Is this Syllogism Valid? 
All widgets are round.
All widgets are smooth.
Therefore, some smooth things are round.

My intuitive answer is YES.
The provided answer is "NO". Please explain.
 
 
 
Why I think the answer is "YES".

 A: This is exactly why Venn diagrams have an advantage!
Start with a basic 3-set Venn diagram:

Take the first premise:

All widgets are round

This means that there cannot be any widgets that are not round. So, we shade the area inside Widget and outside Round, which is our way of saying: "there is nothing here":

Take the second premise:

All widgets are smooth

This has the same logical form as the previous claim, so shade the appropriate area:

OK, the conclusion is:

Some smooth things are round

OK, for there to be smooth things that are round, we need something in the intersection of Smooth and Round.  Let's see if there is something there:

No, we can't say that there is definitely something there, since the areas are 'blank', and a 'blank' area means that there could be something there, but you don't know (as a mnemonic: you 'draw a blank'!). So, the conclusion is not necessarily true. Hence, the argument is invalid.
Now, if you knew that there are widgets (if, for example, you make the Assumption off Categorical Existential Import, which says that for any class/set/category of objects, there is always one element/member ... an assumption logic typically does not make ... but if you do:), then there is only one place for those widgets to be:

(The 'X' means: there is at least one thing here')
And now of course the argument would follow.  As such, this kind of argument is sometimes called 'conditionally valid'. But unconditionally, it is invalid.
