Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples - Aptitude Questions & Answers
Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples is one of the most important topics in Logical Reasoning Explained: Concepts, Tricks, and Practice Questions. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.
Syllogism: Complete Guide with Venn Diagrams & Tick-Cross Method
What is Syllogism?
Syllogism is a form of logical reasoning where conclusions are drawn from given statements (premises). It involves analyzing relationships between different categories or sets to determine what conclusions necessarily follow from the given information.
Example: From \"All dogs are mammals\" and \"All mammals are animals\", we can conclude \"All dogs are animals\".
Standard Answer Options
In syllogism problems, you typically encounter these 5 standard answer choices:
2. Only conclusion II follows
3. Either conclusion I or II follows
4. Neither conclusion I nor II follows
5. Both conclusions I and II follow
Venn Diagram Method
Basic Approach
The Venn diagram method involves drawing circles to represent sets and their relationships, then checking conclusions against all possible diagrams.
Step-by-Step Process:
- Draw circles for each category mentioned
- Represent the given statements in the diagram
- Consider all possible cases (minimum and maximum possibilities)
- Check each conclusion against all possible diagrams
- A conclusion follows only if it\'s true in ALL possible diagrams
Types of Statements and Their Representations
1. All A are B
A is completely inside B (possible: A = B, but shown as A ⊆ B)
2. No A are B
A and B are completely separate circles (no overlap)
3. Some A are B
A and B have some overlap (at least one element common)
4. Some A are not B
At least one A is outside B (shown with star in non-overlapping area)
Examples Using Venn Diagram Method
Example 1: Both Conclusions Follow
Statements: All laptops are wireless devices. No wireless device is a desktop computer.
Conclusions: I. Some laptops are not desktop computers. II. No desktop computer is a laptop.
Analysis:
- Laptops are completely inside wireless devices
- Wireless devices and desktop computers have no overlap
- Therefore, laptops and desktop computers have no overlap
Conclusion I: True - Since no laptop is desktop computer, some laptops are not desktop computers
Conclusion II: True - No desktop computer is a laptop
Example 2: Either-Or Case
Statements: Some cars are electric vehicles. Some electric vehicles are bikes.
Conclusions: I. No bike is a car. II. Some cars are bikes.
Case 1: Minimum Possibility
In Case 1: Cars and bikes don\'t overlap → Conclusion I true, Conclusion II false
Case 2: Maximum Possibility
In Case 2: Cars and bikes overlap → Conclusion I false, Conclusion II true
Analysis Table:
| Case | Conclusion I | Conclusion II |
|---|---|---|
| Case 1 | True | False |
| Case 2 | False | True |
Rule: When one conclusion is true in one case and false in another, and the other conclusion shows the opposite pattern, we have an \"either-or\" situation.
Tick and Cross Method (✔ and ✗)
Understanding the Notation
This is a quick method where we use ✔ for positive/affirmative statements and ✗ for negative statements.
Representation Rules:
| Statement | First Term | Second Term | Type |
|---|---|---|---|
| All A are B | ✔ | ✗ | Universal Positive |
| No A are B | ✔ | ✔ | Universal Negative |
| Some A are B | ✗ | ✗ | Particular Positive |
| Some A are not B | ✗ | ✔ | Particular Negative |
Single Statement Conclusions:
- All A are B: Some A are B, Some B are A
- Some A are B: Some B are A
- No A are B: Some A are not B, Some B are not A, No B are A
- Some A are not B: No conclusion
Rules of Tick and Cross Method
Rule 1: Common Term Requirement
Two statements should have 3 terms total, with one common term. The common term must have at least one ✔.
Example: \"All cats are mammals\" (cats: ✔, mammals: ✗) and \"All birds are animals\" (birds: ✔, animals: ✗)
No common term → No conclusion
Rule 2: Two Particular Statements
If both statements are particular (Some/Some not), no conclusion follows.
Example: \"Some cars are trucks\" (✗✗) and \"Some trucks are vans\" (✗✗)
Both particular → No conclusion
Rule 3: Two Negative Statements
If both statements are negative (No/Some not), no conclusion follows.
Example: \"No pens are books\" (✔✔) and \"No books are buckets\" (✔✔)
Both negative → No conclusion
Rule 4: Particular Conclusion Rule
If one statement is particular, the conclusion must be particular.
Example: \"All A are B\" (✔✗) and \"Some B are C\" (✗✗)
Conclusion must have \"Some\" → \"Some C are B\" follows
Rule 5: Tick-Cross Propagation
If a term has ✗ in statements, it must have ✗ in conclusion. If it has ✔, it can have ✔ or ✗ in conclusion.
