Logical syllogism — determine which conclusions follow beyond doubt. Statements: 1) All the phones are scales. 2) All the scales are calculators. Conclusions: (1) All the calculators are scales. (2) All the phones are calculators. (3) All the scales are phones. (4) Some calculators are phones.

Introduction / Context: Two universal premises describe a chain: phones ⊆ scales ⊆ calculators. We must find the conclusions that necessarily follow from this chain of inclusion.

Given Data / Assumptions:

• All phones are scales.
• All scales are calculators.
• Standard existence assumption for “Some …” based on presence of the subject class.

Concept / Approach: Transitivity of subset: if A ⊆ B and B ⊆ C, then A ⊆ C. Converse statements such as “All C are B” do not follow. From “All A are C,” we may infer “Some C are A” when A is non-empty, which these tests typically assume.

Step-by-Step Solution: From the chain, phones ⊆ calculators, so conclusion (2) holds. Because phones exist, some calculators are phones (those very phones), so conclusion (4) holds. Conclusion (1) claims the converse of Statement 2 for calculators→scales, which is not implied; it can be false. Conclusion (3) claims scales ⊆ phones, also not implied.

Verification / Alternative check: Example model: phones = {p1}, scales = {p1, s1}, calculators = {p1, s1, c1}. Then (2) and (4) are true, but (1) and (3) are false. Thus only (2) and (4) must follow.

Why Other Options Are Wrong: Any option including (1) or (3) assumes a converse or reversal of inclusion not supported by the premises.

Common Pitfalls: Believing “All B are C” implies “All C are B”. Always check direction of inclusion.

Final Answer: Only (2) and (4)