Translate every premise into notation, draw the smallest diagram you can, then check whether the conclusion is forced.
Guide page2 of 7
Syllogisms give you two or three premises about categories, then ask whether a conclusion must follow. The golden rule: never add anything that is not in the premises. Only draw what the arrows force.
1 — The four statement types
All A are B
Write this
A ⇒ B
double arrow = all of A goes into B
A sits completely inside B. Every member of A is also a member of B.
✗ Not reversibleAll B are A is not implied — B can contain members outside A.
No A are B
Write this
A ✕ B
✕ = totally separate, no overlap
A and B are completely separate. No member of A is in B.
✓ ReversibleNo B are A is equally true — exclusion is symmetric.
Some A are B
Write this
A → B
→ = some of A links to B
At least one A is also a B. The circles partially overlap.
✓ ReversibleSome B are A is equally true — if one item is in both sets, that works both ways.
Some A are not B
Write this
A →✕ B
→✕ = some A are cut off from B
At least one A (the dot) falls outside B. Part of A extends beyond the B circle.
✗ Not reversibleTells us nothing from B's perspective — never reverse this.
Also valid
A = B
When can you write A = B?
Only when you know both "All A are B" and "All B are A" — meaning the two sets are completely identical. A = B means every A is a B and every B is an A, so you can swap them freely in any direction.
A ⇒ B alone is not enough — that only tells you A goes into B, not that B goes into A. Use = only when a question explicitly tells you the sets are identical, which is rare.
2 — Chaining premises
Put premises end-to-end. If you can trace an unbroken path from start to finish, the conclusion follows. If the chain breaks or arrows don't connect, it does not follow.
All + All → All✓ FOLLOWS
• All nurses are healthcare workers.
• All healthcare workers have DBS checks.
Nurses⇒Healthcare⇒DBS checks
∴ All nurses have DBS checks ✓
All + No → No✓ FOLLOWS
• All cats are mammals.
• No mammals are fish.
Cats⇒Mammals✕Fish
∴ No cats are fish ✓
Some + All → Some✓ FOLLOWS
• Some doctors are researchers.
• All researchers have PhDs.
Doctors→Researchers⇒PhDs
∴ Some doctors have PhDs ✓
Reversed All — converse error✗ INVALID
• All accountants are graduates.
• Sara is a graduate.
Accountants⇒Graduates← Sara is here, but...
✗ Sara does not have to be inside Accountants. A ⇒ B ≠ B ⇒ A.
3 — Worked example
EXAMPLEThree-premise chain — 5 conclusions to judge
Premises
P1Reptiles ⇒ Cold-bloodedAll reptiles are cold-blooded
P2Cold-blooded ✕ Warm-bloodedNo cold-blooded animals are warm-blooded
P3Lizards → ReptilesSome lizards are reptiles
The chain
Lizards→Reptiles⇒Cold-blooded✕Warm-blooded
How to read the chain
⇒
All — one direction only
Reptiles ⇒ Cold-blooded: you can travel right from Reptiles and always reach Cold-blooded. You cannot go left — not every cold-blooded thing is a reptile.
✕
None — blocks both ways
Cold-blooded ✕ Warm-blooded: the cross blocks both directions. No cold-blooded thing can be warm-blooded, and no warm-blooded thing can be cold-blooded.
→
Some — partial bridge
Lizards → Reptiles: only some lizards cross over. Those that do inherit everything on the right side of the chain — but you cannot say ALL lizards do.
Judge each conclusion
Some lizards are cold-blooded.Yes
P3 says Lizards → Reptiles: some lizards cross into the Reptiles group. P1 says Reptiles ⇒ Cold-blooded: all reptiles are cold-blooded. Those lizards that are reptiles must therefore be cold-blooded. Yes — some lizards are cold-blooded.
Some lizards are not warm-blooded.Yes
From the first conclusion, some lizards are cold-blooded. P2 says Cold-blooded ✕ Warm-blooded — the cross blocks both ways, so cold-blooded things cannot be warm-blooded at all. Those lizards that are cold-blooded are therefore definitely not warm-blooded.
All reptiles are warm-blooded.No
Follow the chain: Reptiles ⇒ Cold-blooded, then Cold-blooded ✕ Warm-blooded. The chain carries reptiles into the cold-blooded group and then hits a full block. Reptiles cannot reach warm-blooded — the conclusion is the opposite of what the premises show.
All cold-blooded animals are reptiles.No
P1 says Reptiles ⇒ Cold-blooded: the arrow goes right from Reptiles. You cannot reverse it. Cold-blooded ⇒ Reptiles is a converse error. There could be cold-blooded animals (fish, frogs) that are not reptiles — P1 says nothing about them.
No lizards are warm-blooded.Does not follow
P3 says SOME lizards are reptiles — the → is partial. The lizards that are reptiles cannot be warm-blooded (via the chain), but the remaining lizards that are not reptiles are not bound by the chain at all. We have no information about them, so we cannot say NO lizards are warm-blooded.
4 — Rules to remember
A ⇒ B ≠ B ⇒ A
The converse is not valid. All A are B does not mean All B are A. This is the most common mistake in UCAT syllogisms.
A ✕ B = B ✕ A
Exclusion works both ways. No A are B also means No B are A — you can always reverse a 'no' statement.
A → B = B → A
Some A are B also means Some B are A. Overlap works both ways.
A →✕ B ≠ B →✕ A
Some A are not B tells you nothing about B's perspective. Never reverse a 'some are not' statement.
Some means at least one
'Some' does not mean 'not all'. It means one or more, and could theoretically include all.
Existence is never guaranteed
All X are Y does not prove any X exist. Universal statements work on what is given to exist in the question.
One counterexample defeats must
If you can draw a valid arrangement where the conclusion is false, it does not must follow. Draw it and move on.
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