Setup and Rule Diagrams

[marissasalazar9899]

Hello, for question #12 I was confused as to how C is correct, because I thought C violated rule number 4. I was under the impression that T always needed to be placed between H and V. Is that not the case? Did i misunderstand the rule?

[Adam Tyson]

The thread for #12 is here, marissasalazar9899: viewtopic.php?f=1164&t=11241

Check that out for a full explanation of that question. You're right that T must be somewhere between H and V, but it doesn't have to be immediately between them. Answer C does have T somewhere after H and somewhere before V!

[npant120]

Hi smaani!

Rule #4 states "The table is auctioned on an earlier date than the harmonica or on an earlier date than the vase, but not both."

There are really two parts to this rule:
1.) T is earlier than H or V
2.) T is not earlier than both H and V

Based on this, we know that T must be earlier than either H or V, but not both.

With an "or" rule like this, one of the options must happen. So in this case we must have T earlier than H or V. If the rule did not include "but not both" then it would have left open the possibility that T was earlier than both H and V, but it still would have had to have been earlier than at least one of them.

Hope this helps!

Best,
Kelsey

Hello,

I want to make sure I am understanding this correctly. The way you chose your 3 templates was as follows: one was the V-T-H scenario and here you knew that H-L could not happen because that would force S into spot 1 so you knew L had to be 1st and all the other variables fall behind S. The other two templates were for the H-T-V scenario where in one template you had H-L, triggering the sufficient condition from the given rule and in the other template you had the sufficient condition not happening (aka L-H) and these two options cover all the possible scenarios for H-T-V. Is this correct? Or is there anything else to keep in mind. Also, would the way we approach the templates change at all if the rule that S can't be 1st didn't exist? I have a lot of trouble with the different scenarios we need to draw out when thinking about conditional rules (other than the sufficient happening and the necessary not happening), so just want to make sure I get this down right! Thank you

[Jeff Wren]

Hi npant120,

While I wouldn't want to speak for Kelsey, based on her templates, your assessment looks correct to me.

Another way to think of it is using templates for each of the combinations of (H - L) and (L - H) along with the combinations of (H - T - V) and (V - T - H), although only the 3 combinations listed work. As you correctly pointed out under the (V - T - H) branch, L must be ahead of H.

While the reason that you gave about forcing S into 1 is one way of thinking about it, that isn't the most direct way I would think about it. Once you know that only H or L can be first (a key inference definitely worth showing on your main diagram), you immediately know that L is first under the (V - T - H) branch. With L first, it is obviously ahead of H since first is ahead of everyone by definition.

Yes, in another game you may have consider another template if (H - L) were possible under the (V - T - H) branch.

[lemonade42]

Hello,
I'm confused on in the setup where it says rule 2 allows us to make this inference
"Allowed: M --- L --- H (this is also possible when H is not before L)"

If the rule is:
H - L ---> H and M are both before L
and the contrapositive of this is:
H and M are both after L ----> L - H
How would we able to get M --- L --- H from this rule? Because I can only see L being on the far right or far left, not the middle.

[Rachael Wilkenfeld]

Hi lemonade!

Great question. You've zeroed in on one of the tricky parts of contrapositives. What happens when we logically negate the term "both"?

Our rule says that if H is before L, then M is before L. That tells us that if H is before L, then both M and H are before L.
We need to find the logical opposite of "both M and H are before L." It's not "L is before both M and H." That's the polar opposite not the logical opposite. We need a statement that encompasses all the worlds in which M and H are not before L. We know there aren't ties.

So we need at least one of M or H to be after L. Our second conditional (L is before M, then L is before H and M) is triggered once L is before M. When M is after L, H is dragged along after too. But what happens when M is before L? The first conditional isn't triggered. Our second conditional isn't triggered. So we could have the situation M---L---H. It doesn't trigger our first conditional. It doesn't trigger our second conditional. It's compliant with the rules.

Hope that helps!

[AnaSol]

The third game appears at first glance to be a standard Basic Linear game, with six auction items--H through V--placed sequentially into six spots, 1-6. Simple enough. The numbers match for a 1:1 distribution, there's no excess or empty spaces, no recycled variables...if anything the scenario here is the easiest on the test thus far.

The rules however proved to be anything but simple.

