🧠 Syllogism - Complete Theory

Master logical deduction - the most scoring reasoning topic!

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🎯 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.

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📐 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

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🎨 Venn Diagram Method

Why Venn Diagrams?

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

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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”

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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

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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

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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

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🔗 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

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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

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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!

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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)

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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)

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

Combined Diagram:

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

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

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💡 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

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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

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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

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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

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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

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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

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⚡ 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)

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📊 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

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⚠️ 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 ✓

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🎯 Decision Table

Statement 1	Statement 2	Possible Conclusions
All A are B	All B are C	All A are C ✓
All A are B	No B are C	No A are C ✓
All A are B	Some B are C	No definite conclusion
No A are B	All B are C	Some C are not A ✓
No A are B	No B are C	No definite conclusion
Some A are B	All B are C	Some A are C ✓
Some A are B	No B are C	Some A are not C ✓
Some A are B	Some B are C	No definite conclusion

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🔄 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 ✓

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💡 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

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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

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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

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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

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📝 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

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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

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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

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🎯 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)

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🔗 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

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