You have built a truth table. The final column is full of T's and F's. A question then asks: "Is this expression a tautology, a contradiction, or a contingency?" The recognition rule is immediate:
- All T's in the final column → tautology.
- All F's → contradiction.
- Mixed (at least one T and at least one F) → contingent (also called contingency).
Three buckets, exhaustive and mutually exclusive. Every compound propositional expression falls into exactly one.
Why the three categories matter
- Tautology. The expression is true in every possible scenario. It carries no factual content about any specific situation — it is a logical law. Examples: p \lor \lnot p, (p \Rightarrow q) \lor (q \Rightarrow p), De Morgan written as a biconditional. Tautologies are the warrants behind proof techniques and the templates for algebraic rewrites. See logic-and-propositions-how-is-tautology-useful for why.
- Contradiction. The expression is false in every possible scenario. It cannot hold in any interpretation. Example: p \land \lnot p. Contradictions are the signal you look for inside a proof by contradiction — derive one, and your starting assumption must be wrong.
- Contingency. The expression sometimes holds, sometimes fails — depending on the truth values of the component propositions. Example: p \lor q, p \Rightarrow q, p \Leftrightarrow q. Contingent statements carry real information: their truth depends on the facts, and the facts can be asked about.
The recognition cue
The instant you see any of these in a JEE or board stem, reach for the classification rule:
- "Which of the following is a tautology?" — build truth tables, look for the all-T final column.
- "The statement X is a contradiction if and only if..." — same, but looking for all-F.
- "X is a contingency because..." — you must justify that the final column has both T and F, i.e. it is neither all-T nor all-F.
- "Classify: p \land (\lnot p \lor q)" — build the table, read the column, report the bucket.
The mechanical workflow
- Count variables. n variables → 2^n rows. For n \le 4, a hand-built table is always feasible.
- Build the table. Columns for each variable, then progressively for each sub-expression, ending with the full expression.
- Scan the final column.
- Every entry T → tautology.
- Every entry F → contradiction.
- Anywhere a T meets an F → contingent. Stop scanning.
- Report the bucket. Job done.
The third step is a single-pass check. If you find one F, it is not a tautology. If you find one T, it is not a contradiction. If you find both, it is contingent.
Walked examples
Example 1 — a tautology
Classify p \lor \lnot p.
| p | \lnot p | p \lor \lnot p |
|---|---|---|
| T | F | T |
| F | T | T |
Final column: T, T. All T's → tautology. (This is the law of excluded middle.)
Example 2 — a contradiction
Classify p \land \lnot p.
| p | \lnot p | p \land \lnot p |
|---|---|---|
| T | F | F |
| F | T | F |
Final column: F, F. All F's → contradiction.
Example 3 — a contingency
Classify p \Rightarrow q.
| p | q | p \Rightarrow q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Final column: T, F, T, T. Mixed — at least one T, at least one F. → contingent.
Example 4 — a trickier tautology
Classify (p \land q) \Rightarrow p.
| p | q | p \land q | (p \land q) \Rightarrow p |
|---|---|---|---|
| T | T | T | T |
| T | F | F | T |
| F | T | F | T |
| F | F | F | T |
Final column: T, T, T, T. All T's → tautology. (Formally: "conjunction elimination" — if both parts hold, either part must hold.)
Example 5 — a trickier contradiction
Classify p \Leftrightarrow \lnot p.
| p | \lnot p | p \Leftrightarrow \lnot p |
|---|---|---|
| T | F | F |
| F | T | F |
Final column: F, F. → contradiction. (A proposition cannot be equivalent to its own negation.)
The short-circuit test
For many expressions you do not need the entire table. The moment you see one F, the expression is no longer a candidate for tautology. The moment you see one T, it is no longer a candidate for contradiction. In an exam, you can stop as soon as you find both a T and an F — the verdict is contingent and no further rows matter.
This is especially useful with n = 4 or n = 5 variables where the table has 16 or 32 rows. As soon as one F appears, tautology is ruled out. As soon as one T appears, contradiction is ruled out. You only need to keep scanning until both have appeared or you have finished all rows without one of them appearing.
Two subtle traps
- Trusting intuition over the table. "It feels like a tautology" is not a valid classification. Build the table. A good candidate for a subtle JEE trap is (p \Rightarrow q) \lor (q \Rightarrow p), which looks contingent and is in fact a tautology. Only the truth table confirms it.
- Forgetting the final column is what matters. Intermediate columns might be all-T or all-F; that tells you about the sub-expression, not the whole thing. Classification is strictly about the column corresponding to the entire compound proposition.
The exam reflex
- See "tautology / contradiction / contingent" in the stem.
- Build the minimum truth table for the expression.
- Scan the final column.
- All T → tautology. All F → contradiction. Mixed → contingent.
- Mark the matching option.
No memory tricks, no pattern recognition beyond the scan. The truth table is mechanical, and the final-column readout is the answer.
Related: Logic and Propositions · Tautology, Contradiction, or Contingent? The Live Detector · Truth Table vs Equivalence Rules · If a Tautology Is Always True, How Can It Possibly Be Useful?