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LSAT LR Bible 2016 - "either/or... but not both" pg 198

Rnmakrinos
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Hi. I just wanted to confirm that 'either/or... but not both' is supposed to yield a :dblline: construct. For example on page 198 of the 2016 LR Bible, "Either Cindy or Clarice will attend the party, but not both" is supposed to yield the following construct - right?

—Ci :arrow: Cl
—Cl :arrow: Ci
>
Ci :arrow: —Cl
Cl :arrow: —Ci
>
Ci :dblline: Cl
>
Therefore, at least one of the two (Ci, Cl) but not both can be selected.

Thanks!
Adam Tyson
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Thanks for the question, Rnmakrinos, and your analysis is good but only gets you halfway there. The use of the double-not-arrow ( :dblline: ) indicates that the two things on either side of the arrow cannot both occur. In your diagram of Ci :dblline: Cl, we would read that as "you cannot have both Cindy and Clarice". The part that's missing there is the other half of the rule, and that;s that you must have one of them. Here's how you show that:

Ci :dblline: Cl

It's the double-not-arrow again, and it still means that the two things on either side of it cannot both occur, but here those two things that cannot both occur are Cindy being "out" and Clarice also being "out". The first one you did means they can't both be in, and the second one here means they can't both be out. NOW we know that we will have exactly one of the two of them in and the other out. Does that make sense?

There are other ways to diagram a rule like this, especially in a grouping game where you might fill up a slot in your base with "Ci/Cl", indicating that one of them must fill that space, and then perhaps a :dblline: to indicate not both, but for LR purposes, if you must diagram the claim, use BOTH double-not-arrows to cover both sides of that rule.

Let us know if you need further clarification, as this is an area where a lot of folks struggle! We're here to help whenever you need us.
Adam M. Tyson
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Rnmakrinos
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Im not totally understanding bc I'm used to the examples from the book. So if I understand you correctly, the statements and their respective constructs would be:

"Either Cindy or Clarice will attend the party."
Ci :dblline: Cl

"Either Cindy or Clarice will attend the party, but not both."
:longline: Ci :dblline: :longline: Cl

Is that accurate?
Adam Tyson
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Still not quite there, but close! "Either Cindy or Clarice will attend the party" gives us this version of the double-not-arrow:

Ci :dblline: Cl

In other words, you cannot have a situation where Cindy does not attend AND Clarice does not attend. If at least one of them must attend (and that's what "or indicates - at least one, but perhaps both), then this double-not-arrow is one way to show that.

"Either Cindy or Clarice will attend the party, but not both" cannot be captured by a single use of the double-not-arrow, just as it cannot be captured by a single use of a standard conditional arrow with its contrapotive (as shown in the book at the bottom of page 198 and the top of page 199). Instead, to get this "exactly one" rule completely represented by conditional arrows, you can either diagram all four of those conditional statements (again, see those two pages), or else you have to use both double-not-arrows, like this:

Ci :dblline: Cl (must have at least one)
and
Ci :dblline: Cl (cannot have both)

Only by doing both together do you cover both aspects of the rule. The first one above requires at least one but allows for both, while the second one above prevents us from having both and allows us to have neither. Put them together and you get "exactly one in and the other out".
Adam M. Tyson
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Rnmakrinos
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Thanks for your help so far!... I hope it's ok if I ask a few more questions...

So, how would diagram "If Gomez runs for president, then Hong will not run for president"? Would you say that that's diagrammed in the same way that you diagram "Either Cindy or Candice will attend the party" below - ie with the negative S & N and the double-not arrow?
Adam Tyson
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Nope, in that case I would do one of the following:

G :arrow: H (the standard conditional arrow, indicating here that if G is in, H is out)

or

G :dblline: H (the "standard" double-not-arrow, with the conditions in their positive state, indicating that they cannot both be in, but allowing that both could be out)

In other words, I can have one or the other, or I can have neither, but I cannot have both.

The double-negated version, where they are both crossed out, means I can have one or the other or I can have both, but I can't have neither. At least one, and maybe both, are in. See the difference?

Again:

G :dblline: H means "not both"

G :dblline: H means "at least one"

Keep on asking, that's why we are here!
Adam M. Tyson
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Rnmakrinos
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Thanks! So that's exactly where I'm stuck...

