Consider the following three facts: Fact 1: All hats have brims. Fact 2: There are black hats and blue hats. Fact 3: Baseball caps are hats. Based on these facts alone, which of the following statements must also be a fact?

Introduction / Context:
This question is a precise test of logical inference. We are given three facts about hats, their colours and baseball caps. The candidate statements mix partial information with extra claims. We must determine which, if any, of the candidate statements is forced by the three given facts. The correct answer is based on what must be true, not what might be true in some situations.

Given Data / Assumptions:

• Fact 1: All hats have brims.
• Fact 2: There are black hats and blue hats.
• Fact 3: Baseball caps are hats.
• We do not know anything else about caps in general except that baseball caps form a subset of hats.

Concept / Approach:
Logical necessity means that in every possible world where the three facts are true, the candidate statement must also be true. If there exists at least one scenario in which all three facts hold but the candidate statement fails, then that candidate is not logically necessary. We will test each candidate statement against this standard, imagining alternative situations consistent with the facts.

Step-by-Step Solution:
Step 1: From Fact 1, we know that any object that qualifies as a hat has a brim.Step 2: From Fact 3, baseball caps are hats, so they must have brims. Thus it is impossible for a baseball cap to have no brim while Fact 1 remains true.Step 3: Fact 2 states that there are hats of two colours, black and blue. It does not say anything about whether those coloured hats are baseball caps or some other kind of hat.Step 4: Consider statement I, All caps have brims. The given facts only guarantee that baseball caps have brims, because they are hats. They say nothing about other kinds of caps that might not be hats, such as swimming caps. Therefore, it is possible that some caps which are not hats have no brim. So statement I is not forced.Step 5: Consider statement II, Some baseball caps are blue. Although Fact 2 says some hats are blue, we are not told that these blue hats are baseball caps. They could all be different hat types. So some baseball caps are blue is not a necessary fact.Step 6: Consider statement III, Baseball caps have no brims. This directly contradicts Fact 1 combined with Fact 3, so it definitely cannot be a fact in the presence of the given information.Step 7: Since none of the individual candidate statements I, II or III is forced by the facts, no option that asserts any of them as a must be fact can be correct.

Verification / Alternative check:
Construct a scenario: suppose all blue hats are wide brimmed sun hats, and all baseball caps happen to be black. The facts are still true; there are blue hats, all hats have brims, and baseball caps are hats.In this scenario, some baseball caps are blue is false, showing that it is not necessary.We can also imagine caps such as shower caps that are not hats and do not have brims, showing that not all caps must have brims.

Why Other Options Are Wrong:
Option a assumes every cap in existence is a hat, which is not supported by the facts.Option b assumes at least one baseball cap shares the colour blue, but the facts do not connect colour distribution specifically to baseball caps.Option c bundles all three candidate statements, including the one that contradicts the given facts, and is therefore impossible.Option e asserts that baseball caps have no brims, which directly contradicts Fact 1 and Fact 3.

Common Pitfalls:
Assuming that all caps are hats without any explicit statement to that effect.Treating might be true as must be true, especially for colour or category assignments.Overlooking that the question asks which statement must also be a fact, not which statements could possibly be consistent.

Final Answer:
Since none of the candidate statements is guaranteed by the three given facts, the correct choice is None of the given statements must be a fact.