Key takeaways
- A tautology is true in every row of its truth table, a contradiction is false in every row, and a contingency is true in at least one row and false in at least one.
- Two statements are logically equivalent exactly when the biconditional between them is a tautology.
- An argument is valid exactly when the conditional from the conjunction of its premises to its conclusion is a tautology.
- To show a statement is not a tautology you need only one row that makes it false; to show it is one, you must rule out every row.
- For statements with many variables, a short proof using known equivalences, or a search for a falsifying row, is faster than a full table.
Every compound statement in propositional logic falls into exactly one of three classes, and the final column of its truth table tells you which.
| Class | Final column | Simplest example |
|---|---|---|
| Tautology | T in every row | p ∨ ¬p |
| Contradiction | F in every row | p ∧ ¬p |
| Contingency | At least one T and at least one F | p → q |
MathWorld defines a tautology as a sentence whose truth table contains only T, and a contingency as one whose table contains at least one T and at least one F. Applied Discrete Structures defines a contradiction as an expression that is false in all cases, and adds a convenient notation: it writes 1 for a tautology and 0 for a contradiction, which is how the same ideas look in Boolean algebra.
Classifying a statement, step by step
- Build the truth table, with a column for each connective.
- Look only at the final column.
- All T: tautology. All F: contradiction. Anything else: contingency.
Example 1: [(p → q) ∧ p] → q.
| p | q | p → q | (p → q) ∧ p | [(p → q) ∧ p] → q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | F | T |
| F | F | T | F | T |
Every row is T, so this is a tautology. It is the statement form of modus ponens: if p implies q and p holds, then q holds. That connection is not an accident, and it is explained below.
Example 2: (p ∨ q) ∧ (¬p ∧ ¬q).
| p | q | p ∨ q | ¬p ∧ ¬q | (p ∨ q) ∧ (¬p ∧ ¬q) |
|---|---|---|---|---|
| T | T | T | F | F |
| T | F | T | F | F |
| F | T | T | F | F |
| F | F | F | T | F |
Every row is F: a contradiction. It claims that at least one of p and q is true and that both are false. By De Morgan's laws, ¬p ∧ ¬q is the negation of p ∨ q, so the statement has the form r ∧ ¬r. Applied Discrete Structures gives the same statement, grouped slightly differently, as one of its examples of a contradiction.
Example 3: (p → q) → (q → p).
| p | q | p → q | q → p | (p → q) → (q → p) |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | T |
| F | T | T | F | F |
| F | F | T | T | T |
Row 3 is F and the others are T, so this is a contingency. In words: "if a conditional holds, its converse holds" is not a law of logic. Row 3 is the counterexample, where q is true and p is false.
Why tautologies matter: equivalence and validity
Tautologies are not just a classification exercise. Two of the central ideas in logic are defined through them.
Equivalence. Statements r and s are logically equivalent when r ↔ s is a tautology; that is the definition Applied Discrete Structures uses. Saying the biconditional is always true is the same as saying r and s always have the same truth value. The guide to proving logical equivalence uses this directly.
Implication and valid arguments. The same text says r implies s when r → s is a tautology. So an argument with premises P₁, …, Pₙ and conclusion C is valid exactly when (P₁ ∧ … ∧ Pₙ) → C is a tautology. Example 1 is modus ponens written this way. Every rule in the rules of inference corresponds to a tautology of this shape, which is why those rules can be trusted.
Faster than a full table
A table with five variables has 32 rows. Two shortcuts save time.
Hunt for a falsifying row. To test whether a conditional is a tautology, ask what would make it false: the hypothesis true and the conclusion false. Then see whether that is possible.
Take (p ∧ q) → (p ∨ r). For it to be false, p ∧ q must be true, so p is true. But then p ∨ r is true, so the conclusion cannot be false. No falsifying row exists: tautology. That took two lines instead of eight rows.
Take (p ∨ q) → (p ∧ q). Make the hypothesis true with p true and q false; the conclusion is false. One falsifying row is enough: not a tautology, and since the row p = q = T makes it true, it is a contingency.
Simplify with known laws. p ∨ (q ∨ ¬p) rearranges, by the commutative and associative laws, to (p ∨ ¬p) ∨ q. Since p ∨ ¬p is always true, the whole statement is a tautology without any table at all.
Facts worth remembering
- The negation of a tautology is a contradiction, and vice versa. The negation of a contingency is a contingency.
- A contradiction implies any statement, and any statement implies a tautology: 0 → p and p → 1 are both tautologies. This is why a set of premises that contradict each other "proves" anything. It is also why a proof by contradiction can stop as soon as it reaches one: if assuming ¬P leads to a statement that is always false, ¬P cannot be true.
- All tautologies are equivalent to one another, a point Applied Discrete Structures makes explicitly, and likewise all contradictions.
Classify these statements
Classify these, then check your answers with a table: p → (q → p); (p → q) ∧ (p ∧ ¬q); (p ↔ q) ∨ (p ↔ ¬q). (Answers: tautology, contradiction, tautology.) Logic puzzles such as knights and knaves are another way to practise, because every solution is a search for the one consistent row.
The truth-table generator in Discrete Math: Study & Quiz builds a truth table for you, which makes checking these quick once you have tried them by hand. Find more in the discrete math hub and among our education and brain-training apps.
Good to know
Frequently asked questions
What is a tautology in logic?
What is the difference between a contradiction and a contingency?
How do you show something is not a tautology?
Is the negation of a tautology a contradiction?
Sources
- Tautology (opens in a new tab)Wolfram MathWorld — accessed
- Contingency (opens in a new tab)Wolfram MathWorld — accessed
- Equivalence and Implication (Applied Discrete Structures) (opens in a new tab)discretemath.org — accessed
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