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 land are hard. II. No stone is land. Conclusions: I. Some hard are land. II. Some stone are hard.
Only conclusion I follows
Understanding Syllogism Statements
This question asks us to analyze two given statements and determine which of the subsequent conclusions logically follows from them, assuming the statements are true.
Let's break down the statements provided:
- Statement I: Some land are hard.
- Statement II: No stone is land.
These statements describe relationships between three categories: 'land', 'hard', and 'stone'.
Analyzing the Relationships
Statement I tells us there is at least one entity that is both 'land' and 'hard'. This is a partial overlap between the categories 'land' and 'hard'.
Statement II tells us there is absolutely no entity that is both 'stone' and 'land'. The categories 'stone' and 'land' are entirely separate from each other.
Visualizing with Venn Diagrams (Optional but helpful)
We can visualize these relationships using Venn diagrams. Imagine three overlapping circles representing 'land', 'hard', and 'stone'.
- For Statement I ('Some land are hard'), the overlapping area between the 'land' circle and the 'hard' circle has something in it.
- For Statement II ('No stone is land'), the overlapping area between the 'stone' circle and the 'land' circle is empty. The 'stone' circle and the 'land' circle do not touch or overlap.
Now, let's look at the conclusions.
Analyzing the Syllogism Conclusions
We need to see if each conclusion is definitely true based *only* on the two given statements.
Conclusion I: Some hard are land.
Let's consider Statement I: "Some land are hard". This means there exists at least one thing which is a 'land' and is also 'hard'. If something is 'land' and 'hard', it is also true that it is 'hard' and 'land'. Therefore, if "Some land are hard" is true, then "Some hard are land" must also be true. This is the concept of the converse of an 'I' type statement (Some A are B ⇒ Some B are A), which is always valid.
Based on Statement I, Conclusion I logically follows.
Conclusion II: Some stone are hard.
Statement II tells us that 'stone' and 'land' have no common elements. Statement I tells us that the 'hard' category overlaps with the 'land' category. We know that the 'hard' items exist (because some land are hard), but we don't know *where else* 'hard' items might be. They could overlap with 'stone', or they might not. There is no information in the statements that connects 'stone' directly to 'hard'. The statements only link 'stone' to 'land' and 'land' to 'hard'. We cannot conclude anything definite about the relationship between 'stone' and 'hard' from these specific statements.
Since we cannot definitively say that "Some stone are hard" is true based on the given information, Conclusion II does not logically follow.
Determining Which Conclusion(s) Follow
Based on our analysis:
- Conclusion I: Some hard are land - Logically follows from Statement I.
- Conclusion II: Some stone are hard - Does not logically follow from the statements.
Therefore, only Conclusion I follows from the given statements.
Revision Table: Syllogism Components
| Component | Description | Example from Question |
|---|---|---|
| Statement | A proposition taken as true for the argument. | Some land are hard. No stone is land. |
| Conclusion | A proposition derived from the statements. | Some hard are land. Some stone are hard. |
| Syllogism | A type of logical argument where a conclusion is inferred from two statements. | The whole structure of statements and conclusions. |
Additional Information: Converses in Logic
Understanding the converse of statements is important in syllogism. The converse of a statement is formed by swapping the subject and predicate terms.
- A-type statement (All S are P): Converse is "All P are S". This is not necessarily true. Example: All dogs are mammals. Converse: All mammals are dogs (False).
- E-type statement (No S is P): Converse is "No P is S". This is always true. Example: No cats are dogs. Converse: No dogs are cats (True).
- I-type statement (Some S are P): Converse is "Some P are S". This is always true. Example: Some students are smart. Converse: Some smart people are students (True).
- O-type statement (Some S are not P): Converse is "Some P are not S". This is not necessarily true. Example: Some animals are not dogs. Converse: Some dogs are not animals (False).
In this question, Conclusion I ("Some hard are land") is the valid converse of Statement I ("Some land are hard"), which is an I-type statement.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
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:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
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. All A are S.
II. No D is A.
Conclusions:
I. Some S are A.
II. All S are D.
III. No A is D.
In the question two statements are given, followed by three conclusions, I, II and III. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : Some cars are scooters.
Statement 2 : All scooters are buses.
Conclusion I : Some scooters are cars.
Conclusion II : Some buses are cars.
Conclusion III : All cars are buses.
