P.330 Logic Game Pure Grouping Super Drill Question

[Obiwonwasabi]

I am having trouble understanding the contrapositive of the rule "If B does not occur, then either H or I but not both occur."

B -> H or I which can be B -> [HI] block.

The contrapositive would be:
H and I -> B or [HI] -> B Why would this be the same? If not H and not I, how would that become H and I? Am I missing something here?

[Emily Haney-Caron]

Hi Obiwonwasabi,

I think I understand your question, but if I'm missing something please let me know and I'll try again. What I think you're asking is how the two contrapositives you listed are the same thing. If that's what you're asking, the answer is, they're not! From that one rule, you really have two separate rules: if no B, then either H or I; and if no B, then not both H and I. As a result, you also have to take the contrapositive of each of those aspects of the rule. The first is, if neither H nor I, then B. The second is, if both H and I, then B. Does that make sense?

[jgray]

Do you have another example to help illustrate?

[Dave Killoran]

Hi J,

Well, there are millions we could make up, such as "C does not occur, then D or G but not both occur." That produces the same type of diagram, which I don't think is super helpful to repeat. So, let's instead discuss what's going on with the rule and hopefully we can add some depth there The basic format is:


If a variable does not occur then one and exactly one of two variables occurs.


Thus, the sufficient condition is negative, which I've written elsewhere is extremely dangerous. The second part is the unusual necessary condition, which specifies that exactly one of two variables must occur. To negate that necessary condition (as would occur in a contrapositive), requires denying that exactly one of the two occurs. Since there are two variables in play, that means two outcomes can occur when exactly one isn't happening: both occur or neither occur. In other words, with 2 variables you have three possible states:

• None occur = 0
  One occurs = one or the other (so two outcomes, actually) = 1
  Both occur = 2

So, if you take out (or, in other words, negate) that middle scenario, you have:

• None occur = 0
  One occurs = one or the other (so two outcomes, actually) = 1
  Both occur = 2

That's why you get two separate diagrams there for the contrapositive—which we do because it makes it easier to understand, not because we have to have two diagrams; we could have just one but it would be harder to work with for most people.

Please let me know if that helps. Thanks!

[jgray]

Still having issues with it....

From a previous post... "if no B, then either H or I; and if no B, then not both H and I. As a result, you also have to take the contrapositive of each of those aspects of the rule. The first is, if neither H nor I, then B. The second is, if both H and I, then B." I understand this statement, but not proficient enough to replicate.

I also get the one-in, one-out, or both out.... A B
I also get the one-in, one-out, or both in.... Not A B

Moving on to #8 explanation...
"...B should be kept because if B occurs, then both H and I can occur ..." Where do we see that if B occurs then H&I occur? I can see having H&I occur bringing B in, but that doesn't seem to replicate the rule.

Thanks

[Dave Killoran]

Hi J,

I'm not following you in the first part—is there a question there that I'm not seeing? I'm just asking to make sure I'm not missing something If so, go back to my last post, and combine the "one and only one" discussion with that breakdown of possibilities you review. That's the pathway to understanding this.

In #8, via the contrapositive, you have this result:

• one and exactly one of two variables occurs = (Both of HI) OR (Neither of HI) B

So, if we have neither or both of H and I, then we have to have B. Well, you get to play LSAT god here, and since you want to maximize the variables that occur, selecting H, I, and B is the outcome you want. So, choosing B allows for that to happen (because if you don't select B, then you get one and only of of H and I). What was being said there was that if you make sure the necessary condition occurs, then that allows you to do as you choose with the sufficient condition (it was not a literal reversal; this was a discussion of what makes other things possible).

Hopefully that helps a bit here. Thanks!