Isn't this wrong: if `all A are B` and `all B are C` then `some A are C`?

Question

I found that statement on Generalized Quantifier page of the Stanford Encyclopaedia of Philosophy.

all(A,B) # read all A are B
all(B,C)
-------- # therefore
some(A,C)

I'd expect the conclusion to be all(A,C), but I'm also surprised that error went unnoticed since Winter 2019.

Can anyone confirm what's the correct one?

Answer 1

The article is not in error. Please read the article carefully. It says that since Aristotle considered the syllogism

$$All \ A \ are \ B$$ $$All \ B \ are \ C$$ $$Some \ A \ are \ C$$

to be valid, he must have assumed that there is at least one $A$. Indeed, if there is at least one $A$, then the syllogism is valid.

More generally, within categorical logic (which is what Aristotle studied) the Assumption of Categorical Existential Import is the assumption that for any class (category) of things you talk about there is at least one representative. So under that assumption, the syllogism in the article is valid. Sometimes we say that this categorical syllogism is conditionally valid, as opposed to some other categorical syllogisms, which are unconditionally valid, i.e. do not need this extra Assumption. An obvious example of the latter is of course:

$$All \ A \ are \ B$$ $$All \ B \ are \ C$$ $$All \ A \ are \ C$$