Syllogism

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What a syllogism is and how to test it

⏱ 6 min read🧩 7 question types🎯 14 practice Q

The idea in one minute

A syllogism gives two or three statements that you must accept as true. Then you judge conclusions. A conclusion follows only if it holds in every picture that fits the statements.

Real-world facts do not matter. Try to break each conclusion with one picture.

What a syllogism asks

You get two or three statements. Accept them as true, even when they sound silly. Then you get conclusions to judge.

A conclusion follows only if it is true in every picture that fits the statements. One picture that breaks it is enough to reject it.

Rule: Assume every group named in the statements has at least one member.

The four statement forms

Statement	Meaning
All A are B	Every A is inside B. B may be bigger.
No A are B	A and B never meet.
Some A are B	At least one member is in both.
Some A are not B	At least one A lies outside B.

All and No describe the whole group. Some only promises that one member exists. That makes conclusions built on 'some' easy to break.

Safe reversals

Statement	Reversal that follows
All A are B	Some B are A
No A are B	No B are A
Some A are B	Some B are A
Some A are not B	none

Watch: 'All A are B' does not give 'All B are A'. 'Some A are not B' does not give 'Some B are not A'.

Chains that work

All A are B and All B are C give All A are C.
Some A are B and All B are C give Some A are C.
All A are B and No B are C give No A are C.
No A are B and Some B are C give Some C are not A.

Chains that do not work

Two 'Some' statements never join. Some A are B and Some B are C give nothing about A and C.
A 'Some-not' does not pass through an All. The A that is outside B may or may not be a C.

Try to break the conclusion

Do not draw a picture that helps the conclusion. Draw the smallest picture that fits the statements. Then ask whether you can move one group to make the conclusion false.

If you can, the conclusion fails. If you cannot, it follows.

Reading the options

Judge conclusion I and conclusion II separately. Then look at the options.

Only I follows: II can be broken.
Only II follows: I can be broken.
Both follow: neither can be broken.
Neither follows: both can be broken.

Never let one conclusion influence the other. Each one has its own test.

A worked example

Statements

Some pens are books.
All books are files.

Conclusions

I. Some pens are files.
II. All pens are files.

Options

(A) Only I follows
(B) Only II follows
(C) Both I and II follow
(D) Neither I nor II follows

Solution

The shared term is books. The pen that is a book is also a file, so I follows.
For II, draw one pen outside books. All statements still hold, so II fails.

Answer: (A) Only I follows.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

Some + All chain (shared member travels)

How to spot it:

'Some A are B' then 'All B are C' (or the other order). One conclusion links A and C.

Method

Find the term that both statements share.
The guaranteed A-and-B member is a B, so it is also a C.
So 'Some A are C' follows.
Test the other conclusion by drawing a counter-picture.

Why it works:

The member we are sure of travels through the All.

Try this

Statements

Some pens are books.
All books are files.

Conclusions

I. Some pens are files.
II. All pens are files.

Options

(A) Only I follows
(B) Only II follows
(C) Both I and II follow
(D) Neither I nor II follows

Show solution

The shared term is books. The pen that is a book is also a file, so I follows.
For II, draw one pen outside books. All statements still hold, so II fails.

Answer

(A) Only I follows

Type 2very common2 practice Q

All-chain with a conversion conclusion

How to spot it:

Two 'All' statements form a chain. One conclusion turns the chain around.

Method

Chain the All statements: A inside B inside C.
'All A are C' follows.
'Some C are A' also follows.
Reject 'All C are A'. The chain never runs backwards.

Why it works:

Containment carries forward, and the reverse needs new information.

Try this

Statements

All rings are coins.
All coins are metals.

Conclusions

I. Some metals are rings.
II. All metals are rings.

Options

(A) Only II follows
(B) Both I and II follow
(C) Neither I nor II follows
(D) Only I follows

Show solution

Chain the All statements: rings sit inside coins, and coins sit inside metals.
So all rings are metals, and the reverse form 'Some metals are rings' follows too.
II runs the chain backwards. A metal that is not a coin breaks it, so II fails.

Answer

(D) Only I follows

Type 3common2 practice Q

No + Some gives Some-not

How to spot it:

'No A are B' and 'Some B are C'. The conclusion is about C and A.

Method

Find the shared class B.
The B-and-C member cannot be an A.
So 'Some C are not A' follows.
Reject stronger claims such as 'No C are A'.

Why it works:

The shared member inherits B's total exile from A.

Try this

Statements

No walls are doors.
Some doors are gates.

Conclusions

I. Some gates are not walls.
II. No gates are walls.

Options

(A) Both I and II follow
(B) Neither I nor II follows
(C) Only I follows
(D) Only II follows

Show solution

The shared term is doors. A gate that is a door cannot be a wall, so I follows.
For II, a gate that is not a door might still be a wall. So II fails.

Answer

Type 4common2 practice Q

Conversion-only pair

How to spot it:

One statement and two conclusions. One is its plain reversal, and the other overreaches it.

