Logical Operators: Negation

In symbolic logic, we have five main logical operators — Negation, Conjunction, Disjunction, Implication, and Biconditional. These operators allow us to build any kind of compound statement from simple ones.

What is Negation?

Negation is the simplest logical operator. If we have a statement p, its negation is simply "not p."

Example

Let p = "It is raining." Then ~p = "It is not raining."

The statement p and its negation ~p are always opposites: one must be true, the other must be false.

Symbol

Negation is denoted by ~ or sometimes by a bar over the statement. In our course, we will use ~p to mean "not p".

Unary Operator

Negation is called a unary operator because it acts on a single statement. It flips its truth value. If p is true, ~p is false. If p is false, ~p is true.

Real-World Examples of Negation

• If p = "I have homework", then ~p = "I do not have homework."
• If p = "The store is open", then ~p = "The store is closed."
• If p = "The light is ON", then ~p = "The light is OFF."

Coding Example:

                    p = true if user is logged in. 
                    if (p = true) print "Hello User"
                    if (~p = true) print "User is not logged in."
                

In this coding example, we use negation to create opposite conditions.

Negation is everywhere in reasoning: whenever someone denies, rejects, or flips the truth of a statement, they are using negation.

What is a Truth Table?

A truth table is a table that explains all the possible combinations a statement can have. A single statement can have two possible states, true or false. So a truth table with a single variable will always only have two rows. The number of rows expand with more variables which we will see in next lessons.