Syllogism chain reasoning: given 'Some kings are queens' and 'All queens are beautiful', determine whether it follows that all kings are beautiful or that all queens are kings

Given data

• Premise 1: Some kings are queens.
• Premise 2: All queens are beautiful.
• Conclusions:
  • I: All kings are beautiful.
  • II: All queens are kings.

Concept/Approach

From 'Some K are Q' and 'All Q are B', the valid derived conclusion is 'Some kings are beautiful'. However, neither of the provided universal conclusions is justified.

Step-by-step reasoning
1) Some K ∩ Q ≠ ∅ and Q ⊆ B → Some K ⊆ B (particular conclusion).2) I (All K ⊆ B) overgeneralizes from 'some' → not valid.3) II (All Q ⊆ K) is converse; not supported.

Verification/Alternative

Countermodel: Let K have members {k1,k2}, Q = {k1}, B contain Q and others. Then 'some K are B' true, I and II false.

Common pitfalls

• Illicit conversion of 'some' to 'all'.
• Assuming symmetry between classes K and Q.

Final Answer
Neither I nor II follows.