# Tautologies and Contradictions

Learn about tautologies and contradictions that are fundamental concepts in mathematical reasoning.

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## Tautologies

A proposition that is always true is called a **tautology**. For example, consider the following compound proposition $p_1$:

$p_1 = q\lor \neg q.$

It is clear that independent of the truth value of $q$, $p_1$ is always true. Therefore, $p_1$ is a tautology.

For any logically equivalent propositions $q_1$ and $q_2$, the bi-implication $q_1 \Leftrightarrow q_2$ is a tautology.

### Examples

Let’s look at some examples.

Take two arbitrary propositions: $q_3$ and $q_4.$ We know,

- $\neg\left(q_3 \Leftrightarrow q_4\right)\equiv q_3 \oplus q_4.$

By using this equivalence, let’s make a tautology.

- $p_2 = \neg\left(q_3 \Leftrightarrow q_4\right)\Leftrightarrow q_3 \oplus q_4.$

Using the following truth table, let’s verify that $p_2$ is a tautology.

$\left(q_3,q_4\right)$ | $\neg\left(q_3 \Leftrightarrow q_4\right)$ | $q_3 \oplus q_4$ | $\neg\left(q_3 \Leftrightarrow q_4\right)\Leftrightarrow q_3 \oplus q_4$ |
---|---|---|---|

(T, T) | F | F | T |

(T, F) | T | T | T |

(F, T) | T | T | T |

(F, F) | F | F | T |

It is clear by the last column that no matter what the truth value of $q_3$ and $q_4$ is $p_2$ is always true.

Here’s another example:

Take,

- $p_3 = \left(q_3 \Rightarrow q_4\right)\lor\left(q_4 \Rightarrow q_3\right).$

The proposition $p_3$ is a tautology. Let’s verify it using the following truth table:

$\left(q_3,q_4\right)$ | $q_3 \Rightarrow q_4$ | $q_4 \Rightarrow q_3$ | $\left(q_3 \Rightarrow q_4\right)\lor\left(q_4 \Rightarrow q_3\right)$ |
---|---|---|---|

(T, T) | T | T | T |

(T, F) | F | T | T |

(F, T) | T | F | T |

(F, F) | T | T | T |

## Contradictions

A proposition that is always false is called a **contradiction**. For example, consider the following compound proposition $p_4$:

$p_4 = q \land \neg q.$

It is clear that independent of the truth value of $q$, $p_4$ is always false. Therefore, $p_4$ is a contradiction.

### Examples

Let’s see some examples.

For some arbitrary propositions $q_5$ and $q_6$, we make $p_5$ as follows:

- $p_5 = \left(q_5 \lor q_6\right) \land \left(q_5 \lor \neg q_6\right) \land \left(\neg q_5 \lor q_6\right)\land \left(\neg q_5 \lor \neg q_6\right).$

The proposition $p_5$ comprises four clauses connected with conjunction operation. If $q_5$ and $q_6$ are true, then the last clause of $p_5$ is false. If $q_5$ and $q_6$ both are false, then the first clause of $p_5$ is false. In the remaining two cases, either the second or third clause is false. For $p_5$ to be true, all of its four clauses should be true. Let’s look at the following truth table to verify that $p_5$ is a contradiction

$\left(q_5, q_6\right)$ | $\left(q_5\lor q_6\right)$ | $\left(q_5\lor \neg q_6\right)$ | $\left(\neg q_5\lor q_6\right)$ | $\left(\neg q_5\lor \neg q_6\right)$ | $p_5$ |
---|---|---|---|---|---|

(T,T) | T | T | T | F | F |

(T,F) | T | T | F | T | F |

(F,T) | T | F | T | T | F |

(F,F) | F | T | T | T | F |

It is evident from the truth table above that no matter what the truth value of $q_5$ and $q_6$ is, $p_5$ is false.

For any logically equivalent propositions $q_1$ and $q_2$, the bi-implication $q_1 \Leftrightarrow q_2$ is a tautology. If we take the negation of any tautology, it will become a contradiction. So, $\neg \left(q_1 \Leftrightarrow q_2\right)$ is a contradiction.

## Contingencies

A proposition that is neither a tautology nor a contradiction is called a **contingency**.
That means contingency is a typical proposition that, in some instances, is true and false in others.

### Examples

Let’s take a look at some examples.

Consider the following compound proposition $p_6$:

- $p_6 = \left(q_7 \land q_8\right)\lor \neg q_9.$

It is clear that the truth value of $p_6$ is dependent on the truth values of $q_7, q_8$, and $q_9$. It can be true if $q_9$ is false, and it can be false if $q_7$ is false and $q_9$ is true. Therefore, $p_6$ is a contingency.

Following is another example of contingency,

Consider:

- $p_7 = \left(q_7 \Rightarrow q_8\right) \land q_9.$

One more example of contingency is as follows.

Take:

- $p_8 = \left(\neg q_7 \oplus q_8\right)\lor\left(\neg q_8 \land q_9\right).$

Observe that $p_8$ is true if $q_7$ and $q_8$ are true. But $p_8$ is false, if $q_7$ is false and $q_8$ is true. Therefore, $p_8$ is a contingency.

Note that every proposition is a tautology, contradiction, or contingency. The following illustration shows this mutually exclusive classification of the propositions discussed in this lesson.

## Quiz

Test your understanding of tautologies and contradictions.

Select the right category for the following statement:

$\neg \left(p \lor q\right) \land \left(p \land \neg q\right).$

It is a tautology.

It is a contingency.

It is a contradiction.

It is not a proposition.