Grouping Games Rule Diagramming Drill #6

[Jhernandez]

I have a question about the 'Grouping Games Rule Diagramming Drill' on page 255 of the 2013 edition of the Logic Games Bible.

The last question states "Either F or G, but not both, must be selected." The answer shows that the Double Not Arrow should be used and states that exactly one must be selected and exactly one must not be selected. I found this to be really confusing because I thought that the Double Not Arrow meant either of the selections OR neither of the selections would be allowed. Therefore, the Double Not Arrow would show that either F or G HAD to be selected.

If I go over the material enough times, I have been able to understand everything so far, but this has me stumped. If you could help, I would greatly appreciate it.

Jennifer

[Nikki Siclunov]

Hi Jennifer,

Thanks for your question!

Before I discuss the drill you asked about, let me clarify the implications of linking two variables with a Double Not Arrow: the arrow simply indicates that the two variables cannot both be selected (when both those variables are positive, meaning they do not have slashes). The Double Not Arrow is basically a shortcut of the following rule (and its contrapositive):

A NOT B

B NOT A

So, A B

This allows for the possibility of selecting A without B, or B without A, and it also allows us to select neither A nor B. What the Double Not Arrow prohibits is the possibility of selecting both A and B.

Regarding your question, a rule stating "Either F or G, but not both, must be selected" has to parts to it. Let's examine them separately:

• 1. "Either F or G must be selected."

  
  
  In other words, if we don't select F, we must select G, and vice versa: at least one of F or G must be selected. This clause, in and of itself, does not preclude the possibility of selecting both F and G, which is why a Double Not Arrow should not be used to represent it. However, the clause disallows us from selecting neither of them:
  
  NOT F G
  
  NOT G F
  
  Or, you can simply write down:
  
  F or G (in the group of selected variables)
  

  2. "...but not both."

  
  
  This is where the Double Not Arrow comes from. If we can't select both F and G, that means that if we do select F, then we cannot select G. And by the contrapositive, if we select G, we cannot select F. Thus, the rule should be represented as:
  
  F NOT G
  
  G NOT F
  
  Shortcut: F G

Now, let's combine the two rules. One way of doing that is by using the Double Arrow, because each variable is both sufficient and necessary for the absence of the other:

F NOT G

G NOT F

The implication of this is that, at all times, either F or G must be selected; however, both cannot be selected. Simply linking the two variables with the Double Not Arrow (F G) is by all means accurate, but somewhat incomplete way of representing the full implications of the rule. This is because the Double Not Arrow, by itself, does not suggest that at least one of the variables must be selected: it simply disallows both from being selected together (i.e. it captures the implications of the second clause above, but not of the first). So, if you'd like to use that arrow, the rule should look like this:

F or G (in the group of selected variables)

and

F G

It's a tough rule to wrap your head around, but once you do - and you practice it for a bit - it will become intuitive. It's frequently used on Grouping Games, so it's imperative that you understand what it entails (and what it does not).

Let me know if this makes sense