Nested Quantifiers

It is sometimes difficult to see how universal and existential quantifies interact when interpreting a first-order logic sentence. The problem is best understood by looking at examples. In order to more easily display this on a web page, I will use FORALL to represent the universal quantifier (instead of the upside down A symbol) and EXISTS to represents the existential quantifier (instead of the backwards E symbol).

Consider the statement:

FORALL x EXISTS y hasParent(x,y)
This says that for every x, there exists a y (someone) who is their parent, or in other words everyone has a parent. This is in fact a true statement.

This is in contrast to:

EXISTS y FORALL x hasParent(x,y)
This says that there exists a y such that for every x, y is the parent of x. In other words this says that there is a person who is everyone's parent. This of course would not be true.

Note that syntactically, the only difference between these two statements is the ordering of the quantifiers, but that the semantics (or meanings) are very different. The key is to maintain the ordering when translating the sentences into English. This issue is also discussed in the text book on page 192, under the heading "Nested quantifiers."