Conditional Reasoning Diagramming Drill - Ties

[Stephanie Oswalt]

We recently received the following question from a student. An instructor will respond below. Thanks!

Good morning,

I recently purchased the LG Bible (2018 edition) via Amazon and it is excellent!

I have a question regarding the conditional reasoning diagramming drill on page 67.

The instructions say, "Assume a Basic Lineral 1-to-1 setup is in effect, with no ties possible."

I'm not sure what "no ties" means in this circumstance. I reread the LG Bible up to this point and haven't found an explanation for what a tie is.

Then, the answer key says for number 3, the answer is:

original: (F -- M) --> (L -- H)
contrapositive: (L -- H) --> (F -- M)

This made sense, but then for answer 8 (among other similar answers):

original: (P -- R) --> P5
contrapositive: P5 --> (R --P)

I thought the answer would be P5 --> (P -- R).

The Bible's explanation says the reasoning it is written (R -- P) rather than (P -- R) is because there are no ties. I don't understand exactly what this means, and why this problem (number 8) is written like this, while the contrapositive for number 3 (above) is written (L -- H) --> (F -- M)instead of (H -- L) --> (M -- F).

I have been puzzling over this for a while today, so any insight into this reason would be greatly appreciated.

Thank you so much!

Kind regards,

Sarah

[Jonathan Evans]

Sarah,

Thanks for the excellent question and for joining us here.

"Ties" means variables could be selected for the same slot on the diagram. Here are a couple examples:

• A professor meets with eight students, A B C D E F G H, at least one per day, Monday through Friday. The following restrictions apply.
  
  A student is registering for classes in Biology, Chemistry, Geology, and History in the upcoming fall, spring, and summer semesters. The student will register for exactly two classes each semester. The following restrictions apply.

In the example in the Logic Games Bible, since there are no ties, we may infer that (P R) is equivalent to (R P). "P doesn't come before R" is equivalent to "R comes before P."

Does this make sense?

[Stephanie Oswalt]

The student, Sarah, asked a follow up question below.

Thank you so much for your prompt response!

I see that "ties" means variables selected for the same slot on the diagram, thank you for clearing that up!

I am still a little confused why it's better to write (R P) in the contrapositive rather than (P R)?

Is it simply easier to understand because it's an affirmative rule rather than a negative one? If so, why is the Question #3 I mentioned earlier written like this:

original: (F -- M) --> (L -- H)
contrapositive: (L -- H) --> (F -- M)
rather than like this:
original: (F -- M) --> (L -- H)
contrapositive: (H-- L) --> (M -- F)

Is it just a preference or is there a reason behind this?

Thank you so much for all of your help!

Sarah

[Jonathan Evans]

Hi, Sarah,

Thanks for the follow-up!

Is it simply easier to understand because it's an affirmative rule rather than a negative one? If so, why is the Question #3 I mentioned earlier written like this:

[...]

Is it just a preference or is there a reason behind this?

Yes, in general, it is easier to understand the affirmative rule rather than the negative one. On question 3, we illustrate it in the negative to introduce the concept. Later, on question 8, we translate the negation into the affirmative rule to show how that is possible. This is a matter of preference, but I would prefer to use the format as illustrated on question 8 if you can. Remember that (R P) and (P R) are equivalent to one another and interchangeable here.

Does this make sense?