Syllogism

Statement-conclusion logical deductions

111 practice questionsPage updated 27 Mar 2026Review metadata pendingPractice questions

Syllogism is a form of logical reasoning where conclusions are drawn from two or more premises (statements). It is a fundamental topic in logical reasoning tests, requiring you to determine whether a given conclusion logically follows from the provided statements. Understanding the relationships between categorical statements is essential for solving these problems accurately.

Key concepts

Types of Categorical Statements

There are four standard types of categorical statements:

1. Universal Affirmative (A-type): All A are B
- Every member of set A is also a member of set B
- Example: All dogs are animals

2. Universal Negative (E-type): No A are B
- No member of set A is a member of set B
- Example: No fish are mammals

3. Particular Affirmative (I-type): Some A are B
- At least one member of set A is a member of set B
- Example: Some students are toppers

4. Particular Negative (O-type): Some A are not B
- At least one member of set A is not a member of set B
- Example: Some fruits are not sweet

Venn Diagram Representation

Venn diagrams are the most reliable method for solving syllogisms:

'All A are B' -> Circle A completely inside circle B
'No A are B' -> Circles A and B completely separate
'Some A are B' -> Circles A and B overlap partially
'Some A are not B' -> Part of circle A outside circle B

Always draw Venn diagrams based on the given premises, then check if the conclusion must be true in all possible valid diagrams.

Immediate Inferences (Conversion)

Conversion is drawing a valid conclusion by interchanging the subject and predicate:

'All A are B' -> 'Some B are A' (valid)
'No A are B' -> 'No B are A' (valid)
'Some A are B' -> 'Some B are A' (valid)
'Some A are not B' -> No valid conversion

Note: 'All A are B' cannot be converted to 'All B are A' - this would be the fallacy of illicit conversion.

Distribution of Terms

A term is 'distributed' when the statement makes an assertion about every member of that class:

'All A are B': A is distributed, B is undistributed
'No A are B': Both A and B are distributed
'Some A are B': Neither A nor B is distributed
'Some A are not B': A is undistributed, B is distributed

Rules for valid syllogisms:
1. The middle term must be distributed at least once
2. If a term is distributed in the conclusion, it must be distributed in the premise

Worked examples

Example 1

Statements:
1. All pens are books
2. All books are boxes

Conclusions:
I. All pens are boxes
II. Some boxes are pens

Solution

Step 1: Draw the Venn diagram based on statements.
- 'All pens are books' -> Circle P inside Circle B
- 'All books are boxes' -> Circle B inside Circle BX

Step 2: This creates: Circle P (pens) inside Circle B (books) inside Circle BX (boxes)

Step 3: Check Conclusion I: 'All pens are boxes'
Since all pens are books and all books are boxes, all pens must be boxes. CONCLUSION I FOLLOWS.

Step 4: Check Conclusion II: 'Some boxes are pens'
Since all pens are boxes (subset relationship), at least some boxes (those that are pens) are indeed pens. CONCLUSION II FOLLOWS.

Answer: Both I and II follow

Example 2

Statements:
1. Some cats are dogs
2. No dog is a cow

Conclusions:
I. No cat is a cow
II. Some cats are not cows

Solution

Step 1: Draw Venn diagram.
- 'Some cats are dogs' -> Circles C and D overlap
- 'No dog is a cow' -> Circle D and Circle CW are separate

Step 2: Check Conclusion I: 'No cat is a cow'
The overlapping region between cats and dogs has no connection to cows. However, cats that are NOT dogs could potentially be cows. So this is NOT DEFINITELY TRUE. CONCLUSION I DOES NOT FOLLOW.

Step 3: Check Conclusion II: 'Some cats are not cows'
The cats that are dogs (overlap region) are definitely not cows (since no dog is a cow). Therefore, some cats (at least those that are dogs) are not cows. CONCLUSION II FOLLOWS.

