Negating Statements

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:

  1. Translate the statement from English to a fully symbolic form
  2. Apply "negation rules" to find the (symbolic) negation of the symbolic form of the original statement.
  3. Translate the symbolic negation back to (perhaps awkward, stilted) English.
  4. Rewrite the translation in plain everyday English, taking care not to change the meaning.
We will start with some statements which are so simple that using this method is not really necessary, in order to illustrate the technique; the go on to more complex cases.

Example 1.

Find a negation of the statement, "Some of the babies are crying."

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.

Example 2.

Find a negation of the statement, "All apples are green."

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

Since our statement contains the word all, we know that there is a universal quantifier in it.
So we can write our statement in symbolic form as
        (∀x) (A(x) → G(x))

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.

Example 3.

Find a negation of the statement, "In the upcoming primary election, every incumbent has a challenger."

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

Since our statement contains the word every, we know that there is a universal quantifier in it.
We can also spot an existential quantifier. We can write our statement in symbolic form as
        (∀x) (I(x) → (∃y) (C(y, x)))

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.