12.3.3 Tautology, Contradiction, and Contingency

. . Tautology, Contradiction, and Contingency Definition . A statement is said to be a tautology if its truth value is always T irrespective of the truth values of its component statements.

It is denoted by 𝕋 . Definition . A statement is said to be a contradiction if its truth value is always F irrespective of the truth values of its component statements. It is denoted by 𝔽 .

Definition . A statement which is neither a tautology nor a contradiction is called contingency Observations . For a tautology, all the entries in the column corresponding to the statement formula will contain T . .

For a contradiction, all the entries in the column corresponding to the statement formula will contain F . . The negation of a tautology is a contradiction and the negation of a contradiction is a tautology. .

The disjunction of a statement with its negation is a tautology and the conjunction of a statement with its negation is a contradiction. That is p ∨¬ is a tautology and p ∧¬ is a contradiction . This can be easily seen by constructing their truth tables as given below. Discrete Mathematics Example for tautology ¬p ∨∨¬¬ Table .

Since the last column of p ∨¬ contains only T, p ∨¬ is a tautology. Example for contradiction ¬ p p ∧∧ ¬ p Table . Since the last column contains only F , p ∧¬ is a contradiction. Note All the entries in the last column of Table .

are F and hence ) ∧ ∨¬ ) is a contradiction. Example for contingency ↔ ↔ ¬ q p → → ¬ q ¬ ( p ¬ q ) ↔ ↔ ∧∧ ¬ ( p ¬ q ) Table . In the above truth table, the entries in the last column are a combination of T and F . The given statement is neither a tautology nor a contradiction.

It is a contingency.