Its assurance follows identity‑through‑time criteria (CC‑PHA‑1..3) and Γ_time ordering (B.1.4).
  • MemberOf remains non‑mereological (CC‑MEM‑2). When modelling a collection‑as‑whole for assurance purposes, the constructive basis is Γ_m.set; no ComponentOf inferences follow from MemberOf.

Choosing the right relation (decision table)

You want to say…UseWhy
“This is a piece of the same stuff (lower amount/extent).”PortionOfGoverned by a measure μ and conservation (Σ‑additive).
“This is a discrete part that sits inside the whole.”ComponentOfStructural parthood; boundary‑respecting, not measured by μ.
“This is a logical part in a conceptual whole.”ConstituentOfSections, lemmas, clauses, conceptual assembly.
“This is the same thing during a sub‑interval.”PhaseOfTemporal slicing with identity continuity.
“This item belongs to that collection/collective.”MemberOfNot a building block of the whole; set‑as‑whole via C.13 (Γ_m.set), collective action via B.1.7/A.15.
“This system plays a Role or position.”playsRole (A.15)Roles are contextual masks, never parts.

Firewall reminder. If your sentence is about who does what, how it is done, or what happened when (role/method/run), you are likely in A.15. If it is about the document/artifact as a carrier (its pages/sections/versions), you may still be in A.14 (Episteme mereology).

Archetypal grounding (System / Episteme)

RelationU.System exampleU.Episteme example
PortionOf50 L from a 200 L fuel tank (μ = volume).Pages 1–10 from a 120‑page report (μ = page/token count).
ComponentOfImpeller ComponentOf PumpUnit.Figure 2 ComponentOf Poster Layout (physical artefact).
ConstituentOfControl law ConstituentOf Controller Design.Lemma A ConstituentOf Theorem Proof.
PhaseOfPumpUnit#3 before/after calibration (same serial).Spec v1 → v2 (same document lineage).
MemberOf (for reference)“is an element of a collection/collective”; use when a grouping is explicitly treated as a whole set, without implying component integration. Not a building block of the whole; constructive aggregation is handled in C.13 Compose‑CAL (Γ_m.set). If the grouping is expected to act, model a collective system (A.15).

Conformance Checklist & type guards (normative)

Global firewall and scope

IDRequirementPurpose
CC‑A14‑0No U.Role or U.Method MAY occur as a node in any partOf chain.Keeps masks/ways‑of‑doing outside mereology (see A.15).
CC‑A14‑0aU.MethodDescription / U.WorkPlan and other describing epistemes MAY participate in partOf only as U.Episteme nodes (e.g., ConstituentOf, text PortionOf, version PhaseOf); they MUST NOT be asserted as ut:StructPartOf of any U.System.Allows document structure/versioning without smuggling Methods into structure.
CC‑A14‑0bMemberOf MUST NOT imply, entail, or be auto‑rewritten into any partOf sub‑relation.Separates collections/collectives from parthood.
CC‑A14‑0cSerialStepOf / ParallelFactorOf MUST NOT appear in any partOf chain or table in A.14; model order via A.15 (Γ_ctx/Γ_method).Prevents the “order‑as‑structure” category error.

PortionOf guards

IDRequirementPurpose
CC‑POR‑1 (Domain)PortionOf(x,y) is valid only if the modelling scope declares at least one extensive measure μ for y (mass, volume, token count, byte size, wall‑time budget, etc.).Prevents “portion” without a measure.
CC‑POR‑2 (Kind)x and y SHALL share the same μ‑kind and compatible units (or an explicit conversion).Prevents apples‑to‑oranges addition.
CC‑POR‑3 (Monotone additivity)For disjoint portions x ⟂ z with PortionOf(-,y): μ(x ⊔ z) = μ(x)+μ(z).Secures Σ‑reasoning and Γ_sys proofs.
CC‑POR‑4 (Boundary)For physical systems, the whole’s boundary encloses the union of portions; cross‑boundary flows are not portions.Distinguishes stock vs flow.
CC‑POR‑5 (Non‑replacement)“Replacing 20% of y by v” MUST be modelled as PortionOf removal + Component/Constituent insertion, not as a single PortionOf rewrite.Avoids silent identity change.

