PIATRA . INSTITUTE

order-theoretic ontology

build ontologies as posets and see where they leak

Ontology Graph

6 entities, 5 relations, 1 meta-relations

mode: soft poset | layout: layered

envelops

grounds

depends on

causes

contrasts

is about

visible edges: 5closure overlays: 0reduction removed: 0

Tip: closure overlay edges are dashed and reduction keeps only essential envelop arrows.

Selection

No active selection.

Rule Checks

No envelop self-loopsok

Acyclic envelop graphok

Antisymmetryok

Single-parent nestingok

Poset Diagnostics

Envelop relations: 3

Multi-parent nodes: 0

Cycle detected: no

Antisymmetry violations: 0

Repair Panel

No repair needed.

Snapshot Diff

Current → Current

Nodes +0 / -0 / ~0

Relations +0 / -0 / ~0

Meta +0 / -0 / ~0

Env delta: +0

Multi-parent delta: +0

Antisym delta: +0

Cycle changed: no

Envelope Tightness

Meta-relations

Order-Theoretic Ontology

The playground treats the relation $A \preceq B$ as an "envelops" claim and checks whether your ontology still behaves like a partial order. Structural leaks appear as cycles, multi-parent chains, and antisymmetry violations.

A \preceq B \wedge B \preceq C \Rightarrow A \preceq C

We compute transitive closure over "envelops" edges and inspect it for contradictory pairs where two distinct entities reach each other.

Leak Metrics

For each envelope $P \supseteq C$ , a local tightness score compares internal relations to boundary crossings.

\operatorname{tightness}(P,C)=\frac{\operatorname{internal}}{\operatorname{internal}+\operatorname{boundary}}

Low tightness means the proposed envelope leaks heavily into the outside graph. Meta-relations then let you represent second-order claims like "this relation contradicts that relation".

Workflow Upgrades

Use strict/soft/free ontology modes depending on whether edits should be blocked.
Toggle closure and reduction to inspect implied versus essential envelop structure.
Apply repair suggestions to remove minimal contradictions.
Save named snapshots and compare structural diffs across revisions.
Switch layout mode between layered and force-directed graph views.

Model Version

claude opus 4.8February 2026·first cut. an editable ontology workbench that treats the "envelops" relation as a partial order A <= B and checks whether the structure still behaves like a poset: no self-loops, acyclicity, antisymmetry, and single-parent nesting. Adds transitive closure and reduction views, an envelope-tightness leak metric, second-order meta-relations, strict / soft / free modes, snapshots with structural diffs, and minimal repair suggestions.

Calibration

Assumptions

Model Changelog

v1.0February 2026

order model: the "envelops" edge is read as the partial-order relation; reflexivity is implicit, self-loops are forbidden, and the displayed graph is the strict cover order.
rule checks: no envelop self-loops, acyclicity via DFS, antisymmetry via mutual reachability in the transitive closure, and single-parent nesting.
transitive closure (Floyd-Warshall) and transitive reduction (cover edges only) exposed as separate toggles.
envelope-tightness leak metric: internal edges over internal plus boundary edges, scoring how self-contained each proposed envelope is.
meta-relations let users assert second-order claims (supports, refines, contradicts, explains) between two relations.
strict / soft / free modes: strict blocks any edit that would violate the poset rules; soft surfaces violations without blocking; free treats the structure as a plain graph.
snapshots with structural diffs and minimal repair suggestions that remove or downgrade the smallest set of edges to restore the order.
calibration added: six hand-checkable posets verify partial-order validity, antisymmetry, acyclicity, transitive closure, multi-parent detection, and transitive reduction against the textbook answers.

read the research companion →