Model-theoretic semantics, rooted in Tarski's theory of truth for formal languages, interprets linguistic expressions relative to a model: a mathematical structure consisting of a domain of entities and an interpretation function that assigns denotations to basic expressions. A sentence is true in a model if the state of affairs it describes obtains in that model. This approach provides the standard framework for formal semantics, enabling precise definitions of truth, entailment, validity, and other core semantic notions.
Models and Interpretation
D = domain of entities
W = set of possible worlds
I = interpretation function
I maps constants to elements of D
I maps predicates to functions from W to subsets of D
M, w ⊨ P(a) iff I(a) ∈ I(P)(w)
M, w ⊨ phi ∧ psi iff M, w ⊨ phi and M, w ⊨ psi
M, w ⊨ ∀x.phi iff for all d ∈ D, M[x→d], w ⊨ phi
In a basic first-order model, the interpretation function maps individual constants to entities in the domain and predicate constants to sets of entities (or sets of tuples for relations). Truth is defined recursively: an atomic sentence P(a) is true iff the denotation of a is in the denotation of P. Complex sentences are interpreted through the standard logical connectives and quantifiers. For intensional semantics, the model is enriched with a set of possible worlds, and denotations become functions from worlds to extensions.
Entailment and Validity
Model-theoretic semantics provides precise definitions of key semantic relations. A sentence phi entails psi iff every model that makes phi true also makes psi true. A sentence is valid iff it is true in every model. Two sentences are equivalent iff they are true in exactly the same models. These definitions capture the intuitive notions of logical consequence, tautology, and synonymy in a mathematically rigorous way, and they underpin natural language inference tasks in computational linguistics.
In computational linguistics, model-theoretic evaluation is used in semantic parsing and question answering. A natural language question is parsed into a logical form, which is then evaluated against a model (typically a database or knowledge base). For example, "Which countries border France?" is parsed to a lambda expression and evaluated against a geographic database acting as a model. The FraCaS test suite evaluates NLI systems against model-theoretic entailment judgments, providing a bridge between formal semantics and NLP evaluation.
Limitations and Extensions
Standard model-theoretic semantics has been extended in several directions. Situation semantics (Barwise and Perry) replaces possible worlds with partial situations, addressing issues of hyperintensionality and information flow. Discourse Representation Theory (Kamp) introduces discourse referents and conditions, providing a dynamic alternative that handles anaphora and temporal reasoning. Possibility semantics and truthmaker semantics offer alternative foundations that avoid some problems of the possible-worlds framework while retaining model-theoretic rigor.
For computational applications, the main challenge of model-theoretic semantics is the open-world assumption: natural language describes an unbounded world, while computational models are necessarily finite. Knowledge graphs and databases provide finite models for evaluation, but handling negation, quantification over absent entities, and the open-ended nature of natural language meaning requires careful engineering. Despite these challenges, model-theoretic evaluation remains the gold standard for semantic correctness in NLP systems.