PhaseOf guards

IDRequirementPurpose
CC‑PHA‑1 (Carrier identity)PhaseOf(x,y) requires an explicit identity criterion for y valid over the union of phases (e.g., serial number, legal identity, theorem statement).Prevents re‑identification by stealth.
CC‑PHA‑2 (Coverage & non‑overlap)The lifetime of y equals the union of its maximal, non‑overlapping phases (on the same aspect).Enables Γ_time composition and audit.
CC‑PHA‑3 (Aspect clarity)If two temporal slices of y overlap, they MUST be phases of different aspects (e.g., mechanical‑state vs software‑state), or else be different carriers.Avoids paradoxical overlaps.
CC‑PHA‑4 (Escalation)If identity criteria fail during change, declare a Meta‑Holon Transition (B.2) instead of PhaseOf.Makes re‑identification explicit.
CC‑PHA‑5 (MethodDescription & Work)Versions of MethodDescription and generic time‑slices of Work SHALL use PhaseOf (A.15/A.15.1); Work‑specific refinements (episodes/retries/concurrency) are modelled in A.15.1. PhaseOf never applies to U.Role or U.Method.Aligns temporal slicing with design/run bindings.

Anchoring & validation (normative)

IDRequirementPurpose
CC‑ANCH‑1Every ut:StructPartOf edge MUST carry a tv:groundedBy link to a valid Γ_m constructor trace (Compose‑CAL).Makes A.10 executable; ensures extensional identity.
CC-ANCH-2For epistemic edges (ut:EpiPartOf and its sub-types), tv:groundedBy is OPTIONAL; instead supply ev:evidence and set validationMode ∈ {axiomatic, postulate, inferential}.Harmonises evidence treatment for epistemic edges.
CC‑ANCH‑3The public query Standard remains ?x ut:PartOf+ ?y; internally it is realised via CT2R‑aliases grounded by Γ_m traces.Preserves the “one query” UX while tightening semantics.

Note. Property names and trace semantics are defined in the CT2R‑LOG / Compose‑CAL.

CT2R‑LOG handshake (Working‑Model → Assurance)

IDRequirementPurpose
CC-A14-10For structural edges on the Working-Model surface, authors SHALL set validationMode=axiomatic and attach **`tv:groundedBy → Γ_m.sumset
CC‑A14‑11PhaseOf edges SHALL NOT use Γ_m for grounding. Authors SHALL provide identity criteria and non‑overlap per CC‑PHA‑1..3 and reference Γ_time when ordering matters.Keeps temporal parthood distinct from construction; preserves the plane firewall.

CT2R‑LOG handshake (Working‑Model → Assurance)

IDRequirementPurpose
CC-A14-10For structural edges on the Working-Model surface, authors SHALL set validationMode=axiomatic and attach **`tv:groundedBy → Γ_m.sumset
CC‑A14‑11PhaseOf edges SHALL NOT use Γ_m for grounding. Authors SHALL provide identity criteria and non‑overlap per CC‑PHA‑1..3 and reference Γ_time when ordering matters.Keeps temporal parthood distinct from construction; preserves the plane firewall.

Validation patterns (author’s decision procedure)

Step 0 — Firewall check. If your sentence is about who does what, how it is done (role/method), or what happened when (run/work), you are not in mereology; go to A.15 (Role–Method–Work). If it is about the carrier episteme (pages/sections/versions of an SOP/algorithm/spec), you may still be in A.14.

Step 1 — Is it measured stuff? If yes, pick PortionOf. Confirm μ is declared (CC‑POR‑1/2). Test additivity on a toy split (CC‑POR‑3). If flows cross a boundary, remodel as interactions, not portions (CC‑POR‑4).

Step 2 — Is it a discrete inside part? If yes, pick ComponentOf (physical) or ConstituentOf (conceptual). Do not use PortionOf here.

Step 3 — Is it the same carrier at a time slice? If yes, pick PhaseOf. Verify identity criteria and non‑overlap (CC‑PHA‑1/2/3). If criteria break, escalate to B.2 (CC‑PHA‑4).

Step 4 — Is it a membership statement? Use MemberOf only; avoid any part‑inferences (CC‑MEM‑2). If you need a collection as a whole, use C.13 (Γ_m.set) for constructive grounding. If you need collective action, defer to A.15.

Quick spot‑tests (repair kit).

SmellLikely errorFix
“20% of the chassis”Treating structure as stuffUse ComponentOf; if truly laminar material, PortionOf applies to material stock, not the assembled chassis.
“Chapter 2 is 15% of the book”Mixing measures and constituentsUse ConstituentOf; the 15% is length‑of‑text as a separate statement.
“Spec v2 overlaps v1”Overlapping phases on same aspectUse PhaseOf(Spec_v2, Spec) with non‑overlap; represent drafting as Work episodes (A.15) rather than overlapping specs.
“Team is part of the project”Member vs part confusionUse MemberOf(Team, ProjectCollective), not partOf.

