Negation
Hi, and welcome to this video about negation! In this video, we will explore what negation means, how it works, and how to work with multiple negations. Let’s get started!
First, let’s review a bit. Negation is a part of propositional logic. Remember that the primary purpose of propositional logic is to determine whether declarative statements – arguments – are true or false. Here is a quick reminder of some of the symbols that we use:
| Symbol | Meaning |
|---|---|
| Lower case letter | variable |
| Upper case letter | predicate |
| \(∀\) | universal quantifier “for every” |
| \(∃\) | “there exists” |
| : | “such that” |
For example, the statement \(\exists x:x^{2}= 9\) is true, since 3 is an \(x\)-value that makes the equation true. If \(x\) is our variable and \(P(x)\) is the predicate \(x^{2}= 9\), then our statement looks like: \(\exists xP(x)\).
On the other hand \(\forall x:x^{2}=9\), read “For every \(x\)-value, \(x^{2}= 9\).” is clearly false, because there are plenty of \(x\)-values that make the equation false. Fully symbolic, this statement reads: \(\forall xP(x)\).
Now, what exactly does negation mean? Negation means basically what it sounds like – to make a statement negative. Any statement can be negated. The word not is perhaps most commonly used to negate a statement, but other words and phrases can be used as well, such as
- No
- It is false
- It is not the case
- It is not true
When a statement is negated, its truth value is the opposite of what it was. Sometimes a truth table can be a helpful illustration:
| \(P(x)\) | \(¬P(x)\) |
|---|---|
| True | False |
| False | True |
How Negation Works
Let’s revisit the statement: “There exists an \(x\)-value such that \(x^{2}=9\),” which we’ve said is true. Negating it should produce a false statement, which it does: “There does not exist an \(x\)-value such that \(x^{2}=9\).”
Note that negation of this statement is not “There exists an \(x\)-value such that \(x^{2}\neq 9\),” because \(x^{2}\neq 9\) is a different predicate for a different time. We are testing \(x\)-values that make \(x^{2}=9\) true or false, not \(x\)-values that make \(x^{2}\neq 9\) true or false.
Likewise, the statement: “For every \(x\)-value, \(x^{2}=9\),” which we’ve said is false, is made true by negation: “It is not true for every \(x\)-value that \(x^{2}=9\)”
The symbol for negation is ¬, let’s add it to our list:
| Symbol | Meaning |
|---|---|
| Lower case letter | variable |
| Upper case letter | predicate |
| \(∀\) | universal quantifier “for every” |
| \(∃\) | “there exists” |
| : | “such that” |
| ¬ | “not,” negation |
Now, back to our examples. We negated the statement “There exists an \(x\)-value such that \(x^{2}=9\),” by saying “There does not exist an \(x\)-value such that \(x^{2}=9\).”
Symbolically, we originally said \(\exists xP(x)\), which is true. It can be negated by saying
- \(¬\exists xP(x)\) (There does not exist an \(x\)-value such that \(x^{2}=9\)) or
- \(\forall x¬P(x)\), (For all \(x\), \(x^{2}\) never equals 9)
Both the negations have the same meaning and are truth-equivalent (in this case, both false).
In our second example, we began by saying “For every \(x\)-value, \(x^{2}=9\),” which we know is false, and negated it by saying “It is not true for every \(x\)-value that \(x^{2}=9\).”
Symbolically, we originally said \(\forall xP(x)\) , which is false. It can be negated by saying
- \(¬\exists xP(x)\) (There exists an \(x\) such that \(x^{2}\) is not equal to 9) or
- \(\forall x¬P(x)\) (Not every \(x\)-value makes \(x^{2}=9\) true)
Again, both negations have the same meaning and are truth-equivalent (in this case, both true).
Also, in this case, the term counterexample comes to mind. For every \(x\)-value, \(x^{2}=9\). False, here’s a counterexample: Let \(x=2\), for instance.
Let’s try some practice: Evaluate each statement and show its negation has the opposite value.
Every voter is at least 18 years old. This is true, according to the law.
Not every voter is at least 18 years old. And this is false, because the voting age is 18.
Every composite number is even. False, some composite numbers are odd, such as 15.
Not every composite number is even. True.
Multiple Negations
Statements can be negated multiple times. Since one negation flips the truth value of a statement, a second negation flips it back. Let’s expand our truth table:
| \(P(x)\) | \(¬P(x)\) | \(¬¬P(x)\) | \(¬¬¬P(x)\) |
|---|---|---|---|
| True | False | True | False |
| False | True | False | True |
Therefore, saying “There exists an \(x\)-value such that\(x^{2}=9\),” is truth equivalent to “There does not not exist an \(x\)-value such that \(x^{2}=9\).” \(\exists xP(x)=¬¬\exists xP(x)\).
Now I want you to try some on your own! Pause the video and complete the chart.
| Statement | Symbolic | Truth |
|---|---|---|
| All integers are rational numbers. | \(\forall xP(x)\) | T |
| Not all integers are rational numbers. | \(¬\forall xP(x)\) | F |
| It is not the case that not all integers are rational numbers. | \(¬¬\forall xP(x)\) | T |
| There exist integers that are not rational numbers. | \(\exists x¬P(x)\) | F |
| It is false that it is not the case that not all integers are rational numbers. | \(¬¬¬\forall xP(x)\) | F |
| There do not exist integers that are rational numbers. | \(¬\exists xP(x)\) | F |
I hope that this video increased your understanding of negation! Thanks for watching, and happy studying!
