Three statements are given, followed by four conclusions numbered I, II, III and IV. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) Statements: 1. Some students are boys. 2. All boys are honest. 3. Some honest are toppers. Conclusions: I. Some toppers are boys. II. Some honest are boys. III. Some honest are students. IV. Some toppers are students.
Only conclusions II and III follow
This question is a classic example of a syllogism problem, which falls under logical reasoning. We are given three statements and four conclusions. Our task is to assume the statements are true and determine which of the conclusions logically follow from these statements.
Understanding the Syllogism Statements
Let's break down the given statements:
- Statement 1: Some students are boys. This means there is an overlap between the group of students and the group of boys.
- Statement 2: All boys are honest. This is a universal positive statement. The entire group of boys is included within the group of honest people.
- Statement 3: Some honest are toppers. This indicates an overlap between the group of honest people and the group of toppers.
Analyzing Conclusions with Syllogism Rules or Venn Diagrams
We can use Venn diagrams or the rules of syllogism to evaluate each conclusion.
Let's represent the groups:
- S = Students
- B = Boys
- H = Honest
- T = Toppers
Statements in symbolic form:
- 1. Some S are B (⋅⋅ there is at least one element common to S and B)
- 2. All B are H (⋅⋅ every element of B is in H)
- 3. Some H are T (⋅⋅ there is at least one element common to H and T)
Now let's examine each conclusion:
Conclusion I: Some toppers are boys.
This links T and B. From Statement 2, we know all B are H. From Statement 3, we know some H are T. The honest people who are toppers might or might not be the honest people who are boys. There is no direct logical connection established between boys and toppers through these statements that guarantees an overlap. Therefore, this conclusion does not necessarily follow.
Conclusion II: Some honest are boys.
This links H and B. Statement 2 says "All boys are honest". If every boy is honest, then it logically follows that some members of the group of honest people are boys. This is a direct implication (conversion) of a universal affirmative statement. Thus, this conclusion follows.
Conclusion III: Some honest are students.
This links H and S. Statement 1 says "Some students are boys". Statement 2 says "All boys are honest". If some students are boys, and every single boy is honest, then the students who are boys must also be honest. This means there is an overlap between the group of students and the group of honest people. Therefore, some students are honest, which is equivalent to some honest are students. This conclusion follows.
Conclusion IV: Some toppers are students.
This links T and S. We know Some S are H (from Statements 1 and 2 combined) and Some H are T (from Statement 3). Having "Some A are B" and "Some B are C" does not guarantee "Some A are C" in syllogism. The overlap between S and H, and the overlap between H and T, might not involve the same members of H. There is no necessary overlap between students and toppers based on the given statements. Therefore, this conclusion does not necessarily follow.
Summary of Conclusion Analysis
| Conclusion | Statements Involved | Logical Follows? | Reasoning |
|---|---|---|---|
| I. Some toppers are boys. | 2, 3 (All B are H, Some H are T) | No | No guaranteed overlap between B and T. |
| II. Some honest are boys. | 2 (All B are H) | Yes | Direct implication of "All B are H". |
| III. Some honest are students. | 1, 2 (Some S are B, All B are H) | Yes | Transitivity: S ↔ B, B → H ∴ S ↔ H. |
| IV. Some toppers are students. | 1, 2, 3 (Some S are B, All B are H, Some H are T) | No | Some S are H, Some H are T does not guarantee Some S are T. |
Based on this analysis, only conclusions II and III logically follow from the given statements.
Revision Table: Syllogism Basics
| Statement Type | Form | Conversion |
|---|---|---|
| Universal Affirmative | All A are B | Some B are A |
| Universal Negative | No A are B | No B are A |
| Particular Affirmative | Some A are B | Some B are A |
| Particular Negative | Some A are not B | Not valid by simple conversion |
Additional Information: Types of Propositions in Syllogism
Syllogisms deal with propositions, which are statements that assert or deny something. In categorical syllogisms, propositions are classified by quantity (universal or particular) and quality (affirmative or negative).
- Universal Affirmative (A): All S are P. (e.g., All boys are honest)
- Universal Negative (E): No S are P. (e.g., No students are lazy)
- Particular Affirmative (I): Some S are P. (e.g., Some students are boys)
- Particular Negative (O): Some S are not P. (e.g., Some students are not toppers)
Understanding these types helps in analyzing how statements can be combined and what conclusions can be drawn.
Important Questions from Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
