Model theory is the study of the relationships between formal languages (signatures and sentences) and mathematical structures (models) that interpret them: it analyses satisfaction relations, classes of structures that satisfy given theories, and connections between syntactic properties of theories and semantic properties of their models (e.g., completeness, elementary equivalence, embeddings), with attention to how changes in language or signature affect what can be expressed and satisfied.

Principle

Truth of a sentence is relative to a model via a satisfaction relation; model-theoretic methods characterize which sentences or theories hold in which structures and how syntactic manipulations of theories translate into semantic consequences across classes of models.

Demonstration

Illustrative scenario — Situation: A language has a unary predicate P and a structure interprets the domain and P as a subset. Recognition: Evaluate a sentence like '∃x P(x)' by checking whether the subset interpreting P is nonempty in the structure. Action: Determine which sentences of the theory are true in that structure. Consequence: The analysis locates which models satisfy the theory and may identify non-isomorphic models that satisfy the same sentences (elementary equivalence) or embeddings between models that preserve truth of formulas.

Misapplication

Assuming existence of a formal model implies empirical instantiation or ontological commitment; the error is conflating mathematical existence of a model (a structure satisfying axioms) with claims about the empirical world without separate justificatory steps.

Consequence

Model-theoretic analysis can classify theories by their model behavior, reveal limits of expressibility in a given language, and produce constructions (e.g., expansions, reducts, ultraproducts) showing that certain semantic phenomena are possible or impossible; these consequences inform what can be captured deductively and how semantics constrains syntactic methods.

Reversal

Model-theoretic conclusions depend on the chosen language/signature, the allowed class of structures, and the logic (first-order versus higher-order); results in one setting (e.g., compactness in first-order logic) may fail or change meaning in others, so applicability must be qualified by these choices.

Boundary

Clearly within: investigation of which structures satisfy a given first-order theory and study of elementary embeddings, equivalence, and definability. Boundary case: applying model-theoretic tools to informal 'models' in social science where the mathematical satisfaction relation is metaphoric. Clearly outside: purely proof-theoretic or computational analyses that do not consider structures interpreting the language.

Semantic Tension

Expressiveness ↔ Model Existence — richer languages permit finer distinctions but may make desirable semantic properties (like compactness or completeness) fail or alter model behavior; selecting a language is a trade-off between expressive power and preservation of meta-properties.

Synthesis

Model theory connects syntactic axioms to classes of semantic structures, making explicit which aspects of a theory are determined by form and which vary across interpretations; its insights depend critically on the language and logic chosen and so must be read as system-relative mappings between syntax and possible structures.