Answer: Only II follows

Example 3

Statements:
1. All mountains are hills
2. Some hills are rivers
3. All rivers are lakes

Conclusions:
I. Some mountains are lakes
II. Some hills are lakes
III. No mountain is a lake

Solution

Step 1: Draw comprehensive Venn diagram:
- M (mountains) inside H (hills)
- H overlaps with R (rivers)
- R inside L (lakes)

Step 2: Check Conclusion I: 'Some mountains are lakes'
Mountains are inside hills. Some hills are rivers (which are inside lakes). But the 'some hills that are rivers' may or may not include the mountain region. This creates two valid possibilities. CONCLUSION I DOES NOT DEFINITELY FOLLOW.

Step 3: Check Conclusion II: 'Some hills are lakes'
Some hills are rivers, and all rivers are lakes. Therefore, those hills that are rivers must also be lakes. CONCLUSION II FOLLOWS.

Step 4: Check Conclusion III: 'No mountain is a lake'
Same reasoning as I - we cannot definitively say this. The mountains could potentially overlap with the river region. CONCLUSION III DOES NOT FOLLOW.

Answer: Only II follows

Example 4

Statements:
1. No table is a chair
2. Some chairs are stools
3. All stools are desks

Conclusions:
I. Some desks are not tables
II. Some stools are not tables
III. All desks are tables

Solution

Step 1: Draw Venn diagram:
- T (tables) and C (chairs) are separate
- C overlaps with S (stools)
- S inside D (desks)

Step 2: Check Conclusion I: 'Some desks are not tables'
The stools that are chairs are inside desks. Since chairs are not tables, and these stools (which are chairs) are inside desks, some desks (at least those containing stool-chairs) are not tables. CONCLUSION I FOLLOWS.

Step 3: Check Conclusion II: 'Some stools are not tables'
Some stools are chairs, and no chair is a table. Therefore, those stools that are chairs are definitely not tables. CONCLUSION II FOLLOWS.

Step 4: Check Conclusion III: 'All desks are tables'
This contradicts Conclusion I which we've proven true. Also, desks contain stools which contain chairs which are not tables. CONCLUSION III DOES NOT FOLLOW.

Answer: Both I and II follow

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

Statements:
All cats are dogs.
All dogs are birds.

Conclusions:
I. All cats are birds.
II. Some birds are cats.

Which conclusion(s) follow?

Only I follows
Only II follows
Both I and II follow
Neither I nor II follows

Answer: A. Only I follows

Explanation

Cats are a subset of dogs, and dogs are a subset of birds, so all cats must be birds and conclusion I follows.

Sources and review notes

This is an AISEA-authored practice question.

Review status: verified · Reviewed 2026-08-13 · structure and answer-key checks, worked-answer review, human editorial review, duplicate screening

Question 2

Statements:
All roses are flowers.
Some flowers are red.

Conclusions:
I. Some roses are red.
II. All roses are red.

Which conclusion(s) follow?

Only I follows
Only II follows
Both I and II follow
Neither I nor II follows

Answer: D. Neither I nor II follows

Explanation

Step 1: Draw Venn diagram - Roses Flowers, Red intersect Flowers !=
Step 2: The red flowers may or may not include roses
Step 3: We cannot conclude that any rose is red
Step 4: Neither conclusion follows from the given statements
Answer: Neither I nor II follows

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

Statements:
No pen is pencil.
All pencils are erasers.

Conclusions:
I. No pen is eraser.
II. Some erasers are not pens.

Which conclusion(s) follow?

Only I follows
Only II follows
Both I and II follow
Neither I nor II follows

Answer: B. Only II follows

Explanation

Step 1: Draw Venn diagram - Pen intersect Pencil = , Pencil Eraser
Step 2: Since pencils are erasers and no pen is pencil -> Some erasers (those which are pencils) are not pens
Step 3: But erasers may include pens (not excluded) -> 'No pen is eraser' cannot be concluded
Answer: Only II follows

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Syllogism

Key concepts

Types of Categorical Statements

Venn Diagram Representation

Immediate Inferences (Conversion)

Distribution of Terms

Syllogism Rules and Fallacies

Three-Statement and Multi-Premise Syllogisms

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Key concepts

Types of Categorical Statements

Venn Diagram Representation

Immediate Inferences (Conversion)

Distribution of Terms

Syllogism Rules and Fallacies

Three-Statement and Multi-Premise Syllogisms

Worked examples

Example 1

Example 2

Example 3

Example 4

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.