Tautology or Contradiction? — Which statements are tautologies, and which are contradictions?

p ∨ ¬p and p → p are always true (tautology). p ∧ ¬p and p ∧ q ∧ ¬p are always false (contradiction).

Tautology and Contradiction

There are logical expressions that are true in every case, and others that are false in every case. These are called tautologies and contradictions, respectively.

Tautology

We call those logical statements tautologies that are true in every case, regardless of the truth values taken by the component statements.

This statement is always true: either p is true, or ¬p is true; there is no third possibility.

Contradiction

We call those logical statements contradictions that are false in every case.

This statement can never be true: p and ¬p cannot both be true at the same time.

Examples

"It will rain tomorrow or it won't rain." → tautology.
"The sun is shining and not shining at the same time." → contradiction.

Why are they important?

Tautologies are often the foundations of logical proofs: they are always valid. Contradictions are useful for immediately filtering out flawed arguments.

Summary

A tautology is always true, a contradiction is always false. These logical expressions help understand how inference and proof work.

Practice Exercise

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