1133 LearnersLast updated on June 18th, 2025
Tautology
Tautology is a fundamental concept when learning about logic and reasoning. It demonstrates whether a statement is true or false and is often represented using truth tables. In this topic, we are going to talk about tautology and show how truth tables are represented.
What is Tautology?
Tautology is a logical statement that is always true, irrespective of the truth values of its components. This is an important concept in propositional logic (which deals with statements that can be true or false). For e.g., the statement "it will either rain today or no," can be logically expressed as p ∨ ¬p. This statement will always be true, and therefore is a tautology.
In order to know more about tautology, we need to know the basic logical operations that are used to present compound statements. Take a look at the different symbols used in tautology statements.
| Logic | Symbols | Representation |
| AND | ∧ | A ∧ B |
| OR | ∨ | A ∨ B |
| NOT | ¬ | ¬ A |
| If and only if | ⇔ | A⇔B |
| Implies or if-then | → | A → B |
| Is equivalent to | = | A = B |
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Truth Table for Tautology
Making a truth table is mandatory to find out if a statement is a tautology or not. A truth table lists all the possible combinations of truth values and results in the truth value of the logical expression for each combination.
Let us consider a simple expression: P ∨ ¬P (P OR NOT P). The truth table will be:
| P | ¬P | P ∨ ¬ P |
| T | F | T |
| F | T | T |
The statement P ∨ ¬ P is a tautology because it is true under all possible truth values.
Let us take another common example, (P ∧ Q) ∨ (¬P ∨ ¬Q):
| P | Q | ¬P | ¬Q | P ∧ Q | ¬P ∨ ¬Q | (P ∧ Q) ∨ (¬P ∨ ¬Q) |
| T | T | F | F | T | F | T |
| T | F | F | T | F | T | T |
| F | T | T | F | F | T | T |
| F | F | T | T | F | T | T |
As you can see in the truth table, the statement (P ∧ Q) ∨ (¬P ∨ ¬Q) is a tautology because the statement is true in all possible combinations of P and Q.
Tautology, Contradiction, and Contingency
Along with tautology, there are other statements that tells us whether a statement is false or true. Here, we will learn the difference between tautology, contingency, and contradiction.
| Tautology | Contradiction | Contingency |
| A statement that is always true regardless of the scenario | A contradiction is a statement that is always false regardless of the circumstances. | When a statement's truth value changes based on specific conditions. |
| In a truth table, all values are true (T) | All rows in the truth table are false (F) | In this truth table, some rows are true, and some are false |
| It is either wet or not wet | It is wet and it is not wet | It is wet and it is cold or it is |
| wet or it is cold | ||
| P ∨ ¬P | P ∧ ¬P | P ∧ Q or P ∨ Q (P or Q) |
Common Mistakes and How to Avoid Them in Tautology
Learning tautology can get tricky, especially for beginners. However, learning about some common mistakes and ways to avoid them can go a long way in mastering tautology. So let us take a look at some of the common mistakes:
Mistake 1
Confusing Tautology with Contradiction
Students may confuse tautology with contradiction. They must remember that a tautology is a statement that is always true, and contradictions are statements that are always false
Mistake 2
Assuming statements are tautology
When finding whether a statement is a tautology or not, students must use truth tables or logical equivalences to confirm whether the statement is true or false.
Mistake 3
Incorrectly constructing truth tables
When constructing a truth table, students may make errors in filling out truth tables. Carefully list all possible truth values of variables and check each row
Mistake 4
Misinterpreting the logical operations
It is very common for students to mix up logical operations like AND (∧) with OR (∨). Learn the exact meanings of the operations and keep practicing with examples to get a basic understanding of the operations.
Mistake 5
Not checking all possible values in the truth table
When filling the truth table with values, students may sometimes wrongly assume that the statement is a tautology. However, it could be a contingency statement as only the first three rows of a truth table could be true, while the last row could be false. Therefore, it is important to check all possible values.
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Solved examples on Tautology
Problem 1
Is the statement P ∨ ¬P a tautology?
Yes, it is a tautology
Explanation
The statement means P OR NOT P.
If P is true, P ∨ ¬P = T ∨ F = True.
If P is false, P ∨ ¬P = F ∨ T = True.
Since it is always true, it is a tautology.
Problem 2
Is ¬(P ∧ ¬P) a tautology?
Yes, this statement is a tautology.
Explanation
The expression means NOT P AND NOT P.
If P is true, then ¬P is false, making P ∧ ¬P = False.
If P is false, then ¬P is true, making P ∧ ¬P = False.
In both cases, ¬(P ∧ ¬P) = True.
Problem 3
Is the statement P → (P ∨ Q) a tautology?
Yes, it is a tautology
Explanation
P → (P ∨ Q) means "If P is true, then P OR Q is true.
If P = True, then P ∨ Q = True, so the statement is true.
If P = False, then an implication (False → anything) is always true.
Since all cases are true, the statement is a tautology.
Problem 4
Is P ∨ P a tautology?
No, it is not a tautology
Explanation
P ∨ P = P, meaning it depends on the truth value of P.
If P is false, P ∨ P = False.
Since it is not always true, it is not a tautology.
Problem 5
Is (P ∧ (P ∨ Q)) ↔ P a tautology?
Yes, it is a tautology.
Explanation
P ∧ (P ∨ Q) simplifies to P, because if P is true, then the whole expression is P.
The biconditional (↔) checks for equality, and both sides are always equal.
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FAQs on Tautology
1.How do you determine if a statement is a tautology or not?
2.Where do we use tautology in real life?
3.Could a tautology be false in any situation?
4.Can a compound statement like (P∨Q)∨¬(P∨Q) be a tautology?
5.What is considered the opposite of tautology?
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Jaipreet Kour Wazir
About the Author
Jaipreet Kour Wazir is a data wizard with over 5 years of expertise in simplifying complex data concepts. From crunching numbers to crafting insightful visualizations, she turns raw data into compelling stories. Her journey from analytics to education ref
Fun Fact
: She compares datasets to puzzle games—the more you play with them, the clearer the picture becomes!
