Syllogism – Determine which conclusions logically follow Statements: • Some pictures are frames. • Some frames are idols. • All idols are curtains. Conclusions to test: I. Some curtains are pictures. II. Some curtains are frames. III. Some idols are frames.

Introduction / Context:
This problem tests standard categorical logic (syllogism) using set relations such as “all,” “some,” and “no.” You must decide which conclusions are guaranteed by the three given statements without injecting extra assumptions. Visualizing with simple Venn-style overlaps or set inclusion helps prevent common traps.

Given Data / Assumptions:

• Some pictures are frames (P ∩ F is non-empty).
• Some frames are idols (F ∩ I is non-empty).
• All idols are curtains (I ⊆ C).
• “Some” asserts existence; “all” indicates subset containment.

Concept / Approach:
Convert each English statement into set relations and propagate inclusion: when “all idols are curtains,” every idol element also belongs to curtains. When “some frames are idols,” the very same elements are frames and idols simultaneously; by inclusion, they are also curtains. Be careful: two different “some” statements need not refer to the same specific elements.

Step-by-Step Solution:
Check II: “Some curtains are frames.” From “some frames are idols” and “all idols are curtains,” the frame elements that are idols are automatically curtains. Therefore at least one element is both curtain and frame. II follows.Check III: “Some idols are frames.” This is the direct converse of “some frames are idols.” With “some,” conversion is valid: if some F are I, then some I are F. III follows.Check I: “Some curtains are pictures.” We know some frames are idols and thus curtains, and (separately) some pictures are frames. But these two “some” groups need not be the same frames. There is no forced overlap between pictures and idols. Hence I is not guaranteed.

Verification / Alternative check:
Create a model where one subset of frames overlaps idols (becoming curtains) and a different subset of frames overlaps pictures, with those subsets disjoint. All premises hold while I fails, confirming that only II and III must follow.

Why Other Options Are Wrong:

• I and II: includes I, which is not compelled.
• I and III: again relies on unforced overlap between pictures and curtains.
• All follow: overly strong; I fails in valid models.
• None of these: incorrect because II and III do follow.

Common Pitfalls:
Assuming two separate “some” statements point to the same elements; assuming transitivity for “some” across different pairs without evidence.

Final Answer:
Only II and III follow