Is "only A are B" the same as "all B are A"?

Question

I have two syllogisms which I think are pretty similar. Correct me if I am wrong but I believe "Only A are B" is the same as "All B are A".

Only A are B (Can be rewritten as "All B are A")
Only B are C
Therefore, only A are C.

I believe this syllogism to be valid because if all B are A and all C are B (which are As), it follows that all C are A.

All A are B
All C are A
Therefore, some C are B.

I believe this syllogism to be valid as well (I think all C are B) because I think all A are B and C as well (and all B are A and C...).

Am I correct for both syllogisms? I'm having a bit of a hard time trying to understand the "only" and "all" words used in these syllogisms.

Answer 1

In the first one, it is valid to go from All B are A; all C are B; therefore all C are A. In predicate logic this would be called a hypothetical syllogism. Aristoteleans would call it a syllogism in Barbara.

In the second one, you are concluding only that some C are B. In Aristotelean logic this is OK, because "all C are B" is understood to have existential import (i.e. it presupposes there are some C's) and so it is OK to move from there to some C are B. In standard predicate logic, one is allowed to say all C are B even if there are no C's, so this is not valid.

Answer 2

When translating these logical relations into set theory, "Only A are B" translates into "The set of B's is a subset of the set of A's". I shall use the shorthand "B is a subset of A".

On the other hand, "All A are B" translates into "A is a subset of B".

ad 1: B is a subset of A and C is a subset of B implies C is a subset of A. Hence: Only A are C.

ad 2: A is a subset of B and C is a subset of A implies C is a subset of B. Hence: All C are B. But herefrom you cannot conclude: Some C are B.

Because C may be the empty set, i.e., there are no C's at all. But Some C are B means: At least one C is B.