5.3 Model-theoretic semantics

Foundations of Model-Theoretic Semantics

Model-theoretic semantics gives you a mathematical way to pin down what sentences mean by defining the conditions under which they're true. Instead of relying on intuition, you build a formal model of the world and then check whether a sentence holds true in that model.

This matters because it makes semantic analysis precise and testable. You can use it to explain why certain sentences entail others, why some pairs contradict each other, and how the meaning of a whole sentence builds up from its parts. The trade-off is that it works best for truth-conditional meaning and can struggle with things like implicature and context-dependence.

Principles of Model-Theoretic Semantics

Two core ideas drive this framework:

• Truth conditions define meaning. The meaning of a sentence is the set of conditions under which it's true or false. If you know exactly when The cat is on the mat is true, you know what it means.
• Compositionality. The meaning of a complex expression is determined by the meanings of its parts and the rules used to combine them. So if you know what cat, on, and mat mean individually, plus the syntactic rules combining them, you can derive the meaning of the whole sentence.

Together, these principles let you build a systematic account of meaning: you assign meanings to basic expressions, then compose them step by step. This also gives you the tools to study semantic relations like entailment (sentence A guarantees the truth of sentence B) and contradiction (A and B can't both be true).

Components of a Semantic Model

A model has three main parts. Think of them as the building blocks you need before you can evaluate any sentence.

• Domain of discourse (D): The set of entities the language is "about." This could include individuals (like specific people), events (like a wedding), times, or any other relevant objects. The domain is finite for a given model, and you decide what's in it based on what the language needs to talk about.
• Interpretation function (I): This maps the basic, non-logical expressions of the language to things in the domain. For example, $I(\text{John})$ might map to a specific individual $j$ in $D$, and $I(\text{tall})$ might map to the set of individuals in $D$ who are tall. It's what connects words to the world the model represents.
• Assignment function (g): This maps variables to elements in the domain. Variables show up when you're dealing with quantified expressions like someone or every student. The assignment function lets you evaluate open formulas (ones with unbound variables) by temporarily assigning a value to each variable.

Applying Model-Theoretic Semantics

Truth Conditions in Natural Language

To evaluate whether a sentence is true in a given model, you follow a compositional process:

1. Set up the model. Identify the domain $D$, the interpretation function $I$, and the assignment function $g$.
2. Break the sentence into parts. Parse it into its constituent expressions: subject, predicate, quantifiers, connectives, etc.
3. Interpret the basic expressions. Use $I$ to map names and predicates to their denotations in the model. For instance, $I(\text{student})$ gives you the set of students in $D$.
4. Compose and evaluate. Apply the semantic rules for each syntactic combination (quantifiers, connectives, predication) to compute the truth value of the whole sentence.

Here's a concrete example. Take the sentence Every student passed the exam. You'd translate it into logical form as:

$\forall x (\text{Student}(x) \rightarrow \text{PassedExam}(x))$

This is true in a model if and only if, for every individual $x$ in the domain $D$, whenever $x$ is in the set $I(\text{student})$, $x$ is also in the set $I(\text{passed the exam})$. If even one student in the domain didn't pass, the sentence comes out false.

Strengths and Limitations

Strengths:

• Provides a precise, unambiguous framework for representing meaning. Two people working with the same model will reach the same truth-value judgments.
• Makes semantic rules explicit, so you can formulate and test them rigorously.
• Gives you clear definitions of logical relations. Entailment, for instance, becomes: sentence A entails sentence B if every model that makes A true also makes B true.

Limitations:

• It focuses on truth-conditional meaning, so it doesn't naturally capture non-truth-conditional phenomena like presuppositions or conversational implicatures. (Those are more the territory of pragmatics.)
• The model assumes a fixed domain and a fixed interpretation, which makes it hard to handle context-dependence directly. Indexicals like here or now, and dynamic phenomena like anaphora across sentences, require extensions to the basic framework.
• Models are idealizations. Real-world meaning involves vagueness, gradience, and speaker intentions that a simple set-theoretic model won't fully capture.

Despite these limitations, model-theoretic semantics remains the standard foundation for formal work in semantics. Many of the extensions you'll encounter later (dynamic semantics, possible-worlds semantics) build directly on this framework rather than replacing it.