Syllogism Statements Analysis

The question asks us to analyze a set of statements and identify which conclusions can be logically deduced. This type of problem falls under deductive reasoning or syllogism. We will evaluate each statement and conclusion systematically.

Understanding the Given Statements

Let's represent the categories:

• Tables (T)
• Shelves (S)
• Chairs (C)
• Benches (B)

The provided statements are:

1. Some tables are shelves. (T ↔ S, with some overlap)
2. Some shelves are chairs. (S ↔ C, with some overlap)
3. All chairs are benches. (C ⊂ B, meaning every chair is also a bench)

Evaluating Each Conclusion

We will now check each conclusion against the given statements to determine its validity.

Conclusion i) At least one bench is a table

Let's trace the connections:

• "Some tables are shelves" (T ↔ S)
• "Some shelves are chairs" (S ↔ C)
• "All chairs are benches" (C ⊂ B)

From (S ↔ C) and (C ⊂ B), we can deduce that some shelves are benches. However, knowing "Some tables are shelves" and "Some shelves are benches" does not guarantee that some tables are benches. The "some shelves" that are tables might be different from the "some shelves" that are benches. There is no definite overlap established between tables and benches through these statements.

Deduction Status: Cannot be deduced.

Conclusion ii) At least one shelf is a bench

Let's connect the relevant statements:

• "Some shelves are chairs." (S ↔ C) — This means there is at least one item that is both a shelf and a chair.
• "All chairs are benches." (C ⊂ B) — This means every single chair is also a bench.

If there is an item that is a shelf and also a chair, and every chair is a bench, then that specific item must also be a bench. Therefore, that item is both a shelf and a bench.

Deduction Status: Can be deduced.

Conclusion iii) At least one chair is a table

Let's look at the connections:

• "Some tables are shelves." (T ↔ S)
• "Some shelves are chairs." (S ↔ C)

Similar to conclusion (i), the "some shelves" that are tables might be distinct from the "some shelves" that are chairs. When two 'some' statements share a common term (shelves), it does not automatically lead to a conclusion about the first and last terms (tables and chairs). There is no guaranteed overlap between tables and chairs.

Deduction Status: Cannot be deduced.

Conclusion iv) All benches are chairs

The original statement is "All chairs are benches" (C ⊂ B). This means the set of chairs is a subset of the set of benches. The conclusion "All benches are chairs" (B ⊂ C) is the converse of the original statement.

The converse of an "All A are B" statement is not necessarily true. For example, if "All dogs are animals," it does not mean "All animals are dogs" (there are other animals like cats, birds, etc.). Similarly, there can be benches that are not chairs.

Deduction Status: Cannot be deduced.

Summary of Conclusions

Based on our analysis, only conclusion (ii) can be logically deduced from the given statements. Therefore, the correct option is "Only ii".