Truth Tables - garden

What is a Truth Table?

A truth table is a systematic way to figure out when a statement is true or false.

It lists every possible combination of truth values, and shows what the whole statement evaluates to.

Why Do We Need Them?

With simple statements like $p \land q$ , you can reason it out in your head.

But what about something like this?

$\neg(p \land q) \lor r$

When is this true? When is it false?

A truth table answers this systematically - no guessing required.

How Many Rows?

Each variable can be true or false (2 options).

So with $n$ variables, you need $2^n$ rows to cover every case.

Variables	Rows needed
1 (just $p$ )	$2^1 = 2$
2 ( $p$ and $q$ )	$2^2 = 4$
3 ( $p$ , $q$ , $r$ )	$2^3 = 8$
4 variables	$2^4 = 16$

Pattern: Every new variable doubles the number of rows.

Building a Truth Table

Let’s work through an example step by step.

Statement: $\neg p \lor q$

“NOT p, OR q”

Step 1: List All Combinations

We have 2 variables, so we need $2^2 = 4$ rows.

$p$	$q$
T	T
T	F
F	T
F	F

How to Fill the Columns

There’s a shortcut. Start from the rightmost column and work left.

Rightmost column: alternate every row (T, F, T, F…)
Each column to the left: double the block size

That’s it. The pattern is: 1, 2, 4, 8, 16…

Start with blocks of 1, then double each time you move left.

Step 2: Evaluate Intermediate Steps

Before we can compute $\neg p \lor q$ , we need to know $\neg p$ .

$p$	$q$	$\neg p$
T	T	F
T	F	F
F	T	T
F	F	T

Just flip each value of $p$ .

Step 3: Evaluate the Final Expression

Now compute $\neg p \lor q$ .

Remember: OR is true when at least one is true.

$p$	$q$	$\neg p$	$\neg p \lor q$
T	T	F	T
T	F	F	F
F	T	T	T
F	F	T	T

Step 4: Read the Result

Look at the final column. $\neg p \lor q$ is:

True in 3 cases (rows 1, 3, 4)
False in only 1 case (row 2)

When does it fail? Only when $p$ is true AND $q$ is false.

A More Complex Example

Statement: $(p \land q) \to r$

“If (p AND q), then r”

We have 3 variables, so we need $2^3 = 8$ rows.

$p$	$q$	$r$	$p \land q$	$(p \land q) \to r$
T	T	T	T	T
T	T	F	T	F
T	F	T	F	T
T	F	F	F	T
F	T	T	F	T
F	T	F	F	T
F	F	T	F	T
F	F	F	F	T

Reading This Table

The statement $(p \land q) \to r$ is:

True in 7 out of 8 cases
False in only 1 case

When does it fail? Only when $p$ AND $q$ are both true, but $r$ is false.

In plain English: “If both conditions are met, then the result must follow.”

The Process

To build any truth table:

Count variables $\to$ you need $2^n$ rows
List all combinations of T/F for each variable
Work inside out - evaluate parentheses first
Add columns for each intermediate step
Final column is your answer

Key insight: Truth tables are mechanical. Follow the steps, and you’ll always get the right answer.

Common Patterns to Recognize

After building enough truth tables, you’ll notice patterns:

Final column	What it means
All T’s	Tautology - always true, no matter what
All F’s	Contradiction - always false, no matter what
Mix of T’s and F’s	Contingent - depends on the inputs

Summary

Truth tables list every possible case
With $n$ variables, you need $2^n$ rows
Work inside out (parentheses first)
The final column tells you when the statement is true or false

Truth tables are the brute force method of logic - slow but always works.