Two Rule Inference Drill #8

[cecilia]

Hi there and question on the last drill on page 17 of the Logic Games workbook:

Just to be clear, J and L could be together, correct? And that's why one can't draw any sort of inference from the two rules?

Thanks in advance for any help.

[Dave Killoran]

Hi Cecilia,

Thanks for the message. Yes, you are correct: J and L could be together.

If the rules had been that: J K <-- L, then J and L could not have been together, but because the arrow between K and L goes the "wrong way," there is no restriction.

Please let me know if that helps. Thanks!

[cecilia]

Hey Dave, thanks so much. That helps.

And the other scenario: "If the rules had been that: J <--|--> K <-- L..."

Sorry to be so dim, but broken down that means:

J <--|--> K
L---->K

And so therefore:

J<--I-->L
( or in English, J can't be with L in this instance)

Is that right?

[Dave Killoran]

Yes, that is correct!

When you look at this relationship:

J K <-- L

It is exactly the same as this, more familiar looking relationship:

L K J

The two are identical, just presented in reverse order, and both yield the same inference: J L .

Thanks!

[cecilia]

Thank you so much! Great explanation.

[jgray]

Doesn't J K L mean, in English, If J, then /if not K, then L?

Seems to conflict with rule 2 "if K then L".... k L.

How do you link the not K to L when L reqires a K?

[Dave Killoran]

Hey J,

Thanks for the question!

What that statement means if you read it in that fashion is: if J, then NOT K; then, the second part is if K, then L. So, you can see there's an issue there with Not K and K. How is that we connect them? So, the two statements that are connected there are:

• J K
  
  K L

This can be done because of how the double-not arrow works. The double-not arrow is incredibly useful because it allows us to keep K in "positive" form, and it transfers the negative into the symbol between J and K, which more fully capture the meaning that K and J can't go together. Then, because K is positive, we can connect the two statements.

When I look at those two statements, what I think is:

• 1. J and K can't go together, so if you have one of the two, you can't have the other (which really means J K, and K J )
  
  2. K L

We can go a lot deeper on this (and discuss the inference here as well), but I'm going to stop here and you let me know if this makes sense so far. If it doesn't, we'll look at it again.

Thanks!

[praxis]

Hi! MP here. I was just wondering if someone could shed light on why there aren't two sufficient conditions in this drill? I am confused on the comparison to item #1 in the same drill set. To me, the rules look exactly the same, with K being the necessary condition for one rule and the sufficient condition for the other rule.

Why is K the sufficient condition for both rules?

Thanks,
MP

[Adam Tyson]

K is not a necessary condition in either rule in this drill, MP, because there is nothing that proves that K must occur. Instead, if J happens then K CANNOT occur. That makes "not K" a necessary condition. When you do the contrapositive of that relationship, K becomes sufficient to prove "not J." Thus, K is sufficient to prove that L occurs, and it is also sufficient to prove that J does not occur, but there is nothing sufficient to prove that K does occur.