The assignment of semantic values, domain elements, and structural relations to the symbols of a formal language or model that establishes truth‑conditions or satisfaction relations relative to a specified structure, thereby linking the formal system to a target domain or class of structures.

Principle

A model-theoretic interpretation determines the truth or satisfaction of formal statements by specifying a structure (domain and interpretation function) against which formulas are evaluated; different interpretations yield different semantic consequences even when syntax is unchanged.

Demonstration

Situation: A logician has a formal language for graphs and wants to test whether a formula expressing "existence of a path" holds for a social network. Recognition: They fix an interpretation mapping predicate symbols to nodes and adjacency relations to edges in a particular dataset. Action: They evaluate the formula in that structure to determine whether the network satisfies it. Consequence: The result is a semantic judgment about that interpretation, not a universal fact about all possible social networks.

Misapplication

Confusing a model with the real-world system it represents, or assuming a single interpretation is uniquely intended; such reification or unwarranted uniqueness treats semantic fit as identity or empirical proof rather than as one mapping among many.

Consequence

Explicit model-theoretic interpretation makes formal reasoning applicable to empirical or conceptual targets and clarifies underdetermination when multiple structures satisfy the same formal sentences.

Reversal

In contexts that impose an intended semantics (for example, a fixed standard model in a specification), interpretation may be uniquely constrained; in non-classical logics or alternative semantics, the satisfaction relation and truth-conditions themselves may differ.

Boundary

Clearly within: first-order or modal semantics where a structure and interpretation function are explicitly given. Boundary case: informal or metaphorical 'models' used in exposition that lack precise satisfaction conditions. Clearly outside: rhetorical uses of 'model' that do not supply a mapping from symbols to domain elements.

Semantic Tension

Formal Precision (explicit mappings and truth-conditions) ↔ Empirical Adequacy (fit of interpretation to messy real-world data): formal semantics demands exactness that empirical interpretation must approximate.

Synthesis

Model-theoretic interpretation separates syntactic derivation from semantic commitment: to apply formal results to a domain, one must make the interpretation explicit and assess which structures are being represented.