🧠 Syllogism - Complete Theory

Master logical deduction - the most scoring reasoning topic!

🎯 What is Syllogism?

Syllogism is logical reasoning where you draw conclusions from given statements (premises).

Structure:

Statement 1: All A are B
Statement 2: All B are C
Conclusion: All A are C (Valid!)

Key Point: You must assume statements as TRUE (even if absurd in real life) and then check if conclusions logically follow.

📐 Basic Elements

1. Statements (Premises)

The given facts you must accept as true.

2. Conclusions

Logical inferences you need to verify.

3. Types of Statements

A - Universal Affirmative (All)

All A are B
Example: All dogs are animals

E - Universal Negative (No)

No A are B
Example: No cats are dogs

I - Particular Affirmative (Some)

Some A are B
Example: Some students are girls

O - Particular Negative (Some…not)

Some A are not B
Example: Some birds are not crows

🎨 Venn Diagram Method

Why Venn Diagrams?

• Visual representation makes logic clear
• 100% accuracy in drawing conclusions
• Works for all question types

Rule 1: All A are B

Diagram:

   ┌─────────────┐
   │      B      │
   │   ┌─────┐   │
   │   │  A  │   │
   │   └─────┘   │
   └─────────────┘

Meaning: A is completely inside B

Example: All cats are animals

• Every cat is an animal
• Circle of “cats” inside circle of “animals”

Rule 2: No A are B

Diagram:

   ┌─────┐     ┌─────┐
   │  A  │     │  B  │
   └─────┘     └─────┘

Meaning: A and B are completely separate

Example: No men are women

• Circles don’t touch

Rule 3: Some A are B

Diagram:

   ┌─────┐
   │  A ╱│╲ B  │
   │   ╱ │ ╲   │
   └──────┴───┘

Meaning: At least one A is B (intersection exists)

Example: Some students are girls

• Circles overlap
• Intersection = students who are girls

IMPORTANT: “Some” means:

• At least one
• Could be all (but we don’t know for sure)
• Minimum = 1

Rule 4: Some A are not B

Diagram:

   ┌─────┐
   │  A ╱│╲ B  │
   │   ╱ │ ╲   │
   └──────┴───┘

Meaning: At least one A is outside B

IMPORTANT: Same diagram as “Some A are B” because:

• If some are B, then some are not B (could be)
• We show both possibilities

🔗 Combining Two Statements

Pattern 1: All A are B + All B are C

Diagrams:

Statement: All A are B

   ┌─────────────┐
   │      B      │
   │   ┌─────┐   │
   │   │  A  │   │
   │   └─────┘   │
   └─────────────┘

Statement: All B are C

   ┌───────────────┐
   │       C       │
   │  ┌─────────┐  │
   │  │    B    │  │
   │  │ ┌─────┐ │  │
   │  │ │  A  │ │  │
   │  │ └─────┘ │  │
   │  └─────────┘  │
   └───────────────┘

Valid Conclusions: ✅ All A are C ✅ Some A are C ✅ Some C are A

Pattern 2: All A are B + No B are C

Combined Diagram:

   ┌─────────────┐           ┌─────┐
   │      B      │           │  C  │
   │   ┌─────┐   │           └─────┘
   │   │  A  │   │
   │   └─────┘   │
   └─────────────┘

Valid Conclusions: ✅ No A are C ✅ No C are A ✅ Some A are not C

Pattern 3: All A are B + Some B are C

Combined Diagram:

   ┌───────────────┐
   │       B       │
   │  ┌─────┐  ╱─╲ │
   │  │  A  │ ╱ C ╲│
   │  └─────┘╱─────╲
   └───────────────┘

Valid Conclusions: ✅ Some A are C (Possible, but NOT definite) ✅ Some C are B (Already given)

IMPORTANT: Can’t conclude “Some A are C” for sure!

Pattern 4: No A are B + All B are C

Combined Diagram:

              ┌───────────┐
              │     C     │
   ┌─────┐    │  ┌─────┐  │
   │  A  │    │  │  B  │  │
   └─────┘    │  └─────┘  │
              └───────────┘

Valid Conclusions: ✅ Some C are not A (Definite, because B is part of C and no B are A)

Pattern 5: Some A are B + All B are C

Combined Diagram:

        ┌───────────────┐
        │       C       │
   ╱─╲  │  ┌─────────┐  │
  ╱ A ╲─┼──│    B    │  │
  ╲───╱ │  └─────────┘  │
        └───────────────┘

Valid Conclusions: ✅ Some A are C (Definite) ✅ Some C are A (Definite)

Pattern 6: Some A are B + No B are C

Combined Diagram:

   ╱─╲──┬─────┐           ┌─────┐
  ╱ A ╲ │  B  │           │  C  │
  ╲───╱ └─────┘           └─────┘

Valid Conclusions: ✅ Some A are not C (Definite)

