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 g_lawyered
  • Posts: 213
  • Joined: Sep 14, 2020
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#89452
Hi P.S.,
As a follow up post to my question about my inferences. I tried to "hurdle the uncertainty" as your lesson book explains by thinking about the numbers in this game (especially since it's an UNDEFINED game). I realized that the game only provide a minimum of 1 variable but realized the rules lead to more than just 1 variable.
I came up with these inferences that throughout the game weren't as helpful as the chain inferences (from my previous post). Can someone correct if the following inferences are correct or incorrect?

From rule 1 because of the conditional statement & it's contrapositive that from positive :arrow: negative as:
Positive rule: M :arrow: L OR S
Negative contrapositive: NOT L and NOT S :arrow: NOT M
I inferred that at most 2 of the variables can go (either M with L or M with S- but not both- like the rule states). Meaning that 1 of them must be out.
Does this affect the minimum # of variables that can go in?
I wrote: 7 (total # of variables) - 1 (either M with L or S) = 6 variables. A minimum of 6 variables can go in (1 of L or S must be out).

From rule 4 because of the conditional statement & it's contrapositive that from positive :arrow: negative as:
Positive rule: G :arrow: H AND Z
Negative contrapositive: NOT Z OR NOT H :arrow: NOT G
I inferred that at most 1 of the variables can go .
Does this affect the minimum # of variables that can go in?
7 (total # of variables) - 3 (contrapositive states that all 3 could be out) = 4 variables. A minimum of 4 variables can go in (these 3 could be out).

As I mentioned, these inferences weren't as helpful in the questions as I thought it would be. Instead my inference were more helpful. But I wanted to know if these inferences are worth or correct to make (before starting the questions). :-?

Thanks in advance! Appreciate your time and help
 g_lawyered
  • Posts: 213
  • Joined: Sep 14, 2020
|
#89564
Hello,
Can someone please get back to me on my previous posts.
Thanks in advance!
 Robert Carroll
PowerScore Staff
  • PowerScore Staff
  • Posts: 1819
  • Joined: Dec 06, 2013
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#89592
GGIBA003,

First, I want to note that a discussion of the contrapositive is already present, in a post a few above where your first post is:

viewtopic.php?p=62619#p62619

There, Jay says:
Here's the CP:

(Literature and Sociology) or (~Literature and ~Sociology) --> ~Mathematics

If Math, we need 1 from L/S. If either 0 or 2 from L/S, we cannot have math.
The first rule is a complicated conditional, and while you can apply the rule for turning "and" into "or" and vice versa, the first rule is actually two conditionals:

"If M, then L or S"

"If M, then not both L and S"

...each of which has to be contraposed. I think Jay's quoted post simplifies what otherwise could be overly complicated. Paraphrasing him:

"If L and S are both in, or L and S are both out, M is out"

It ends up that, because of conflicting requirements regarding P, L and S can't both be in anyway, so we don't need to worry much about that aspect of the contrapositive. It's logically true, but practically won't come up, because L and S will never be found together in any valid group.

M :arrow: L is not a proper inference, which is why we don't show it, and why our chain does not start with M. If M is in, we don't know which of L and S is in, so no inference can be made there for certain. I don't think it's very clear or helpful to write M :arrow: L as an "inference" when it's not necessarily true. You're instead claiming something like two hypotheticals, or even templates: M with L, or M with S. That's fine. To complete that, of course, you'd want to account for a situation where M is out entirely.

The rules don't lead to more than 1 variable. I think it's possible the discussion of the previous paragraph is necessary to avoid that mistake - M does not have to be in, so none of the conditionals have to trigger at all. H could be in with nothing else, for instance. So the minimum group size is 1.

The maximum group size is 5. Every variable except S and P could be used. Attempting to use more than 5 will cause a problem - L and P can't both be in, for instance, so we can't use all 7, but trying to use 6 variables will force everything EXCEPT L and P in, with ONE of those in, which forces S and G together. Leaving P and S out, as mentioned, avoids this problem, so 5 is the maximum.

The minimum and maximum values you're looking at aren't minimum or maximums at all because you're considering them conditionally. Our inference about minimum and maximum group sizes should be global, not local under certain assumptions - "If X is in, our minimum is 3" doesn't tell us what out global minimum is, for instance.

Robert Carroll

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