Negation and the Not Connector

In the previous post, we have discussed compound propositions. We learned that compound proposition is a proposition formed from simple propositions using some logical connectors. The first logical connector that we are going learn is about “not” which is used for negation.

The negation of a proposition p denoted by ~p (read as “not p”) and is defined by the following truth table. As we can see, if p is true, then ~p is false. If p is false, then ~p is true.

Consider the following negations.

Example 1
q: I am going to the party.
~q: I am not going to the party.

Example 2
r: John is wealthy.
~r: John is not wealthy.

Example 3
s: This pen is not expensive.
~s: This pen is expensive.

Example 4
$p_1$: Five is a prime number.
$~p_1$: Five is not a prime number.

Example 5
$p_2$: 3 is odd
$~p_2$: 3 is not odd

In example 5, since integers can be classified into odd or even only, we can also write ~p2 as follows:
~p2: 3 is even.

In some cases, a statement can have more than one negation such as shown in the next two examples.

Example 6
$p_3: 3 + 5 = 8$
$~p_3: 3 + 5 \neq 8$
$~p_3: 3 + 5 > 8$
$~p_3: 3 + 5 < 8$

As you can see from the examples, negation is usually done by either adding or removing the word “not.”

One of the common misconceptions in negation is changing the word to its antonym. For instance, in Example 2,

r: John is wealthy.
~r: John is poor. (Wrong!).

As we can see, John may not be wealthy, but John may also be not poor. So, the statement ~r is wrong. The general rule is adding/removing the word “not” to/from the sentence.

In the next post, we are going to discuss the and connector. Meanwhile, here are some exercises.

Exercise: Negate the following statements:

1. ) I am going to finish reading this book.

2.) The number 5 is odd.

3.) 7 < 5

4.) 12 + 3 = 7

5.) Six is not a prime number.

6.) He is 6-foot tall.

In the next post, we are going to learn about the “or” connective. Keep posted.