Statements with Multiple Quantifiers

Now we will work on the rules associated with statements containing more than one quantifiers. That is, consists of more than one predicate variables and usually denoted as P(x, y), P(x, y, z), Q(x, y, z, ...) etc.

The negation of 

∀ x, ∃ y such that P(x, y)

is logically equivalent to 

∃ x such that ∀ y, ~P(x, y)

For example, consider a statement:

∀ integer n, ∃ an integer k such that n = 2k.

The negation of the above statement is:

∃ an integer n such that ~(∃ an integer k such that n = 2k).

∃ an integer n such that ∀ integer k such that n = 2k.

The negation of

∃ x such that ∀ y, P(x, y)

is logically equivalent to

∀ x, ∃ y such that ~P(x, y).

For example, consider a statement:

∃ a person x such that ∈ people y, x loves y.

The negation of the above statement is:

∈ people x, ~(people y, x loves y).

∈ people x, ∃ a person y such that x does not love y.

Variants of Universal Conditionals