Aptitude: What are the better ways of solving syllogisms?

Complete Guide to Solving Syllogisms

What is a Syllogism?

A syllogism has:

• Two premises (statements assumed to be true)
• One conclusion (what follows from the premises)
• Three terms (each appears in exactly two statements)

Example:

• Premise 1: All dogs are animals
• Premise 2: Some animals are pets
• Conclusion: Some dogs are pets (?)

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Method 1: VENN DIAGRAMS (Most Reliable)

Basic Rules:

1. Draw three overlapping circles for the three terms
2. For "All" statements: Shade out the region that's empty
3. For "No" statements: Shade out the overlapping region
4. For "Some" statements: Put an 'X' where at least one element exists
5. Always draw universal statements (All/No) before particular statements (Some)

Example 1: Valid Syllogism

Premises:

• All cats are mammals
• All mammals are animals

Conclusion: All cats are animals

Solution:

Step 1: Draw three circles (Cats, Mammals, Animals)

Step 2: "All cats are mammals" 
→ Put the entire Cats circle inside Mammals

Step 3: "All mammals are animals"
→ Put the entire Mammals circle inside Animals

Step 4: Check conclusion
→ Cats circle is now completely inside Animals
→ Conclusion VALID ✓

Example 2: Invalid Syllogism

Premises:

• All doctors are educated
• Some educated people are wealthy

Conclusion: Some doctors are wealthy

Solution:

Step 1: Draw circles (Doctors, Educated, Wealthy)

Step 2: "All doctors are educated"
→ Put Doctors entirely inside Educated

Step 3: "Some educated people are wealthy"
→ Put X in the overlap of Educated and Wealthy
→ But the X could be outside the Doctors circle!

Step 4: Check conclusion
→ We cannot prove some doctors MUST be wealthy
→ Conclusion INVALID ✗

Method 2: DISTRIBUTION METHOD

Understanding Distribution:

A term is distributed when the statement refers to ALL members of that category.

Statement Type
Subject
Predicate
All A are B
Distributed
Undistributed
No A are B
Distributed
Distributed
Some A are B
Undistributed
Undistributed
Some A are not B
Undistributed
Distributed

Rules for Valid Syllogisms:

1. Middle term rule: The middle term (appears in both premises but not conclusion) must be distributed at least once
2. Illicit process: If a term is distributed in the conclusion, it must be distributed in the premise
3. Negative premise rule: If one premise is negative, the conclusion must be negative
4. Two negatives rule: If both premises are negative, no valid conclusion
5. Two particulars rule: If both premises are particular (Some), no valid conclusion
6. Existential import: At least one premise must be particular if the conclusion is particular

Example 3: Using Distribution

Premises:

• All politicians are speakers (Middle term = speakers, distributed)
• Some speakers are liars

Conclusion: Some politicians are liars (?)

Analysis:

• Middle term "speakers": distributed in premise 1 ✓
• "Politicians" in conclusion (undistributed) vs premise 1 (distributed) ✓
• Both premises affirmative, conclusion affirmative ✓
• BUT: This doesn't guarantee overlap!

Using Venn to verify:

• Politicians inside Speakers
• X somewhere in Speakers∩Liars
• X might not be in Politicians region
• INVALID ✗

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Method 3: STANDARD FORM PATTERNS

Major Valid Forms (to memorize):

Form 1: Barbara (All-All-All)

• All M are P
• All S are M
• ∴ All S are P ✓

Example: All birds are animals. All sparrows are birds. ∴ All sparrows are animals.

Form 2: Celarent (No-All-No)

• No M are P
• All S are M
• ∴ No S are P ✓

Example: No reptiles are mammals. All snakes are reptiles. ∴ No snakes are mammals.

Form 3: Darii (All-Some-Some)

• All M are P
• Some S are M
• ∴ Some S are P ✓

Example: All engineers are graduates. Some women are engineers. ∴ Some women are graduates.

Form 4: Ferio (No-Some-Some not)

• No M are P
• Some S are M
• ∴ Some S are not P ✓

Example: No fish are mammals. Some sea creatures are fish. ∴ Some sea creatures are not mammals.

