Three statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All p are r.
Some r are c.
All c are d.
Conclusions:
I. Some d are p.
II. Some c are p.
- AEither conclusion I or II follows
- BOnly conclusion II follows
- CNeither conclusion I nor II follows
- DOnly conclusion I follows
Solution & Step-by-step Explanation
Let's analyze the statements using a Venn diagram:
"All p are r" → The entire circle of p is enclosed inside r.
"Some r are c" → There is a partial overlap between r and c. This overlap does not necessarily contain any portion of p.
"All c are d" → The entire circle of c is enclosed inside d. Since c overlaps with r, d will also naturally overlap with r.
Let's evaluate the conclusions based on definite truth:
Conclusion I: "Some d are p." → Since the circle of c (and therefore d) only definitely intersects with r, it is possible but not certain that it intersects with p. Thus, it does not definitely follow.
Conclusion II: "Some c are p." → Similarly, there is no mandatory overlapping region defined between c and p. Thus, it does not definitely follow.
Since neither conclusion follows with absolute certainty, Neither conclusion I nor II follows.
"All p are r" → The entire circle of p is enclosed inside r.
"Some r are c" → There is a partial overlap between r and c. This overlap does not necessarily contain any portion of p.
"All c are d" → The entire circle of c is enclosed inside d. Since c overlaps with r, d will also naturally overlap with r.
Let's evaluate the conclusions based on definite truth:
Conclusion I: "Some d are p." → Since the circle of c (and therefore d) only definitely intersects with r, it is possible but not certain that it intersects with p. Thus, it does not definitely follow.
Conclusion II: "Some c are p." → Similarly, there is no mandatory overlapping region defined between c and p. Thus, it does not definitely follow.
Since neither conclusion follows with absolute certainty, Neither conclusion I nor II follows.