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.
Both conclusions III and IV follow
Understanding Statements and Conclusions Logic
This question tests your ability to draw logical inferences based on given statements, even if those statements contradict real-world facts. We need to analyze each statement and see which of the given conclusions logically follows.
Analyzing the Given Statements
We have three statements:
- All lemons are plums. (Statement 1)
- All plums are dates. (Statement 2)
- Some dates are mangoes. (Statement 3)
Analyzing the Given Conclusions
We need to evaluate four conclusions:
- Some lemons are mangoes. (Conclusion I)
- Some mangoes are plums. (Conclusion II)
- All lemons are dates. (Conclusion III)
- Some mangoes are dates. (Conclusion IV)
Step-by-Step Evaluation of Conclusions
Evaluating Conclusion I: Some lemons are mangoes
From Statement 1, we know that the set of 'lemons' is entirely contained within the set of 'plums'. From Statement 2, we know that the set of 'plums' is entirely contained within the set of 'dates'. This means the set of 'lemons' is also entirely contained within the set of 'dates'.
Statement 3 tells us 'Some dates are mangoes'. This indicates an overlap between the set of 'dates' and the set of 'mangoes'. However, this overlap might occur in the part of 'dates' that are NOT 'lemons' (or 'plums'). There is no direct or indirect link established between 'lemons' and 'mangoes' that guarantees an overlap between these two sets.
Therefore, Conclusion I does not necessarily follow.
Evaluating Conclusion II: Some mangoes are plums
We know from Statement 2 that 'All plums are dates'. Statement 3 says 'Some dates are mangoes'. This means there is an intersection between 'dates' and 'mangoes'. However, this intersection might be in the part of 'dates' that are NOT 'plums'. For example, if 'Dates' includes categories A and B, and 'Plums' are only in category A, 'Mangoes' could overlap only with category B of 'Dates'.
There is no information that forces the overlap between 'dates' and 'mangoes' to include the specific subset of 'dates' that are 'plums'.
Therefore, Conclusion II does not necessarily follow.
Evaluating Conclusion III: All lemons are dates
Statement 1: All lemons are plums.
Statement 2: All plums are dates.
If every lemon is a plum, and every plum is a date, then it logically follows that every lemon must also be a date. This is a basic principle of syllogistic reasoning (specifically, Barbara in AAA-1 form, though knowing the name isn't necessary, the logic is key).
Representing this:
- Lemons <span>⊂</span> Plums
- Plums <span>⊂</span> Dates
- Therefore, Lemons <span>⊂</span> Dates
This means 'All lemons are dates' logically follows from Statements 1 and 2.
Evaluating Conclusion IV: Some mangoes are dates
Statement 3 says: Some dates are mangoes.
In logic, the statement 'Some A are B' implies that 'Some B are A'. If it is true that 'Some dates are mangoes', then it must also be true that 'Some mangoes are dates'. This is the rule of conversion for 'I' type propositions (Particular Affirmative).
Therefore, Conclusion IV logically follows directly from Statement 3.
Summary of Conclusions
- Conclusion I (Some lemons are mangoes): Does NOT follow.
- Conclusion II (Some mangoes are plums): Does NOT follow.
- Conclusion III (All lemons are dates): FOLLOWS.
- Conclusion IV (Some mangoes are dates): FOLLOWS.
Based on our analysis, only Conclusions III and IV logically follow from the given statements.
Revision Table: Statement Types and Conversion Rules
| Statement Type | Format | Conversion (if valid) |
|---|---|---|
| A (Universal Affirmative) | All A are B | Some B are A |
| E (Universal Negative) | No A is B | No B is A |
| I (Particular Affirmative) | Some A are B | Some B are A |
| O (Particular Negative) | Some A are not B | Not Valid (generally) |
Additional Information: Syllogisms and Logic
Statements and Conclusions questions often relate to syllogisms, which are a form of logical argument where a conclusion is drawn from two or more premises (statements). Key points to remember:
- Always assume the given statements are true, regardless of real-world knowledge.
- The conclusion must logically follow *only* from the statements provided.
- 'All A are B' means the set A is a subset of set B.
- 'Some A are B' means there is at least one element common to sets A and B.
- Drawing Venn diagrams can be helpful to visualize the relationships between the categories in the statements.
- Mastering the rules of inference and conversion helps in quickly evaluating conclusions.
In this problem, the combination of 'All lemons are plums' and 'All plums are dates' is a classic example demonstrating transitivity, leading to the conclusion 'All lemons are dates'. The conversion rule directly supports 'Some mangoes are dates' from 'Some dates are mangoes'.
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 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.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. Some L are R.
II. Some A are R.
Conclusion:
I. All A are L.
II. All R are L.
