Tautology and Contradiction:
A compound statement in logic is a tautology if it is always true for all possible truth values of its component statements.
Tautologies are true by virtue of their logical structure. Their truth is independent of the truth values of the individual components and also independent of anything to which the sentences might refer. Let us consider the sentence ‘It is right or it is wrong’. In symbolic form, the sentence may be denoted by p ∨ ~p where p stands for ‘It is right’. The truth table for the sentence is shown below. As the last column contains T everywhere, the given proposition is a tautology. This tautology is known as the ‘Law of Excluded Middle’.
| p | ~ p | p ∨ ~ p |
|---|---|---|
| T | F | T |
| F | T | T |
A compound statement in logic is called a contradiction if it is always false for all possible truth values of its component statements. A contradiction is also known as a fallacy. A negation of any tautology is a contradiction and a negation of any contradiction is also a tautology.
Example 1- Using truth tables determine whether the given statement is a tautology or not.
(i) (p ∧ ~ p) ⇒ q
Solution-
| p | q | ~ p | p ∧ ~ p | (p ∧ ~ p) ⇒ q |
|---|---|---|---|---|
| F | F | T | F | T |
| F | T | T | F | T |
| T | F | F | F | T |
| T | T | F | F | T |
Hence (p ∧ ~ p) ⇒ q is a tautology.
(ii) (p ∧ q) ⇒ (p ∨ q)
Solution-
| p | q | p ∧ q | p v q | (p ∧ q) ⇒ (p v q) |
|---|---|---|---|---|
| F | F | F | F | T |
| F | T | F | T | T |
| T | F | F | T | T |
| T | T | T | T | T |
Hence (p ∧ q) ⇒ (p v q) is a tautology.
(iii) (p v q) ⟺ (~ p ∧ ~ q)
Solution-
| p | q | ~ p | ~ q | p v q | ~ p ∧ ~ q | |
| c1 | c2 | c1 ⟺ c2 | ||||
| F | F | T | T | F | T | F |
| F | T | T | F | T | F | F |
| T | F | F | T | T | F | F |
| T | T | F | F | T | F | F |
Hence (p v q) ⟺ (~ p ∧ ~ q) is not a tautology.
(iv) (p ⇒ q) ⇒ [~ q ⇒ (~ p ∧ ~ q)]
Solution-
| p | q | ~ p | ~ q | p ⇒ q | ~ p ∧ ~ q | ||
| c1 | c2 | ~ q ⇒ c2 | |||||
| c3 | c1 ⟺ c3 | ||||||
| F | F | T | T | T | T | T | T |
| F | T | T | F | T | F | T | T |
| T | F | F | T | F | F | F | T |
| T | T | F | F | T | F | T | T |
Hence (p ⇒ q) ⇒ [~ q ⇒ (~ p ∧ ~ q)] is a tautology.
(v) [(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r)
Solution-
| p | q | r | p ⇒ q | q ⇒ r | p ⇒ r | ||
| c1 | c2 | c1 ∧ c2 | |||||
| c3 | c4 | c3 ⇒ c4 | |||||
| F | F | F | T | T | T | T | T |
| F | F | T | T | T | T | T | T |
| F | T | F | T | F | F | T | T |
| F | T | T | T | T | T | T | T |
| T | F | F | F | T | F | F | T |
| T | F | T | F | T | F | T | T |
| T | T | F | T | F | F | F | T |
| T | T | T | T | T | T | T | T |
Hence [(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r) is a tautology.
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