How do universal and existential quantifiers work in predicate logic?

In predicate logic, quantifiers are used to express the quantity of individuals that satisfy a given property or relation. They allow for the construction of complex propositions beyond simple subject-predicate statements.12

Universal and Existential Quantifiers

• Universal Quantifier ($\forall x$): This symbol expresses that a formula holds for all values of the variable $x$. For example, $(\forall x)\varphi x$ translates to "for all $x$, $x$ is $\varphi$" or "everything is $\varphi$."12
• Existential Quantifier ($\exists x$): This symbol expresses that a formula holds for at least one value of the variable $x$. For example, $(\exists x)\varphi x$ translates to "there is an $x$ such that $x$ is $\varphi$" or "something is $\varphi$."12

Logical Scope and Interdependence

Quantifiers are often used in combination, and their scope—indicated by parentheses—determines how variables are bound and their relative logical priority. When multiple quantifiers are present, their order can change the meaning of a statement. For instance, $(\forall x)(\exists y)\varphi xy$ asserts that "everyone loves someone," whereas $(\exists y)(\forall x)\varphi xy$ asserts that "there is someone who is loved by everyone."32

Logically, these quantifiers are interconnected: asserting that something exists is equivalent to denying that everything lacks that property, allowing one to be defined in terms of the other.2