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Intersecting particulars problem: given ‘‘Some books are tables’’ and ‘‘Some tables are mirrors’’ decide whether mirrors overlap books or whether books and mirrors are disjoint.

Difficulty: Medium

Correct Answer: Neither I nor II follows

Explanation:

Both premises are particular and speak only about overlaps with Tables.

  • Premise 1: Books ∩ Tables ≠ ∅.
  • Premise 2: Tables ∩ Mirrors ≠ ∅.
  • Conclusions: I. Some mirrors are books. II. No book is mirror.

Concept/Approach
Two separate overlaps with the same middle set do not force a direct overlap, nor do they preclude it. Both a positive and a negative relation between Books and Mirrors remain possible.
Testing conclusion I
There need not be a common element across all three sets. The book table items and the table mirror items could be different elements. Hence I is not necessary.
Testing conclusion II
The premises do not rule out Books ∩ Mirrors. Therefore claiming disjointness is not justified; II does not follow.
Verification/Alternative
Model A: Tables = {x, y}, Books = {x}, Mirrors = {y} gives neither overlap nor disjoint proof of I or II. Model B: Tables = {x}, Books = {x}, Mirrors = {x} makes I true and II false. Since both patterns are possible, neither conclusion is forced.
Common pitfalls
Assuming transitivity for particular statements with the same middle term.
Final Answer
Neither I nor II follows.
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