How do existential and universal quantifiers represent claims about individuals?

In logic, quantifiers are used to express claims about individuals by defining the scope of a predicate over a specific group of objects, known as the universe of discourse.

• Universal Quantifier ($\forall x$): Symbolized as $(\forall x)$, this represents a claim that every individual in the universe of discourse possesses a certain property. For example, $(\forall x)\varphi x$ asserts that "everything is $\varphi$" or "for all $x$, $x$ is $\varphi$."123
• Existential Quantifier ($\exists x$): Symbolized as $(\exists x)$, this represents a claim that at least one individual in the universe of discourse possesses a certain property. For example, $(\exists x)\varphi x$ asserts that "something is $\varphi$" or "there is an $x$ such that $x$ is $\varphi$."123

These quantifiers function by binding individual variables, which act like anaphoric pronouns. By using these symbols, logicians can construct complex statements about relations between individuals, such as "everyone loves somebody" or "somebody loves everyone." Philosophically, these bound variables are central to ontological commitment, as they define the entities a theory assumes to exist.234