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# Truth Table

Written by Forrest Stroud

A truth table is a logically-based mathematical table that illustrates the possible outcomes of a scenario. The truth table contains the truth values that would occur under the premises of a given scenario. As a result, the table helps visualize whether an argument is logical (true) in the scenario.

The rows of a basic truth table contain the Boolean logic true or false values, while the columns list the premises of a scenario as well as the conclusion. A simple truth table contains a single scenario and lists the valid statement and its negation.

## How to Create a Truth Table

The first step in creating a truth table is to determine the number of variables and rows needed for the table and then write down all the possible combinations (typically diagrammed as “p” and “q“).

For creating a basic “And” (conjunction) truth table, we’ll use the following example:

“If you get accepted into State University, you’ll get a six-figure job once you graduate.”

In this example, “p” represents the first premise in which you get accepted into State University and “q” represents getting a six-figure job upon graduating.

The truth table will have a column for each of these premises and a third for the logical conclusion, with each row containing a logical result from the combination of the two premises, as shown in the illustration below:

Simple Truth Table
p q Result
T T T
T F F
F T F
F F F

## The Five Basic Operations in Truth Tables

Truth tables use five basic operations:

1. Conjunction: An “and” operation where both arguments must be true in order for the statement itself to be true

2. Disjunction: An “or” operation where both arguments must be false in order for the statement itself to be false

3. Negation: A “not” operation is one that is the opposite (or complement) of the original value

4. Conditional: An “if – then” operation where the statement is false only when the first premise is true and the second is false

5. Bi-Conditional: An “if and only if” operation wherein the statement is true only when the premises share the same truth value (they’re both either true or false)