In the question below are given some statements followed by some conclusions based on those statements. Take the given statements to be true even if they seem to be different from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements. Statements: I. Some P are T. II. Some T are S. III. No J is P. Conclusion: I. Some S are not J. II. Some T are not J. III. Some J are not T.
Only II
Analyzing Statements and Conclusions
This question requires us to analyze the given statements and determine which of the provided conclusions logically follow from them. We must assume the statements are true, even if they contradict common knowledge.
Given Statements:
I. Some P are T.
II. Some T are S.
III. No J is P.
Given Conclusions:
I. Some S are not J.
II. Some T are not J.
III. Some J are not T.
Step-by-Step Analysis:
We will examine each conclusion based on the logical relationships established by the statements. We can think of these relationships in terms of sets or categories.
Analyzing Conclusion I: Some S are not J.
- From Statement II, we know there is an overlap between the set T and the set S (Some T are S).
- From Statement III, we know that the set J and the set P have absolutely no overlap (No J is P).
- Statement I tells us there is an overlap between P and T (Some P are T).
- The statements establish a relationship between P and T, T and S, and J and P.
- However, there is no direct statement linking J and S, or indirectly through P and T that forces some S to be outside of J.
- Consider the part of S that overlaps with T. Statement II says "Some T are S". This part of S could potentially also be J. We only know J is separate from P. S could overlap with J completely outside of P.
- Therefore, based on the given statements, we cannot definitively conclude that some S must be outside of J. This conclusion does not logically follow.
Analyzing Conclusion II: Some T are not J.
- From Statement I, we know that some entities belong to both set P and set T (Some P are T). Let's call this overlapping group 'X'. So, X is part of P, and X is also part of T.
- From Statement III, we know that the set J has no overlap with the set P (No J is P). This means no entity that is in P can also be in J.
- Since the group 'X' is part of P (from Statement I), and no P can be J (from Statement III), it follows that no entity in group 'X' can be J.
- Group 'X' is also part of T (from Statement I).
- So, the entities in group 'X' are T and are not J.
- Since Statement I guarantees that such a group 'X' exists (because 'Some P are T' means at least one entity is both P and T), we can definitively say that there are some entities that are T and are not J.
- Therefore, the conclusion "Some T are not J" logically follows from the statements.
Analyzing Conclusion III: Some J are not T.
- From Statement III, we know that set J and set P are disjoint (No J is P).
- Statement I and II involve sets P, T, and S, establishing overlaps between P and T, and T and S.
- The statements do not provide a direct or indirect necessary connection between J and T that would require some part of J to be outside of T.
- It is possible to draw diagrams where J is completely outside of T, in which case "All J are not T" would be true, and "Some J are not T" would also be true.
- However, it is also possible to draw diagrams where J overlaps with T in a region that is not P. Or, if the statements allowed it, J could even be a subset of T, provided J is disjoint from P.
- The statements only forbid J from overlapping with P. They do not restrict J's relationship with T enough to guarantee that some J must be outside T.
- Therefore, based on the given statements, we cannot definitively conclude that some J must be outside of T. This conclusion does not logically follow.
Summary of Conclusions:
- Conclusion I: Some S are not J. — Does NOT follow.
- Conclusion II: Some T are not J. — FOLLOWS.
- Conclusion III: Some J are not T. — Does NOT follow.
Only Conclusion II logically follows from the given statements.
Important Questions from Statements and Conclusions
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 B.
II. Some B are C.
Conclusions:
I. Some A are C.
II. Some A are not C.
III. Some A are B.
In the question below are given some statements followed by some conclusions based on those statements. Take the given statements to be true even if they seem to be different from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements.
Statements:
I. Some P are Q.
II. All P are N.
Conclusion:
I. All Q are N.
II. Some P are not Q.
In the question below are given some statements followed by some conclusions based on those statements. Take 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 conclusions logically follows from the given statements.
Statement :
I. All A are B.
II. No B is D.
III. Some D are G.
Conclusion :
I. No A is D.
II. Some G are not B.
III. Some A are not G.
In the question below are given some statements followed by some conclusions based on those statements. Take 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 conclusions logically follows from the given statements.
Statements:
I. All A are B.
II. Some B are C.
III. All C are D.
Conclusion:
I. Some A are D.
II. Some C are A.
III. Some B are D.
In the question below, some statements followed by some conclusions based on those statements are given. Take 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 conclusions logically follows from the given statements.
Statements:
I. All P are Q.
II. Some P are C.
Conclusions:
I. Some Q are C.
II. Some P are not C.