💡 Solved Examples

Example 1: Basic Pattern

Statements:

I. All books are notebooks
II. All notebooks are papers

Conclusions:

I. All books are papers
II. Some papers are books

Solution:

Draw Venn diagrams:

Books ⊂ Notebooks ⊂ Papers

Conclusion I: All books are papers → TRUE ✓
(Books completely inside Papers)

Conclusion II: Some papers are books → TRUE ✓
(At least some portion of papers contains books)

Answer: Both I and II follow

Example 2: Negative Statement

Statements:

I. All cats are dogs
II. No dogs are rats

Conclusions:

I. No cats are rats
II. Some cats are rats

Solution:

Cats ⊂ Dogs, Dogs ∩ Rats = ∅

Since all cats are dogs, and no dogs are rats:
→ No cats can be rats

Conclusion I: No cats are rats → TRUE ✓
Conclusion II: Some cats are rats → FALSE ✗

Answer: Only conclusion I follows

Example 3: Some Statement

Statements:

I. Some apples are oranges
II. All oranges are fruits

Conclusions:

I. Some apples are fruits
II. Some fruits are apples

Solution:

Apples ∩ Oranges ≠ ∅, Oranges ⊂ Fruits

Since some apples are oranges, and all oranges are fruits:
→ Those apples (which are oranges) must be fruits

Conclusion I: Some apples are fruits → TRUE ✓
Conclusion II: Some fruits are apples → TRUE ✓

Answer: Both I and II follow

Example 4: Tricky Case

Statements:

I. Some pens are pencils
II. Some pencils are erasers

Conclusions:

I. Some pens are erasers
II. No pen is eraser

Solution:

Pens ∩ Pencils ≠ ∅, Pencils ∩ Erasers ≠ ∅

But we don't know relation between Pens and Erasers!

Possible cases:
- Some pens might be erasers
- No pen might be eraser

Conclusion I: Some pens are erasers → Can't say ✗
Conclusion II: No pen is eraser → Can't say ✗

Answer: Neither I nor II follows

Example 5: Complementary Pair

Statements:

I. All mangoes are fruits
II. All fruits are sweet

Conclusions:

I. All mangoes are sweet
II. Some sweet are mangoes

Solution:

Mangoes ⊂ Fruits ⊂ Sweet

Conclusion I: All mangoes are sweet → TRUE ✓
(Mangoes completely inside Sweet)

Conclusion II: Some sweet are mangoes → TRUE ✓
(At least the mango portion is sweet)

Answer: Both I and II follow

Example 6: Either-Or Case

Statements:

I. All phones are gadgets
II. No gadget is a toy

Conclusions:

I. Some phones are toys
II. No phone is a toy

Solution:

Phones ⊂ Gadgets, Gadgets ∩ Toys = ∅

Since all phones are gadgets, and no gadget is toy:
→ No phone can be a toy

Conclusion I: Some phones are toys → FALSE ✗
Conclusion II: No phone is a toy → TRUE ✓

Answer: Only II follows

⚡ Quick Rules & Shortcuts

Rule 1: Two Particular Statements = No Conclusion

Some A are B + Some B are C = NO definite conclusion about A and C

Rule 2: I + A = I Type Conclusion

Some A are B + All B are C = Some A are C ✓

Rule 3: A + A = A Type Conclusion

All A are B + All B are C = All A are C ✓

Rule 4: E + A = E + O Type Conclusions

No A are B + All B are C = No A are C ✓ + Some C are not A ✓

Rule 5: Complementary Pair

If "All A are B" is true, then:
- "Some A are B" is also true ✓
- "No A are B" is false ✗
- "Some A are not B" is false ✗

Rule 6: Conversion

All A are B → Some B are A (Always true)
No A are B → No B are A (Always true)
Some A are B → Some B are A (Always true)

📊 Standard Question Patterns

Pattern 1: Only One Follows

• Most common in IBPS
• Verify each conclusion separately
• Only one is logically valid

Pattern 2: Both Follow

• Both conclusions are valid
• Use Venn diagrams to verify

Pattern 3: Either I or II Follows

• Both can’t be true together
• Both can’t be false together
• They form complementary pair

Example of Complementary Pair:

I. All A are B
II. Some A are not B

Only one can be true (they contradict each other)

Pattern 4: Neither Follows

• Both conclusions are invalid
• No logical connection

⚠️ Common Mistakes

❌ Mistake 1: Real-World Logic

Wrong: "All books are notebooks" is false in reality, so I reject it ✗
Right: Accept ALL statements as TRUE for the question ✓

❌ Mistake 2: Possibility = Definite

Wrong: "Some A are B" could mean "All A are B", so conclusion follows ✗
Right: Only use what's DEFINITE from the diagram ✓

