How do existential and universal quantifiers work in logic?

In logic, quantification involves attaching signs of quantity to the subject or predicate of a proposition to indicate the scope of a variable.1

Universal and Existential Quantifiers

• Universal Quantifier ($\forall x$): This symbol expresses that a given formula holds true for all values of the specified variable. For example, the statement "For all x, if x is a whale, then x is a mammal" is represented as $(\forall x)(W(x)\supset M(x))$.12
• Existential Quantifier ($\exists x$): This symbol asserts that a formula holds for at least one value of the variable. For instance, "There is an x such that x is a vampire" is written as $(\exists x)(V(x))$.132

Scope and Logical Priority

In first-order logic, the scope of a quantifier defines the segment of a formula where the variable is bound. Scopes serve two primary functions:2
1. Binding scope: Indicates the range of the variable.2
2. Priority scope: Indicates the relative logical priority of quantifiers.2

The order of these quantifiers significantly changes meaning. For example, $(\forall x)((\exists y)(x\text{ loves }y))$ means "Everybody loves someone," whereas $(\exists y)((\forall x)(x\text{ loves }y))$ asserts the existence of a single person loved by everyone.2