Universal Conditional

The most important form of proposition in logic and mathematics is the universal conditional proposition:

∀ x if, P(x) then Q(x)

For example,

∀ x ∈ R, if x > 2 then x2 > 4

Equivalent Forms of Universals and Existentials

Consider the following propositions:

The above two propositions mean the same thing. That is,

All integers are rational.

In general, a proposition of the form:

∀ x ∈ U, if P(x) then Q(x)

can be written as

∀ x ∈ D, Q(x)

by narrowing the set U to be the domain D consisting of all values of the variable x that make P(x) true.

Conversely, a proposition of the form:

∀ x ∈ D, Q(x)

can be written as

∀ x, if x is in D then Q(x)

For example, the universal proposition:

∀ polygons p, if p is a square, then p is a rectangle.

is equivalent to:

∀ squares p, p is a rectangle.

The existential proposition:

∃ x ∈ U such that P(x) and Q(x)

is equivalent to:

∃ x ∈ D such that Q(x)

For example, the existential proposition:

∃ a number n such that n is prime and n is even.

is equivalent to 

∃ a prime number n such that n is even.

Negation of Universals

The negation of a proposition of the form:

∀ x in D, Q(x)

is logically equivalent to a proposition of the form:

∃ x in D such that ~Q(x)

Symbolically,

~(∀ x in D, Q(x)) ≡ ∃ x in D such that ~Q(x)

Therefore, the negation of a universal proposition ("all are") is logically equivalent to an existential proposition ("some are not").

For example, consider the proposition:

All mathematician wear glasses.

The negation is:

One or more mathematicians do not wear glasses.

or 

Some mathematician do not wear glasses.

Another example, consider the universal proposition:

∀ primes p, p is odd.

By applying the rule for the negation of a ∀ proposition, we have

∃ a prime p such that p is not odd.

or equivalently,

∃ a prime p such that p is even.

Negation of Existentials

The negation of a proposition of the form:

∃ x ∈ D such that Q(x)

is logically equivalent to a proposition of the form:

∀ x in D, ~Q(x)

Symbolically,

~(∃ x ∈ D such that Q(x)) ≡ ∀ x in D, ~Q(x)

Therefore, the negation of an existential proposition ("some are") is logically equivalent to a universal proposition ("all are not").

For example, consider the proposition:

Some fish breath air.

The negation is:

No fish breath air.

Another example, consider the existential proposition:

∃ a triangle T such that the sum of the angles of T equals 200 degrees.

By applying the rule for the negation of a ∃ proposition, we have

∀ triangles T, the sum of the angles of T does not equal 200 degrees.

Negation of Universal Conditionals

By definition of the negation of a 'for all' proposition,

~(∀ x, P(x) → Q(x)) = ∃ x such that ~ (P(x) → Q(x)) ____(1)

But the negation of an if-then proposition is logically equivalent to an "and" proposition, i.e.,

~(P(x) → Q(x)) = P(x) ∧ ~Q(x) _____(2)

Substituting 2 in 1, gives

~(∀x, P(x) → Q(x)) = ∃ x such that P(x) and ~Q(x)

For example, consider the proposition:

∀ people p, if p is blond then p has blue eyes.

A formal negation is:

∃ a person p such that p is blond and p does not have blue eyes.

Example 2: 

Example 3:

Example 4:

Example 5: