Tautologies and Contradictions

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.

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

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

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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:

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Take,

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

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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.

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

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

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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.

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Consider the following compound proposition $p_6$:

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

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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,

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Consider:

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

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One more example of contingency is as follows.

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Take:

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

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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