NoShips » June 24th, 2017, 2:42 am wrote:Good question, Positor! (it crossed my own confused mind too)
In ordinary language, "
not all" seems clearly to imply "
some".
In formal syllogistic logic, I dunno how this works. Do you?
If not, any logicians out there who can help?
I am but a humble bartender, but I've done my fair share of wasting my youth. In predicate logic we can interpret
Not all bartenders do logic
As either
~(Ax(Bx > Lx))
("it is not the case that anyone who is a bartender does logic") or as
Ex(Bx & ~Lx)
("there exist people who are bartenders who do not do logic"), but natural language gives us the ability to talk about differences among various non-existent bartenders, which most formal logics don't do. "Free logics" are an exception. Anyway, if there are no bartenders then as far as most mathematical logicians are concerned, that makes "bartenders" = the empty set, 0. Which means that under some logics "all bartenders do logic" and "all bartenders do not do logic" are both true, and under other logics they are both meaningless.
So this sort of thing depends on which formal logic we are going with. The short answer is that "not all" does get treated by many logicians as meaning "some", which leads to some unusual outcomes, because we are introducing existential quantifiers where they might not belong. It allows us, for instance, to prove that it is logically necessary that something exists:
| ~Ex(x=x)
| Ax(x≠x)
Ex(x=x)
(Hypothesise that it is not the case that something exists which is equal to itself. Therefore everything is not equal to itself. Hypothesis proven false by contradiction. Therefore something exists which is equal to itself. QED.)