Tautologies and contradictions

Tautologies and contradictions

Tautologies and contradictions are special types of logical formulas that are always true or always false, regardless of the truth values of their components.

Tautologies

A tautology is a statement that evaluates to true under all possible interpretations. Tautologies are useful in proofs and as logical identities.

Examples

• p ∨ ¬p (Law of excluded middle)
• (p → q) ∨ (q → p)
• (p ∨ q) → (q ∨ p)

Contradictions

A contradiction is a statement that evaluates to false under all possible interpretations. Contradictions represent impossible situations.

Examples

• p ∧ ¬p (Law of non-contradiction)
• (p ∧ q) ∧ ¬(p ∧ q)
• (p → q) ∧ (p → ¬q) ∧ p

Mixed Example

Some formulas are neither tautologies nor contradictions but depend on the truth values of their components. For instance, p ∧ q can be true or false depending on p and q.

Truth Table Illustration

p	¬p	p ∨ ¬p (tautology)	p ∧ ¬p (contradiction)
T	F	T	F
F	T	T	F