Syllogisms

Syllogisms are a crucial part of logical reasoning in competitive exams. Let’s tackle the three types of syllogism questions with solved examples. Each example includes step-by-step solutions, diagrams where applicable, and tips and tricks.

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Type 1: Based on Statements and Conclusions

Example 1

Statements:

1. All apples are fruits.

2. Some fruits are sweet.

Conclusions:

1. Some apples are sweet.

2. All fruits are apples.

Solution:

Step 1: Analyze the Statements

• "All apples are fruits" implies every apple belongs to the fruit category.

• "Some fruits are sweet" implies a part of the fruit category overlaps with the sweet category.

Step 2: Venn Diagram

We represent the categories using three circles.

Diagram:

Step 3: Test Conclusions

1. Some apples are sweet:

   • The diagram does not show any direct overlap between Apples (A) and Sweet (S).

   • Conclusion: False.

2. All fruits are apples:

   • The statement "All apples are fruits" does not imply the reverse (All fruits are apples).

   • Conclusion: False.

Answer: Both conclusions are false.

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Example 2

Statements:

1. No cat is a dog.

2. All dogs are animals.

Conclusions:

1. No cat is an animal.

2. Some animals are cats.

Solution:

Step 1: Analyze the Statements

• "No cat is a dog" means Cats and Dogs have no overlap.

• "All dogs are animals" means Dogs are a subset of Animals.

Step 2: Venn Diagram

Diagram:

Step 3: Test Conclusions

1. No cat is an animal:

   • The diagram shows Cats (C) and Animals (A) have no definitive overlap or exclusion.

   • Conclusion: False.

2. Some animals are cats:

   • There is no overlap shown between Cats and Animals.

   • Conclusion: False.

Answer: Both conclusions are false.

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Example 3

Statements:

1. All pencils are pens.

2. Some pens are erasers.

Conclusions:

1. Some pencils are erasers.

2. All pens are pencils.

Solution:

Step 1: Analyze the Statements

• "All pencils are pens" means Pencils are entirely within Pens.

• "Some pens are erasers" implies part of the Pens category overlaps with Erasers.

Step 2: Venn Diagram

Diagram:

Step 3: Test Conclusions

1. Some pencils are erasers:

   • There is no direct overlap between Pencils and Erasers.

   • Conclusion: False.

2. All pens are pencils:

   • "All pencils are pens" does not imply the reverse.

   • Conclusion: False.

Answer: Both conclusions are false.

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Type 2: "Only a Few" Type of Syllogisms

Key Tip:

"Only a few" implies some overlap and some exclusion between two categories.

Example 1

Statement: Only a few cats are dogs.
Conclusions:

1. Some cats are dogs.

2. All dogs are cats.

Solution:

• "Only a few cats are dogs" implies partial overlap.

Diagram:

Test Conclusions

1. Some cats are dogs:

   • True as per the statement.

   • Conclusion: True.

2. All dogs are cats:

   • The statement specifically limits overlap to "a few."

   • Conclusion: False.

Answer: Conclusion 1 is true, Conclusion 2 is false.

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Example 2

Statement: Only a few chairs are tables.
Conclusions:

1. All chairs are tables.

2. Some chairs are not tables.

Solution:

• "Only a few chairs are tables" implies some chairs overlap with tables, and some do not.

Diagram:

Test Conclusions

1. All chairs are tables:

   • The statement explicitly limits overlap to "a few."

   • Conclusion: False.

2. Some chairs are not tables:

   • The statement supports this as only "a few" chairs are tables.

   • Conclusion: True.

Answer: Conclusion 1 is false, Conclusion 2 is true.

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Example 3

Statement: Only a few flowers are plants.
Conclusions:

1. Some flowers are not plants.

2. Some plants are flowers.

Solution:

• "Only a few flowers are plants" implies partial overlap between flowers and plants.

Diagram:

Test Conclusions

1. Some flowers are not plants:

   • True because only "a few" overlap, implying the rest do not.

   • Conclusion: True.

2. Some plants are flowers:

   • True as "a few flowers" are plants, so this overlap exists.

   • Conclusion: True.

Answer: Both conclusions are true.

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Type 3: Possible Conclusions

Key Tip:

"Possible conclusions" test if a conclusion can logically occur based on the given statements. It does not need to always occur.

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Example 1

Statements:

1. All mangoes are fruits.

2. Some fruits are yellow.

Conclusions:

1. Some mangoes are yellow.

2. All yellow things are fruits.

Solution:

• Analyze possibilities based on the given statements.

  • "All mangoes are fruits" places mangoes fully within the fruits category.

  • "Some fruits are yellow" shows a partial overlap between fruits and yellow things.

Diagram:

Test Conclusions

1. Some mangoes are yellow:

   • There is no explicit overlap shown between Mangoes (M) and Yellow (Y).

   • Conclusion: False, but possible.

2. All yellow things are fruits:

   • The diagram does not support this as Yellow (Y) could extend beyond Fruits (F).

   • Conclusion: False.

Answer: Conclusion 1 is possibly true, Conclusion 2 is false.

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Example 2

Statements:

1. Some pens are pencils.

2. All pencils are stationary.

Conclusions:

1. Some pens are stationary.

2. Some stationary items are pens.

Solution:

• Analyze the relationships.

  • "Some pens are pencils" means partial overlap between Pens and Pencils.

  • "All pencils are stationary" places Pencils fully within Stationary.

Diagram:

Test Conclusions

1. Some pens are stationary:

   • Pens overlap with Pencils, and all Pencils are stationary.

   • Conclusion: True.

2. Some stationary items are pens:

   • True because some Pens (Pn) overlap with Stationary (S).

   • Conclusion: True.

Answer: Both conclusions are true.

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Example 3

Statements:

1. No birds are animals.

2. All animals are mammals.

Conclusions:

1. Some mammals are birds.

2. No birds are mammals.

Solution:

• Analyze the relationships.

  • "No birds are animals" excludes any overlap between Birds and Animals.

  • "All animals are mammals" places Animals fully within Mammals.

Diagram:

Test Conclusions

1. Some mammals are birds:

   • No overlap exists between Birds (B) and Mammals (M).

   • Conclusion: False.

2. No birds are mammals:

   • True as Birds (B) and Mammals (M) have no overlap.

   • Conclusion: True.

Answer: Conclusion 1 is false, Conclusion 2 is true.

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