How can one formal system have more than one interpretation?

A formal system can have more than one interpretation because it is initially constructed as an uninterpreted or purely formal system, consisting only of a symbolic apparatus, formation rules, and rules for manipulating formulas.1

When a system is treated apart from any specific meaning, it functions as a logical calculus that focuses on validity and satisfiability rather than truth or falsity. Because the system's internal structure—such as its axioms and theorems—is defined independently of what those symbols represent, the same formal structure can be applied to various domains.21

An interpretation is determined by assigning specific meanings to the system's primitive symbols, such as denoting specific objects, relations, or functions within a chosen domain. Since the formal system provides a structure common to many potential applications, a logician can assign different meanings to the same set of symbols, effectively creating multiple models for the same underlying formal language.31