Syllogism — Advanced Exam Problems

🎯 Learning Objective

Chain four statements into one diagram, mix definite and possibility conclusions, and catch the Either-Or pair under exam pressure.

💡 Concept

• Four statements = one picture: chain the All statements first (All+All=All), attach the No walls, and keep every Some flexible
• Definite conclusion → must survive EVERY diagram; possibility → needs just ONE valid diagram — read each conclusion's wording before choosing the test
• A forced No between two sets kills every 'they can overlap' possibility along that pair
• Either-Or checklist: neither conclusion follows alone + same subject and predicate + one positive and one negative (Some ↔ No, or All ↔ Some-not)
• Conversions still run the show: All A are B → Some B are A; No A is B → No B is A; Some-not never reverses

🧮 Key Formulas

All+All=All | Some+All=Some | All+No=No | Some+No=Some-not

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Definite → all diagrams; possibility → at least one diagram

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Either-Or = neither alone + complementary pair (Some ↔ No, All ↔ Some-not)

✏️ Easy Example

Q. Statements: All keys are locks. No lock is a chain. Conclusion: No key is a chain. Does it follow?

1. Draw keys fully inside locks (All keys are locks)
2. Draw chains completely separate from locks (No lock is a chain)
3. Keys live inside locks and locks never touch chains, so keys can never touch chains — All + No = No

Answer: Yes, it follows

🇮🇳 Real-Life Example

A railway recruitment notice: every ALP post needs ITI, and no ITI seat goes to under-18s — so no under-18 gets an ALP post. Policy documents chain rules exactly like 4-statement syllogisms.

📝 Exam-Level Example

Q. Statements: All files are folders. All folders are boxes. Some boxes are trunks. No trunk is a drawer. Conclusions: I. Some boxes are files. II. Some trunks are folders. III. No drawer is a trunk. Which follow?

1. Chain the two All statements through their shared middle terms: files sit inside folders, folders inside boxes — four statements shrink into one picture
2. Conclusion I: every file is a box, so those file-boxes prove 'Some boxes are files' — the valid reverse of an All → follows
3. Conclusion II: the trunk overlap sits somewhere in boxes, but nothing forces it to touch the folders region — redraw with trunks overlapping only the non-folder part of boxes and II breaks → does not follow
4. Conclusion III: 'No trunk is a drawer' converts both ways because No always reverses → 'No drawer is a trunk' follows
5. Verdict: I and III follow; II fails on the alternative diagram

Answer: Only conclusions I and III follow

📝 Exam-Level Example

Q. Statements: Some engines are coaches. All coaches are wagons. No wagon is a trolley. Conclusions: I. Some engines are not trolleys. II. All engines being wagons is a possibility. III. Some trolleys are coaches. Which follow?

1. Fix the definite parts first: coaches fully inside wagons, trolleys fully outside wagons; the engine circle overlaps coaches somewhere — the Some stays flexible
2. Conclusion I: the engine-coach overlap sits inside wagons, and wagons never touch trolleys — so at least those engines are NOT trolleys → definite, follows (a Some + All + No chain)
3. Conclusion II is a possibility, so ONE legal diagram is enough: slide the whole engine circle inside wagons — no statement objects → follows
4. Conclusion III: coaches live inside wagons and wagons never touch trolleys, so a trolley-coach overlap is impossible in EVERY diagram → fails
5. Verdict: I and II follow; III is blocked by the forced No

Answer: Only conclusions I and II follow

📝 Exam-Level Example

Q. Statements: All signals are lamps. Some lamps are gates. Conclusions: I. Some gates are signals. II. No gate is a signal. Which follow?

1. Draw signals inside lamps and let gates overlap lamps — the gate region MAY touch signals in one diagram and miss them in another
2. Conclusion I alone: not guaranteed, because gates can avoid signals → fails alone; conclusion II alone: not guaranteed either, because gates can touch signals → fails alone
3. Now inspect the pair: both talk about gates and signals — same subject, same predicate — one positive (Some) and one negative (No): a complementary pair
4. In any single diagram the two circles either touch or they do not — exactly one of I and II must be true, and no third case exists
5. Neither follows alone + complementary pair → the answer is Either-Or

Answer: Either conclusion I or conclusion II follows

🪄 Memory Trick

Timebox the diagram: twenty seconds for the forced picture (Alls chained, Nos walled). Then test definite conclusions against the WORST diagram you can draw, and possibilities against the FRIENDLIEST legal diagram.

⚠️ Common Mistakes

• ❌ Testing a possibility with the every-diagram rule, or a definite conclusion with the one-diagram rule
• ❌ Declaring Either-Or when the two conclusions have different subjects or predicates
• ❌ Letting real-world truth overwrite the given statements

🏆 Exam Tips

• ✅ Underline the middle term of every adjacent statement pair — a broken chain means no definite conclusion across it
• ✅ Check the Either-Or pair ONLY after both conclusions have individually failed

📌 Summary

• Chain the Alls, wall off the Nos, keep the Somes flexible
• Definite = all diagrams; possibility = one legal diagram
• A forced No kills the matching possibility
• Either-Or = both fail alone + complementary pair