Ryan S. answered • 03/26/19
PhD in Philosophy with more than 6 years of experience teaching logic
The universal and existential quantifiers in first-order logic are logical operators used to translate sentences from ordinary language into logical notation. The universal quantifier indicates that something is being said about every/any member of a group. So for example, if you said that "all cats are animals", then you'd be saying something about every/any member of a group (cats), and so we would translate that sentence using the universal quantifier. The sentence basically says that, for any x, if x is a cat, then x is an animal. The "for any x" part is the universal quantifier, and we could represent this in logical notation as: A(x) (x is a cat --> x is an animal), where the A(x) is the universal quantifier indicating that, for any x, what follows is true of it (the universal quantifier is usually represented by an upside-down A next to an x in parentheses). Any sentence that says something about all x, no x, every x, or any x would be translated with the universal quantifier.
The existential quantifier, by contrast, indicates that something exists (or is supposed to exist). When I say "some cats are black" I'm indicating that there are black cats, and so we would translate this with the existential quantifier: E(x) (x is a cat and x is black). This literally reads as "There exists an x such that x is a cat and x is black," and the existential quantifier represents the "there exists an x such that..." part. It should be noted that the existential quantifier is usually represented as a backwards E followed by the x in parentheses.
