How does an interpretation assign meanings to symbols in a formal language?

In metalogic, an interpretation assigns meaning to a formal language by establishing a relationship between its symbols and a specific domain of objects. This process involves several key steps:

• Domain Specification: An interpretation is formulated by defining a domain of objects to which the language refers.1
• Denotation Assignment: The interpretation stipulates which specific objects in the domain are denoted by the language's constants, and which relations and functions are denoted by its predicate letters and function symbols.1
• Atomic Sentences: By interpreting these basic components, the meaning of the atomic sentences is determined.1
• Truth-Value Determination: Once the atomic sentences are interpreted, the truth-value of every sentence in the language is derived based on the standard interpretation of logical connectives (such as conjunction or disjunction).1

Through this assignment, the formal language moves from being an uninterpreted system—which consists only of symbols and syntactic rules—to an interpreted system that can express meaningful principles.2