The statements below are followed by two conclusions labelled 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: All owls are horses. Some horses are hares. Conclusions: I. Some owls are hares. II. Some hares are horses.
Only conclusion II follows.
Understanding Syllogism and Logical Conclusions
This question asks us to analyze a set of statements and determine which of the given conclusions logically follow from them. This type of problem is known as a syllogism, which is a form of logical reasoning where a conclusion is drawn from two or more statements.
Analyzing the Statements
We are given two statements:
- All owls are horses.
- Some horses are hares.
In syllogism problems, we must assume the statements are true, even if they contradict real-world knowledge. We need to find what *must* be true based *only* on these statements.
Evaluating the Conclusions
Now, let's evaluate each conclusion based on the given statements.
Conclusion I: Some owls are hares.
The statements tell us about the relationship between owls and horses, and between horses and hares. Let's think about this:
- "All owls are horses" means the set of owls is entirely contained within the set of horses.
- "Some horses are hares" means there is at least one horse that is also a hare.
Based on this, can we say for sure that there is at least one owl that is also a hare? Not necessarily. The horses that are hares might be only the horses that are *not* owls. There is no information connecting owls directly to hares, or stating that the overlap between horses and hares must include the part of horses that contains owls.
Think of it using sets:
- Let O = Owls, H = Horses, R = Hares.
- Statement 1: $\text{O} \subseteq \text{H}$ (O is a subset of H)
- Statement 2: $\text{H} \cap \text{R} \neq \emptyset$ (The intersection of H and R is not empty)
Conclusion I asks if $\text{O} \cap \text{R} \neq \emptyset$. Based on $\text{O} \subseteq \text{H}$ and $\text{H} \cap \text{R} \neq \emptyset$, we cannot definitively say that $\text{O} \cap \text{R} \neq \emptyset$. It's possible that the part of H that intersects with R is outside O.
Therefore, Conclusion I does not logically follow from the statements.
Conclusion II: Some hares are horses.
The second statement is "Some horses are hares". This statement asserts that there is an overlap between the set of horses and the set of hares. If some horses are hares, this means there are individuals that belong to both categories.
The relationship "Some A are B" is symmetrical with "Some B are A". If some horses are hares, it automatically means that some hares are horses.
- Statement 2: Some horses are hares.
- This implies there exists at least one individual X such that X is a horse AND X is a hare.
- If X is a hare AND X is a horse, then it is true that some hares are horses.
This conclusion directly and logically follows from the second statement.
Summary of Analysis
Based on our evaluation:
- Conclusion I (Some owls are hares) does not logically follow.
- Conclusion II (Some hares are horses) logically follows.
| Statement / Conclusion | Analysis | Logically Follows? |
|---|---|---|
| Statement 1: All owls are horses. | Owls set is inside Horses set. | N/A |
| Statement 2: Some horses are hares. | Horses and Hares sets overlap. | N/A |
| Conclusion I: Some owls are hares. | Does the overlap between Horses and Hares necessarily include the Owls part of Horses? No. | No |
| Conclusion II: Some hares are horses. | If some Horses are Hares, then some Hares must be Horses (symmetry of 'some'). | Yes |
Therefore, only conclusion II follows from the given statements.
Revision Table: Basic Syllogism Rules
| Statement Type | Relationship | Inference |
|---|---|---|
| All A are B | A is entirely within B | Some A are B, Some B are A (if A is not empty) |
| No A are B | A and B are separate | No B are A, All A are not B, All B are not A |
| Some A are B | A and B overlap | Some B are A |
| Some A are not B | Part of A is outside B | No direct simple symmetrical inference. |
Additional Information on Logical Reasoning
Syllogisms are a fundamental part of deductive reasoning. They help test the ability to draw valid conclusions strictly from given premises. The structure typically involves two statements (premises) and a third statement (conclusion).
Key points to remember:
- Always assume the given statements are absolutely true, regardless of real-world facts.
- The conclusion must necessarily follow from the statements; it cannot be just a possibility.
- Venn diagrams can be a helpful visual tool to represent the relationships described in the statements and check the validity of conclusions.
- Pay close attention to qualifying words like "All", "Some", and "No". These words define the extent of the relationship between the categories.
- For "Some A are B", the minimum is one, and the maximum could be all A or all B (or both), but we only know for sure that there is *at least one* in the intersection.
Mastering syllogisms improves logical thinking and analytical skills, which are crucial for many competitive exams.
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 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.
In the question below are given three statements followed by two conclusions. You have to take each of the given statements to be true. Read the conclusions and then decide which of the conclusions can be logically derived.
Statements:
Some schools are colleges.
No college is university.
All universities are hospitals.
Conclusions:
I. Some schools are universities.
II. At least some hospitals are colleges.
