January 12, 2016 at 3:05 pm #1322
Could any LG masters help me understand the conditional logic and contrapositive of the last rule? I wrote the original rule as: VS/m –> WN/m. I then wrote the contrapositive as WN/f –> VS/f. I was able to break up the game board and diagram both situations, and the subsequent inferences, but while answering Q1 and reviewing my work, I realized that I assumed that VS and WN will always have a gender match– which isn’t the case. I’m starting to understand why there could be VSf and WNm without breaking the rules, but could someone explain more clearly why I can’t assume a gender match between WN and VS? Also what lessons about contrapositives and conditional logic can I learn from this game that could apply to other games?
January 12, 2016 at 4:19 pm #1323dannypearlbergParticipant
The more general lesson here is that when given the conditional “If A then B” [in this case, if males are assigned to Veblen South then males are assigned to Wisteria North] there is one good inference (the contrapositive) and two bad inferences.
Here is the good inference: If not B then not A.
Here is the first bad inference: If not A then not B.
Here is the second bad inference: If B then A.
So it sounds like you are asking for additional clarification as to why the second bad inference is a bad inference. Here’s why: Imagine that ‘A’ stands for ‘Danny’s head gets chopped off”, and ‘B’ stands for ‘Danny is dead’. If Danny’s head gets chopped off, then Danny is dead. But there are lots of possible ways for Danny to die. So if we are given the conditional that If Danny’s head gets chopped off, then Danny is dead, and then we are told that Danny is dead, we cannot infer that Danny’s head got chopped off- maybe Danny was poisoned? Had a heart attack? Who knows? All we know is that if Danny’s head got chopped off, then Danny is dead, which also means that if Danny is not dead, then Danny’s head did not get chopped off (this is the contrapositive). If Danny is dead it is of course possible that Danny’s head got chopped off, but it isn’t necessarily the case.
Back to the game: The conditional tells us that If males are assigned to Veblen South, then males are assigned to Wisteria North. So, if males are assigned to Wisteria North, it is of course possible that males are assigned to Veblen South, but it isn’t necessarily the case.
January 14, 2016 at 6:09 am #1328
Hey — great q —
Just to add on to some of what Danny mentioned —
You’ve hit upon one of the great “mental tongue twisters” that the LSAT commonly presents — it’s important for you to get your alignment correct on this —
The short answer and specific answer to your q is this: you understood the conditional statement correctly, and you were right in how you thought of the contrapositive. The issues came when you created split boards — your boards did not represent all the possibilities of the game.
Black and white takeaway / general advice: It’s rarely a good idea to use conditional statements to split your board —
Longer explanation —
I promise I’ll return to the game you actually asked about, but to focus on the actual conditional rules, if you don’t mind, let’s start by discussing a much simpler scenario —
Imagine that there is a game where, out of a certain number of students, some get picked for a team and some don’t.
So, basically you have an “in” or “out” situation.
Imagine getting one of three different rules:
Either M or N, but not both, will be selected.
If M is selected, N will not be.
If M is not selected, N will be.
These three rules tell you three different things — can you figure out the different consequences each can provide?
Per the first rule, there are just two options: M selected and N not, or N selected and M not.
The second rule just tells us something specific that happens when M is selected, which is that N will not be. But notice, this doesn’t tell us the same info as the first rule, because it doesn’t prevent us for having both M and N not be selected.
The third rule just tells us something specific that happens when M is not selected, which is that N will be. This isn’t the same as the first rule because it doesn’t prevent us from having both M and N selected.
General Takeaways —
Conditional rules are meant to allow us to infer additional truths when we know something — that is, we should bring them into our process when the trigger part “If M is selected, etc.” is “activated.”
They are not meant to give us, in a direct way, “exhaustive” consequences (in the way that a rule like “Either M or N, but not both, will be selected” does) and so you almost always do not want to use them to create an exhaustive representation of the game.
When you are creating split boards, you are not just saying “these are two ways the game can play out” but rather “these are THE two ways the game can play out,” and, again, conditional rules are not, per the way they are designed, intended for giving you that info. (You can figure out exhaustive info from them, as we’ll talk about in just a sec, but again, my point is that it’s not what they are naturally meant for and that shouldn’t be your primary plan.)
Back to the specific game —
You are right that the original rule states:
If males are in VS -> males are in WN.
and that the contrapositive would be:
If females are in WN -> females are in VS.
