Complementary statements about wealth and poverty: From 'No gentleman is poor' and 'All gentlemen are rich', decide whether the conclusions (I) 'No poor man is rich' and (II) 'No rich man is poor' necessarily follow.
Verbal Reasoning
Logical Deduction
Difficulty: Medium
Choose an option
Answer
Correct Answer: Neither I nor II follows
Explanation
Given data
- Premise 1: Gentleman ∩ Poor = ∅.
- Premise 2: Gentleman ⊆ Rich.
- Conclusions: (I) No Poor is Rich. (II) No Rich is Poor.
Concept/Approach
The premises restrict relations involving the subset 'Gentleman', not the entire sets 'Rich' or 'Poor'. Without information about non-gentlemen, we cannot generalize to all rich or all poor.
Step-by-step evaluation
1) From Premise 1 & 2: Gentlemen are rich and not poor.2) But there may exist rich non-gentlemen who could or could not be poor (no info given).3) Hence both universal claims (I) and (II) are not derivable.Verification/Alternative
Countermodel: Let Gentlemen = {g1}, Rich = {g1, r2}, Poor = {r2}. Premises hold (g1 rich, not poor), yet (I) is false (r2 is both rich and poor in a definitional sense) and (II) is false. Therefore no conclusion is necessary.
Common pitfalls
- Overextending results from a subset (Gentleman) to the whole set (Rich/Poor).
Final AnswerNeither I nor II follows.