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 conclusions logically follow(s) from the statements. Statements: Some spices are medicines. All medicines are chemicals. No chemical is cheap. Conclusions: I. Some medicines are cheap. II. Some chemicals are spices. III. No medicine is cheap.
Only conclusions II and III follow.
Analyzing Statements and Conclusions in Logical Reasoning
This problem requires us to evaluate which conclusions logically follow from a given set of statements, assuming the statements are true. This type of question falls under the category of logical reasoning, specifically syllogisms.
Understanding the Statements
We are given three statements:
- Statement 1: Some spices are medicines.
- Statement 2: All medicines are chemicals.
- Statement 3: No chemical is cheap.
Evaluating the Conclusions
We need to check each conclusion based on the given statements.
Conclusion I: Some medicines are cheap.
Let's analyze this based on the statements:
- Statement 2 says "All medicines are chemicals."
- Statement 3 says "No chemical is cheap."
If all medicines are chemicals, and none of the chemicals are cheap, then it logically follows that none of the medicines can be cheap. Therefore, the conclusion "Some medicines are cheap" directly contradicts the information derived from the statements.
Conclusion I does not follow.
Conclusion II: Some chemicals are spices.
Let's analyze this based on the statements:
- Statement 1 says "Some spices are medicines." This means there is at least one spice that is also a medicine.
- Statement 2 says "All medicines are chemicals." This means everything that is a medicine is also a chemical.
Since there are some spices that are medicines (Statement 1), and all medicines are chemicals (Statement 2), these 'some spices' that are medicines must also be chemicals. Therefore, some spices are chemicals. If some spices are chemicals, it logically follows that some chemicals are spices.
Conclusion II follows.
Conclusion III: No medicine is cheap.
Let's analyze this based on the statements:
- Statement 2 says "All medicines are chemicals."
- Statement 3 says "No chemical is cheap."
If every single medicine belongs to the category of chemicals (Statement 2), and nothing in the category of chemicals is cheap (Statement 3), then nothing in the category of medicines can be cheap either. This means "No medicine is cheap."
Conclusion III follows.
Summary of Conclusions
Based on the analysis:
- Conclusion I: Some medicines are cheap. (Does not follow)
- Conclusion II: Some chemicals are spices. (Follows)
- Conclusion III: No medicine is cheap. (Follows)
Therefore, only conclusions II and III logically follow from the given statements.
Revision Table: Syllogism Analysis
| Statement/Conclusion | Relationship | Analysis based on given Statements | Follows? |
|---|---|---|---|
| Statement 1 | Some Spices are Medicines (S → M) | Given | - |
| Statement 2 | All Medicines are Chemicals (M → C) | Given | - |
| Statement 3 | No Chemical is Cheap (C → Not Cheap) | Given | - |
| Conclusion I | Some Medicines are Cheap (M → Cheap) | All M are C, No C is Cheap → No M is Cheap. Contradicts conclusion. | No |
| Conclusion II | Some Chemicals are Spices (C → S) | Some S are M, All M are C → Some S are C → Some C are S. | Yes |
| Conclusion III | No Medicine is Cheap (M → Not Cheap) | All M are C, No C is Cheap → No M is Cheap. | Yes |
Additional Information on Syllogisms
Syllogisms are a form of deductive reasoning where a conclusion is drawn from two or more premises (statements). Understanding the relationship between categories described in the statements is key.
- Universal Affirmative (All A are B): Every member of category A is also a member of category B.
- Universal Negative (No A is B): No member of category A is a member of category B.
- Particular Affirmative (Some A are B): At least one member of category A is also a member of category B. 'Some' means 'at least one' and can include 'all'.
- Particular Negative (Some A are not B): At least one member of category A is not a member of category B.
Visual aids like Venn diagrams can often help represent the relationships between categories and verify conclusions. In this problem, the relationships are:
- Some Spices overlap with Medicines.
- The entire set of Medicines is inside the set of Chemicals.
- The set of Chemicals and the set of Cheap things have no overlap.
Combining these: Since Medicines are inside Chemicals, and Chemicals don't overlap with Cheap, Medicines cannot overlap with Cheap. Also, since Some Spices overlap with Medicines, and Medicines are inside Chemicals, that overlap of Spices and Medicines must also be within Chemicals, meaning Some Spices are Chemicals, and thus Some Chemicals are Spices.
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.