❌ Mistake 3: Reverse Logic

Wrong: "All A are B" means "All B are A" ✗
Right: "All A are B" only converts to "Some B are A" ✓

❌ Mistake 4: Ignoring “Some”

Wrong: "Some A are B" means exactly half ✗
Right: "Some" means at least one, could be all ✓

❌ Mistake 5: Two Particulars

Wrong: Some A are B + Some B are C = Some A are C ✗
Right: No definite conclusion possible ✓

🎯 Decision Table

🔄 Either-Or Cases

When Either-Or Applies:

Conclusions must be:

1. Complementary pairs (opposite meanings)
2. Both individually false when checked separately
3. At least one must be true logically

Example:

Statements:
I. Some dogs are cats
II. Some cats are rats

Conclusions:
I. All dogs are rats
II. No dog is a rat

Check individually:
- Conclusion I alone: FALSE ✗
- Conclusion II alone: FALSE ✗

But they form complementary pair (All vs No)
So: Either I or II follows ✓

💡 More Solved Examples

Example 7: Complex Three Terms

Statements:

I. All squares are rectangles
II. All rectangles are polygons
III. No polygon is a circle

Conclusions:

I. No square is a circle
II. Some polygons are squares

Solution:

Squares ⊂ Rectangles ⊂ Polygons, Polygons ∩ Circles = ∅

Conclusion I: No square is a circle → TRUE ✓
(All squares are polygons, no polygon is circle)

Conclusion II: Some polygons are squares → TRUE ✓
(At least the square portion of polygons)

Answer: Both I and II follow

Example 8: Negative Chain

Statements:

I. No teacher is a student
II. All students are learners

Conclusions:

I. No teacher is a learner
II. Some learners are not teachers

Solution:

Teachers ∩ Students = ∅, Students ⊂ Learners

Conclusion I: No teacher is a learner → FALSE ✗
(We don't know about teachers and learners)

Conclusion II: Some learners are not teachers → TRUE ✓
(At least students are learners who are not teachers)

Answer: Only II follows

Example 9: Double Some

Statements:

I. Some doctors are engineers
II. Some engineers are artists

Conclusions:

I. Some doctors are artists
II. All artists are doctors

Solution:

Doctors ∩ Engineers ≠ ∅, Engineers ∩ Artists ≠ ∅

No connection between Doctors and Artists

Conclusion I: Some doctors are artists → Can't say ✗
Conclusion II: All artists are doctors → Can't say ✗

Answer: Neither I nor II follows

Example 10: Possibility Question

Statements:

I. All laptops are computers
II. Some computers are tablets

Which conclusions are POSSIBLE?

I. Some laptops are tablets
II. No laptop is a tablet
III. All tablets are laptops

Solution:

For possibility questions, check if conclusion CAN be true (not must be true)

I. Some laptops are tablets → POSSIBLE ✓ (not contradicted)
II. No laptop is a tablet → POSSIBLE ✓ (also not contradicted)
III. All tablets are laptops → POSSIBLE ✓ (could be true)

Answer: All three are possible

📝 Practice Problems

Level 1 (Easy):

1. Statements:

All roses are flowers
All flowers are plants

Conclusions:

I. All roses are plants
II. Some plants are roses

2. Statements:

No car is a bike
All bikes are vehicles

Conclusions:

I. No car is a vehicle
II. Some vehicles are not cars

Level 2 (Medium):

3. Statements:

Some Indians are engineers
All engineers are professionals

Conclusions:

I. Some Indians are professionals
II. All professionals are Indians

4. Statements:

All birds can fly
Some flying objects are kites

Conclusions:

I. Some birds are kites
II. Some kites can fly

Level 3 (Hard):

5. Statements:

Some actors are singers
No singer is a dancer

Conclusions:

I. Some actors are not dancers
II. No dancer is an actor

6. Statements:

All cups are glasses
Some glasses are plates
No plate is a spoon

Conclusions:

I. Some cups are not spoons
II. No glass is a spoon

🎯 Exam Strategy

Time Allocation:

• Per question: 30-40 seconds
• For 5 syllogism questions: 2.5-3 minutes total

Quick Approach:

1. Read statements (5 sec)
2. Draw Venn diagram (10 sec)
3. Check conclusions (15 sec)
4. Mark answer (5 sec)

Priority:

• ✅ Direct All/No combinations (easiest, 20 sec)
• ✅ Some with All/No (moderate, 30 sec)
• ⏭️ Double Some statements (tricky, 45+ sec - attempt last)

🔗 Related Topics

Uses Concepts From:

• Logic and reasoning fundamentals
• Set theory (Venn diagrams)

Related Reasoning Topics:

• Data Sufficiency - Logical analysis
• Coded Inequalities - Similar logical chains

Practice:

• Syllogism Questions

Master Syllogism - Draw diagrams, don’t assume! 🧠