A mathematical structure that assigns meanings to the symbols of a formal language (a signature)—a domain together with interpretations of constants, functions, and relations—such that sentences of the language can be evaluated as true or false in that structure.

Principle

A model realizes the syntax of a language semantically: each constant denotes an element, each function symbol denotes a concrete function on the domain, and each relation symbol denotes a relation; satisfaction of sentences is determined by these interpretations and the usual Tarski-style semantics.

Demonstration

A group (G, ·, e) considered as a first-order structure has domain G, interprets the binary symbol · by the group operation and the constant e by the identity element; sentences stating associativity or existence of inverses evaluate to true in this model.

Misapplication

Confusing the model with its underlying set or treating isomorphic models as identical without regard to the interpretation of symbols; another misuse is assuming a homomorphism of algebraic structures must preserve truth of all first-order formulas (it need not—only elementary embeddings do).

Consequence

Models allow direct semantic questions: whether a sentence holds, whether types are realized, and how structures relate via homomorphisms, embeddings, and isomorphisms; correct use yields concrete realizations of theories and counterexamples to conjectures.

Reversal

Inversion emphasizes the syntactic perspective: instead of a concrete structure one may work with the theory of that structure (the set of sentences true in it); this reverses the direction from semantic object to syntactic description.

Boundary

A model presupposes a fixed signature and first-order semantics; it excludes nonstandard semantics unless explicitly construed (e.g., many-sorted signatures or second-order models require extension of the notion), and the term is not synonymous with 'interpretation' in other logical frameworks.

Semantic Tension

Tension exists between algebraists' notion of a structure (focused on operations and equations) and model-theorists' emphasis on definability and satisfaction of formulas; the same underlying set with different interpretations yields different models.

Synthesis

A model is the semantic realization of a formal language: a domain equipped with concrete interpretations of symbols so that first-order sentences acquire definite truth values, enabling the study of theories through their concrete instances.