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How do existential and universal quantifiers affect the truth conditions of a formula?

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In logic, quantifiers determine the truth conditions of a formula by specifying the scope and range of variables within a given universe of discourse. They function as follows:

Universal Quantifier ( $\forall$ ): A formula prefixed with a universal quantifier is true if and only if the following expression holds for every individual within the specified domain. For example, $(\forall x) (R x)$ is true only if everything in the universe of discourse possesses the property $R$ .12
Existential Quantifier ( $\exists$ ): A formula prefixed with an existential quantifier is true if and only if there exists at least one individual in the domain for which the following expression holds. Thus, $(\exists x) (R x)$ is true if at least one thing is $R$ .12

Beyond simple range, quantifiers establish dependencies between variables. When multiple quantifiers are used, the truth of the formula often depends on the order and scope of these dependencies, which can be represented by Skolem functions. In such cases, the truth of a sentence is defined by the existence of a full array of functions that satisfy these dependency relations.3