Difficulty: Medium
Correct Answer: None of the conclusions follows.
Explanation:
Introduction / Context:
This syllogism question involves four sets: blades, hammers, knives and axes. Each statement gives a "some are" relationship between two sets, and you must decide whether any of the proposed conclusions are logically forced by these partial overlaps. The challenge is to avoid assuming that all overlaps intersect each other automatically.
Given Data / Assumptions:
Concept / Approach:
Each "some" statement merely guarantees that an intersection between two sets is non empty. It does not say that the same items appear in multiple intersections. Therefore, three separate "some" statements about pairwise overlaps do not necessarily create a common element shared by three or four sets. We must check whether the conclusions require such shared elements and if the premises force that situation.
Step-by-Step Solution:
Step 1: Represent the statements as intersections:
• B ∩ H ≠ ∅ (Some blades are hammers).
• H ∩ K ≠ ∅ (Some hammers are knives).
• K ∩ A ≠ ∅ (Some knives are axes).
Here B = blades, H = hammers, K = knives, A = axes.
Step 2: None of these statements indicate that the particular hammer which is also a blade is the same as the hammer which is also a knife, or that the knife which is also a hammer is the same as the knife which is also an axe.
Step 3: Test Conclusion 1: "Some axes are hammers."
For this to be necessarily true, the intersection A ∩ H must be non empty in every possible diagram. However, we can construct a diagram in which:
• One object is both blade and hammer (B ∩ H).
• A different object is both hammer and knife (H ∩ K).
• A third object is both knife and axe (K ∩ A).
If all three objects are distinct, then no axe is a hammer, so Conclusion 1 fails even though all statements are true.
Step 4: Similarly, for Conclusion 2: "Some knives are blades," we would need K ∩ B ≠ ∅. Our constructed diagram does not require that; the knife involved with hammers and the blade involved with hammers can be different, so knives and blades need not overlap.
Step 5: For Conclusion 3: "Some axes are blades," we would need A ∩ B ≠ ∅. Nothing in the premises forces this, and in our example with three distinct overlapping objects, no axe is a blade.
Verification / Alternative check:
The key verification is the counterexample: choose three distinct items x, y and z such that:
• x is a blade and a hammer (B ∩ H).
• y is a hammer and a knife (H ∩ K).
• z is a knife and an axe (K ∩ A).
Then all three original statements hold, but A ∩ H, B ∩ K and A ∩ B are all empty. Because a single consistent diagram exists where none of the conclusions hold, we know that none of them is logically necessary.
Why Other Options Are Wrong:
Options B, C and D each claim that at least one conclusion must follow, but we have demonstrated a valid scenario where all conclusions fail while all statements remain true. Therefore, these options are incorrect. Only Option A, which states that none of the conclusions follows, matches the logical analysis.
Common Pitfalls:
Many students intuitively chain "some" statements as if they were "all" statements, assuming that overlaps must pass through. For example, they think that if some blades are hammers and some hammers are knives, then some blades must be knives. This is not logically required, because the hammer involved in the first overlap can be different from the hammer in the second. Always remember that "some" does not imply that all relationships share the same individuals.
Final Answer:
None of the proposed conclusions is guaranteed by the statements, so none of the conclusions follows.
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