Syllogisms & Statements

Snapshot

• Syllogisms give you two or more statements ("All A are B", "Some B are C") and ask which conclusions necessarily follow. The trap is everyday intuition — the exam only rewards what must be true, not what is likely.
• The reliable method is Venn diagrams: draw the statements, then test each conclusion against every way the diagram could be drawn. A conclusion follows only if it holds in all cases.
• This guide gives you the three statement types, the Venn method, the rules of thumb, and the "possibility / either-or" cases — with worked examples.
• Exam reality: +5 / −1. Draw the circles; never reason from real-world knowledge.

Part 1 — The three statement types

Every syllogism statement is one of these, and each has a fixed Venn picture:

• All A are B — circle A sits entirely inside circle B.
• Some A are B — circles A and B overlap (at least one common element).
• No A are B — circles A and B are completely separate.
• Some A are not B — there is a part of A outside B.

Part 2 — The Venn method

1. Draw the statements exactly as given (use the pictures above).
2. Try to draw the diagram in a way that makes the conclusion FALSE. If you can, the conclusion does not follow. If every possible drawing keeps it true, it follows.
3. A conclusion that is only sometimes true does not follow — "possibly" is not "necessarily".

Part 3 — Quick rules of thumb

• Two "Some" statements never give a definite conclusion. "Some A are B + Some B are C" proves nothing about A and C.
• Two negative statements give no definite conclusion.
• "All + All" can chain: All A are B + All B are C ⇒ All A are C.
• "All + No": All A are B + No B are C ⇒ No A are C.
• A definite conclusion can never be more certain than its weakest statement — a "Some" anywhere caps the conclusion at "Some".

Part 4 — Possibility & "Either-Or" cases

When neither of two conclusions follows on its own but together they cover all cases, the answer is "either I or II follows". This happens classically when the two conclusions are a direct pair like "Some A are B" and "No A are B" drawn from the same uncertain statements — one of them must be true, but you cannot say which, so it is either-or.

Part 5 — Speed techniques

1. Always draw — never reason verbally for syllogisms.
2. Test the conclusion by trying to break it (draw it false). If you can't, it follows.
3. Spot "two Some" or "two No" instantly — usually "no conclusion".
4. Watch for Either-Or when two conclusions are complementary and neither alone is certain.
5. Ignore real-world truth — "All dogs are cats" must be treated as true if the statement says so.

Part 6 — Worked examples

1. All pens are books. All books are red. Conclusion: All pens are red? Chain of two "All" ⇒ follows.

2. Some cats are dogs. All dogs are animals. Conclusion: Some cats are animals? The cats that are dogs must be animals ⇒ follows.

3. Some A are B. Some B are C. Conclusion: Some A are C? Two "Some" ⇒ does not follow.

4. All roses are flowers. No flower is red. Conclusion: No rose is red? All + No ⇒ follows.

5. Some books are pens. Conclusion I: Some pens are books (converse of "Some" is always valid) ⇒ follows.

6. No A is B. Conclusions: I. Some A are not B II. Some B are not A. Both immediate inferences of "No" ⇒ both follow.

7. All A are B. Conclusion: Some B are A? "All A are B" guarantees at least those A's are B ⇒ Some B are A follows.

8. Some A are B. Conclusions: I. All A are B II. Some A are not B. Neither is certain alone, but together they exhaust the possibilities ⇒ either I or II follows.

Part 7 — Common traps

• Reasoning from real life instead of the diagram.
• "Possibly true" ≠ "follows" — it must be certain in every diagram.
• Forgetting the valid converse of "Some" — Some A are B always gives Some B are A.
• Missing Either-Or when two complementary conclusions appear.
• Two "Some" or two "No" rarely yield a definite conclusion.

Part 8 — How to use this page

Lock the three Venn pictures and the "draw it false" test, re-solve the eight examples by sketching circles, then attempt the practice set and the timed test.

One-line revision: draw every statement as circles, a conclusion follows only if it is true in every possible diagram, two "Some" usually give nothing, and complementary conclusions that exhaust the cases give "either-or".