Negation -- NOT--
Simple Statements

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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)


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.



Under negation, TRUE becomes FALSE - or - FALSE becomes TRUE.