always apart - why not double not arrow?

Christina Walton

In a rule like "Either J or K are selected, but not both are selected" in an In Out Game, why is it represented as J<->not K and not J double not arrow K?? Wouldnt J double not arrow K mean that both are not selected?

Law and Yoda

"Either J or K, but not both " means, one or the other, never both. Not both has a minimum of 0 and maximum of 1. So the rule can have 3 possible worlds;

/J /K (both out, minimum of 0 satisfied) World 1
/J K (J out, K in, maximum of 1 satisfied) World 2
J /K (J in, K out, maximum of 1 satisfied) World 3

The world that doesn't work is;
J K (J in, K in) This exceeds the maximum

If you draw /J<->/K, that represents world 1 but it might cause you to forget that you can also have world 2 and 3. So typically when you see a "Not Both" you should immediately recognize that the sufficient is not negated and the necessary is negated, this will help you remember the 3 possible worlds.

bruhhhhhh

Just to clarify, in the case above of "either J or K, but not both", wouldn't world 1 also not be a viable option?

The "either or" rule requires at least one of J or K to be present, so only world 2 and world 3 remain as possible worlds --- please correct me if I'm overlooking something!

Law and Yoda

@bruhhhhhh Because the word not both is tagged onto the end, it becomes a not both rule, not an "or" rule. Check out these lessons to remember what worlds are possible for each rule! If it is still confusing let me know and I'll try to explain it in a different way!

https://7sage.com/lesson/not-both-v-or-truth-tables-longer-explanation/
https://7sage.com/lesson/not-both-v-or-truth-tables/

Also this forum explains the rules a bit more in depth and may be helpful
https://7sage.com/discussion/#/discussion/10622

bruhhhhhh

@"Law and Yoda" Thanks for the reply! hmm but wouldn't OP's rule qualify as an "or, but not both" rule? In that case, please refer to the below lesson:

https://7sage.com/lesson/or-but-not-both/

In light of the lesson above, the Lawgic for "Either J or K are selected, but not both are selected" translates into J <--> /K (biconditional). The either or rule must be taken into account as well as the not both, and thus at least one, either J or K, must be present, which is not the case for world 1.

This would also address @"Christina Walton"'s original question, as to why J <--/--> K (J double not arrow K) cannot represent "either or, but not both".

In J <--/--> K, exactly 1 outcome is impossible -- (1) both J and K present. J <--/--> K does not mean always apart, since they can still be absent together, but simply not present together.

In J <--> /K, exactly 2 outcomes are impossible -- (1) both present and (2) both absent, therefore always apart, which correctly represents all the possibilities for "Either J or K are selected, but not both are selected".

Hope that made sense!

Law and Yoda

@bruhhhhhh Hey you're totally right! I was overlooking the "or but not both", so you're definitely correct in stating that world 1 is not possible. Thank you