An interpretation (a domain together with assignments for the nonlogical symbols of a language) that makes a particular formula, set of formulas, or an entire theory true; commonly described as a structure that satisfies those sentences.

Principle

A structure is a model of a sentence or theory exactly when every sentence in the given set is true under the interpretation induced by the structure; satisfaction is defined relative to the language and signature.

Demonstration

The group axioms (binary operation, identity, inverses, associativity) are satisfied by the structure (Z, +) of integers with addition, so (Z, +) is a model of the theory of groups; likewise, the natural numbers with usual zero and successor form a model of Peano axioms (standard model).

Misapplication

Calling an arbitrary structure a model of a theory without checking all axioms (for example, verifying associativity or existence of inverses) is a misuse that can falsely claim consistency or counterexamples.

Consequence

Models provide concrete realizations of abstract theories; existence of a model shows satisfiability and relative consistency, while witnesses of nonexistence or countermodels show independence or falsity of conjectures.

Reversal

The opposite perspective focuses on theories rather than structures: instead of asking which structures satisfy a theory, ask which sentences are true in all structures of a given class (validity).

Boundary

Being a model is relative to a language/signature and to which sentences are being considered; a structure may be a model of one theory but not of another, and modelhood does not imply uniqueness or intendedness.

Semantic Tension

Tension exists between 'intended models' (canonical structures motivating a theory) and 'nonstandard models' (unexpected or larger structures that still satisfy the axioms); both are legitimate models but play different explanatory roles.

Synthesis

A model is an interpretation that satisfies a chosen body of sentences; it instantiates abstract axioms as concrete structure, enabling semantic evaluation, counterexamples, and the study of satisfiability and entailment.