∎ QED
Propositional Logic · step 2 of 13

Tautology, contradiction & contingency

Every propositional formula falls into exactly one of three classes. A tautology is true under every valuation (p ∨ ¬p), a contradiction is false under every valuation (p ∧ ¬p), and a contingency is true under some and false under others. The classification matters because a tautology is exactly a formula that can be proved from no assumptions, and a contradiction is exactly one whose negation is a tautology.

Practise tautology, contradiction & contingency free → All Propositional Logic subtopics

✓ Unlimited questions · marked criterion by criterion · no card needed

Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

  1. Try to falsify the formulaRather than building all 2ⁿ rows, assume the whole formula is false and push that requirement inward. If the demand is contradictory, you have proved it is a tautology in a few lines.
  2. For an implication, target the one false caseA → B is false only if A is true and B is false. Set those and see whether the variable assignments can be made consistent.
  3. Produce a witness for a contingencyOne valuation making it true and one making it false is a complete answer. Nothing more is needed, and nothing less will do.
  4. Fall back on the full table when stuckFor two or three variables the table is short and self-checking. The final column tells you the class immediately.

Worked example

Classify ((p → q) ∧ (q → r)) → (p → r).

  1. Suppose it is false. Then the antecedent (p → q) ∧ (q → r) is true and p → r is false.
  2. p → r false forces p = T and r = F.
  3. With p = T and p → q true, q = T.
  4. But then q → r is T → F, which is false — contradicting the assumption that the antecedent held.

Answer. No valuation falsifies it, so the formula is a tautology. (It is the transitivity of implication, also called hypothetical syllogism.)

Where marks get dropped

These are the specific errors that cost credit on tautology, contradiction & contingency questions — QED's rubric penalises each of them separately.

Practise this until it is automatic

Unlimited fresh questions

QED generates new tautology, contradiction & contingency problems on demand at warm-up, exam and challenge level, so you can drill this one skill until it stops costing you marks.

Marked like an examiner

Every answer is scored against a point-by-point rubric with partial credit, so you see exactly which step of the method broke down — not just a tick or a cross.

Answer in real notation

A one-tap symbol palette, a visual equation editor and a truth-table builder — or photograph your handwritten working and QED converts it to LaTeX.

Saved to your library

Every question you generate is kept and re-takeable as a timed exam, and your Propositional Logic mastery is tracked so you know when this is exam-ready.

Tautology, contradiction & contingency — frequently asked questions

What is the fastest way to prove a formula is a tautology?

Assume it is false and derive a contradiction. For implications this pins down several variables immediately, so you often finish in three or four lines instead of sixteen table rows.

Is ¬(tautology) always a contradiction?

Yes. If A is true under every valuation, ¬A is false under every valuation, and vice versa. This duality lets you convert any tautology question into a satisfiability question.

Can a formula be neither of the three?

No — the three classes are exhaustive and mutually exclusive. Every formula is either always true, always false, or somewhere in between.

The rest of Propositional Logic

Connectives, truth tables, equivalences and normal forms. Each subtopic below has its own method, worked example and mark-losing traps.

Ready to make tautology, contradiction & contingency exam-proof?

Generate your first questions free — no card, no setup, no personal data stored. Practise until the method is second nature.

Start practising free →