Step 1: Understanding the Question
We are given a statement that establishes a fact ("Douglas is my cousin") and provides a definition for the term "cousin". The definition includes three distinct possibilities. We are then given two conclusions, each presenting one of these possibilities. We need to determine if conclusion I, conclusion II, both, neither, or either of them is a logically certain deduction from the statement.
Step 2: Key Formula or Approach
The problem tests our ability to distinguish between possibility and certainty. The statement "A may be X, Y, or Z" means that if something is A, it could be X, it could be Y, or it could be Z. We cannot be certain which one it is without more information. A conclusion is only considered 'true' or 'valid' if it is a necessary consequence, not just a possible one.
Step 3: Detailed Explanation
Let's break down the information:
Fact: Douglas is my cousin.
Definition: Cousin = (Sister's son) OR (Uncle's son) OR (Brother's son).
Now, let's evaluate the conclusions based on this:
Conclusion I: Douglas is my sister's son. This is one of the three possibilities listed in the definition of a cousin. Since Douglas is a cousin, it is *possible* that he is my sister's son. However, it is not *certain*. He could equally be my uncle's son or my brother's son. Since it is not a guaranteed truth, we cannot accept this conclusion as logically true.
Conclusion II: Douglas is my uncle's son. Similar to the first conclusion, this is another of the three possibilities. It is *possible* that Douglas is my uncle's son, but it is not *certain*. He could be one of the other two options. Therefore, we cannot accept this conclusion as logically true either.
Let's analyze the options based on our evaluation:
(A) and (B) are incorrect because neither conclusion is certainly true on its own.
(C) is incorrect because we cannot be sure that both (or even one) are true. In fact, they cannot both be true simultaneously in this context.
(E) "If either conclusion I or conclusion II is true" is tempting, but also incorrect. This option implies that the only possibilities are I and II. However, the statement provides a third possibility: "brother's son". Since Douglas could be a brother's son, it is not necessary that he must be *either* a sister's son *or* an uncle's son. Therefore, the 'either/or' construct is not logically sound here.
(D) "If neither of the conclusions is true" is the correct choice. This means that we cannot logically prove conclusion I, and we cannot logically prove conclusion II. They are mere possibilities, not certainties.
Step 4: Final Answer
Since neither Conclusion I nor Conclusion II can be definitively proven from the given statements, they are not valid logical deductions. They represent possibilities, not certainties. Therefore, the correct option is that neither of the conclusions is true (i.e., can be established as a definite fact).