Syllogism – Mix of “some” and “all” with existence caution: Statements: (i) Some papers are pens. (ii) All pencils are pens. Conclusions: I) Some pens are pencils. II) Some pens are papers. Identify which conclusions necessarily follow.

Introduction / Context:The trap here is drawing an existential conclusion from a universal statement. We carefully separate what is guaranteed from what is merely possible.

Given Data / Assumptions:

• Some Papers are Pens (∃ Ppr∩Pen).
• All Pencils are Pens (Pcl ⊆ Pen).

Concept / Approach:A particular premise (“some”) certifies existence; a universal premise (“all”) does not by itself yield a “some” conclusion unless the subject class is known to be non-empty.

Step-by-Step Solution:Conclusion II: “Some pens are papers.” This follows directly from premise (i); the very elements witnessing (i) ensure Pens∩Papers is non-empty.Conclusion I: “Some pens are pencils.” From “All pencils are pens” we cannot assert that pencils exist. Without existential import, this conclusion is not necessary.

Verification / Alternative check:Countermodel for I: Imagine there are no pencils at all. Premises remain true, but “Some pens are pencils” becomes false. Hence I does not necessarily follow.

Why Other Options Are Wrong:

• Both follow: invalid because I can fail under a valid model.
• Only I follows: ignores the direct particular given in (i).
• Either I or II follows: too weak; II specifically follows while I may fail.

Common Pitfalls:Assuming that “All A are B” implies “Some B are A” without separately establishing that A exists.

Final Answer:Only II follows.