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

In formal logic, the natural-language verb "is" is considered multiply ambiguous, as it can be used to express predication, identity, class-inclusion, or existence. To avoid this ambiguity, formal logic distinguishes these senses by using specific symbolic structures rather than the word "is" itself.12

When "is" is used to express existence—such as in the sentence "Zeus is" or "Zeus exists"—it is represented using the existential quantifier $(\exists x)$. In this context, the statement is formalized as $(\exists x)(x=z)$, which translates to "There exists an $x$ such that $x$ is identical to Zeus".13

This approach aligns with the view that existence is not a simple predicate but is instead codified by the existential quantifier. By treating existence as a value of a bound variable, formal logic provides a precise way to represent ontological commitments, ensuring that the logical form of a statement is clearly distinguished from other uses of "is," such as identity ($c=s$) or predication ($B(t)$).312