Diagramming Either/Or

[Madison5941]

Hello,

I am currently on page 216 of the LR Bible and I do not understand how to diagram either/or statements.

I will use the example on page 216, "Either Cindy or Clarice will attend the party, but not both."

The "either/or" phrase gives us the following diagram:
Not Cindy --> Clarice
or the contrapositive: Not Clarice --> Cindy

The "but not both" phrase gives us the following diagram:
Cindy --> Not Clarice
or the contrapositive: Clarice --> Not Cindy.

I am confused because I don't understand how and why either/or is diagrammed to negate the first term (Cindy) and how "but not both" negates the second term (Clarice). I understand that either/or means the possibility of having both but I don't understand how the diagram reflects that idea. Why couldn't "either/or" be diagrammed like

Cindy --> Not Clarice

What is the real difference between the two diagrams and what do they mean?

[Jonathan Evans]

Hi, Madison!

One fundamental issue here is we effectively have two different statements:

1. Either Cindy or Clarice will attend the party.
2. Cindy and Clarice will not both attend the party.

You do a good job providing the conditional diagram for both statements. The conditional diagram for statement one implies that at least one of them is there. In other words, if one of them isn't there, then the other must be. This is how we arrive at the first conditional:

Cindy attends -----> Clarice attends
Clarice attends -----> Cindy attends

If Cindy isn't there, we know for sure Clarice must be. If Clarice isn't there, we know for sure Cindy must be. This is the diagram of the either/or part.

However, looking at this diagram, we still leave open the possibility that both could be there together. There's nothing in that conditional that tells us they can't both attend. When we add the "but not both" we have to rule out this possibility.

If we know that both Clarice and Cindy won't attend together, we need to add the conditional:

Cindy attends -----> Clarice attends
Clarice attends -----> Cindy attends

Using PowerScore's double-not arrow we can shorten this as Cindy <--|--> Clarice

We are then left with the following information:
Cindy attends -----> Clarice attends
Clarice attends -----> Cindy attends
Cindy <--|--> Clarice

According to the necessary conditions above, at least Clarice or Cindy must be there. However, according to the double-not arrow, they both can't attend together.

I hope this helps! Please respond with further questions.