Conditional Reasoning Diagramming Drill #8, 10, 11, and 12

[kalifaingold]

Hello, I am having a lot of difficulties diagramming conditionality with sequencing. Specifically in two specific instances.
The first was on questions 8 and 10. For question #8, the contrapositive would be "if R is not published 5th then R is not published after P" but that is not what is shown in the diagram of the answer key. In the answer key, it shows that "if R is not published 5th then R must be published before P- remember no ties!". I don't understand this part. What does it mean by no ties? How can you just reverse the letters in the contrapositive if the contrapositive is already a reverse of the original statement?
I encountered the same problem in question 10 where again, the contrapositive in the answer key is not the contrapositive I drew.
The second instance that causes me to have problems is on questions 11 and 12, where there is sequencing as well as the "unless equation". For question 11, in the original, why is it G-F? Wouldn't it be F-G since "G cannot be cleaned until F is cleaned"? And then again it flips and the contrapositive is F-G.
The same thing happens in question 12. If "the A train cannot arrive until the C train arrives" wouldn't it be C-A. In the answer key, it shows A-C which I just don't understand. And then AGAIN it is flipped in the contrapositive!

So basically I have two main problems, I don't understand why the letters are always switched in the contrapositive and how the letters are arranged within the sequence in an unless equation instance.
please help

[Dave Killoran]

Note to other readers: due to newly added content, in the 2020 Edition this drill is now on page 76.

Hello, I am having a lot of difficulties diagramming conditionality with sequencing. Specifically in two specific instances.
The first was on questions 8 and 10. For question #8, the contrapositive would be "if R is not published 5th then R is not published after P" but that is not what is shown in the diagram of the answer key. In the answer key, it shows that "if R is not published 5th then R must be published before P- remember no ties!". I don't understand this part. What does it mean by no ties? How can you just reverse the letters in the contrapositive if the contrapositive is already a reverse of the original statement?

Hi K,

Take a look at this one again—I think you mistook P and R in your question (because you said "if R is not published 5th" which is wrong). The rule is, with the necessary condition italicized:

• R is published after P only if P is published fifth.

So, that is:

• ( P R ) P5

So far, so good, right?

Well, what does a contrapositive do? It reverses and negates all the terms. That means the P5 will be negated and move over to the sufficient side:

• Sufficient:
  
  P5

That's what is in the book. You mentioned R being there, but that's not the case—it's P.

Continuing on, now we do the same to the other term, which is:

• Necessary:
  
  P R this is the whole relationship negated; forum limitations don't allow it to be shown clearly

However, and this is key, what is P R in actual practice? If P is not published before R, then it has to be published after R, which is R P. And that's what is shown in the book as well.

What is shown in the book is taking the relationship to it's logical conclusion, which is the skill we are attempting to develop here, among others This is a skill you want to have in LG since it makes you see the relationships, and their outcomes, more clearly.

Also, the "no ties" reference comes from the fact that in the drill we mentioned no ties were possible, otherwise we could have had R = P as one possible solution, and that's not something we want to deal with at this point in the book. It's also a good tipoff to always keep an eye out for when ties are possible or not possible!


Do me a favor and look over the others after reading what I've written above. I suspect it will help shed some light on those. Thanks!

[kalifaingold]

Hi Dave, thank you so much for taking the time to answer my questions. Seriously this forum and all the Powerscore staff are so wonderful I am immensely grateful.
I now understand question 8 and 10 after your explanation! I am still unsure about 11 and 12 though because of the word "unless".
In question 11, it states that "G cannot be cleaned until F is cleaned unless F is cleaned second". I came to this diagram:
(F-G)--F2 but this is wrong in the answer key. The answer key flips the F and G and instead the diagram is (G-F)--F2. I don't understand this. Wouldnt it have to be that F goes before G since "G cannot be cleaned until F is cleaned"? Then the contrapositive just further confuses me because it states what I would have originally thought which is (F-G).
I made the same mistake in question 11, in which I put (C-A)--AC but the answer key put (A-C)--AC.
I am very frustrated because I just feel like I was never explained these kinds of questions where there is an "unless" with sequencing and then am expected to know how to do it.
Thank you again.
Kali

[kalifaingold]

Also sorry, last question, can you explain this "ties/no ties" concept. I don't remember reading about it and would really appreciate some clarification.

[Adam Tyson]

The bit about ties is about variable occupying the same space in the diagram, kalifaingold. At the beginning of that drill the instructions said to assume that no ties were possible, meaning that no two variables can be in the same slot. Thus, if X is NOT before Y, then it must be after Y. In a game where ties are possible, then if X is not before Y, it is either after Y or else is AT THE SAME TIME as Y. Most games do not allow ties like this, but some do, and so you must be careful not to assume that "before" and "after" are the only two options. Read carefully to see if a tie is possible!

The trouble you are having with 11 and 12 in that drill is in applying what we call the Unless Equation, which is triggered by the use of four special words - Unless, Except, Until, or Without. When you encounter one of those words, whatever they refer to is your Necessary Condition, but the other thing in the relationship is not Sufficient. Instead, the other thing has to be negated, and that negated condition is the true Sufficient Condition. So in #11, for example, where G cannot be cleaned until F is cleaned unless F is cleaned second, the word "unless" refers to "F is cleaned second," and so F2 becomes your Necessary Condition.

Applying the Unless Equation now, you have to negate the other thing, which was "G cannot be cleaned until F is cleaned." That statement means that F must come before G. The negation of that statement is that F is NOT before G, and since they cannot be at the same time (again, no ties are allowed in this drill), that means G is before F. So, if G is before F, then F must be second, or:

(G F) F2.

Try applying that same series of steps to #12! Remember that whatever unless/except/until/without refers to is Necessary, and the negation of the other thing is Sufficient. And if there are no ties allowed, the negation of X before Y is Y before X!

[momgoingbacktoschool]

Why is question #3 not treated similarly? by the logic used in these questions, shouldn't the contrapositive be (H-L) then (M-F) instead of what it says in the book which is not(L-H) then not (F-M)? I understand the logic behind your conclusions on these questions. But I don't understand why question #3 is not treated in the same way. Thank you!

[Dave Killoran]

Why is question #3 not treated similarly? by the logic used in these questions, shouldn't the contrapositive be (H-L) then (M-F) instead of what it says in the book which is not(L-H) then not (F-M)? I understand the logic behind your conclusions on these questions. But I don't understand why question #3 is not treated in the same way. Thank you!

It actually should be. So it's not technically wrong but it's also not optimally presented. I will take another look at it when I can grab a hardcopy and change it assuming so. Thanks!

[momgoingbacktoschool]

Why is question #3 not treated similarly? by the logic used in these questions, shouldn't the contrapositive be (H-L) then (M-F) instead of what it says in the book which is not(L-H) then not (F-M)? I understand the logic behind your conclusions on these questions. But I don't understand why question #3 is not treated in the same way. Thank you!

It actually should be. So it's not technically wrong but it's also not optimally presented. I will take another look at it when I can grab a hardcopy and change it assuming so. Thanks!

Thanks!