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.
Only conclusion I follows
Understanding Syllogism Statements and Conclusions
The question provides two statements and two conclusions based on those statements. We need to determine which conclusions logically follow from the given statements, assuming the statements are true.
Given Statements:
- Statement I: All P are Q.
- Statement II: Some P are C.
Given Conclusions:
- Conclusion I: Some Q are C.
- Conclusion II: Some P are not C.
Analyzing the Statements and Conclusions
We can use Venn diagrams or logical deduction to analyze the relationship between the sets P, Q, and C based on the given statements.
Analysis using Venn Diagrams:
Statement I: "All P are Q" means the entire set P is contained within the set Q. We can represent this by drawing a circle for P completely inside a circle for Q.
Statement II: "Some P are C" means there is at least one element that is both in set P and in set C. We represent this by drawing the circle for C such that it overlaps with the circle for P. The overlapping region represents the elements that are P and C.
Since the set P is entirely inside the set Q, any overlap between P and C must necessarily also be an overlap between Q and C.
Analyzing Conclusion I: Some Q are C.
Conclusion I states that there is at least one element that is both in set Q and in set C. From our Venn diagram analysis, the region where P and C overlap is guaranteed to be non-empty (from Statement II). Since all P are Q (Statement I), this overlapping region (P $\cap$ C) is also within Q. Therefore, the intersection of Q and C (Q $\cap$ C) must contain this non-empty region (P $\cap$ C). This means Q $\cap$ C is non-empty, which is the meaning of "Some Q are C".
Thus, Conclusion I logically follows from the statements.
Analyzing Conclusion II: Some P are not C.
Conclusion II states that there is at least one element that is in set P but not in set C. Statement II tells us "Some P are C", meaning P $\cap$ C is not empty. However, Statement II does not preclude the possibility that all P are C (i.e., P is a subset of C). If all P were C, then there would be no elements that are P but not C. In that specific case, "Some P are not C" would be false.
Since Conclusion II is not necessarily true in all possible scenarios consistent with the statements (specifically, the scenario where all P are C), it does not logically follow from the statements.
Summary of Analysis:
- Conclusion I ("Some Q are C") is logically supported by the statements.
- Conclusion II ("Some P are not C") is not logically supported by the statements because it is possible that all P are C.
Conclusion
Based on the analysis, only Conclusion I logically follows from the given statements.
Therefore, the correct option is the one stating that only conclusion I follows.
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 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.
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.
