Syllogism
🔒 Log in to trackOnly-a-few and definite-case rulings
🔒 Log in to track'Only a few A are B' means two things at once: some A are B and some A are not B. 'A few' and 'Many' mean plain 'Some'.
Split the statement first. Then treat it like any other syllogism.
What only a few really says
Modern papers use three phrases.
- Some A are B: at least one A is a B. Nothing more.
- A few and Many A are B: exactly the same as Some.
- Only a few A are B: two facts at once. Some A are B, and some A are not B.
So 'Only a few cups are plates' guarantees a cup outside plates. It says nothing about plates that are not cups.
Rule: Split 'only a few' into two arrows. Feed both into your picture.
What follows at once
From Only a few A are B:
- Some A are B follows.
- Some A are not B follows.
- All A are B is false.
- No A are B is false.
- Anything about B outside A is undecided.
Combining with All
Take 'Only a few A are B' and 'All B are C'.
- Some A are C follows. The A that is a B is also a C.
- Some A are not C does not follow. The A outside B may still be a C.
Watch: Only the A-and-B member travels through the All. The A-outside-B member does not.
Combining with No
Take 'Only a few A are B' and 'No B are C'. Then some A are not C follows, because the A that is a B cannot be a C.
But 'All A are C' is impossible in this case. It would put that A into C.
Two only-a-few statements
'Only a few A are B' and 'Only a few B are C' do not join. The A-and-B members and the B-and-C members can be different members.
Three phrases side by side
| Phrase | Some A are B | Some A are not B |
|---|---|---|
| Some A are B | Yes | Not fixed |
| A few / Many A are B | Yes | Not fixed |
| Only a few A are B | Yes | Yes |
Only the last phrase guarantees the outside part. That is the whole difference between them. Do not read 'A few' or 'Many' as 'only a few'. They promise the overlap and nothing else.
Checking with a picture
Draw the outside A member as a dot beside the B circle. Then ask whether the dot has to sit inside C. If it can sit either way, the conclusion is not forced.
Watch which side each conclusion is about. 'Some A are not B' talks about the A side. 'Some B are not A' talks about the B side.
'Only a few A are B' fixes the first and leaves the second open.
A worked example
Statements
- Only a few cups are plates.
- All plates are bowls.
Conclusions
- I. Some cups are bowls.
- II. Some cups are not bowls.
Options
- (A) Neither I nor II follows
- (B) Only I follows
- (C) Only II follows
- (D) Both I and II follow
Solution
- The cup that is a plate is also a bowl. So I follows.
- The cup that is not a plate may still be a bowl. So II fails.
Answer: (B) Only I follows.
Question types you will see
Each type: how to recognise it, the method step by step, and one question to try.
Only-a-few direct consequences
'Only a few A are B', and conclusions quoting Some, Some-not, All or No on A and B.
Rewrite as Some A are B and Some A are not B.
All A are B and No A are B are false.
Anything about B outside A is undecided.
Only-a-few welds two facts about the A side and says nothing about B.
Statements
- Only a few pens are pencils.
Conclusions
- I. Some pens are not pencils.
- II. Some pencils are not pens.
Options
- (A) Only II follows
- (B) Both I and II follow
- (C) Neither I nor II follows
- (D) Only I follows
Show solutionHide solution
Split the statement: some pens are pencils, and some pens are not pencils.
I is the second half, so it follows.
II is about pencils outside pens. The statement says nothing about that side. II fails.
(D) Only I follows
Only-a-few + All chain
'Only a few A are B' and 'All B are C'. Conclusions are about A and C.
The A-and-B member is also a C, so 'Some A are C' follows.
The A outside B is not tied to C.
So 'Some A are not C' does not follow.
Only the B member travels through the All.
Statements
- Only a few cups are plates.
- All plates are bowls.
Conclusions
- I. Some cups are bowls.
- II. Some cups are not bowls.
Options
- (A) Neither I nor II follows
- (B) Only I follows
- (C) Only II follows
- (D) Both I and II follow
Show solutionHide solution
The cup that is a plate is also a bowl. So I follows.
The cup that is not a plate may still be a bowl. So II fails.
(B) Only I follows
'A few / Many' = plain Some
The statement says 'A few A are B' or 'Many A are B'. Students over-read it as only-a-few.
Treat it exactly as 'Some A are B'.
No outside part is guaranteed.
Chain the Some member through any All.
'A few' and 'Many' only promise that the overlap exists.
Statements
- Some doctors are singers.
- All singers are dancers.
Conclusions
- I. Some doctors are dancers.
- II. Some doctors are not singers.
Options
- (A) Only I follows
- (B) Only II follows
- (C) Both I and II follow
- (D) Neither I nor II follows
Show solutionHide solution
Treat 'A few' like 'Some'. Some doctors are singers, and every singer is a dancer.
So some doctors are dancers. I follows.
II needs a doctor outside singers. 'A few' does not guarantee it, so II fails.
(A) Only I follows
Only-a-few with possibility and No statements
An only-a-few statement, a No statement and a mix of definite and possibility conclusions.
Split only-a-few into Some and Some-not.
Trace the guaranteed member through the No statement.
Test each conclusion by its own standard.
The No statement can cut off the guaranteed member from a third group.
Statements
- Only a few birds are pets.
- No pets are stones.
Conclusions
- I. Some birds are not stones.
- II. All birds being stones is a possibility.
Options
- (A) Both I and II follow
- (B) Neither I nor II follows
- (C) Only I follows
- (D) Only II follows
Show solutionHide solution
Some birds are pets. A pet is never a stone. So some birds are not stones. I follows.
II asks for all birds to be stones. The bird that is a pet would have to be a stone. II fails.
(C) Only I follows
Two only-a-few statements
'Only a few A are B' and 'Only a few B are C', with conclusions about A and C.
Split both statements.
The A-and-B members and B-and-C members may differ.
Draw the two overlaps apart.
Neither Some nor Some-not is forced.
Each only-a-few links one pair. Two links do not join.
Statements
- Only a few trees are plants.
- Only a few plants are shrubs.
Conclusions
- I. Some trees are shrubs.
- II. Some trees are not shrubs.
Options
- (A) Only II follows
- (B) Both I and II follow
- (C) Neither I nor II follows
- (D) Only I follows
Show solutionHide solution
Each 'only a few' gives some and some-not. The two links do not join up.
The trees that are plants may all be shrubs, or none of them may be. Neither claim is forced.
(C) Neither I nor II follows
Shortcuts that save time
Rewrite every 'Only a few A are B' as two lines: Some A are B, and Some A are not B. Then solve as normal.
Mistakes to avoid
Where most students lose marks on this subtopic.
Reading 'A few' or 'Many' as 'only a few'.
They mean plain 'Some'. No outside part is guaranteed.
Assuming 'only a few A are B' says something about B outside A.
It only fixes the A side. B outside A is undecided.
Passing the outside-A member through 'All B are C'.
Only the A-and-B member travels. The outside member may or may not be a C.
Thinking 'only a few' allows 'All A are B'.
It guarantees an A outside B, so 'All A are B' is false.
Joining two only-a-few statements.
The overlaps can be different members. Nothing joins.
Quick revision
Read this the night before the exam.
Only a few A are B = Some A are B + Some A are not B.
A few and Many = plain Some. Nothing is guaranteed outside.
Only the A-and-B member travels through All B are C.
Only a few A are B makes 'All A are B' and 'No A are B' false.
Nothing is fixed about B outside A.
Practice: 12 questions
Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.
Topic test · 12 questions
Suggested time 11 min · wrong answers go to your mistake notebook automatically.