Syllogism

Here is a detailed explanation of Syllogism as presented in your attached PDF, complete with step-by-step examples and practice questions to help you master this reasoning topic¹²³.

What is Syllogism?

Syllogism is a form of logical reasoning where you use two or more statements (premises) to reach a valid conclusion. In competitive exams, you are usually given two or three statements and a set of conclusions. You must decide which conclusions logically follow from the statements, treating the statements as 100% true even if they contradict common knowledge¹³.

Key Concepts and Rules

1. Types of Statements in Syllogism

• All A are B: Every A is included in B.
• Some A are B: At least one A is also B.
• No A is B: No overlap between A and B.
• Some A are not B: At least one A is not B.

2. Venn Diagram Approach

• All A are B:
  • Diagram: Circle A entirely inside Circle B.
• Some A are B:
  • Diagram: Circles A and B overlap.
• No A is B:
  • Diagram: Circles A and B do not touch.
• Some A are not B:
  • Diagram: Part of Circle A is outside Circle B.

3. Rules for Definite Conclusions

• A conclusion must be 100% true to be accepted.
• If a conclusion is false in any possible Venn diagram, it is considered false.
• Negative conclusions are only valid if there is a negative statement in the premises.
• If all statements are positive, negative conclusions are false.

4. Either-Or Case

• Formed when two conclusions are complementary and neither is definitely true.
• Complementary pairs:
  • Some + No
  • All + Some not
• Both conclusions must have the same subject and predicate (can interchange for “Some + No”).

5. Possibility Case

• If a conclusion is possible (not definitely true or false), it is only accepted if the question asks for possibility.
• Keywords: “can be”, “possibility”, “may be”

6. New Keywords

• All, Each, Every, Only, 100%: Use “All”.
• Some, Almost, Little, Few, At most, At least, Many, Mostly, 1-99%: Use “Some”.
• None, 0%: Use “No”.

Solved Examples

Example 1: Definite Conclusion

Statements:

• No panda is dog.
• Some panda is crow.
• All dogs are animal.

Conclusions: I. Some dog are crow. II. Some crow are panda. III. Some animal are not panda.

Solution:

• Conclusion I: False (no direct relation).
• Conclusion II: True (some panda are crow).
• Conclusion III: True (all dogs are animal and no dog is panda).

Answer: Only II and III follow¹.

Example 2: Either-Or Case

Statements:

• Some wicket are green.
• Some wicket are dry.
• No green is pitch.

Conclusions: I. Some dry are pitch. II. Some pitch are not green. III. No pitch is dry.

Solution:

• Conclusion I: False (no direct relation).
• Conclusion II: True (no green is pitch, so some pitch are not green).
• Conclusion III: False (no direct relation).

Either-Or: I and III are complementary (“Some + No”). Answer: Either I or III, and II follow¹.

Example 3: Possibility Case

Statements:

• Only a few doctors are teachers.
• No teacher is a lawyer.
• No lawyer is a scientist.

Conclusions: I. Some teachers can be scientist is a possibility. II. Some doctors are not lawyers.

Solution:

• Conclusion I: True (possibility).
• Conclusion II: True (some doctors are teachers and no teacher is lawyer).

Answer: Both I and II follow¹.

Example 4: New Keywords

Statements:

• Some tables are chairs.
• Only a few chairs are sofas.
• No sofa is a cupboard.

Conclusions: I. Some chairs are not sofa. II. All table can be cupboard is a possibility. III. No chair is a cupboard.

Solution:

• Conclusion I: True (“only a few” means some are, some are not).
• Conclusion II: True (possibility).
• Conclusion III: False (not definite).

Answer: Only I and II follow¹.

Example 5: Reverse Syllogism

Conclusions: I. No real is fake. II. All fake can be a man.

Statements:

1. All papers are fake. No fake is pen. All real is fake. All pens are man.
2. All papers are fake. No fake is pen. All real are pen. All pens are man.
3. Some papers are fake. No fake is man. All real are pen. All pens are man.
4. Some papers are fake. No fake is pen. All real is fake. All pens are man.
5. All papers are fake. No fake is man. All real are pen. All pens are man.

Solution:

• Statement 2:
  • Conclusion I: True (all real are pen, no fake is pen).
  • Conclusion II: True (possibility).

Answer: Both conclusions can be derived from statement set 2¹.

Example 6: Coded Syllogism

Statement: Ww $ Pp !Uu * Rr ~ Tt Meaning:

• “A$B” = All A are B
• “A!B” = All B are A
• “A*B” = Some A are B
• “A~B” = No A is B

Decoded: All Ww are Pp. All Uu are Pp. Some Uu are Rr. No Rr is Tt.

Conclusions: I. Some Ww are Tt. II. All Uu are Tt. III. Some Pp are Rr.

Solution:

• Conclusion I: False (no direct relation).
• Conclusion II: False (some Uu are Rr, no Rr is Tt).
• Conclusion III: True (some Uu are Rr, Uu are Pp).

Answer: Only III follows¹.

Practice Questions

Directions: In each question below are given three statements followed by two conclusions. You have to take the given statements to be true even if they seem to be at variance with commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements.

Practice Question 1

Statements:

• All liquids are solids.
• Some solids are gases.
• All gases are clouds.

Conclusions: I. Some clouds are solids. II. Some clouds are liquids. III. Some gases are liquids. IV. Some solids are clouds.

A. Only I and IV follow B. Only I and II follow C. Only III and IV follow D. None follows

Answer: A. Only I and IV follow⁴.

Practice Question 2

Statements:

• All K are H.
• No S is K.

Conclusions: I. No K is S. II. All H are K.

A. Only I follows B. Only II follows C. Both follow D. None follows

Answer: A. Only I follows².

Practice Question 3

Statements:

• All flowers are beautiful.
• Vaidehi is beautiful.

Conclusions: I. Vaidehi is a flower. II. Some beautiful are flowers.

A. Only I follows B. Only II follows C. Both follow D. None follows

Answer: B. Only II follows².

Practice Question 4

Statements:

• All Z are D.
• No Y is Z.

Conclusions: I. Some D are not Y. II. Some Z are not Y. III. Some D are Z.

A. All follow B. Only I and II follow C. Only II and III follow D. Only I and III follow

Answer: A. All follow².

Practice Question 5

Statements:

• Some schools are houses.
• Some colleges are schools.

Conclusions: I. Some colleges are houses. II. Some colleges are not houses.

A. Only I follows B. Only II follows C. Both follow D. None follows

Answer: D. None follows².

Summary Table

Tips for Solving Syllogism Questions

• Always assume the statements are 100% true.
• Draw Venn diagrams for clarity.
• Check for definite and possibility cases.
• Look for complementary pairs for either-or cases.
• Practice with different types of syllogism questions.

This comprehensive guide, with explanations, examples, and practice questions, will help you master syllogism reasoning for any competitive exam¹²³.