Truth Values & NOT

The Two-Valued Universe

In everyday reasoning, we often deal with uncertainty: "maybe", "probably", "I'm not sure". Boolean logic is different. It operates in a closed universe where every statement must be exactly one of two values: true or false.

There is no "maybe". There is no "unknown". If something isn't true, it must be false. If it isn't false, it must be true.

T and F Notation

Writing "true" and "false" over and over can be tedious. In many contexts, we abbreviate these values using single letters.

T stands for true, and F stands for false.

Binary Representation

In computer science and digital circuits, Boolean values are often represented as numbers:

• 1 represents true
• 0 represents false

This is why we call it "binary" - there are exactly two digits.

The NOT Operation

The NOT operation is the simplest operation in Boolean logic. It takes a single truth value and returns the other one. If the input is true, NOT returns false. If the input is false, NOT returns true.

NOT is also written as $\neg$ (logical negation symbol) or ! (in programming).

Notation for NOT

In logic and mathematics, we use the symbol $\neg$ (the negation sign) to represent NOT. So instead of writing "NOT P", we write $\neg$P.

In programming languages like C, Java, and JavaScript, we use ! (the exclamation mark, sometimes called "bang") instead. So "NOT x" becomes !x.

Both symbols mean exactly the same thing - they negate the value.

Applying NOT with Symbols

Now let's practice evaluating NOT expressions using symbolic notation.

Remember: {{symbol}} flips the value.

• {{symbol}}true = false
• {{symbol}}false = true

What is a Truth Table?

A truth table is a table that shows all possible inputs to a Boolean operation and what output each input produces.

For NOT, the truth table has two rows (one for each possible input):

Using the NOT Truth Table

The truth table for NOT shows how the operation maps each input to its output.

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Double Negation

What happens when you negate something twice? If NOT flips a value to "the other one", then applying NOT again should flip it back.

This is called double negation: NOT(NOT(x)).

Triple Negation

We know that double negation ($\neg$$\neg$) returns the original value.

What about triple negation ($\neg$$\neg$$\neg$)?

English to Logic

In everyday language, we express negation in many ways:

• "It is not raining"
• "The light is not on"
• "I am not hungry"

To translate to logic, we first identify the base statement, then apply $\neg$.

Logic to English

We can also go the other direction - from logical notation back to English.

If {{varName}} means "{{statement}}", then $\neg${{varName}} means "{{negatedStatement}}".

Origins of Boolean Logic

Boolean logic was developed in the mid-1800s by George Boole, an English mathematician. He created a mathematical system for reasoning about logical statements, treating them as algebraic expressions.

Boole's work was primarily theoretical until nearly a century later, when Claude Shannon recognized that Boolean algebra could describe electrical switching circuits - laying the foundation for all modern digital computing.

NOT in Programming Languages

Different programming languages use different symbols for the NOT operation:

The symbol changes, but the meaning is always the same: flip the truth value.

Nested Negation Challenge

When negations are nested deeply, you must work from the inside out.

Each $\neg$ flips the truth value:

• $\neg$T = F
• $\neg$$\neg$T = $\neg$F = T
• $\neg$$\neg$$\neg$T = $\neg$$\neg$F = $\neg$T = F
• $\neg$$\neg$$\neg$$\neg$T = $\neg$$\neg$$\neg$F = $\neg$$\neg$T = $\neg$F = T

Pattern: An even number of negations returns the original value.

An odd number of negations returns the opposite.