Method

Recall the safe reversals: All gives Some. Some and No reverse both ways.
The reversal follows.
The other conclusion adds a new claim. Reject it.

Why it works:

A reversal only swaps the terms. It never adds new information.

Try this

Statements

Some tables are chairs.

Conclusions

I. Some chairs are tables.
II. All chairs are tables.

Options

(A) Neither I nor II follows
(B) Only I follows
(C) Only II follows
(D) Both I and II follow

Show solution

'Some' converts freely, so I follows.
II adds a new claim about every chair. The statement never says that.

Answer

(B) Only I follows

Type 5common

Three-statement chain

How to spot it:

Three statements link four groups in a line, mostly All and No.

Method

Chain the All statements from the left.
Carry the chain through the group named in the No statement.
Anything inside that group is also cut off from the last group.
Remember to reverse a No statement freely.

Why it works:

Everything inside a group inherits that group's No relation.

Try this

Statements

All cups are plates.
All plates are bowls.
No bowls are jars.

Conclusions

I. No cups are jars.
II. No jars are plates.

Options

(A) Only I follows
(B) Only II follows
(C) Both I and II follow
(D) Neither I nor II follows

Show solution

Chain the first two: cups sit inside plates, and plates sit inside bowls.
No bowl is a jar, so nothing inside bowls can be a jar. Cups and plates are inside bowls.
That gives both I and II, because 'No' reads the same in both directions.

Answer

Type 6common

Some-not does not travel

How to spot it:

'Some A are not B' with 'All B are C'. The conclusions are about A and C.

Method

Find the outside member: an A that is not a B.
Ask whether it must be a C. Nothing forces it in or out.
Draw it outside C, then draw it inside C.
Neither conclusion is forced.

Why it works:

The All statement covers B only. It says nothing about members outside B.

Try this

Statements

Some birds are not pets.
All pets are animals.

Conclusions

I. Some birds are not animals.
II. No birds are animals.

Options

(A) Only II follows
(B) Both I and II follow
(C) Neither I nor II follows
(D) Only I follows

Show solution

The bird outside pets is the only bird we are sure of.
That bird may or may not be an animal. Nothing links it to animals.
So I and II both fail. A 'Some-not' does not travel through an All.

Answer

Type 7common

Two Some statements

How to spot it:

'Some A are B' and 'Some B are C'. Conclusions link A and C.

Method

Note that the two overlaps may involve different members of B.
Draw them apart.
Test both conclusions in that picture.
Neither follows.

Why it works:

A 'Some' promises one member. Two promises can point to different members.

Try this

Statements

Some boxes are bags.
Some bags are sacks.

Conclusions

I. Some boxes are sacks.
II. Some sacks are boxes.

Options

(A) Both I and II follow
(B) Neither I nor II follows
(C) Only I follows
(D) Only II follows

Show solution

The box that is a bag and the bag that is a sack can be different bags.
Draw the two overlaps apart. Boxes and sacks never meet.
Two 'Some' statements never join. Neither conclusion follows.

Answer

(B) Neither I nor II follows

Formula sheet

Transitivity of All

A \subset B \wedge B \subset C \Rightarrow A \subset C

'All A are B + All B are C → All A are C' is the only freely chained rule.

Conversion of All

\text{All } A \subset B \Rightarrow \text{Some } B \subset A

Valid (classes are assumed non-empty): 'Some B are A' follows.

Conversion of No / Some

\text{No } A \cap B \Leftrightarrow \text{No } B \cap A;\quad \text{Some } A \cap B \Leftrightarrow \text{Some } B \cap A

Both convert symmetrically.

Shortcuts that save time

⚡ Find the shared term

The conclusion must join the two outer terms. Find the term that appears in both statements, and ask what travels through it.

Example

All pens are books. Some books are files. Does 'Some pens are files' follow?

Show solution

The shared term is books.
The files that are books may not be pens.
Draw the file-books outside pens.

Answer

No.

⚡ One counter-picture kills it

You do not need to prove a conclusion. You need only one picture that fits the statements and breaks it.

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Rejecting a statement because it is false in real life.

Accept every statement as true. Reason only from the statements.

Mistake 02

Reading 'All A are B' as 'All B are A'.

All does not reverse. Only 'Some B are A' follows.

Mistake 03

Joining two 'Some' statements into a conclusion.

Two 'Some' statements never join. Draw the overlaps apart to test.

Mistake 04

Passing a 'Some-not' through an All.

The A outside B may or may not be a C. Nothing follows.

Mistake 05

Drawing the picture that supports the conclusion.

Draw the smallest picture that fits the statements. Then try to break the conclusion.

Quick revision

Read this the night before the exam.

Accept the statements as true. A conclusion follows only if it holds in every picture.
All converts to Some. No and Some convert both ways. Some-not does not convert.
Chains that work: All+All, Some+All, All+No, No+Some (gives Some-not).
Two 'Some' statements never join. A 'Some-not' does not pass through an All.
Draw the smallest picture and try to break the conclusion with one counter-picture.
Assume every named group has at least one member.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 9 min · wrong answers go to your mistake notebook automatically.

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