Negation of "All" and "Some"

Student...

I'm studying the curriculum and found the negation of "all" a little confusing.
I understand that the logical opposite of "some" is just "none."
But isn't "not all" the negation of "all"?
In other words, does "some... not..." mean the same thing as "not all"?

Thank you very much!

Student...

Here's the link to the lesson:
http://7sage.com/lesson/advanced-negate-all-statements/?ss_completed_lesson=1055

Other people expressed the same confusion in the comments too.
Thanks again.

Q.E.D

Gotta follow the negation or else it doesn't make any sense.

'ALL ravens are black.' = 'NOT SOME (not any) ravens are NOT black.'
'NOT ALL ravens are black.' = 'NOT NOT SOME (or just some) ravens are NOT black.'
'SOME ravens are black.' = 'NOT ALL ravens are NOT black.'
-in all cases, one quantifier is negated and one predicate is negated

One quantifier ('all' or 'some') is definable in terms of the other, negated with a negated predicate. I think it confuses people that you have to rope in the predicate ('black' above). If you're hazy on this, you might benefit from a quick study in the object-predicate structure of language, i.e. predicate logic. The notation is really handy. It makes it easy to see the principle at work above.

For example, ∃xPx iff ~∀x~Px. Here you see that '∃x' does the work of '~∀x~'.

But it might be easier just to think of what would make a quantified statement true/false using the other quantifier. If you just think of what it means that all ravens are black, namely that every single one of them is black, then it follows that you couldn't possibly have a not-black one. If you had some raven that was not-black, it would be false that all ravens are black (hence not all...). Again, if some cat is furry, surely not all cats are not furry, because clearly at least one cat is indeed furry. But If no cat is furry, you can be sure all of them are not furry.

It may help to check out of the interdefinability of quantifiers here:
https://en.wikipedia.org/wiki/Quantifier_(logic)

Check out "Equivalent Expressions". Not to worry, simpler than it looks. If want a thorough run-down, go here:
http://plato.stanford.edu/entries/quantification/

Good luck mang

nicole.hopkins

Wow.

Student...

Wow. Thanks.

Q.E.D

meh, I talk too much

I want to clarify my meaning in “follow the negation”: When the ‘not’ crosses the quantifier, the quantifier changes. Observe:

(i) Not…some…
(ii) --------> Not  pulling that ‘not’ over to this side
(iii) All...-->Not… and the quantifier changed
(iv) Not<--------- and we’re headed back
(v) Not…some… and the quantifier changed back

Think of it as a choice about where to negate. You use a different quantifier depending on where you choose, but you’ll be using equivalent expressions, Also remember that ‘not some’ is weird logician’s locution for ‘not any’ or ‘none’.

Now concrete examples:
(i) ‘ <b>Not all cats are furry.’ = ‘Some cats are not furry’
(ii) ‘ Not any cats are evil.’ = ‘All cats are not evil’.

In both cases, I dropped the negation through to the sentence and the quantifier switched. Pulling the negation back out of the sentence and in front of the quantifier has the same effect. Notice that ‘not any’ (not…some) in (ii) belongs with ‘all…not’. Likewise, ‘some…not’ in (i) belongs with ‘not all’; so yes, they're different.

I think you got the hang of it, but I’m sorry if this only confused you more. I guess there’s a reason we turn to folks like Ping for assistance.

Good luck crushing the LSAT!

Q.E.D

bold on (i) fail, life goes on

apublicdisplay

I could not get through the dense fog that is Q.E.D.'s comment but I thought "Some...Not" (0-99) is equivalent to "Not All" (0-99).

Pacifico

@apublicdisplay said:
I could not get through the dense fog that is Q.E.D.'s comment

That's it... Bertrand Russell's new nickname is "The Fog"...

Q.E.D

That's it... Bertrand Russell's new nickname is "The Fog"...

meh, guess I had that coming

Anyway, props for being able to identify Russell.