In these examples, we use a very systematic method to find the negation of an English statement containing one or more quantifiers. The method consists of four steps:

- Translate the statement from English to a fully symbolic form
- Apply "negation rules" to find the (symbolic) negation of the symbolic form of the original statement.
- Translate the symbolic negation back to (perhaps awkward, stilted) English.
- Rewrite the translation in plain everyday English, taking care not to change the meaning.

**Step 1:** Translate into symbolic form. If we take our "universe of
discourse" to be all the babies in the day-care facility, then we have one
predicate on this set: "is crying".
Let

*C*(*x*) be
"*x* is crying".

Since our statement contains the word *some*, we
know that there is an existential quantifier in it.

So we can write our
statement in symbolic form
as

(∃*x*)
(*C*(*x*))

**Step 2:** Find the negation of the symbolic form. The negation
is

~(∃*x*)
(*C*(*x*))

Apply the rule for negating an existential
quantifier:

(∀*x*)
~(*C*(*x*))

**Step 3:** Translate the negation back into
English:

For every baby in
the facility, that baby is not crying.

**Step 4:** Rewrite this in good
English:

None of the babies
are crying.

**Step 1**: Translate into symbolic form. We can take the domain of the
predicates to be the set of all fruits. Then there are two predicates in this
statement: "is an apple" and "is green". Lets let

*A*(*x*) be "*x*is an apple"*G*(*x*) be "*x*is green"

So we can write our statement in symbolic form as

(∀

**Step 2**: Find the negation of the symbolic form. The negation
is

~(∀*x*)
(*A*(*x*) → *G*(*x*))

First, apply the rule for negating
a universal
quantifier:

(∃*x*)
~(*A*(*x*) → *G*(*x*))

Now we must apply the rule for
forming the negation of a conditional statement. Recall that the negation of
*p* → *q* is *p* ∧ ~*q*. We
get

(∃*x*)
(*A*(*x*) ∧ ~ *G*(*x*))

**Step 3**: Translate the negation back into
English:

There exists a fruit
that is an apple and that is not green.

**Step 4**: Rewrite this in good
English:

There is an apple
that is not green.

**Step 1**: Translate into symbolic form. We can take the domain of the
predicates to be the set of all candidates who have filed for office in this
primary. Then there are two predicates in this statement: "is an incumbent" and
"is challenging". Note that "is challenging" is a function of two variables: who
is challenging whom! Lets let

*I*(*x*) be "*x*is an incumbent"*C*(*x*,*y*) be "*x*is challenging*y*"

We can also spot an existential quantifier. We can write our statement in symbolic form as

(∀

It may be helpful to note that this symbolic form can be read, "For every candidate, if that candidate is an incumbent, then there is some candidate that is challenging her." That is not very good English writing, but it clearly has the same meaning as the original sentence.

**Step 2**: Find the negation of the symbolic form. The negation
is

~(∀*x*)
(*I*(*x*) → (∃*y*) (*C*(*y*, *x*)))

First,
apply the rule for negating a universal
quantifier:

(∃*x*)
~(*I*(*x*) → (∃*y*) (*C*(*y*, *x*)))

Now we
must apply the rule for forming the negation of a conditional statement. Recall
that the negation of *p* → *q* is *p* ∧ ~*q*. We
get

(∃*x*)
(*I*(*x*) ∧ ~(∃*y*) (*C*(*y*, *x*)))

Finally,
we apply the rule for forming the negation of an existential
quantifier:

(∃*x*)
(*I*(*x*) ∧ (∀*y*) (~*C*(*y*, *x*)))

**Step 3**: Translate the negation back into
English:

There is a candidate
who is an incumbent, such that no candidate is challenging her.

**Step 4**: Rewrite this in good
English:

There is an
unopposed incumbent in this election.