Consider the following statements as true and decide which of the given conclusions can definitely be drawn. Statements: Politicians marry only beautiful girls. X is beautiful. Conclusions: (1) X will marry a politician. (2) X will not marry a politician.

Introduction / Context:
This question is about logical inference from given statements. The statement about politicians and beautiful girls specifies a condition on whom politicians marry, and then you are told that X is beautiful. You must determine which conclusions about X's future marriage follow logically. This type of problem tests understanding of the difference between one way and two way conditions in logic.

Given Data / Assumptions:

• Statement: Politicians marry only beautiful girls.
• Statement: X is beautiful.
• Conclusion 1: X will marry a politician.
• Conclusion 2: X will not marry a politician.
• You must decide which conclusions follow necessarily from the statements.

Concept / Approach:
The key phrase marry only beautiful girls means: if someone is a politician and marries, then the person he marries is beautiful. Symbolically, Politician spouse implies Beautiful. This does not say that every beautiful girl marries a politician. That would require the reverse condition: if someone is beautiful, then she marries a politician, which is not stated. When a statement is one directional, you cannot reverse it. Therefore, knowing that X is beautiful does not guarantee that X will marry a politician or that X will not marry a politician.

Step-by-Step Solution:
Step 1: Translate the first statement. Politicians marry only beautiful girls means: For any politician, his wife is beautiful.Step 2: Note that this restriction is about politicians and their choice of spouse, not about all beautiful girls and the husbands they will marry.Step 3: The second statement simply tells us that X is beautiful.Step 4: Conclusion 1 states that X will marry a politician. For this to follow, we would need the rule that all beautiful girls marry politicians. We have no such rule. Many beautiful girls may marry non politicians.Step 5: Therefore conclusion 1 does not logically follow.Step 6: Conclusion 2 states that X will not marry a politician. This also cannot be deduced, because X might marry a politician or might not. The given statements do not restrict X's choice in that way.Step 7: As we cannot determine whether X will or will not marry a politician, neither conclusion 1 nor conclusion 2 definitely follows from the statements.

Verification / Alternative check:
Consider two different possible worlds that fit the given statements. In World A, X is beautiful and later marries a politician. This is allowed because politicians marry only beautiful girls. In World B, X is beautiful and marries a doctor who is not a politician. This is also allowed because there is no statement that restricts whom beautiful girls may marry. Since both worlds are consistent with the premises but give opposite outcomes for X’s spouse, no definite conclusion about X’s marriage partner can be drawn.

Why Other Options Are Wrong:

• Option b claims only conclusion 2 follows, suggesting X definitely will not marry a politician, which is not supported by the premises.
• Option c suggests that one of the two conclusions must hold, but we cannot determine which; however, the question is about what can be definitely concluded from the premises, not about what must happen in real life.
• Option d claims conclusion 1 follows, wrongly treating the statement as if all beautiful girls must marry politicians.
• Option e claims both conclusions follow, which is impossible since they contradict each other.

Common Pitfalls:

• Reversing the logical direction of the statement politicians marry only beautiful girls and treating it as beautiful girls marry only politicians.
• Assuming that being beautiful completely determines whom X will marry.
• Forgetting that logical follow means guaranteed by the premises, not just possible or plausible in real life.

Final Answer:
Neither conclusion 1 nor conclusion 2 follows from the given statements.