How can the word “is” express identity in formal logic?

In formal logic, the word “is” can function in multiple ways, but when it expresses identity, it signifies that two terms refer to the same object. While natural language uses “is” for both predication (e.g., “the dish is warm”) and identity (e.g., “Mark Twain is Samuel Clemens”), formal logic distinguishes these roles to maintain clarity.123

To represent identity, logicians replace the word “is” with the symbol “=”. This symbol is treated as a logical constant that denotes numerical identity. The logical properties of this identity are defined by two primary axioms:14

• Reflexivity: $x=x$, meaning every object is identical to itself.1
• Leibniz’s Law (Indiscernibility of Identicals): $(x=y)\supset (Fx\equiv Fy)$, which states that if $x$ is identical to $y$, then any property $F$ true of $x$ must also be true of $y$.1

By using the “=” sign, logicians ensure that statements of identity are structurally precise, allowing for valid inferences that are independent of the specific subject matter.14