Difficulty: Medium
Correct Answer: None of the conclusions follows.
Explanation:
Introduction / Context:
This syllogism question deals with mountains and three related categories: hillocks, rivers and valleys. You are told that some mountains belong to each of these categories and then asked whether stronger conclusions about all mountains or about intersections between rivers and valleys follow logically. The key is to distinguish between facts about "some" and claims about "all" or about additional intersections.
Given Data / Assumptions:
Concept / Approach:
The "some" statements tell us that there is at least one mountain in each overlapping category. They do not say anything about the remainder of the mountains or about the relationships among hillocks, rivers and valleys themselves. To claim that all mountains belong to one of these three categories, or to assert that valleys and rivers always or sometimes intersect, would require additional information that we do not have.
Step-by-Step Solution:
Step 1: Evaluate Conclusion 1: "All mountains are either hillocks or rivers or valleys."
We know only that some mountains are hillocks, some are rivers and some are valleys. It is still possible that many mountains do not belong to any of these three categories. Hence, the statement about "all" mountains is not justified. Conclusion 1 does not follow.
Step 2: Evaluate Conclusion 2: "No valley is a river."
The original statements say nothing about whether valleys and rivers can intersect. It is entirely possible that a particular mountain is both a river and a valley, or that they are always separate. Since the premises do not specify this relationship, we cannot deduce that no valley is a river.
Step 3: Evaluate Conclusion 3: "Some rivers are valleys."
Similarly, we are not told that any mountain which is a river is also a valley. We only know that there exists at least one river mountain and at least one valley mountain, which could be completely different mountains. Therefore, we cannot conclude that there is a non empty intersection between rivers and valleys.
Step 4: Since none of the three conclusions is forced by the given information, the correct answer is that none follows.
Verification / Alternative check:
Construct a diagram where the set of mountains overlaps with each of hillocks, rivers and valleys at three distinct points, but with no overlap between rivers and valleys and with many mountains outside all three subcategories. This diagram satisfies all three original statements but makes Conclusion 1 false (not all mountains are in those categories), Conclusion 2 false (if we allow some valley to also be a river in another diagram) or at least not guaranteed, and Conclusion 3 false in our specific construction. The existence of such a diagram shows that none of the conclusions must hold in every case.
Why Other Options Are Wrong:
Option B claims that all mountains fall into one of the three categories, which overgeneralises "some" into "all". Option C and Option D claim relationships between rivers and valleys that are not supported by the premises. The original statements give no direct information about how rivers and valleys intersect, so such conclusions are speculative, not necessary.
Common Pitfalls:
A common mistake is to treat multiple "some" statements as if they automatically covered the whole set or forced overlaps between the secondary sets. Always remember that "some" only guarantees existence, not completeness, and says nothing about the rest of the elements or about additional intersections unless explicitly stated.
Final Answer:
None of the conclusions is logically guaranteed, so none of the conclusions follows from the given statements.
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