As you guessed, and as Danny mentioned, this rule doesn’t, in any way, prevent a situation where females are in VS and males are in WN. Notice that that situation doesn’t violate either the original rule as written or the contrapositive.
To see it from a different perspective — without any rules given to us, the game originally starts out with four possibilities for how males and females can be assigned to those 2 areas:
Notice that the rule allows us to eliminate one of the four possibilities (the second one), but leaves the other three. By creating just two diagrams off of your condiitonal, you failed to account for the other possibility.
I feel I’ve now gone on too long, and perhaps said a bit too much (woke up too early and had too much coffee) but I hope you were able to get out of that what you needed–
1. Your understanding of the conditional rule was correct, but where I think you may have gotten into trouble is in confusing conditional information with information about all possible options.
2. Conditional rules are useful for yielding conditional inferences, but you generally want to avoid using to them to think of all the options for a game; when you create split boards, you are laying out all the options for a game; thus, conditional rules are not great for splitting your board.
Sorry for the length — if you have any follow up feel free to ask — Mike
January 14, 2016 at 7:35 am #1331dannypearlbergParticipant
Great explanation Mike!!! Two quick comments:
(1) If you are very comfortable working with conditionals and with diagramming split boards, then actually I think it is often helpful to use a conditional to create split boards. For example, in this game, we can use the last rule to divide the game into two scenarios: One where males are assigned to Veblen South, and one where males are not assigned to Veblen South. In the scenario where males are assigned to Veblen South we now know everything except the assignments in Richards South and Tuscarora South- one of them is males and the other is females, but we don’t know which one is which. That’s a lot to know about one of two scenarios. In the other scenarios, the only additional piece of information we have is that males are not assigned to Veblen South- i.e. females are assigned to Veblen South. If you are very comfortable working with conditionals and with diagramming split boards, you will know that this doesn’t tell us anything about who gets assigned to Weblen North.
Another reason why it is often helpful to use a conditional to create split boards if because you are, in effect, drawing the rule into your master diagram(s), which makes it much less likely that you will forget about the rule.
(2) If you have a biconditional (for example, the following rule is a biconditional: males are assigned to Veblen South if but only if males are assigned to Wisteria North), you should definitely use it to diagram split boards. The biconditional generally tells us that two things are either in together or out together, so that creates THE two scenarios: The case where the two things are in together, and the case where the two things are out together.
January 14, 2016 at 8:04 am #1332
Thanks for adding your thoughts Danny —
As I mentioned, in general I don’t recommend that students use conditionals to split boards, but that’s definitely a subjective decision, and I think your advice is totally valid as well, and I can certainly see how thinking about it as you described would work great for splitting off conditionals if a student did want to do that –
And I definitely support point #2 — especially if it’s a biconditional that involves 3 or more elements, it can provide for great “splittage” 🙂 — mk
January 14, 2016 at 3:55 pm #1334
Thanks Danny and Mike for the explanations! I understand more clearly now.
Also Mike, if you don’t mind, could I ask a few more questions about this game, and related to The Trainer?
So if I were to avoid splitting boards by conditionals, for this specific game, would you have made one game board and stuck with that instead? I redid the game without splitting it– and it seemed to work fine (granted I’ve read over the rules many times by now). Then as a general rule of thumb, according to the way The Trainer taught, should I spilt game boards if I know a certain rule restricts the number of possibilities there are? i.e. What you said earlier about my confusion between conditional statements vs. rules that give us exhaustive consequences.
Also, this question is a bit off topic, but I was curious about this as I redid the game. In The Trainer you mention that if we have to constantly draw new boards throughout a single game, then we probably missed an opportunity to make additional inferences at the beginning. For this game, as I reworked the questions I found myself redrawing boards (*I’m also not sure if I should be writing in the additional conditions given in the questions next to the master game board or just draw an entirely new board near the question). Is this ideal, or am I missing a more efficient way to visualize the new conditions given in the questions?
January 15, 2016 at 3:27 pm #1335
In terms of splitting boards / splitting this board —
Yeah, I would have just created one board and worked off of that — I just tried this game on my own, and that’s how I solved it — and even though I missed a pretty big inference ( exactly 1 F,F pairing) I still had no trouble w/the problems and they went fairly quickly.
Again, this is not to say that Danny’s method isn’t a great one — it’s very clever, and would have certainly made it easier and faster to go through the q’s — however, in general, I do recommend splitting boards less often than most other instructors do. But it’s a gray area, and it’s an issue I constantly think about, so it’s definitely something you want to try out a variety of ways to see what works best for you —
The basic rule of thumb that I suggest is to split boards when you see that all of the possibilities for a game can be cleanly split into a few different groupings, and you see that splitting boards in this way has an impact on how you manage the other given rules.
