Double Not Arrow

[yongjook]

Hi,

I was reading up on pg.382 part C. Double not arrow.

It says "If you are a T, then you are not a V", which could be translated as

T->not V and its contrapositive V->not T
therefore V T.

Does "no Xs are Ys" work the same way?

not X- > Y and not Y ->X

Therefore X Y. Would this mean that if exactly one of the term is negative then it would be double not arrow?

Thank you

[Dave Killoran]

Hi Yongjook,

Thanks for the question! This is an important question, and one that comes up on this Forum frequently, including just the other day. There's two issues in your question, so let's address both.

The first is that "no Xs are Ys" is diagrammed X Y (if you are an X then you are not a Y), so it does become X Y, but only because it contains a positive sufficient condition and a negative necessary condition.

The second issue is what happens when we have a statement such as "if you are not an X, then you are a Y?" That is indeed X Y, but that does NOT then equate to X Y. It actually equates to X Y, which means the same as at least one is present, possibly both.

This is a tricky concept, and so I'd suggest reading the post I made on Monday, at http://forum.powerscore.com/lsat/viewto ... 809#p19809. It's about a game rule, but just focus on the rule part (which is in the form of J S). I made a related response at http://forum.powerscore.com/lsat/viewto ... 625&p=6880 (read my indented portion).

Please check those out, and if they don't help clear up the idea, please let me know and I'll go into greater detail.

Thanks!

[HarmonRabb]

Hi Dave,
Above you say

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if you are not an X, then you are  a Y... 
That is indeed /X--->Y but that does NOT then equate to X<---|--->Y

But on pg 225 of the 2024 LR book you say

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.. .when G<---|--->H is in effect, then several possible scenarios can still occur:
...
/G and H
...

To me, these statements seem contradictory. Can you help me make sense of them (or resolve the paradox if you will :P)

[Adam Tyson]

Hey Harmon, let me help! Note that in Dave's explanation in that section of the LR Bible, he says that there are several possibilities when G and H have a double-not arrow, and that G out and H in is just one of those possibilities, the others being that G is in and H is out, and that they are both out.

There's no contradiction here. The double-not relationship between G and H does not mean that whenever G is out, H must be in. That's why it's possible that they are both out. But it does ALLOW for that possible scenario.

So, if G is in, H is out, means that they can never both be IN. Either they are both out, or one is in and the other is out.

And the rule that if X is out, Y is in means that they can never both be OUT. They can both be in, or you can have one in and one out.

When the Sufficient Condition is positive and the Necessary Condition is negative, you get the relationship shown here between G and H. But when the Sufficient Condition is negative and the Necessary Condition is positive, you get the relationship illustrated by X and Y.

Check it again and see if that clears up your confusion!

[HarmonRabb]

Adam, thanks for the detailed response. I have a couple follow up questions.

If we want to convey both terms cannot both be OUT is that /A<----|---->/B ?
i.e. If Mac is not prosecuting the case, then Harm is (negative sufficient condition, positive necessary condition)

If we want to convey that both terms cannot both IN, is that A<----|---->B ?
i.e. If Mac is trying the case, Harm cannot be (positive sufficient condition, negative necessary condition)

[Adam Tyson]

Yes, exactly! Although I don't personally use the double-not arrow in the first case, where the sufficient condition is negative and the necessary condition is positive. Just for myself, that gets a little confusing. But it's accurate, because it means the two things on either said of that arrow cannot both simultaneously occur. In that case, they cannot both be out.

[HarmonRabb]

Thank you so much!