Grouping Setup Practice Drill #6

[kristinaroz93]

1) This was a drill problem, but I am stuck on one of the diagram features of the drill.

The problem goes like " Chris is choosing flowers for his garden from among the following choices..."

As the asnwer key shows, the fifth and six rules can be combined to give

not H--> L <---/---> M
Which then yields the following inference:
not H --> not M

Is it because
not H<---/---> M = not H --> not M ?

2) And on a seperate issue, what would for istance be the implications of this kind of condition:
"If N is reduced, then either R is not reduced or S is not reduced."
Does that just mean you can't have both r and s present with N, but you can have one of the two present?
i.e. NS okay
NR okay
NSR not okay

Thanks in advance!

[Nikki Siclunov]

Hi kristinaroz93,

Thanks for your questions! Let me briefly address each one of them individually.

1. Regarding your first question, yes - you are absolutely correct. Let me elaborate on each of the two rules whose combination yields the inference you are asking about:

1. NOT H L

(Contrapositive: NOT L H)

Notice that the sufficient condition here is negative, while the necessary condition is positive. Essentially, the rule tells us that if either H or L does not occur, the other one has to occur. In other words, at least one of H or L must occur. Under this rule, there are three possible outcomes:

H occurs but L does not occur
L occurs but H does not occur
both H and L occur

It is critical to recognize that this rule does not prohibit both H and L from occurring together. Just because at least one of them has to occur does not mean that both of them cannot occur at the same time.

Now, compare this rule to the rule about L and M:

2. L NOT M
Contrapositive: M NOT L

The Double-Not arrow (L M) is a shortcut that we use to represent a relationship between two variables, wherein the sufficient condition is positive and the necessary condition is negative. The arrow captures both the original rule and its contrapositive; however, sometimes I just use the regular arrow and negate the necessary condition. It does not matter how you represent this relationship, as long as you know what it allows, and what it disallows.

Under this rule, the following outcomes are possible:

L occurs but M does not occur
M occurs but L does not occur
neither L nor M occurs

Unlike the previous rule, this one allows for the possibility that neither M nor L occurs; however, it prohibits M and L from occurring together - at least one of them does not occur.

Note that the distinction between these two rules was also discussed by Dave Killoran on this recent blog:

http://blog.powerscore.com/lsat/the-mos ... -rule-lsat

The two rules above can be combined, and their combination yields the following conditional chain:

NOT H L NOT M
Contrapositive: M NOT L H

Shortcut: NOT H L M

Whichever way you end up representing these relationships, make sure your arrows go in one direction As should be clear from the chain above, the absence of H requires the absence of M (not H not M), and - by the contrapositive - M requires H:

M H

Hope this answers your first question!

Onto your second question:

"If N is reduced, then either R is not reduced or S is not reduced."

N NOT R or NOT S
Contrapositive: R and S NOT N

(according to De Morgan's laws of propositional logic, "or" becomes "and" in the contrapositive form, and vice versa)

According to this rule, if N is reduced, it is impossible that both R and S reduced (either R or S is not reduced). Note, however, that the either/or conjunction is inclusive. In other words, it is possible that neither R nor S is reduced! The following outcomes are possible:

Nothing is reduced.
Only N is reduced.
Only R is reduced.
Only S is reduced.
Both N and R are reduced.
Both N and S are reduced.
Both R and S are reduced.

Indeed, the only possibility prohibited here is the one where all three variables are reduced.

Let me know if this clears things up!

[kristinaroz93]

Dear Nikki,

Your explanation was amazing and just what I was looking for. Thanks so much for clearing it all up! I appreciate it very much!

[anabil@umich.edu]

Hello,

For rule 5.

"If he does not choose H, then he chooses L"

Rule is (Not H) --> L
Is the controposotive true?
(Not L) --> H.

I don't believe it is. However how come a double not arrow was not used for an inference? To include the possibility of Not H and Not L.

