The given premises are: All film stars are playback singers, All film directors are film stars.
The conclusions are: All film directors are playback singers, Some film stars are film directors.
We have to show how we arrive at these conclusions using rules of inference. I was able to prove the 1st conclusion using hypothetical syllogism but am having trouble with the second conclusion. I know it's true but I can't seem to express my thinking into rules of inference.
On
That second conclusion would only be true if we know that there is at least one film director. For in that case, that film director would have to be a film star, and hence there is something that is both a film director and a film star. By commutation, that mean that that something is both a film star and a film director, and hence some film stars are film directors.
In fact it's not possible to prove the second conclusion using what is given. The premises are consistent with the possibility that there do not exist any film stars, nor any playback singers, nor any film directors at all. And if that's the case then the statement "some film stars are film directors" is false.