If "If Gomez runs for president, then Hong will not run for president" is G :dblline: H, and that means "not both"... Then why isn't "Either Cindy or Candice will attend the party, but not both" diagrammed as Ci :dblline: Ca bc the double-not diagram means "not both"? Why is the latter diagrammed as a negative sufficient and necessary with a double-not arrow?

Is there a number I can call to talk this through?
Adam Tyson
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The rule about Cindy and Clarice goes further than the one about Gomez and Hong - it requires that one of the girls attend, in addition to preventing them both from doing so. If Gomez runs, Hong won't, but what if Gomez doesn't run? Then we don't really know anything - maybe Hong runs, maybe not, right? The sufficient condition of Gomez running doesn't occur, so the rule simply isn't triggered.

Not so in the case of Cindy and Clarice, though - if Cindy attends, Clarice does not, and if Cindy does not attend then Clarice must. As long as we know what one of them does, we know the other does the opposite!

Let's take this to another place for a second. Let's go to the movies. I'm buying, which means I get to pick the movie.

There's a giant megaplex theater near me, 16 screens, all the latest releases plus some older stuff and even a Bollywood film. Now, I tell you "if we go see Ghost in the Shell then we will not go see Beauty and the Beast". That makes sense, right? If we choose one film then we won't see the other. Going to both is not an option. But that rule doesn't prevent us from going to see Fate of the Furious, does it? Of course not. In fact, we could also choose to see Baby Boss or Kong: Skull Island or whatever else is playing. Just remember - if we choose to see Ghost in the Shell, then we cannot go see Beauty and the Beast, and that also means that if we see Beauty and the Beast we cannot also go see Ghost in the Shell.

But now I change the rules, and I tell you "we are going to see either Ghost in the Shell or Beauty and the Beast, but not both." This changes things a lot! Now I know that no matter what, we are going to see one of those two films, and we are NOT going to see the other. That's that - it's exactly one of those two films. Maybe we will also go see Baby Boss (please don't make me, but it's possible), but for sure we are going to see one and only one of Ghost and Beauty.

The first situation is what Gomez and Hong are in. If one runs, the other doesn't, but maybe neither does. The second situation is Cindy and Clarice - exactly one will attend while the other does not attend. Others might also, but only and exactly one of those two.

Third situation here for a second. What if I say that we are going to see Beauty or Ghost, and that's all I say? Now I have to see at least one of them, but I could, in this case, see both, a double feature. Why is that allowed? Because I didn't say it wasn't - I didn't say "but not both".

Don't worry if these aren't coming to you easily just yet. Some people never adopt the use of the double-not-arrows, either because they don't get them or because they actively dislike them. Other folks end up embracing them and use them all the time. I am in the "love them" camp, especially in grouping games, but if you never get to that point then just use the regular conditional arrows and you'll be fine. They are a tool, a shortcut, but they are not mandatory.

There is a number to call if you are in one of the full length courses that gives you access to the Homework Hotline. If not, you can continue to post questions here, and we will continue to help.
Adam M. Tyson
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Rnmakrinos
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No, I NEED to get this. Let me sit on this for a bit and go over it. Will respond back if I EITHER get it OR I don't. ;)
Dave Killoran
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Rnmakrinos wrote:Then why isn't "Either Cindy or Candice will attend the party, but not both" diagrammed as Ci :dblline: Ca bc the double-not diagram means "not both"? Why is the latter diagrammed as a negative sufficient and necessary with a double-not arrow?


Hi Rn,

I wanted to answer just this part of your question, and I realize Adam may have done so as well, but it never hurts to hear it from more than one person :-D


    "Either/or" by itself generates just the negative sufficient and necessary with a double-not arrow, to use your description. "Either/or" simply means "at least one," and thus you can never have a situation where both are absent, which is what that diagram shows.

    "Either/or but not both" generates the negative sufficient and necessary with a double-not arrow AND the regular double-not arrow. It does so because it's two statements combined in one. This is why it's so tricky—you have to analyze two things concurrently and they have meanings that are separate but overlap.

This is also a question that has come up a lot previously, and so it's been written about in various places. I'm going to link you to several of those discussions, and they might help shed additional light on how this all works:



Take a look at those two pieces and let me know if it further helps or not. If not, we'll keep discussing it until it makes perfect sense. the good news is that it's all logical underneath, so if we keep chipping away at the meaning we will arrive at a point where you can see how it all relates.

Thanks!
Dave Killoran
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