Models in Logic

In logic, models describe how we interpret a given formal language. Syntax provides the symbols and rules, while semantics (models) define what they mean.

Syntax and Semantics

Syntax specifies how a correct formula can be built (for example, p ∧ q, ¬p, p → q). Semantics tells us what truth value these formulas receive in a given model.

What is a Model?

A model is a specific interpretation: we assign truth values (true or false) to logical variables, and in first-order logic, meanings to individuals, functions, and relations.

Example: Suppose p = 'It is raining', q = 'The road is wet'. In one model, p can be true, q true. In another model, p true, but q false. The value of the formula p → q varies according to the model.

Satisfiability with Models

A formula is satisfiable if there is a model in which it is true. If there is no such model, the formula is a contradiction. If true in every model, it is a tautology.

This notation shows that the φ formula is true in a model.

Examples of Models

• Propositional logic: we simply assign true/false values to variables.
• First-order logic: constants, functions, and relations receive meanings (e.g., '<' relation on natural numbers).
• Mathematical structures: models can be number sets (ℕ, ℤ, ℝ), graphs, set systems.

Why Are Models Important?

• They help understand when a formula is true or false.
• They enable comparison of logical systems.
• They are the foundation of mathematical logic, formal proofs, and algorithm analysis.
• They create a connection between purely syntactic formulas and real-world interpretations.

Summary

Logical models are carriers of semantics: they specify how we interpret a formal language. A formula can be satisfiable, a tautology, or a contradiction, depending on which models it is true in.

Practice Exercise