Syllogism
🔒 Log in to trackEither-or (complementary pairs)
🔒 Log in to trackWhen neither conclusion follows individually, check whether the two form a complementary pair — two statements that together cover every case and never fail simultaneously:
- (All A are B) + (Some A are not B)
- (Some A are B) + (No A is B)
Both pairs are exact negations of each other. If neither follows on its own but the statements cannot break both at once, the answer is 'Either I or II follows'. Note the trap: 'Some A are not B' pairs with 'All A are B' — NOT with 'No A is B'.
Detailed notes
What an either-or pair is
Sometimes neither conclusion follows on its own, but the two conclusions are exact opposites of each other. In that case one of them must be true in every possible picture — the exam answer is "Either I or II follows". This is not generosity; it is logic. If two statements are negations, no valid diagram can make both false.
The two legal complementary pairs
About the same two terms, in any order:
| Pair | Why they are negations |
|---|---|
| All A are B + Some A are not B | either every A is inside B, or at least one A is outside — nothing else can happen |
| Some A are B + No A is B | either the classes meet, or they never do — nothing else can happen |
The three-step test
- Check the pair: same two terms, and one of the two legal shapes above? If not, either-or is impossible.
- Test each conclusion separately with the counter-diagram method. If either one follows alone, answer "Only I/II follows" — never either-or.
- If neither follows alone, answer "Either I or II follows".
Step 2 before step 3 — the commonest error is jumping to either-or when one conclusion was actually provable.
Traps that break the pair
- Different terms. "All cups are bowls" + "Some cups are not plates" look like the (All, Some-not) shape but the second pair is cups–plates, not cups–bowls. Both can fail at once → neither follows.
- No + Some-not is NOT a pair. "No pencil is an eraser" and "Some pencils are not erasers" can both be false — when all pencils are erasers. They are not negations.
- Reversed terms are fine. "Some A are B" and "No B is A" are still negations (both links are symmetric), so the pair still works.
- One conclusion contradicts a statement. It then fails immediately, but the other still needs checking — the pair rule only applies when neither follows alone.
Worked examples
- Statements: Some flowers are red. No flower is blue. Conclusions: I. All flowers are red. II. Some flowers are not red. Same pair (flowers–red), legal (All, Some-not) shape, neither provable → either-or.
- Statements: All beds are sofas. Conclusions: I. Some sofas are beds. II. No sofa is a bed. I follows at once by conversion — so the answer is "Only I follows", NOT either-or. This is the classic trap.
Why the rule is safe
Take the (Some, No) pair about A–B. In any valid diagram either the A–B overlap is populated (I true) or empty (II true). Since every class is non-empty, there is no third option and no way to lose both. The same argument covers (All, Some-not): A is either fully inside B or it is not.
Quick revision
- Legal pairs only: (All, Some-not) and (Some, No) — same two terms.
- Order of work: check the shape → test each conclusion alone → only then mark either-or.
- If one follows alone, pick it; never mark either-or.
- No + Some-not is a fake pair; different-term pairs are fake.
- Reversed word order does not spoil the pair.
Types of questions asked
Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.
Type 1: (All, Some-not) complementary pairvery common2 practice Q
Conclusions 'All A are B' and 'Some A are not B' — same two terms, neither follows alone.
- Confirm the two terms match.
- Test each alone: neither can be proved when the statements leave A's link with B open.
- Mark 'Either I or II follows'.
Why it works: A is either wholly inside B or at least partly outside — no third option exists.
Example: Some candles are lamps. Conclusions: I. All candles are lamps. II. Some candles are not lamps.
Neither is forced by the single Some statement, and they are exact negations. Either I or II follows.
Type 2: (Some, No) complementary pairvery common2 practice Q
Conclusions 'Some A are B' and 'No A is B' about the same two terms.
- Confirm the pair and the terms.
- Check neither follows alone (the statements leave the A–B overlap undecided).
- Mark either-or.
Why it works: the overlap is either populated or empty — one of the two must hold.
Example: All keys are metal. Conclusions: I. Some keys are iron. II. No key is iron.
Nothing fixes keys against iron, so neither follows alone; the pair is a true negation pair. Either I or II follows.
Type 3: Fake complementary pair (trap)common3 practice Q
The conclusions LOOK like a legal pair but use different terms, or use (No, Some-not), or one of them actually follows alone.
- Check the terms word by word — both conclusions must use the same two classes.
- Reject the (No, Some-not) shape — it is not a negation pair.
- If one conclusion follows by itself, answer that one; never either-or.
Why it works: the either-or shortcut is valid only for exact negations about one pair of terms.
Example: Some pencils are erasers. Conclusions: I. No pencil is an eraser. II. Some pencils are not erasers.
I is contradicted by the statement. II is not forced. And they can both fail (all pencils are erasers) → Neither follows.
Type 4: Either-or with three statementsoccasional2 practice Q
A three-statement puzzle where the two conclusions ignore most of it and just form a negation pair.
- Ignore the extra statement while checking the pair shape — it only matters for step 2.
- Test each conclusion alone (the extra statement may occasionally prove one).
- If neither is provable, mark either-or.
Why it works: the pair argument needs only the two conclusion terms; extra statements cannot create a third option.
Example: All squares are rectangles. Some rectangles are big. No big thing is small. Conclusions: I. Some squares are small. II. No square is small.
Squares may or may not be big, so small links stay undecided; neither follows alone → Either I or II follows.
Shortcut tricks
⚡ Spot the negation pair
Scan the two conclusions for the word pairs (All … , Some … not) or (Some … , No …) about the SAME two terms. If found and neither follows individually, mark either-or — no diagram needed.
Example: Q. Conclusions: I. Some shirts are red. II. No shirt is red.
Sol. Same terms, Some/No pair, neither provable from 'Some shirts are blue'-type statements → either-or.
Complementary pair = exact logical negations about the same two terms.
Where students lose marks
Pairing 'No A is B' with 'Some A are not B' — they are not negations.
Marking either-or when one conclusion already follows on its own.
Practice sets — 12 questions
Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.
Topic test · 12 questions
Suggested time 10 min · wrong answers go to your mistake notebook automatically.