Logical quantifiers are a feature of first- and higher-order logical languages
that allow one to express
general facts about a number of individuals, or ascribe
properties to some individual object, without actually identifying that object by name. There are two main quantifiers used in first-order logic
The Universal Quantifier is represented by the symbol
(If the above did not print on your browser, it looks like an upside-down capital letter A. ) The universal quantifier means "for all" or "for every". For example, you might say in English
, "for every X, if X is an infant
, X cannot speak
." In this statement, the variable
X is universally quantified. The universal quantifier lets us make a statement about all infants without identifying any of them individually.
The Existential Quantifier is represented by the symbol
(If the above did not print correctly, it looks like a backwards capital letter E. ) The existential quantifier means "there exists" or "there is at least one such that". For example, one could say in English "There exists X such that X is rotten and X is in Denmark
." In this statement, the variable X is existentially quantified. This lets us propose the existence of at least one thing that is rotten in Denmark, without actually identifying it.
As you've noticed from the examples above, when a quantifier symbol appears in a logical sentence, it is always associated with, and applies to, a variable. The quantifier affects how the variable is interpreted in the rest of the sentence. Universally quantified variables stand for everything in the universe of discourse, and existentially quantified variables stand for just something -- maybe only one thing, maybe more than that, but not necessarily everything.