I tried the Dinos game on its own the other day and I can't really seem to find an answer to my concern with this game online.

One of the Rules reads "If both the 'L' and the 'U' are included, at least one of them is not Muave."

For this rule: what would the contra-positive be, and is there a simple way of writing it out?

Ignore the post that I deleted.

I would just diagram this whole thing as:



Because effectively it just is saying that both L and U can't be Mauve at the same time.

Is that a "Not both arrow"? Let's say either L or U is included, but not both. Would that particular Dino need to be muave?

This one rule shouldn't make it need to be the case that that one would need to be Mauve.

My issue is that basically the rule reads: If both L and U are included, one or both is Not-Muave. Wouldn't the contrapositive make it similar to how if a rule reads "some are not X."

Honestly, that seems to be the issue for me here. What is the logical opposite of "some are not X"? Thanks

I'm not 100% sure what you're getting at... This is a very strange conditional.

But I think the answer to your latter question is this:

some Y are not X is logically equivalent to some not X are Y (assuming that some Y exist)

Some one else feel free to jump in here if I've said anything wrong... it's been a while.

This is the only conditional statement I have come across that I have not enjoyed forming the contrapositive of. I googled "some are not X" and the logical opposite appears to be "All are X."

So wouldn't L or U need to be Muave if it was by itself? I am confused, lol

Yea, that's correct. I thought you were asking for the logical equivalent, not the opposite.



To your second question, I don't think so.

If you have just:

R
L

Then you don't need to fulfill anything per this conditional. The conditional tells you what to do if you have both L and U, not if you have only one of L or U.

I'm having trouble making a contrapositive, though. I keep getting a contradiction (along the lines of if A and B then not A or not B), so I'm doing something wrong.

Because my first inkling is that the contrapositive should be:

If both L and U are Mauve, then either L or U isn't included. But that's a contradiction.

Which is why I was thinking it should just be diagrammed as not both L and U are Mauve.

Edit: I could just be confused because I'm not looking at this in the context of the game.

Yeah that creates a double not arrow. If L & M --> both not mauve... It's really hard to draw up a contrapositive here, but the first poster has the right idea. If both are included, then they can't be mauve... The contrapositive for this rule would work best with the global accounting question. So, if both are included, and both are mauve, then that is an invalid scenario.

This seems like a rule that you would just really have to commit to memory because it's really hard to form a contarpositive out of. Just know that they cannot both at the same time be mauve. Both of them could be another color. They don't HAVE to be mauve.

I am gonna do the game over again assuming if either one but not the other, it needs to be muave. I am jumproping right now in frustration cuz i thought i had a good handle on formal logic, but i guess not, lol

I don't remember the game (only did it once, and it kicked my ass), but it COULD be that the rule could be manipulated to mean something else depending on the other rules. Remember that what makes some games solvable are the inferences you draw from the other rules.

The problem with this conditional is resolved if you write it out simply:

Oops! See two posts down.

Recommend re-reading the phrase. (note that it's "not Mauve", not "is Mauve")

Right you are, sorry about that! Let's try it again.

If L and U, then ~LM or ~UM
The contra would then be, if both L and U are mauve, then they're not both present. This is a contradiction of terms that can then be summarized as "L and U cannot both be mauve."

L and U --> ~LM or ~UM
LM and UM --> ~(L and U) (i.e, never LM and UM)
Hope that helps!

So if either one is in it needs to be mauve? lol

I feel like if I truly get this one rule ill get a 181 on the LSAT

Nope. How did you get to that?

The rule is basically saying that if they're both in then at least one of them can't be mauve. It only applies when they're both included. And when you see that they're both included you have to make sure that they're not both mauve. So maybe the L is mauve but the U isn't, or the U is mauve but the L isn't, or neither of them are mauve. That all complies with this rule. You have a problem if you have an arrangement where both L and U are present and they're both mauve.

