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