A tautology is a statement that is always true, regardless of the truth values of its components. A contradiction is always false. A contingency is sometimes true and sometimes false. Tautologies are the foundation of valid logical inferences.
Construct truth tables for various formulas and classify the results. Identify tautologies by recognizing patterns like p ∨ ¬p.
When you build a truth table for a compound statement, the rightmost column gives you a verdict for each possible combination of truth values. Three outcomes are possible: the column is all T's, all F's, or a mix. A tautology produces all T's — it is true no matter what. A contradiction produces all F's — it is false no matter what. A contingency produces a mix — its truth depends on the values of its components. These three categories are exhaustive and mutually exclusive.
The simplest tautology is P ∨ ¬P — "P or not P." This is true regardless of whether P is true or false, because exactly one of P and ¬P is always true. Similarly, the simplest contradiction is P ∧ ¬P — "P and not P" — which is always false because P and ¬P can never both be true simultaneously. These feel trivial, but they are the atomic units of a much larger system: any logical equivalence P ≡ Q can be verified by checking that P ↔ Q is a tautology, and any valid argument can be checked by confirming its logical form is a tautology.
Tautologies are the currency of logical inference. When a logician writes an inference rule — "from P and P → Q, conclude Q" — what they are saying is that (P ∧ (P → Q)) → Q is a tautology. The rule is valid precisely because the corresponding conditional is always true, regardless of what P and Q happen to be. This is why your next topics, rules of logical inference and proof by contradiction, build directly on tautologies: a proof step is valid exactly when the underlying logical form is a tautology.
Contradictions play an equally important role in proof by contradiction. That proof strategy works by assuming ¬P (the negation of what you want to prove) and then deriving a contradiction — a statement of the form Q ∧ ¬Q. Since a contradiction is always false, and you derived it from ¬P by valid steps, the assumption ¬P must be false. Therefore P is true. The power of contradiction-detection is that any statement of the form (something) ∧ ¬(something) immediately signals logical impossibility, giving you a route back to the original claim. Recognizing contradictions — especially disguised ones — is one of the key skills that separates routine calculation from genuine mathematical reasoning.