The contrapositive is a little counter-intuitive because it basically says that if they're both mauve then they're not both included. But how can they BOTH be mauve if one of them isn't included. That seems self-contradicting. And it is, in a way (it has to do with the modal relationship between the two conditions, don't worry there's no reason for you to know that). The point is that it gives you a rule to check your arrangement - if you see that both L and U are mauve you know that there's a mistake since you shouldn't have them both in the arrangement in that case.

TopHatToad's treatment is very good and correct. Make sure you understand how s/he worked it out.

And the logical opposite of 'some are not' is 'all'. Think of it this way, 'some are not' basically means anything less than 100% (that includes 0% btw, that's why you can have a situation where neither toy is mauve), so the logical opposite will have to be 100% since that's the only possibility left. For 'some' the logical opposite is 'none'. Some means anything above 0%, so it's opposite is 0%.

Also, contrapositives like this are the reason I tell my students that they should only make contrapositives when they're not too complicated. But my approach doesn't depend on making deductions so it may not work for everyone.

Do you know how you can quickly spot a contra that results in a contradiction? I got 5/6 on this game but the fact that i was unsure of whether either L or U had to be Muave if they were in the In group, and the other was in the out group was in the back of my head.

I see the contradiction and I try to adapt, but I guess that does not work with formal logic (my only class = LGB). When I saw the contradiction, I assumed that it meant that both L and U would be Muave, but with one in the out group (2/7) and one in the in group(5/7), so the rule still held but only one of them could be In group - as Muave..

Do these contradictions in rules appear frequently?

Thanks everyone for the help, btw.

Almost, they can both be mauve - but they have to be in different groups or they both have to be in the out group, and only in those cases. So according to the contrapositive if it's not the case that one of them is not mauve (meaning that they're both mauve) then it's not the case that both L and U are included (meaning they're both out, or one of them is included and the other isn't).

Don't worry about contradictions, that's not going to be a factor; sorry if I was unclear in the last post. Focus on knowing what the negatives are of the different logical qualifiers (some, all, none, etc) and what the rules are for negating when you have the AND and OR conjunctions (the negative of 'A AND B' is ~A OR ~B, etc).

Wait, so, what.. lol

What I deduced was that if they were both muave, then one of them was not in and the other was in, meaning that "either one of them that was IN was muave."

Is this deduction correct?

This game tests our ability to handle the IN group so I guess I could have accounted for the possibility that both could be OUT and muave but can we just be clear that if either one is in the in group (but not the other!) it must be muave, right?

I have a good grasp of quantifiers but what is escaping me is this concept of a contradiction and what we can deduce from this. Can it even be called a contradiction if the rule works in an in-out scenario?

mAUve dude, not mUAve.


How did you get to this deduction, from the contrapositive? That's not what the contrapositive says. It says that if they're both mauve then they're not both included. That means that only one of them is included or they're both excluded. If one is in and the other is out then yes, the one that's in is mauve.


If they're both mauve then yeah, the one that's in the in-group is going to be mauve. But that doesn't mean that any time you find L or U in the in-group and the other one in the out-group then the one in the in-group has to be mauve (only if they're both mauve to begin with). If that's what you're saying then I don't see how you're getting there (if you're not then kudos).


Forget what I said about contradictions, I wasn't being clear enough. There's no contradiction with this rule; don't worry about contradictions in the rules.

1) You know muave looks better mauve.

2) I think we all finally understand the meaning of life.

If this game was designed such that the 7 Dinos were to fit in Display A (included) and Display B (not included) (instead of on display or not on display), and a question dictated conditions: "Both L and U must be mauve" then we could deduce with the aid of the contrapositive of the last condition that at least one of the two needs to be on display B, and the other on either display A or B.

This particular scenario did not present itself, but understanding this albeit convoluted contrapositive could yield some critical information, i think.

Thank you. I may be able to fall asleep tonight!

Agreed.


You're gonna have to watch Monty Python for that.

I think you may be right. My professor played a clip when the town was trying to drown an alleged witch (i think, its been a while).
The rhetoric in the dialogue was beautiful.

I am gonna treat myself to watching the whole move on Saturday evening..............October 6th.... after my first and hopefully last shot at the LSAT, lol.