Syllogisms give you statements ('All A are B') and ask which conclusions necessarily follow. The trap is real-world knowledge — judge only by the logic, drawing Venn diagrams to test each conclusion.
Venn rules
All A are B — circle A sits entirely inside circle B. No A is B — the circles do not touch. Some A are B — the circles overlap; this is reversible (some B are A). A conclusion follows only if it is true in every valid diagram.
- All A are B — circle A sits entirely inside circle B.
- No A is B — the circles do not touch.
- Some A are B — the circles overlap; this is reversible (some B are A).
- A conclusion follows only if it is true in every valid diagram.
How to Approach It
- Translate each statement to a Venn picture — 'All' means one circle sits inside another, 'No' means the circles are separate, and 'Some' means they overlap. Draw exactly what is stated and nothing more.
- Try to break the conclusion — Look for a valid alternative diagram in which the conclusion would fail. If you can draw even one such picture, the conclusion does not follow.
- Apply the reversibility rules — 'Some A are B' always yields 'some B are A', but 'All A are B' never yields 'all B are A'. Use these facts without re-deriving them each time.
- Ignore real-world truth — A conclusion can be perfectly true in life yet still not follow from the statements given. Mark only what the logic actually forces.
Techniques & Methods
- Venn every statement — All = one circle inside another, No = separate, Some = overlap.
- Counter-diagram test — If even one valid diagram breaks a conclusion, it does not follow.
- 'Some' is reversible — 'Some A are B' always yields 'some B are A'.
- 'All' is one-way — 'All A are B' never gives 'all B are A'.
The Edge
"Some" statements are reversible: "some A are B" always gives "some B are A". "All" statements are not — "all A are B" does not give "all B are A". If even one valid diagram breaks the conclusion, it does not follow. Always try to draw a counter-diagram before accepting a conclusion.Worked example
Statements: All cats are animals. All animals are living things. Conclusion: All cats are living things.
- Draw it as nested circles: 'all cats are animals' puts the cat circle entirely inside the animal circle.
- 'All animals are living things' puts the animal circle entirely inside the living-things circle.
- So the cat circle sits inside animals, which sits inside living things — cats are wholly inside living things in every possible diagram.
- Therefore 'all cats are living things' must be true; the conclusion follows.
Answer: The conclusion follows
Worked example
Statements: All dogs are animals. Some animals are wild. Conclusion: Some dogs are wild.
- Place all dogs inside the animal circle, then mark an overlapping 'wild' region somewhere within animals.
- Nothing in the statements forces that wild region to touch the dog part of the circle.
- We can easily draw a valid picture in which every wild animal is a non-dog, so no dog is wild.
- Since a valid diagram breaks it, the conclusion does not follow.
Answer: The conclusion does NOT follow
Worked Drills
Worked example
All A are B. All B are C. Some C are D. Conclusions: (I) Some A are C. (II) Some A are D. (a) only I follows b) only II follows c) both follow d) neither follows)
- A is inside B, which is inside C, so all A are C — hence some A are C: I follows.
- The 'some C are D' region need not touch A.
- So II is not forced; only I follows.
Answer: only I follows — option a)
Worked example
Some books are pens. No pen is a pencil. Conclusions: (I) Some books are not pencils. (II) Some pens are books. (a) only I follows b) only II follows c) both follow d) neither follows)
- The book-pens are pens, and no pen is a pencil, so those books are not pencils: I follows.
- 'Some books are pens' reverses to 'some pens are books': II follows.
- So both follow.
Answer: both follow — option c)
Worked example
All cats are dogs. Some dogs are not cats. Conclusions: (I) Some dogs are cats. (II) All dogs are cats. (a) only I follows b) only II follows c) both follow d) neither follows)
- All cats are dogs, so the cats inside the dog circle are dogs that are cats: I follows.
- 'Some dogs are not cats' directly contradicts II.
- So only I follows.
Answer: only I follows — option a)
Worked example
No A is B. All B are C. Conclusions: (I) Some C are not A. (II) No A is C. (a) only I follows b) only II follows c) both follow d) neither follows)
- All B are C and no A is B, so the B's are C's that are not A — hence some C are not A: I follows.
- Nothing forbids A and C from overlapping elsewhere, so II is not forced.
- So only I follows.
Answer: only I follows — option a)
Worked example
All flowers are red. Some red things are big. Conclusions: (I) Some flowers are big. (II) Some big things are flowers. (a) only I follows b) only II follows c) both follow d) neither follows)
- The big-red things need not include any flowers.
- And flowers need not be big.
- Neither conclusion is forced.
Answer: neither follows — option d)
Worked example
Some A are B. All B are C. All C are D. Conclusions: (I) Some A are D. (II) Some A are C. (a) only I follows b) only II follows c) both follow d) neither follows)
- The A's that are B lie inside B, which is inside C, so some A are C: II follows.
- C is inside D, so those A's are also inside D: I follows.
- So both follow.
Answer: both follow — option c)
Worked example
No cat is a dog; all dogs are pets. 'No cat is a pet': (a) follows b) does not follow c) both d) cannot say)
- No cat is a dog, but cats may still be pets through another route.
- A valid diagram lets cats overlap the pet circle without touching dogs.
- So the conclusion does not follow.
Answer: does not follow — option b)
Worked example
Some A are B. Conclusion 'Some B are A': (a) follows b) does not follow c) both d) cannot say)
- 'Some' statements are reversible.
- 'Some A are B' always yields 'some B are A'.
- So the conclusion follows.
Answer: follows — option a)
⚠ Watch out
- Real-world truth is irrelevant — judge only what the statements force.
- "All A are B" never implies "all B are A".
- "Some" leaves room for the rest to be anything; don't over-read it.
Takeaways
- Draw exactly what each statement says — nothing more.
- A conclusion follows only if it holds in every valid diagram.
- 'Some' reverses freely; 'All' is strictly one-way.
- Try to build a counter-diagram before accepting any conclusion.