In this question, three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusion(s) logically follow(s) from the statements. Statements: All chairs are tables. Some tables are stands. All stands are baskets. Conclusions: I. Some chairs are stands. II. Some tables are baskets. III. All baskets are chairs.
Only conclusion II follows.
Let's analyze the given statements and conclusions using the principles of logical reasoning, specifically syllogisms. We need to determine which conclusions logically follow from the statements provided.
Understanding the Statements and Conclusions
The problem provides three statements that we must assume are true, even if they contradict common sense. We then have three conclusions, and we need to check their validity based *only* on the given statements.
Statements:
- All chairs are tables.
- Some tables are stands.
- All stands are baskets.
Conclusions:
- Some chairs are stands.
- Some tables are baskets.
- All baskets are chairs.
Analyzing Each Conclusion Logically
We can use Venn diagrams or rules of inference to test each conclusion. Let's analyze them one by one.
Analyzing Conclusion I: Some chairs are stands.
This conclusion links 'chairs' and 'stands'. Let's look at the statements involving these terms or terms that connect them:
- Statement 1: All chairs are tables. (Chairs → Tables)
- Statement 2: Some tables are stands. (Tables ∩ Stands ≠ ∅)
Statement 1 tells us that the set of chairs is entirely contained within the set of tables. Statement 2 tells us that there is an overlap between tables and stands. However, the tables that overlap with stands might not be the tables that are also chairs. There is no definite connection established between chairs and stands. Therefore, "Some chairs are stands" does not necessarily follow.
Analyzing Conclusion II: Some tables are baskets.
This conclusion links 'tables' and 'baskets'. Let's examine the relevant statements:
- Statement 2: Some tables are stands. (Tables ∩ Stands ≠ ∅)
- Statement 3: All stands are baskets. (Stands → Baskets)
Statement 2 says there's an intersection between tables and stands. Statement 3 says that the entire set of stands is contained within the set of baskets. If some tables are stands, and all stands are baskets, then those specific tables that are stands must also be baskets. Thus, there must be an overlap between tables and baskets. The conclusion "Some tables are baskets" logically follows.
Analyzing Conclusion III: All baskets are chairs.
This conclusion links 'baskets' and 'chairs' in a universal way. Let's see the relationships:
- Statement 3: All stands are baskets. (Stands → Baskets)
- Statement 2: Some tables are stands. (Tables ∩ Stands ≠ ∅)
- Statement 1: All chairs are tables. (Chairs → Tables)
We know some tables are stands, and all stands are baskets. This means some tables are baskets. We also know all chairs are tables. There's no statement or combination of statements that allows us to conclude that *all* baskets must be chairs. The relationship is indirect and does not cover all baskets. For instance, there might be baskets that are stands, and those stands are tables, but they are not necessarily chairs (as only *some* tables relate to stands, and *all* chairs are just *part* of the tables). Therefore, "All baskets are chairs" does not follow.
Summary of Conclusions
Based on the analysis:
- Conclusion I (Some chairs are stands): Does not follow.
- Conclusion II (Some tables are baskets): Follows.
- Conclusion III (All baskets are chairs): Does not follow.
Only conclusion II logically follows from the given statements.
| Conclusion | Statements Involved | Logical Inference | Follows? |
|---|---|---|---|
| I. Some chairs are stands. | All Chairs are Tables, Some Tables are Stands. | No direct necessary link between Chairs and Stands. | No |
| II. Some tables are baskets. | Some Tables are Stands, All Stands are Baskets. | If Some B are C and All C are D, then Some B are D. | Yes |
| III. All baskets are chairs. | All Stands are Baskets, Some Tables are Stands, All Chairs are Tables. | Cannot conclude All D are A from the given premises. | No |
Revision Table: Key Concepts in Syllogism
| Term | Explanation |
|---|---|
| Statement (Premise) | A proposition assumed to be true for the purpose of the argument. |
| Conclusion | A proposition that is claimed to follow logically from the statements. |
| Syllogism | A form of logical reasoning where a conclusion is derived from two or more statements. |
| Universal Affirmative (All A are B) | States that every member of class A is a member of class B. |
| Particular Affirmative (Some A are B) | States that at least one member of class A is a member of class B. |
Additional Information: Rules for Some-All Combinations
Here are some basic rules for combining statements in syllogisms:
- Some A are B + All B are C → Some A are C (e.g., Some students are tall, All tall people are visible from distance → Some students are visible from distance)
- Some A are B + All C are B → No definite conclusion about A and C.
- All A are B + Some B are C → No definite conclusion about A and C (as seen in Conclusion I).
- Some A are B + All B are C → Some A are C. (This rule does not directly apply to Conclusion II, but the logic is similar: Some Tables are Stands + All Stands are Baskets implies Some Tables are Baskets).
Remember, in syllogisms, you must strictly follow the logical structure and not rely on outside knowledge.
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.
