Syllogism – Three chained “some” statements (watch for non-guaranteed transitivity) Statements: • Some blades are hammers. • Some hammers are knives. • Some knives are axes. Conclusions to test: I. Some axes are hammers. II. Some knives are blades. III. Some axes are blades.

Introduction / Context:
A frequent trap in syllogisms is assuming that three separate “some” overlaps chain together to force an overlap between the first and the last categories. Without a single element proven to belong to all necessary intersections, no such transitive conclusion is guaranteed.

Given Data / Assumptions:

• Bd ∩ Hm ≠ ∅ (some blades are hammers).
• Hm ∩ Kn ≠ ∅ (some hammers are knives).
• Kn ∩ Ax ≠ ∅ (some knives are axes).

Concept / Approach:
Each “some” might refer to different subsets of the middle category. For example, the hammers that are blades could be different from the hammers that are knives. Similarly, the knives that are hammers could be different from the knives that are axes. Without explicit linkage, you cannot assert overlaps like Ax ∩ Hm or Bd ∩ Kn.

Step-by-Step Solution:
I: “Some axes are hammers.” Not compelled; the knives that are axes need not be the same knives that are hammers.II: “Some knives are blades.” Not compelled; the hammers that are knives need not be the same hammers that are blades.III: “Some axes are blades.” This would require tracking an element across all three statements, which is not guaranteed.

Verification / Alternative check:
Construct a counterexample: pick distinct hammer subsets for the first two intersections and distinct knife subsets for the latter two; ensure no single element spans all. All premises remain true while I–III fail, proving none necessarily follow.

Why Other Options Are Wrong:

• Options claiming any single conclusion follow assume a shared element across intersections that is not provided.
• “None of these” is inapplicable because the correct choice is explicitly “None follows.”

Common Pitfalls:
Forgetting that “some A are B” and “some B are C” does not imply “some A are C.”

Final Answer:
None follows