Which one of the following propositions is logically equivalent to the proposition-"Some attorneys are logicians"?

The correct option is: Some logicians are attorneys.

To understand why these two propositions are logically equivalent, we can analyze the statement using categorical logic and set theory.

1. Categorical Logic Analysis:
In classical logic, the proposition "Some attorneys are logicians" is a particular affirmative proposition (conventionally referred to as an I-proposition). It has the general form:
"Some S is P", where S (Subject) is "attorneys" and P (Predicate) is "logicians".

In categorical logic, the process of swapping the subject and predicate terms of a proposition is called conversion. For an I-proposition, conversion is logically valid and yields an equivalent statement. Therefore, converting "Some S is P" gives:
"Some P is S"

Substituting our terms back, this gives "Some logicians are attorneys." Since conversion is valid for particular affirmative propositions, the two statements are logically equivalent.

2. Set Theory Representation:
Let A represent the set of all attorneys, and let L represent the set of all logicians.

The statement "Some attorneys are logicians" means that there is at least one member in set A that also belongs to set L. In set notation, this means the intersection of set A and set L is non-empty:

$$A\cap L\ne \varnothing$$

Because the intersection operation in set theory is commutative (meaning the order of the sets does not change the result), we can write:

$$A\cap L=L\cap A$$

Therefore, it logically follows that:

$$L\cap A\ne \varnothing$$

Translating the set notation $L\cap A\ne \varnothing$ back into English gives "Some logicians are attorneys." Thus, the equivalence is mathematically and logically absolute.