Quantifiers

One way of making sentences out of predicates is by replacing the individual variables by designators. Another way is by using quantifiers.

There are two types of quantifier:

Using an existential quantifier we can translate "Something is a cow" as "$x x is a cow". (Read this as "E x x is a cow", or as "For some x, x is a cow", or as "There is an x such that x is a cow".)

Using a universal quantifier we can translate "Everything is a cow" as ""x x is a cow". (Read this as "For all x, x is a cow".)

In the sentences "$x x is a cow" and ""x x is a cow" the variable "x" is said to be bound by the quantifier. In the predicate "x is a cow" it is said to be free: that is, free to be replaced by a designator. In the sentences the variable is not free to be replaced by a designator. A quantifier with an "x" in it binds only occurrences of "x"; one with "y" in it binds only occurrences of "y"; and so on.

Naturally we can make sentences out of predicates by using a combination of quantifiers and designators. So, from the predicate "x likes y" we can get the sentence "$x x likes Buttercup", which translates "Something likes Buttercup".

Notice, by the way, that "$x x is a cow" means exactly the same as "$y y is a cow". So "[$x x is a cow Ù $y y is a cow]" does not mean that there are two cows. (To say that, we shall need to make use of "="; that comes later.)

Notice  also that "$x x likes x" translates "Something likes itself"; and that "$x$y x likes y" translates "Something likes something"; it does not translate "Something likes something else".