Either/Or and the Double Arrow

[snuggs]

Does an 'either or but not both' statement always diagram as a double not-arrow?

[Dave Killoran]

Hi Snuggs,

Good question! The answer is yes, but that is only half of the story. The phrase you reference is actually two relationships in one, so let's look at both sides.

To start, let's take a statement that says: "Either A occurs or B occurs but not both."

This statement actually has two separate components:

• 1. Either A occurs or B occurs
  
  2. Not both of A and B occurs

Each part has a different diagram (and in fact, polar opposite diagrams):

• 1. Either A occurs or B occurs
  
  This statement means that at least one occurs, which is the same as when one does not occur the other must occur. That can be diagrammed as A B, or the contrapositive, B A. When we turn that into a super-statement using the double-not arrow, it becomes
  
  A B
  
  This statement is one of the toughest in LR or LG, but it simply means both cannot be absent (and thus, at least one must occur).
  
  2. Not both of A and B occurs
  
  This statement is the one I think you were referencing, which means that both cannot occur:
  
  A B

When these two statement are combined, you discover that both cannot be absent (not occur) and that both cannot occur. Thus, one and only one of the two can occur, leaving just two possible outcomes:

• 1. A and B
  
  2. A and B

Essentially, the "but not both" portion closes off one of the possible outcomes that is typically available under and either/or scenario, leaving just two possible scenarios.

Please let me know if that helps. Thanks!

[snuggs]

wow, that must be what it's like to watch the large hadron collider split particles.
thank-you, that definitely helps!

[Dave Killoran]

Lol, I'll take that as a compliment

Thanks!

[a.lsat]

Hi Dave

So how to diagram "either A or B, but not both occurs" in a LR stimulus?

Are we going to draw out both "A <-/->B" and "~A <-/-> ~B" at the same time?
If yes, how to combine other rules if there's any in the stimulus?

Thanks for your help!

[Dave Killoran]

Hi A,

Thanks for the question! Actually, I wouldn't diagram it at all. I'd simply move forward knowing that exactly one of A and B is going to occur, and I'd feel comfortable enough that I wouldn't feel like the diagram had to be drawn.

If you made me diagram it, then yes, it would be those two diagrams

Please let me know if that helps. Thanks!

[Osa]

Hi Dave,
So Either A or B occurs simply means at lease one of A or B must occur. Is it also true that both A and B can occur together? I.e We can have A, B or AB.

If yes, then is the statement Either A or B occur same as Either A or B or both occurs?

Thanks

[Dave Killoran]

Hi Osa,

Yes, that is exactly correct. Very well done!

There is one circumstance that can affect the outcomes there, however. That is when the two items under discussion are known to be mutually exclusive. For example: "In Brazil, it's either day or night right now." What LSAC calls "commonsense" applies here, because we know that it can't be both day and night in the same spot at the same time. So, in those cases, only one of the two can occur at once, not both. That variant comes up pretty rarely, and when it does, it's always fairly obvious (as in the night/day relationship).

So, if you don't see "but not both" language nearby, and if the two variables aren't connected in a mutually exclusive manner, then it is the case that both could occur.

Please let me know if that helps. Thanks!

[elyssa1]

Hi Dave,

I'm stumped on a statement and trying to figure out what the opposite would be.

Rule: "If I occurs, then K occurs, and if K occurs, then I occurs."
If K does not occur, does this mean I does not occur? (and if I does not occur, would K not occur?) How is this negation diagrammed? If the regular rule is written as K I, K not occurring would not be the same as K I. Is it an opposite? Is it a contrapositive?

Thanks!
Elyssa

[nicholaspavic]

Hi Elyssa,

Welcome to the forum!

I can jump in for Dave here. In the case of K I, the contrapositive of a Double Arrow (aka biconditional) would be ~K ~I. In other words [not] K [not] I. And you are correct that the Double Not Arrow would mean that K cannot occur simultaneously with I and I cannot occur simultaneously with K. Thanks for the great question and I hope this helps!