Syllogism – Evaluate conclusions from two premises: Statements: I) Some leaves are apples. II) No apple is an egg. Conclusions: I) All apples are leaves. II) Some eggs are leaves. III) Some leaves are not eggs. IV) All eggs are leaves. Choose the option that identifies which conclusions necessarily follow.

Introduction / Context:
This classic syllogism asks which conclusions must be true given two premises about sets Leaves, Apples, and Eggs. We judge necessity, not possibility.

Given Data / Assumptions:

• Premise A: Some Leaves are Apples (∃ overlap L∩A).
• Premise B: No Apple is an Egg (A ∩ E = ∅).
• We cannot assume anything beyond these statements.

Concept / Approach:
Translate to set relations and test each conclusion for necessity across all models consistent with the premises.

Step-by-Step Solution:
C1: “All apples are leaves.” Not forced. The overlap “Some L are A” does not imply A ⊆ L.C2: “Some eggs are leaves.” Not forced. Eggs could be entirely disjoint from Leaves and Apples.C3: “Some leaves are not eggs.” Yes. The leaves that are apples (from Premise A) cannot be eggs (Premise B), so at least those leaves are outside E.C4: “All eggs are leaves.” Not forced; eggs could be outside both apples and leaves.

Verification / Alternative check:
Construct a model: Take a few elements that are both Leaf and Apple, keep Egg disjoint from Apple and possibly disjoint from Leaves. C3 holds; C1, C2, C4 fail in at least one model, so they are not necessary.

Why Other Options Are Wrong:

• None follow / All follow: Contradicted by the reasoning above.
• Either II or III follows: II does not necessarily follow; III uniquely does.
• Only I and III follow: I does not follow as explained.

Common Pitfalls:
Assuming that “some L are A” implies “all A are L,” or accidentally converting “No A are E” into “No E are L.”

Final Answer:
Only III follows.