The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements. Statements: 1) Some manure are ammonia. 2) Some ammonia are potash. Conclusion: I. Some ammonia are manure. II. Some potash are ammonia.
Both conclusions follow.
Understanding the Syllogism Problem
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. In logic problems like this, we must assume the statements are true, even if they contradict real-world knowledge.
Analyzing the Statements and Conclusions
Let's break down the statements and conclusions:
- Statement 1: Some manure are ammonia.
- Statement 2: Some ammonia are potash.
- Conclusion I: Some ammonia are manure.
- Conclusion II: Some potash are ammonia.
Evaluating Conclusion I: Some ammonia are manure
Statement 1 says, "Some manure are ammonia." This establishes a relationship between the categories "manure" and "ammonia," indicating that there is an overlap between them. If some part of the "manure" category is included in the "ammonia" category, it logically follows that some part of the "ammonia" category must also be included in the "manure" category.
This is based on the principle of conversion in logic for 'Some' statements. The converse of "Some A are B" is "Some B are A", and this conversion is valid.
Therefore, Conclusion I ("Some ammonia are manure") definitely follows from Statement 1 ("Some manure are ammonia").
Evaluating Conclusion II: Some potash are ammonia
Statement 2 says, "Some ammonia are potash." This establishes a relationship between the categories "ammonia" and "potash," indicating an overlap. Similar to the previous conclusion, if some part of the "ammonia" category is included in the "potash" category, it logically follows that some part of the "potash" category must also be included in the "ammonia" category.
This is the valid converse of Statement 2. The converse of "Some A are B" is "Some B are A".
Therefore, Conclusion II ("Some potash are ammonia") definitely follows from Statement 2 ("Some ammonia are potash").
Combining the Analysis
We found that both Conclusion I and Conclusion II logically follow from the given statements.
| Statement/Conclusion | Type | Logical Follows? | Reason |
|---|---|---|---|
| Statement 1: Some manure are ammonia. | Given Statement | N/A | Assumed True |
| Statement 2: Some ammonia are potash. | Given Statement | N/A | Assumed True |
| Conclusion I: Some ammonia are manure. | Conversion of St. 1 | Yes | The converse of "Some A are B" is valid. |
| Conclusion II: Some potash are ammonia. | Conversion of St. 2 | Yes | The converse of "Some A are B" is valid. |
Based on the analysis, both conclusions are logically derived from the statements.
Revision Table: Syllogism Rules
| Statement Type | Structure | Valid Conversion |
|---|---|---|
| A (Universal Affirmative) | All A are B | Some B are A (Partial Conversion) |
| E (Universal Negative) | No A are B | No B are A (Simple Conversion) |
| I (Particular Affirmative) | Some A are B | Some B are A (Simple Conversion) |
| O (Particular Negative) | Some A are not B | No valid conversion |
Additional Information: Types of Syllogisms
The problem presented is a categorical syllogism, which is a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.
- A categorical syllogism consists of three parts: two premise statements and one conclusion.
- Each part is a categorical proposition (A, E, I, or O) which relates two categories (or terms).
- The statements relate three terms: a major term (predicate of the conclusion), a minor term (subject of the conclusion), and a middle term (appears in both premises but not the conclusion). In this specific problem, while it has two statements and two conclusions, each conclusion only involves terms from one statement, making the conversions the key logical step.
- Understanding valid conversions and other immediate inferences (like obversion and contraposition) is crucial for solving syllogism problems effectively.
Important Questions from Conventional 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:
No bank is an office.
All offices are stalls.
Conclusions:
I. No bank is a stall.
II. No stall is a bank.
III. Some stalls are offices.
IV. All the stalls are offices
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 flowers are beautiful.
Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
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:
1. All rugs are blankets.
2. All blankets are pillows.
3. Some blankets are frames.
Conclusions:
I. All pillows are rugs.
II. Some pillows are rugs.
III. All rugs are frames
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 fingers are toes.
Some toes are rings.
Some rings are hands.
Conclusions:
I. Some hands are toes.
II. Some rings are fingers.
III. Some hands are fingers.
V. Some fingers are rings.
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 polygons are angles.
All angles are diagonals.
All cones are cubes.
All cubes are decagons.
No diagonal is a cube.
Conclusions:
I. Some diagonals are polygons.
II. All diagonals are decagons.
III. No polygon is a cone.
IV. Some cubes are angles.
