A systematic correspondence between models (possibly of different signatures or domains) that relates elements, structure, and satisfaction so as to compare or translate semantic content across models.

Principle

An intermodel mapping provides a rule (or family of rules) that assigns elements and relations of one model to elements and relations of another, and specifies how truth of formulas in one model is reflected or translated in the other.

Demonstration

Example: a reduct-expansion mapping sends a model of a richer language to its reduct in a smaller language by forgetting certain symbols; more structured examples include functorial translations between categories of models induced by signature translations or interpretations.

Misapplication

Assuming an intermodel mapping preserves every logical consequence or all semantic properties without verifying which formula classes or satisfaction conditions are respected; confusing syntactic substitution of symbols with a semantic model mapping.

Consequence

A well-defined intermodel mapping enables transferring theories, comparing expressiveness, transporting counterexamples, and defining translations between specifications or database schemas while making precise which truths are preserved or reflected.

Reversal

The converse operation is a back-mapping or inverse translation which may not exist or be unique; reversing an intermodel mapping can require choices, extensions, or additional structure and may introduce new consequences not present originally.

Boundary

Not every relation between models is an intermodel mapping in this sense; legitimate mappings must specify how relations and functions are translated and must make clear whether they preserve satisfaction, preserve only certain formula classes, or are merely correspondences of underlying sets.

Semantic Tension

Tension occurs between mappings that are purely syntactic (relabelings of symbols), homomorphic (structure-preserving maps), and interpretative (which may reencode predicates); distinguishing these is crucial to know which semantic features move across the mapping.

Synthesis

An intermodel mapping is the explicit mechanism linking models across languages or domains: it formalizes how elements and truths correspond, indicates what is preserved or reflected, and supports comparison and translation of semantic content across systems.