How do quantifiers such as “for all” and “there is some” work in formal logic?

In formal logic, quantifiers are symbols used to express the quantity of a variable within a proposition, allowing for statements about entire domains of objects rather than just specific individuals.12

The Two Primary Quantifiers

• Universal Quantifier ($\forall$): Symbolized as $(\forall x)$ or simply $(x)$, it is read as "for all $x$." It asserts that a given formula holds for every value of the variable within the specified domain.13
• Existential Quantifier ($\exists$): Symbolized as $(\exists x)$, it is read as "there is some $x$" or "there exists an $x$ such that." It asserts that the formula holds for at least one value of the variable.13

Logical Relationships

These quantifiers are deeply connected; asserting that something has a property is equivalent to denying that everything lacks it. Consequently, one is often taken as primitive, and the other is defined in terms of it. For example:3
* $(\exists a)\alpha \equiv \neg (\forall a)\neg \alpha$
* $(\forall a)\alpha \equiv \neg (\exists a)\neg \alpha$

Beyond simple assertions, quantifiers indicate dependencies between variables. In complex expressions, the scope and order of quantifiers determine how variables relate to one another, a concept further explored through Skolem functions and independence-friendly logic.4