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How do universal and existential quantifiers express claims about members of a group?

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logic: universal affirmative — Modern logic Representations of the universal affirmative, “All A's are B's” in modern logic.

In modern logic, universal and existential quantifiers are fundamental tools used to express claims about members within a defined group of objects, known as the universe of discourse. These quantifiers "range over" the individuals in this group to determine the scope and truth of a proposition.12

Universal Quantifier ( $\forall x$ ): Symbolized as $\forall$ , this quantifier expresses that a given formula or property holds for every member of the universe of discourse. For example, the expression $(\forall x) (R x)$ translates to "for every $x$ , it is red" or simply "everything is red".31
Existential Quantifier ( $\exists x$ ): Symbolized as $\exists$ , this quantifier expresses that a formula holds for at least one member of the group. The expression $(\exists x) (R x)$ translates to "there is an $x$ such that it is red" or "something is red".31

These quantifiers function similarly to anaphoric pronouns, allowing logicians to systematically represent properties and relations without the ambiguities often found in natural language. While first-order logic restricts these quantifiers to individuals, higher-order logics allow them to range over sets, properties, and relations.12