Either but not both and Double-Not Arrow

[lsathelpwanted]

***This note you're reading is being written after I wrote the body of my post. I THOUGHT I had a grasp on this but as I wrote this post I have just REALLY confused myself to the point I am not even sure what I am asking anymore, so I will try to clarify some main points I THINK I have.

Can I always take the contrapositive of a double arrow ?
Can I take a contrapositive of a double-not arrow ?
(If I can, do I do something funky with negating the not* in the double-not arrow thus making it a double arrow ?
Why when two things are mutually exclusive, and one MUST occur, do you not use a ?


I have two sets of two scenarios I'm not too clear on when I try to compare them.

1.

Either Cindy or Clarice will attend the party, but not both.
versus
If Gomez runs for president, then Hong will not run for president.

I really want to diagram the first as:

Cindy Clarice
Contrapositive
NOT Clarice NOT Cindy

To me the initial diagram says one person WILL attend the party, the other CANNOT be attend the party.
And the contrapositive says if one does NOT go to the party, then the other MUST go to the party. They CANNOT both NOT go. I thought that was what signifies.

Compared to:
If Gomez runs for president, then Hong will not run for president. This IS diagramed as G H
Now I'm confused about the contrapositive! NOT H NOT G.....this doesn't seem right to me. That says that one of them MUST run for president, but G NOT H allows for the possibility of NEITHER running.

Is it because the President example is conditional on whether one runs in the first place? Whereas, the Party example has one person going for certain?




2.

Either Jones or Kim will win the election.
versus
It is either feast or famine.


***I completely understand you do NOT want a double-not arrow because we do not know the ramifications of what they are being elected for and their COULD BE multiple people elected, like the school board the book mentions.

Compared to:

It is either feast or famine.
I want to:
Feast Famine
But, I think I'm wrong because it is not listed as an answer. David addresses that feast and famine are mutually exclusive. So you cannot have both. To me the original statement says that it must be either feast or famine and there is no room in between and my common sense says there can't be both.

My mind is blown. I felt really good about this entire chapter up until this one thing. I think my issue resides in the mutually exclusivity, but I am honestly not even sure anymore. I have reread this section and I am confusing myself even more.

Please do not feel it necessary to respond to each point of this post. If you may be able to explain it in a way I can apply it and figure this out myself, I think that would be really beneficial. But you guys know better than I do!

Thank you so much for any replies.

[Jeremy Press]

Hi lsathelp,

You've asked a lot here, but the good news is that it's relatively simple to boil down!

Let's start with the Double-Not Arrow.

The Double-Not Arrow is used for rules that state that two variables cannot be selected (cannot be IN a group) together. Let's use one of your examples: "If Gomez runs for president, then Hong will not run for president."

The diagram of that rule as stated is: G H.
The diagram of the contrapositive is: H G.

The Double-Not Arrow is a way to simultaneously represent the rule AND its contrapositive: G H. Starting from G and running to H, we can see that G will trigger NOT H (the slash in the middle of the arrow indicates the "not"). But starting from H and running to G, we can see that H will trigger NOT G (again, the slash in the middle indicates the "not").

Long story short: there is no "contrapositive" of a Double-Not Arrow, because the Double-Not Arrow incorporates the contrapositive within it. That's why we like the symbol! It's more efficient!

Now let's talk about the Double Not-Not Arrow

The Double Not-Not Arrow CAN be used (some don't like it, others do) for rules that state that at least one of two variables must be selected for inclusion in a group (i.e. that the variables CANNOT both be OUT). So let's take one part of an example you gave: "Either Cindy or Clarice will attend the party."

The diagram of that rule as stated is: Cindy Clarice
The diagram of the contrapositive of that rule is: Clarice Cindy

The Double Not-Not Arrow is a way to simultaneously represent that rule AND its contrapositive: Cindy Clarice. So, if you use that symbol, it doesn't need another "contrapositive," because the rule and contrapositive are already contained within it.

Rules that take BOTH the Double-Not Arrow AND the Double Not-Not Arrow

Sometimes a rule tells you that at least one variable must be included (the Double Not-Not Arrow), but that the two variables cannot both be included (the Double-Not Arrow). That's a rule like the full Clarice-Cindy rule: "Either Cindy or Clarice will attend the party, but not both."

For such a rule, your diagram either needs to list four separate conditional arrow relationships:
Cindy Clarice
Clarice Cindy
Cindy Clarice
Clarice Cindy

OR, your diagram needs to list the Double Not-Not Arrow, plus the Double-Not Arrow:
Cindy Clarice
Cindy Clarice

Practically speaking (in an "In-Out" sceanrio), that combination-style rule means every solution in the game will either have "Cindy in, Clarice out" or "Clarice in, Cindy out."

Let me know if that clears up confusion!

[lsathelpwanted]

Hi lsathelp,

You've asked a lot here, but the good news is that it's relatively simple to boil down!

Let's start with the Double-Not Arrow.


The Double-Not Arrow is a way to simultaneously represent the rule AND its contrapositive:

Long story short: there is no "contrapositive" of a Double-Not Arrow, because the Double-Not Arrow incorporates the contrapositive within it. That's why we like the symbol! It's more efficient!

Now let's talk about the Double Not-Not Arrow

OR, your diagram needs to list the Double Not-Not Arrow, plus the Double-Not Arrow:
Cindy Clarice
Cindy Clarice

Practically speaking (in an "In-Out" sceanrio), that combination-style rule means every solution in the game will either have "Cindy in, Clarice out" or "Clarice in, Cindy out."

Let me know if that clears up confusion!

Holy cow thank you! Amazing.

The explanation about it actually INCLUDING the contrapositive was perfect. As I was reading along at the end towards your Cindy/Clarice explanation, I anticipated you were going to say we would need BOTH the Double-Not and Double-Not-Not. So I feel great about that now.

Also, "Double Not-Not" is really fun to say.

I truly think your write-up should be included in this chapter in the next edition!! It really ties it all together and even takes it to the next level. (I think I recognized the potential for this but I couldn't wrap my brain around it to get there.) Thank you SO SO SO SO MUCH Jeremy. I tried my best to explain my thought process and the questions (I thought) I had and to be not as confusing as possible. Thank you again!