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.
Neither I nor II
Understanding the Syllogism Question
This question asks us to analyze given statements and determine which of the provided conclusions can be logically derived from them. In syllogism, we must assume the statements are true, even if they contradict real-world facts, and then evaluate the conclusions based solely on these statements.
Analyzing the Statements and Conclusions
Let's list the statements and conclusions given:
Statements:
- Some schools are colleges.
- No college is university.
- All universities are hospitals.
Conclusions:
- Some schools are universities.
- At least some hospitals are colleges. (This is equivalent to "Some hospitals are colleges")
Step-by-Step Logical Analysis
We will examine each conclusion based on the provided statements.
Analyzing Conclusion I: Some schools are universities.
The statements connect schools to colleges, colleges to universities, and universities to hospitals. Let's trace the relationship between Schools and Universities:
- Statement 1: Some Schools are Colleges. This means there is an overlap between the set of Schools and the set of Colleges.
- Statement 2: No College is University. This means there is absolutely no overlap between the set of Colleges and the set of Universities. They are entirely separate.
From these two statements, we know that the part of Schools that are Colleges cannot be Universities. What about the part of Schools that are *not* Colleges? The statements provide no information about the relationship between Schools (specifically the non-college part) and Universities. Therefore, based on the given statements, we cannot definitively conclude that Some schools are universities. This conclusion is not logically derived.
Analyzing Conclusion II: At least some hospitals are colleges.
Let's trace the relationship between Hospitals and Colleges:
- Statement 2: No college is university. (Colleges and Universities are separate).
- Statement 3: All universities are hospitals. (The set of Universities is completely contained within the set of Hospitals).
We know that Universities are inside Hospitals, and Colleges have no overlap with Universities. Does this force Hospitals to overlap with Colleges? Not necessarily.
Imagine Hospitals as a large circle, and Universities as a smaller circle completely inside the Hospitals circle. Colleges are a separate group that cannot overlap with the Universities circle. Colleges *could* overlap with the Hospitals circle in the area outside the Universities circle, or Colleges *could* be entirely outside the Hospitals circle. Both scenarios are consistent with the statements.
Since we can draw a scenario where Colleges and Hospitals do not overlap (while still satisfying the statements), we cannot definitively conclude that At least some hospitals are colleges. This conclusion is not logically derived.
Summary of Conclusions
- Conclusion I: Some schools are universities - Not logically derived.
- Conclusion II: At least some hospitals are colleges - Not logically derived.
Since neither conclusion I nor conclusion II can be logically derived from the statements, the correct answer is that neither conclusion follows.
Checking the Options
Based on our analysis, neither conclusion I nor conclusion II is valid. We look for the option that states this.
- Option 1: Only I - Incorrect.
- Option 2: Either I or II - Incorrect.
- Option 3: Only II - Incorrect.
- Option 4: Neither I nor II - Correct.
| Statement | Type | Relationship |
|---|---|---|
| Some schools are colleges. | Particular Affirmative (I) | S $\cap$ C ≠ ∅ |
| No college is university. | Universal Negative (E) | C $\cap$ U = ∅ |
| All universities are hospitals. | Universal Affirmative (A) | U ⊆ H |
| Conclusion | Equivalent Form | Logically Derived? | Reason |
|---|---|---|---|
| I. Some schools are universities. | S $\cap$ U ≠ ∅ | No | No direct connection between S and U is established that guarantees overlap. C separates S and U, and C has no overlap with U. |
| II. At least some hospitals are colleges. | Some hospitals are colleges. (H $\cap$ C ≠ ∅) | No | U is inside H, and C has no overlap with U. C may or may not overlap with H (outside U). |
Conclusion
Based on the detailed analysis of the statements and conclusions, neither of the given conclusions logically follows. Therefore, the correct option is Neither I nor II.
Revision Table: Syllogism Concepts
| Concept | Description | Key Rule/Idea |
|---|---|---|
| Syllogism | A form of deductive reasoning where conclusions are drawn from given premises (statements). | Assume statements are true. |
| Statement | A premise providing information about the relationship between two categories. | Types: All (A), No (E), Some (I), Some Not (O). |
| Conclusion | A statement that is claimed to follow logically from the premises. | Must be necessarily true if premises are true. |
| Validity | A conclusion is valid if it must be true whenever the premises are true. | Not about real-world truth, only logical necessity. |
Additional Information: Checking Syllogism Conclusions
There are several methods to check the validity of conclusions in syllogisms:
- Venn Diagrams: Draw overlapping circles representing the categories. Shade or mark areas based on the statements. If a conclusion's condition is *always* met in any valid diagram representing the statements, it's valid. If you can draw *at least one* diagram where the conclusion's condition is *not* met, the conclusion is invalid.
- Rules of Syllogism: There are formal rules based on the types and arrangement of statements (e.g., number of negative statements, distribution of terms). These rules can be complex but provide a systematic way to check validity.
- Logical Deduction: Combining statements sequentially to see if they logically lead to the conclusion. For example, if "Some A are B" and "All B are C", you can deduce "Some A are C". However, some combinations yield no valid conclusion between the extreme terms.
In this specific problem involving "Some", "No", and "All" statements, both Venn diagrams and logical deduction help show that neither conclusion is guaranteed.
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.
Read the given statements and conclusions carefully. Assuming that the information in the statement is true even if they appear to be at variance with commonly known facts, choose the conclusion that logically follows from the statement beyond reasonable doubt.
Statement:-
1. Some dogs are cats.
2. All cats are birds.
Conclusion:-
1. Some dogs are birds.
2. No cat is dog.