Interplay with Γ‑flavours (how these relations behave under aggregation)

Γ‑flavourMereological hooks (what A.14 supplies)Key effect
Γ_sys (B.1.2)Treat PortionOf as Σ‑additive stocks; ComponentOf must respect boundary integration; PhaseOf is not aggregated here.Conserves extensive measures and keeps structural WLNK (weakest‑link) on components.
Γ_epist (B.1.3)PortionOf of texts/data uses μ = token/byte count; ConstituentOf composes arguments/sections; PhaseOf versions MethodDescriptions/documents.Preserves provenance and avoids trust inflation by keeping constituents vs portions distinct.
Γ_ctx / Γ_time (B.1.4)PhaseOf provides the legal slicing for time; order/dependencies live in Γ_ctx and method graphs (A.15/B.1.5). PortionOf is orthogonal (quantities inside steps/runs).Ensures chronological consistency and monotone coverage.
Γ_method (B.1.5)Recipes are MethodDescription graphs (not parthood). When a recipe refers to stuff‑like inputs, those are PortionOf statements on resources.Separates recipe composition from structure.
Γ_work (B.1.6)Only Work carries resource deltas; when logging “consumed 5 kg from Tank A”, model it as PortionOf relation to the stock prior to consumption.Makes Σ‑balance explicit; aligns with CC‑POR‑3/4.

Consequences

Benefits

  • Predictable composition. Σ‑additivity for portions and identity‑through‑time for phases make Γ‑proofs straightforward.
  • History without confusion. Temporal slicing is explicit and audit‑ready; no paradoxical overlaps.
  • Cleaner integration with roles and recipes. The firewall prevents “functional object” creep into structure.
  • Compatibility with engineering practice. Mirrors product breakdown (components) vs functional breakdown (roles) vs material stocks (portions) vs versioning (phases).

Trade‑offs / mitigations

  • Modelling energy. Authors must pick μ and declare units; provide a short μ‑catalog per project.
  • More relation names. Two extra sub‑relations increase vocabulary; mitigated by the decision table (§ 6) and spot‑tests (§ 9).
  • Escalation discipline. Deciding PhaseOf vs MHT requires judgement; A.14 provides criteria, and B.2 captures true re‑identification.

Pedagogy aids (non‑normative)

Two‑minute checklist for reviewers

  1. Do I see “process/workflow/policy/script” used to mean enactment? — then A.15. If it names a carrier episteme whose structure/version is being discussed, A.14 may apply.
  2. Does every PortionOf have a declared μ and unit?
  3. Do phases cover a lifetime without overlap for the same aspect?
  4. Are any roles/recipes appearing as parts? If yes, stop and refactor.

Patch map (where to touch in the working file)

  1. Kernel - Holonic Mereology (§ A.1 → A.14) Add sub‑sections “PortionOf” and “PhaseOf” with axioms (POR‑1..5, PHA‑1..5). Move MemberOf note to a minimal semantics paragraph (no composition here).

  2. A.15 (Role–Method–Work) Cross‑link firewall (CC‑A14‑0/0b) and reinforce that versioning uses PhaseOf only on MethodDescription/Work.

  3. B.1.2 Γ_sys / B.1.3 Γ_epist / B.1.4 Γ_ctx/Γ_time / B.1.5 Γ_method / B.1.6 Γ_work Insert a one‑line “A.14 compliance” note: which A.14 sub‑relations each flavour relies on, as in § 10.

  4. Examples & Annexes Refactor any “percentage as part” examples into PortionOf with declared μ; Split any overlapping histories into PhaseOf sequences.

Each edited heading should carry the badge “► decided‑by: A.14 Advanced Mereology”.

Rationale (state‑of‑the‑art alignment, post‑2015)

  • Metrical mereology advances (e.g., recent work on quantity‑based parthood and additivity) motivate PortionOf with explicit μ and Σ‑laws, preventing the classic “stuff as components” fallacy.
  • Temporal parts & identity through change (renewed treatments in analytic metaphysics and formal ontology) motivate PhaseOf with coverage/non‑overlap and escalation when identity criteria fail.
  • Engineering ontologies (BORO lineage, Core Constructional practice, ISO 15926 family) keep a strict separation between functional breakdowns (our Roles) and product breakdowns (our Components), with stock/consumable modelling (our Portions) handled by quantities, not by component trees.
  • Knowledge artefact lifecycles in contemporary MBSE and open‑science workflows use explicit versioning (our PhaseOf) and provenance‑preserving composition (our ConstituentOf).
  • The net effect is a minimal‑sufficient catalogue: two added sub‑relations close real modelling gaps while preserving parsimony, didactic clarity, and Γ‑compatibility across domains.

A.14:End