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Method 4: QUICK ELIMINATION TECHNIQUE

Instant "No Conclusion" Cases:

1. Two particular premises (both "Some")
   • Some A are B + Some B are C = No conclusion
2. Two negative premises (both "No" or "Some not")
   • No A are B + No B are C = No conclusion
3. Particular + Negative = Weak
   • Often leads to no definite conclusion

Example 4: Quick Elimination

Premises:

• Some cats are black
• Some black things are expensive

Quick check: Both particular → NO VALID CONCLUSION ✗

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COMPREHENSIVE WORKED EXAMPLES

Example 5: Complex Problem

Premises:

1. All mangoes are fruits
2. No fruit is a vegetable
3. Some vegetables are green

Question: Which conclusions follow?

• A) Some mangoes are green
• B) No mango is a vegetable
• C) Some green things are not fruits
• D) All of the above

Solution using Venn:

Step 1: Draw circles (Mangoes, Fruits, Vegetables, Green)

Step 2: "All mangoes are fruits"
→ Mangoes entirely inside Fruits

Step 3: "No fruit is a vegetable"
→ Fruits and Vegetables don't overlap

Step 4: "Some vegetables are green"
→ X in Vegetables∩Green overlap

Step 5: Test conclusions:
A) Some mangoes are green
   → Mangoes in Fruits, separated from Vegetables
   → X is in Vegetables, could be in Green
   → But Mangoes can't reach there
   → FALSE ✗

B) No mango is a vegetable
   → Mangoes inside Fruits
   → Fruits don't touch Vegetables
   → TRUE ✓

C) Some green things are not fruits
   → X in Vegetables∩Green
   → Vegetables separate from Fruits
   → So X (green vegetable) is not a fruit
   → TRUE ✓

Answer: B and C are valid

Example 6: Tricky Case

Premises:

• All dancers are artists
• Some artists are rich
• No rich person is sad

Conclusion: Some dancers are not sad (?)

Solution:

Venn Diagram:

- Dancers ⊂ Artists
- X somewhere in Artists∩Rich
- Rich and Sad are separate

Analysis:
- The X (rich artist) is definitely not sad ✓
- BUT: Is the X necessarily a dancer? NO!
- The X could be in the Artists region outside Dancers

Therefore: We cannot prove "Some dancers are not sad"
→ INVALID ✗

However: We CAN prove "Some artists are not sad" ✓

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STEP-BY-STEP STRATEGY FOR ANY SYLLOGISM

5-Step Process:

Step 1: Identify the three terms

• Find what appears twice in premises but not in conclusion (middle term)
• Identify the other two terms

Step 2: Quick elimination check

• Two particulars? → No conclusion
• Two negatives? → No conclusion

Step 3: Draw Venn diagram

• Three overlapping circles
• Process universal statements first (All/No)
• Then particular statements (Some)

Step 4: Verify the conclusion

• Does your diagram definitely show what the conclusion claims?
• Or are there alternative arrangements?

Step 5: Double-check with distribution rules

• Middle term distributed? ✓
• Conclusion terms properly distributed? ✓
• Negative/affirmative matching? ✓

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COMMON MISTAKES TO AVOID

1. Illicit conversion: "All A are B" does NOT mean "All B are A"
2. Assuming existence: "All unicorns are magical" doesn't prove unicorns exist
3. Drawing Some incorrectly: X must be placed carefully - don't assume it's in a specific overlap
4. Ignoring alternatives: Just because something CAN be true doesn't mean it MUST be true

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PRACTICE PROBLEM

Try this:

Premises:

• No teacher is lazy
• Some hard-workers are teachers
• All students admire hard-workers

Questions:

1. Are some hard-workers not lazy?
2. Do some students admire teachers?
3. Are all teachers hard-workers?

Solutions:

1. Some hard-workers are not lazy?
   • Some hard-workers are teachers (given)
   • No teacher is lazy (given)
   • So those hard-working teachers are not lazy
   • TRUE ✓
2. Some students admire teachers?
   • All students admire hard-workers
   • Some hard-workers are teachers
   • So students admire at least those hard-working teachers
   • TRUE ✓
3. All teachers are hard-workers?
   • We know some hard-workers are teachers
   • But this doesn't mean ALL teachers are hard-workers
   • CANNOT BE DETERMINED ✗

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BEST APPROACH FOR EXAMS

1. For accuracy: Use Venn diagrams
2. For speed: Learn to recognize common patterns + quick elimination
3. Practice: 30-50 problems to build intuition
4. Time management: Spend max 60-90 seconds per syllogism