Conditional Chains and A,B,C linkage example

[Tommy2456]

Hello again, I am reading chapter 5 of the LRB. I am total confused about conditional chains. On page 243, how do you know when to use the contrapositive in logic games and also link the inferences together? I am looking at the A,B,C linkage example.

[Dave Killoran]

Note to other readers: the page reference here refers to the 2015 edition. In the 2016 edition, the discussion referenced here is on page 269.

Hi Tommy,

Thanks for the questions! As far as the contrapositive, over time you will come to simply realize what it is automatically, and then use it when needed. to me, the CP is something that's always there. In the LRB, the analogy I use is that a statement and its contrapositive are like two sides of a coin: you may only be looking at the front, but you know the back is always there. So, in a Logic game, when i see a rule like A B, I automatically know that B A is also true. At first, we advise you to write that CP down because it reinforces the idea. But over time you will find that you don't always need to write it down; recognizing it becomes second nature.

When does it come into play? When the condition is triggered. If you were doing a game and all you saw was "A", then you would just worry about A B. But, if you saw a situation where B occurred, then that would trigger the CP, and you'd know that B A was in play.

The same thing is true for lining rules together. Let's say I'm given the following two rules:


A B

B C


Ok, I can see that B is common to both, and it's in a position where I can easily link them together to form the following chain:


A B C


Now let's say that instead I'm given the following two rules:


C D

G D


I can see that D exists in both chains, but in different forms. Wait, if I take the CP of either rule, I can link them together:


C D

D G

Which can then be linked as: C D G


Or, take the CP of the first rule instead:


D C

G D

Which can then be linked as: G D C


The cool thing there is that those chains are contrapositives of each other, which means they are basically identical in functional meaning. So, no matter which one you use to take the CP, you still arrive at the same place


The more you study, the easier all this gets, and you will start to see the CP as an ever-present element in conditional reasoning. You know it's there, but you only have to use it when you see other things that force it to occur.

Please let me know if that helps. Thanks!

[Tommy2456]

That helped tremendously!! I wish you were here to study with me LOL! I am just afraid that I will forget the correct way even though I have been studying a while. But I do know when to use it. My issue is feeling guilty about getting questions wrong when it comes to the games. From experience, should I spend a lot of time on grouping games, these are tough but I am getting through it? How many practice games should I be doing a day and any other advice on how to get these down packed? How can I be as productive as possible before taking the June test? Sorry for all the questions but my nerves are getting the best of me.

[Dave Killoran]

Hey Tommy,

Great, I'm glad that helped! That's what the Forum is for, and in a way, it is like we're there studying with you

I understand the concern over forgetting the right approaches at the right time. But that's why practicing is so valuable—the more you practice, the more it becomes second nature. There's a term you hear in sports sometimes: muscle memory. That's the goal as far as working with LSAT concepts and techniques.

In LG, the key isn't how many games you do each day, it's how much you learn from each one. I'd rather you do 4 games and review them thoroughly than do 12 games and just gloss over the explanations. Every time you do a game, try to wring every ounce of knowledge form it that you can. And, most LSAT questions are so complex that even looking at a game 2-3 times doesn't reveal every single thing about it. So, focus on quality of review, not volume!

As far as focus, Linear and Grouping are the two dominant concepts in LG, so you want to spend the majority of your time on those ideas. Basic Linear, Advanced Linear, Grouping, and Grouping/Linear Combination games make up about 95% of all games, so that's clearly where everyone needs to focus. LSAC can throw in a random game now and then, and they do this to keep people honest (and off-balance). But, if you are solid on Linear and Grouping, you'll be solid on the majority of what you see on each LSAT.

Please let me know if that helps. Thanks!

[Tommy2456]

Thank you so much, I had it backwards, I know quality matters but I was looking at each problem at a glance thinking of quantity. But I am pretty comfortable once I get the concept of the rules.

[amagari]

I don't understand how you can link the conditional chains together with the Double Not arrows and the Double Arrows.

for example, what are the rules governing this? A B C ?

Why not something like A B C D Z?

Thanks

[Luke Haqq]

Hi amagari,

I'd be happy to address both of the diagrams you're asking about. In general, the Double Arrows and Double Not Arrows mean the following (I use "~" to denote "not"):

X Y, means:

1. X Y, AND
2. Y X

In other words, if you have one, then you must have the other.


X Y, means:

1. X ~Y
2. Y ~X

In other words, if you have one of the variables, then you know the other one cannot be present.


To discuss the specific diagrams you mentioned--

A B C

The double arrow is showing that you know (1) A B, and (2) B A. So with that in your above example, you'd additionally know A B C. And from that you could know that A C.

Why not something like A B C D Z?

Why not indeed? If diagraming a setup in that way clearly indicates to you how the variables are interacting with one another, then I see no problems with the hypothetical setup you've given. To mention some inferences--as with the previous one, the A B C allows the inference that A B C.

And given that we also have C D, then we could put that together with the previous diagram and we'd have A D. That is, if A implies B, and B implies C, but C can never be with D, then we know that A also can never be with D.

Also, because D and Z are connected by a double arrow, then any inferences about D in relation to A, B, and C will also be true of Z. So A, B, and C cannot be present with Z, either.

Hope that helps!

[amagari]

I understand it when it's broken apart into the underlying conditional relationships.

Each of the bi-conditional arrows actually encompasses normal conditional relationships. They're just short hand.

But I still feel like I don't have a good handle on all the different types of arrows when they're in a huge chain like the example I created.

How do you know when you can infer something from the arrows? Like you said, we can't say anything about Z to the other variables besides D.

I would only like to diagram arrows where you actually can make inferences, I wasn't trying to suggest I was trying to notate it differently than the book suggests, I just wasn't clear on the rules.

In my made up example, how would you actually represent those rules? Like how would you break those up?

Thanks

[Jon Denning]

Hey amagari - thanks for the question and welcome to the Forum!

Let me take a second to re-examine the hypothetical chain you proposed: A B C D Z

Luke noted in his earlier reply that we don't know about the relationship of Z to A, B, or C. But that's not entirely accurate, and could be causing some of your confusion here.

So what I'll do is explain exactly what we can know from that particular hypothetical, and also amend Luke's reply to include the correct set of relationships

First, it is accurate to say that the connection between A, B, and C still holds, where A C and the contrapositive
C A are both true.

From that however we can make connections with D and Z.

That is, if A, B, and C each remove D (since C and D can't go together), then we also know that A, B, and C would each eliminate Z. How? Because Z requires D from the D Z connection. Inherent in that double arrow is Z D, meaning that as soon as D is knocked out (by any of A, B, or C) then Z cannot be present either.

So that's the correct set of relationships.

If it was simply a single arrow D Z then we could not make A, B, and C connections to Z, but with the double arrow inferences become possible.

I hope that helps!

[andbzav@gmail.com]

Hi,

How do you take the contrapositive of a conditional chain, for example, h -> K -> N?

Thank you