Syllogism — Envelopes, Gums, Seals, and Adhesives Statements: • Some envelopes are gums. • Some gums are seals. • Some seals are adhesives. Conclusions: (1) Some envelopes are seals. (2) Some gums are adhesives. (3) Some adhesives are seals. (4) Some adhesives are gums.

Introduction / Context:This problem asks which conclusions necessarily follow given three “some” statements across four categories. With “some,” overlaps are possible but not guaranteed unless directly stated or logically converted.

Given Data / Assumptions:

• E = envelopes, G = gums, S = seals, A = adhesives.
• Premises: Some E are G; Some G are S; Some S are A.
• Conclusions to test: (1) Some E are S; (2) Some G are A; (3) Some A are S; (4) Some A are G.

Concept / Approach:Only conclusions that must be true in every diagram consistent with the premises are valid. From “Some S are A,” we can apply conversion for “some”: “Some A are S.” Other claimed overlaps require the same individuals to be shared across different “some” statements, which is not forced.

Step-by-Step Solution:

Verification / Alternative check:Construct a counterexample: place one item in E∩G; a different item in G∩S; and another in S∩A. All premises hold while E∩S, G∩A, and E∩A remain empty. Thus only (3) must follow.

Why Other Options Are Wrong:

• (1), (2), (4) each assumes compelled overlap among different “some” statements. This is not logically necessary.
• Only (3) is the unique must-follow conclusion.

Common Pitfalls:Chaining “some” statements as if they behave like “all.” Remember: “some” + “some” does not compel intersection beyond what is explicitly stated.

Final Answer:Only (3)