Well, I should clarify that statement. Two of the rules, #1 and #3, are quite simple: S cannot go in 1, and must go earlier than both M and V. Those should be familiar to even a novice test taker. They also lead to a handful of clear Not Laws: S is not in 1, 5, or 6; M is not in 1 or 2; V is not in 1 or 2. (M and V can't go in 2 because that would put S in 1 and break the first rule)

It's rules 2 and 4 that caused issues. Let's look at them individually:

Rule 2: this is a conditional sequencing rule, where H earlier than L triggers the sequence M before L. That sounds clear enough, but it caused people a lot of problems. Essentially this rule allows for three unique sequences, while only disallowing one:

Allowed: H/M --- L (this is what happens when H is before L: H and M are both before L)

Allowed: L --- H/M (this is what could happen when H is NOT before L: H and M are both after L)

Allowed: M --- L --- H (this is also possible when H is not before L)

Not Allowed: H --- L --- M (this is the only sequence that violates the rule; with H ahead of L, M must also be ahead of L)

So really what that rule does is eliminate a single possibility, H --- L --- M. Beyond that anything goes. L ahead of, or after, both H and M? Fine. L between H and M? Fine, as long as M is first. These possibilities are a killer if you don't catch them all. Be careful with conditional sequencing.

Rule 4: this is also a conditional sequencing rule, albeit less obviously conditional than rule 2. We talk a lot about this exact construct in our books and courses so I'll spare you too much detail here, but suffice it to say that only two possible sequences can occur with H, T, and V:

H --- T --- V

V --- T --- H

Essentially T must always be placed between H and V (it's always earlier than one, but not the other, so it gets sandwiched), whatever H and V's order. That also produces two Not Laws for T: T can't be first or last.

That's great, and leaves us with the following diagram thus far:

game 3-1.JPG
Now, with the rules comfortably in hand, we need to quickly consider additional inferences that might be made, either from ideas we've noted but not explored, or from rule linkage (shared variables in multiple rules that begin to affect one another).

The only thing we've really found so far but not probed for consequences is the outlawed sequence from rule 2, where we can never have H --- L --- M. Is there anywhere we could put L that would force H --- L --- M? Yes. Two places, in fact. If we put L in 2, as you can see from the diagram above, that places H in 1. Meaning H 1, L 2, and M somewhere after L in 3-6. That's the forbidden sequence, so L can never be in 2. What about L in 3? Once again, that puts H in 1, but can we get M ahead of L and avoid the violation? Nope. We know that M is after S from rule 3, and there's no room for S ahead of M if H is 1 and M is 2 (we also have a Not Law saying no M in 2). So M between H and L is impossible with L in 3, meaning we'd end up with the impossible H --- L --- M again, and thus L cannot go in 3 either.

So your final diagram (or at least the one I took to the questions with me), is this:

Game 3-2.JPG

As for the questions, they become much, much easier with the setup above, but there are still a few twists and turns to navigate.

Hi,

The administrator said that:

Rule 2: this is a conditional sequencing rule, where H earlier than L triggers the sequence M before L. That sounds clear enough, but it caused people a lot of problems. Essentially this rule allows for three unique sequences, while only disallowing one:

Allowed: H/M --- L (this is what happens when H is before L: H and M are both before L)

Allowed: L --- H/M (this is what could happen when H is NOT before L: H and M are both after L)

Allowed: M --- L --- H (this is also possible when H is not before L)

Not Allowed: H --- L --- M (this is the only sequence that violates the rule; with H ahead of L, M must also be ahead of L)

Is my understanding correct that: Allowed: M --- L --- H (this is also possible when H is not before L) because the two necessary conditions can happen, and the one not allowed because it refers to the two sufficient conditions?

Thanks!

[Adam Tyson]

That's correct, AnaSol! In a conditional relationship, you typically have three possibilities (as long as none of them conflict with some other rule):

The Sufficient Condition and Necessary Condition both occur (so H - L and M - L)

Neither of those conditions occurs (L - H and L - M)

and

The Necessary Condition occurs and the Sufficient Condition does not (M - L and L - H). You can think of that last one as the case where the rule is never triggered or activated.

Now as it happens, in this game that third possibility won't work, but not because of the conditional rule. It's because of the other rules. If we tried to make it happen, we would have S - M - L - H, with S also being before V. This would mean that either S would have to be first, violating one of the rules, or else T would be first, violating another rule. The only variables that can go first are H and L, so the rule will be activated either way. But in general, your understanding of the three possibilities for a conditional rule, including a conditional sequencing rule, is correct. Good work!