The simplest example I can think of is something like this combination (imagine we know each appears exactly once):
N is either first or last.
R is two spots away from N.
The first of the rules splits up all the possibilities for a game into two neat categories, and it happens to so in a way that allows us to more easily notate and control the second rule.
Another very common example is the biconditional, which Danny brought up initially — especially if you have a biconditional w/ 3 components — something like —
“J is after M if and only if J is before K.”
There are two ways this rule can play out — either you have
M – J – K or K – J – M, and no matter the game, it’s certain that knowing that order of elements, and using it as the focal point of two different diagrams, will make it markedly easier for you to control the other rules and answer the q’s.
So, my recommendation is to start from there in terms of deciding when to split boards (also, you may to review Lesson 14 again, which deals w/some of this stuff) — and then, as you get more and more comfortable, try out splitting boards more and more and see what works best for you —
In terms of drawing new diagrams —
I can’t remember quite where I mentioned what you are referencing, but I have a feeling it either had to do with a certain question type or having to constantly have to draw new diagrams or hypos to assess each of the answer choices within a problem (an example would be something like an unconditional could be true q — most of the time you run into one, if you have set up the game properly, you should be able to use your setup to see why most of the wrong choices must be false — if you can’t, and have to “test out” each answer, that could be a sign of trouble) —
But more importantly, please know there is nothing wrong with drawing new diagrams next to problems when you need to, and in fact you should end up doing it a ton the time — for this game, I personally drew new ones next to each q except #2 —
To me, without making the frames off the conditional upfront, this game doesn’t yield many upfront inferences (or at least I didn’t see them), and so it’s what I characterize in the book as a backend game — it gives us a fairly simple situation, fairly simple rules, and not a lot of inferences to uncover upfront, and the work is going to be done on the backend — at the point of the q’s —
And, you’ll notice that 4 of the 5 problems are conditional ones — meaning they give you a new bit of information that, when combined with what you initially know about a game, sets of a chain of inferences that allows you to get to the right answer.
So, I spent very little time on the setup of the game, and instead solved the problems based on going down a chain of inferences when the q’s asked me to (and in much the same way as I document in the LG solutions in the trainer) —
That’s it — you ask some interesting and broad q’s, and I feel like I can’t give full responses without writing a tome back, so I have to be pretty general, but I hope that helps a bit, and if you have any follow up or want me to flesh anything out just let me know — mk
January 15, 2016 at 4:45 pm #1336LSAT DanParticipant
A good general rule to keep in mind is that a single conditional rule will not cover all of the possible permutations; for instance, it won’t tell you that two variables must always be in the same group, or can never be in the same group. To get that level of certainty, you either need a second rule (a good example of this is the first game in Preptest 54, from June, 2008 – the first rule tells us that If Jaclyn is on stage, then Lorena is offstage, and the second rule tells us that if Lorena is offstage, then Jaclyn is onstage. Only after the second rule do we know that Jaclyn and Lorena must always be separated); OR, barring a second rule, we need an “if and only if” rule, which is really two rules in one single sentence. An example of this is from Preptest 56 (December 2008), the second game: “Grace helps most the sofa if but only if Heather moves the recliner. Now we know that they’re either both working on their respective pieces of furniture, or neither one of them is.
In these two games, if we know about Jaclyn, Lorena, Heather, or Grace, then we automatically know about the other person in their pairing, but it’s only because the rules gave us two pieces of information; if there’s only a single conditional rule, then there will always be ambiguous cases where a piece of information (like “Females are in V South”) doesn’t allow us to draw any further inferences.
January 16, 2016 at 9:29 am #1342
Thanks Mike and Dan for your responses! With all of these explanations, I think I’m definitely getting a solid sense of conditionals and important things to watch out for when they’re included in games 🙂
or having to constantly have to draw new diagrams or hypos to assess each of the answer choices within a problem (an example would be something like an unconditional could be true q — most of the time you run into one, if you have set up the game properly, you should be able to use your setup to see why most of the wrong choices must be false — if you can’t, and have to “test out” each answer, that could be a sign of trouble) —
Ah yes, that was exactly what I had in mind when I asked my question. Seems like I got confused between drawing a new diagram for a q that gives a new condition vs. testing out each answer choice for a q.
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