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Syllogism and logical deduction practice (Venn diagram method): Given the statements 'Some doctors are fools' and 'Some fools are rich', determine which of the conclusions — (I) 'Some doctors are rich' and (II) 'Some rich are doctors' — logically follow beyond doubt.

Difficulty: Medium

Correct Answer: Neither I nor II follows

Explanation:

Given data

  • Premise 1: Some doctors are fools.
  • Premise 2: Some fools are rich.
  • Conclusions to test: (I) Some doctors are rich. (II) Some rich are doctors.

Concept/Approach

In categorical syllogisms, two particular ('Some') premises with the same middle term do not guarantee overlap between the two end terms. Use Venn diagrams or counterexample construction to check necessity versus possibility.

Step-by-step evaluation

1) Draw three sets: Doctors (D), Fools (F), Rich (R).2) Place an 'X' in the intersection D∩F to represent 'Some doctors are fools' (at least one element).3) Place another 'X' in F∩R (not necessarily the same spot) for 'Some fools are rich'.4) There is no compulsion that the two X's coincide; hence there may be no element in D∩R.5) Therefore, neither 'Some doctors are rich' nor its symmetric 'Some rich are doctors' follows with certainty.

Verification/Alternative

Create a countermodel: Let the doctor-fool individual be a (not rich) and the fool-rich individual be b (not a doctor). Both premises are true; D∩R remains empty, so both conclusions fail — proving non-necessity.

Common pitfalls

  • Confusing 'possible' with 'must be true' — both conclusions are possible but not compelled.
  • Assuming transitivity across 'Some' statements; it does not hold.

Final Answer
Neither I nor II follows.

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