Example: \"All trees are plants\" (trees: ✔, plants: ✗) and \"Some living things are trees\" (living things: ✗, trees: ✗)
Common term: trees (has ✔)
Other terms: plants (✗), living things (✗)
Valid conclusion: \"Some living things are plants\" (living things: ✗, plants: ✗)
Tick and Cross Examples
Example 1: Applying Rules
Statements: All trees are plants. Some living things are trees.
Conclusions: I. Some living things are not plants. II. Some plants are not living things. III. Some living things are plants.
Step 1: Assign Tick and Cross
Statement 1: All trees are plants → trees: ✔, plants: ✗
Statement 2: Some living things are trees → living things: ✗, trees: ✗
Step 2: Check Common Term
Common term: \"trees\" has at least one ✔ → Valid for conclusion
Other terms: \"plants\" (✗), \"living things\" (✗)
Step 3: Check Conclusions
Conclusion I: \"Some living things are not plants\" → living things: ✗, plants: ✔ (✗ can become ✔, but ✗ cannot become ✔ for plants)
Conclusion II: \"Some plants are not living things\" → plants: ✗, living things: ✔ (same issue)
Conclusion III: \"Some living things are plants\" → living things: ✗, plants: ✗ (both ✗ remain ✗) → Valid
Example 2: Multiple Statement Combination
Problem: Which combination yields conclusion \"No cats are pets\"?
A. DEC (D: No bird is pet, E: All cats are birds, C: No cats are pets)
Step 1: Assign Tick and Cross
D: No bird is pet → bird: ✔, pet: ✔
E: All cats are birds → cats: ✔, birds: ✗
Step 2: Check Common Term
Common term: \"birds\" has ✔ (from D) and ✗ (from E) → has at least one ✔
Other terms: \"cats\" (✔), \"pet\" (✔)
Step 3: Check Conclusion
C: No cats are pets → cats: ✔, pets: ✔
Since both terms have ✔ in statements, they can have ✔ in conclusion → Valid
Syllogism: Frequently Asked Questions
What is the difference between \"Some A are B\" and \"Some A are not B\"?
\"Some A are B\" means at least one A is B (minimum 1). \"Some A are not B\" means at least one A is not B, but it doesn\'t rule out that all A could be B or some A could be B.
When do we use \"either-or\" conclusion?
When two conclusions are complementary (one says \"all/some\", other says \"none\") and both cannot be true together, but either could be true depending on the case.
Which method is better: Venn diagram or Tick-Cross?
Venn diagram is more intuitive for beginners. Tick-Cross is faster for experienced solvers. Learn both to verify answers in exams.
Can \"All A are B\" and \"Some A are not B\" both be true?
No, they contradict each other. If all A are B, then no A is not B. These cannot be true together.
How to handle \"Many A are B\" statements?
Treat \"Many\" as \"Some\" in syllogism. It means at least some, possibly all.
Practice Problems
Problem 1
Statements: All roses are flowers. Some flowers fade quickly.
Conclusions: I. Some roses fade quickly. II. All flowers that fade quickly are roses.
Problem 2
Statements: No engineers are artists. All dancers are artists.
Conclusions: I. No engineers are dancers. II. Some artists are not engineers.
Problem 3
Statements: Some dogs are pets. All pets are loved.
Conclusions: I. Some dogs are loved. II. All loved are dogs.
Problem 4
Statements: Some phones are smart. No smart device is cheap.
Conclusions: I. Some phones are not cheap. II. No phone is cheap.
Problem 5
Statements: All students are learners. No learner is ignorant.
Conclusions: I. No student is ignorant. II. Some learners are students.
Quick Reference Guide
• All A are B → Some A are B, Some B are A
• No A are B → Some A are not B, Some B are not A, No B are A
• Some A are B → Some B are A
• Some A are not B → No conclusion
• Common term must have at least one ✔
• Two particulars → No conclusion
• Two negatives → No conclusion
• One particular → Conclusion must be particular
• ✗ in statement → ✗ in conclusion
• ✔ in statement → ✔ or ✗ in conclusion
• Conclusions are complementary
• One true in some cases, other true in other cases
• Both not true together
• Answer: Either conclusion I or II follows
Frequently Asked Questions
What is Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples?
Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.
Is Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples important for competitive exams?
Yes, Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.
How to prepare Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples easily?
Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples.
What are the important formulas in Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples?
Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples.
How many questions come from Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples?
Typically 5-10 questions come from Syllogism in Logical Reasoning: Concepts, Rules, and Solved Examples in most competitive exams, making it a high-scoring section.
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