Negation Practice Questions
Let \(\exists xP(x)\) be the statement \(\exists x:x^2=16\). Which of the following is the statement for \(\lnot\exists xP\left(x\right)\) and its truth value?
There does not exist an \(x\)-value such that \(x^2=16\), which is a false statement.
There does not exist an \(x\)-value such that \(x^2=16\), which is a true statement.
There exists an \(x\)-value such that \(x^2\neq16\), which is a false statement.
There does not exist an \(x\)-value such that \(x^2\neq16\), which is a true statement.
The symbolic statement \(\lnot\exists xP(x)\) means to negate the initial portion of the statement, “There exists an \(x\)-value” written symbolically as \(\exists x\). When a statement is negated, its truth value is the opposite of what it was. Negated statements often use the word “not” or “does not” when they are stated. Changing the initial portion of our statement to “There does not exist,” our negated statement, \(\lnot\exists xP\left(x\right)\), is, “There does not exist an \(x\)-value such that \(x^2=16.\)”
To show the negation is false, we can show that \(\exists x:x^2=16\) is true. We can choose an \(x\)-value of 4 to make the predicate true since \(4^2=16\). The symbol \(\exists\) tells us there only needs to exist one \(x\)-value to make the statement true. So, \(\exists xP(x)\) is a true statement. Thus, the negated statement must be false. Therefore, choice A is the correct answer.
Let \(\forall xP(x)\) be the statement “For every value of \(x\), \(x^2=25\)”, which is a false statement. Which of the following is the symbolic negation for the statement, “It is not the case that not every value of \(x\) makes \(x^2=25\),” that also has the same truth value of \(\forall xP(x)\)?
For the symbolic statement \(\forall xP(x)\), \(\forall x\) means “For every value of \(x\),” and \(P(x)\) is \(x^2=25\). We are told this is a false statement.
When a statement is negated, its truth value is the opposite of what it was. Symbolically negating our statement once, we can use \(\lnot\forall xP(x)\), which is true since it can be stated as, “Not every value of \(x\) makes \(x^2=25\).” Use \(x=4\) to show that the square of not every \(x\)-value equals \(25\) because \(4^2=16\neq25\).
We can negate \(\lnot\forall xP(x)\), using \(\lnot\lnot\forall xP(x)\), which becomes an equivalent false statement to \(\forall xP(x)\). It can be stated as, “It is not the case that not every value of \(x\) makes \(x^2=25\).” Since we saw from above that “Not every value of \(x\) makes \(x^2=25\)” was true, its negation must be false so, \(\lnot\lnot\forall xP(x)\) has an equivalent truth statement using \(P\left(x\right)\).
Let \(\forall xP(x)\) be the statement, “All even numbers are integers.” Which of the following is the statement for \(\exists x\lnot P\left(x\right)\) and its truth value?
There does not exist an even number that is an integer, which is a false statement.
There does not exist an even number that is not an integer, which is a true statement.
There exists an even number that is not an integer, which is a true statement.
There exists an even number that is not an integer, which is a false statement.
The symbolic statement \(\exists x\lnot P(x)\) means to negate the conclusive portion of the statement “are integers.” When a statement is negated, its truth value is the opposite of what it was. Negations often use the word “not” when they are stated. Additionally, the symbol \(\exists\) means that there exists. Since our statement is “All even numbers are integers”, our negated statement, \(\exists x\lnot P\left(x\right)\), should state “There exists an even number that is not an integer.”
The set of integers contains the numbers \(\{…,-3,-2,-1,0,1,2,3,…\}\). The set of even numbers contains the numbers \(\{…,-6,-4,-2,0,2,4,6,…\}\) which is a subset of the set of integers. Therefore, all even numbers are also integers. This means the statement, “All even numbers are integers” is a true statement. Thus, the negation \(\exists x\lnot P(x)\) must be false.
The statement, “The minimum voting age for all voters in the United States is 18 years old,” is a true statement. What is the negation of the statement and its truth value?
The minimum voting age for all voters in the United States is not 18 years old, which is a true statement.
The minimum voting age for all voters in the United States is not 18 years old, which is a false statement.
There exists a voter in the United States whose minimum voting age is not 18 years old, which is a false statement.
There does not exist a voter in the United States whose minimum voting age is not 18 years old, which is a false statement.
When a statement is negated, its truth value is the opposite of what it was. We can negate the predicate of a statement by using the word “not” when stating it. Since our predicate states “is 18 years old,” our negated statement will be “The minimum voting age of all voters in the United States is not 18 years old.”
Since it is given that the original statement is true, the negation of the original statement must be a false statement.
The statement, “Only one of my neighbors has exactly two dogs,” is a true statement. What is the negation of the statement?
None of my neighbors do not have exactly two dogs.
There is at least one of my neighbors that does not have exactly two dogs.
All of my neighbors do not have exactly two dogs.
Not all of my neighbors have exactly two dogs.
When a statement is negated, its truth value is the opposite of what it was. We can negate the predicate of a statement by using the word “not” or “does not” when stating it. Since our predicate states, “has exactly two dogs” we can write its negation as, “does not have exactly two dogs.”
Additionally, if you only have two neighbors, then the negated statement, “Only one of my neighbors does not have exactly two dogs,” is still a true statement since it implies the other neighbor does have exactly two dogs. The negated statement needs to be false. To make the negated statement false, we can change the initial part of the negation, “Only one of my neighbors” to “All of my neighbors.” So, the negation becomes, “All of my neighbors do not have exactly two dogs,” which is a false statement.
