How do models help interpret the symbols in a formal system?

In the context of formal logic, a model acts as a structure that provides an interpretation for the symbols within a formal system. While a formal system is often treated as a mathematical construct focused on syntax—dealing with rules for symbol manipulation, axioms, and theorems—a model bridges the gap between these abstract symbols and their meanings.12

Models interpret symbols by mapping them to a specific domain of objects. This process involves:2

• Denotation: Stipulating which objects in a domain are represented by the constants of the language, and identifying which relations and functions are denoted by specific predicate letters and function symbols.2
• Truth-Value Assignment: Once an interpretation is established, the truth-value of sentences within the system is determined based on the standard interpretation of logical connectives (such as conjunction or negation).2

By assigning these meanings, a model allows for the evaluation of truth and falsity, which are semantic concepts distinct from the purely syntactic rules of a formal system. Consequently, model theory provides the framework to study how these interpretations relate to the logical entailment of sentences, ensuring that a system can be analyzed beyond its internal deductive structure.1234