Thanks

[Nikki Siclunov]

Hi anabil,

The original rule and your contrapositive are both correct:

• NOT H L
  NOT L H

This "negative sufficient" rule has been discussed extensively on our blog and Forum, so please take a look at this link:

The Most Dangerous Conditional Rule on the LSAT

Essentially, the rule allows for both H and L to occur, because the sufficient condition is triggered only in the event that one of them does NOT occur. If either H or L does occur, however, we have no way of knowing if the other one may or may not occur. Consequently, both H and L could occur.

Another way of conveying this relationship is to say, "Either H or L, or both, must occur."

Hope this helps!

Thanks,

[anabil@umich.edu]

So my understanding is the double negative arrow only applies when the necessary condition is negated.

[Dave Killoran]

Hi A,

Actually, that's not correct. Where the negative appears in the original statement does have an effect. Let's look at both cases:

• 1. When the negative is on the necessary condition.
  
  This is the one that appears most frequently. A statement such as " J K " results in a situation where one of the two variables appears, the other cannot appear. thus, they can never appear together, a "super statement" that we diagram as:
  
  
  J K
  
  
  2. When the negative is on the sufficient condition.
  
  This is the one that appears less frequently, and is the statement I called the most dangerous conditional rule . A statement such as " J K " results in a situation where when one of the two variables is absent, then the other must appear. Thus, they can never both be absent, and this produces a "super statement" that we diagram as:
  
  
  J K
  
  
  Because this diagram is a generally confusing, I tell most students to just write out the single relationships ( J K, and K J ) instead of using the double-not arrow. But either approach works, so use the one that is most comfortable for you!

Please let me know if that helps. Thanks!

[akanshachandra]

Hello! So I was doing a few setup practice drill problems in the Logic Game bible under the section "grouping". Though I fairly understand the rules, I'm a bit perplexed with number 6 (on page 320). When I was mapping out the rules "If he does not choose H, then he chooses L" I did it the way the book says to in the answer key by doing H(not) L, and in the SAME way I diagrammed the rule "If he chooses M, he does not choose L" as M L(not), how come in the answer key, the diagram for the sixth rule is using double not arrows but for the fifth its a single arrow with a simple negation?

[akanshachandra]

Also, Is it okay that I usually skip the practice problems in the Bibles, and move on to the question type training questions, so as to save some questions for after all my studying and come back to the ones in the Bible? I dont want to exhaust all the questions presented by Powerscore, because after my 2 months of studying, as a review I want to be able to come back to them to practice?

[Adam Tyson]

Double-not-arrows are a useful shortcut, akanshachandra, but they are not the only way to diagram a conditional statement wherein one condition is positive and the other is negative. You could absolutely diagram that 6th rule as:

M L

and the contrapositive of that rule as:

L M

The double-not-arrow captures both the original claim and the contrapositive in one swift, efficient image:

M L

This can be read simply as "you cannot have both M and L together." You can have just M, or just L, or neither, but having both is impossible. Some people love the double-not-arrow as a time- and space-saver in their diagrams. I'm firmly in that camp - they are, to me, fast and clear and easy. Other folks, though, never get to liking them all that much, and so they stick with the simpler conditional diagrams like I did up above.

Now, the fifth rule can also be diagrammed using a version of the double-not-arrow, with both terms negated, like so:

H L

This version causes more confusion for some people, but it's another one that I love and use all the time when I have a rule like this one, where the sufficient condition is negative (out of the group) and the necessary is positive (in the group). It, too, has a shorthand meaning, and in this case that is "you must have at least one of these in the group." You can have H without L, or L without H, or you can have L and H, but you can't have them both out.

The diagram in the book is saying that if H is out, L is in and M is out, but if M is in, then L is out and H must be in. That gives us the additive inference that whenever H is out, M is also out, and the contrapositive of that inference, which is if M is in, H must also be in.

Play around with those double-nots and see if you get used to them and start to like them. If you do, great! If not, no worries, just keep using the slightly longer but, for you and many others, clearer approach. It may mean a little more time drawing, but as long as you are getting right answers and not confusing yourself or wasting time, you should do what works best for you.

I hope that helps!