
In
logic, a negation of a simple statement
(one logical value) can usually be
formed by placing the word "not"
into the original statement. The negation will always have
the opposite truth value of the original statement.
Under negation, what was TRUE, will become FALSE 
or  what was FALSE, will become TRUE. 
Examples
of simple negations:
1. 
Original
Statement: "15 + 20
equals 35." (is true)
Negation: "15 + 20 does not equal 35."
(is false)

2. 
"A dog is
a cat." is a false statement.
"A dog is not a cat." is a
true statement.
"It is not true that a dog is a
cat." is a true statement.*
"It is not the case that it is not true
that a dog is not a cat." is a true statement.* 

* Notice that
there are different ways of inserting the concept of
"not" into a statement. While
we would not usually speak in a manner similar to the last
statement, we must be alert to people who attempt to win
arguments by using several negations at the same time to
cause confusion. 

3. 
"A
fish has gills." is a true statement.
"A fish does not have gills."
is a false statement.
"It is not true that a fish does not
have gills." is a true statement.** 

**
Notice how using TWO negations, returns the truth value of the
statement to its original value. In plain English, this
means that two negations will "undo" one another (or
cancel out one another). 

4. 
Original
statement: "Jedi masters do not
use light sabers."
Negation: "Jedi masters do
not not
use light sabers."
Better Negation:
"Jedi masters
do use light sabers."
Notice:
even though the first negation shows the proper insertion of the
word "not", the second negation can be more easily read
and understood. 

Mathematicians often
use symbols and tables to represent concepts in logic. The use of
these variables, symbols and tables creates a shorthand method for
discussing logical sentences.
Truth
table for negation (not):
(notice the
symbol used for "not" in the table below)


T 
F 
F 
T 

A
truth
table is a pictorial representation of all of the possible
outcomes of the truth value of a sentence. A letter such
as _{
}
is used to represent the sentence or statement. 

REMEMBER: 
Under
negation, TRUE becomes FALSE  or  FALSE becomes TRUE. 
