A formal relation between two models in which there exists a bijective mapping between their elements and relations that preserves the structure of connections and operations; when isomorphism holds, formal properties and inferences valid in one model correspond to formally equivalent properties in the other, independent of differing ontological labels.

Principle

Isomorphism guarantees structural equivalence: a bijection f exists such that for any relation or operation R in model A, R holds of elements in A exactly when the corresponding relation R' holds of f(elements) in model B, enabling transfer of formal results while leaving semantic interpretation contingent on domain assignments.

Demonstration

Illustrative scenario — Situation: Two researchers represent interacting agents using distinct ontologies (nodes labeled differently). Recognition: They construct a bijective mapping between nodes and edges that preserves adjacency and interaction rules. Action: A theorem about propagation of a property in model A is translated via the mapping to model B and tested. Consequence: Because the mapping preserves structure, the formal prediction holds in model B under the mapped conditions; however, domain‑level interpretation (what nodes represent) remains a separate step.

Misapplication

Conflating formal isomorphism with ontological identity — e.g., claiming two theories describe the same real entities because their models are isomorphic; the error is treating structural equivalence as evidence of referential or metaphysical equivalence without independent corroboration.

Consequence

Model isomorphism supports formal transfer, unification of reasoning procedures, and detection of deep equivalences between representations; it does not by itself resolve interpretive questions about what modeled elements stand for in the world.

Reversal

Approximate mappings (homomorphisms, partial isomorphisms) weaken guarantees: preserved properties may be only one‑way or limited. At different scales or with additional constraints, an isomorphism in an abstract formalism may cease to hold when domain‑specific parameters are introduced.

Boundary

Clearly within: a bijective, relation‑preserving mapping between two formal models. Boundary case: a homomorphism that preserves structure in one direction but is not bijective. Clearly outside: informal analogy or similarity without a formal mapping.

Semantic Tension

Formal Equivalence ↔ Ontological Commitment — structural sameness facilitates mathematical transfer while leaving open whether the models commit to the same interpretation of entities or processes.

Synthesis

Model isomorphism identifies formal sameness of structure: it legitimizes transferring formal conclusions across differently labeled representations but requires a separate, explicit step to connect structural results to domain‑level interpretation.