Question about either or, either or not both and bi-cond(s)

[Pepperdine2014]

Hi everyone,

I do not fully grasp the concept of either or, either or not both and bi-conditionals and would appreciate some help, please.

Example of either or: /A --> B. In this relationship, when A is not selected B is, when B is not selected A is, it is possible for both to be selected, but is it possible for both A and B not to be selected? I'm thinking no because the absence of one variable, i.e. A, is going to make the other variable, i.e. B, be selected and vise versa. Is that correct?

Another example of either or: A --> /B. This relationship is different from the above relationship because the negation is on the necessary condition instead of the sufficient condition. So, in this relationship, when A is selected B is not, when B is selected A is not. Is it possible for both not to be selected? Is it possible for both to be selected? I'm thinking that it is possible for both not to be selected because the rule only states what must happen when one of the variables occurs (the other must not be selected). Since both variables are not selected, the rule is not triggered. I'm also thinking it is not possible for both to be selected because the rule states that when one variable is selected the other variable must not be selected. Is that correct?

So even though both examples include the same variables, A and B, a negation on the sufficient condition is going to have a different effect on the relationship as opposed to a negation on the necessary condition, and vise versa.

Example of either or not both: A --> /B and /A--> B. These relationships will always lead to a bi-conditional. So, the bi-conditional relationship will look like this: A <--> /B. These two variables will forever be apart; when A is selected B is not, when B is selected A is not, both cannot be selected together and both cannot be unselected (because they would be together in the same unselected group). Is that correct?

So the difference between an either or relationship and an either or not both relationship is that in the former it is possible for the variables to be selected together, but in the latter, a bi-conditional relationship is formed where the variables are forever apart. Is that correct? But I want to add that there are bi-conditionals such as: A <--> B where the variables are forever together.

What would the contrapositive for the bi-conditional, A <--> /B, be? I'm thinking it is: /A <--> B. And what would be the contrapositive for A <--> B be? I'm thinking /A <--> /B. Are those answers correct?

Does anyone know where I can find materials that test on either or, either or not both and bi-conditionals? I would really like to practice these.

Thank you for taking the time to read this.

[KDLMaj]

I don't have time to address everything you just wrote, but here's a quick trick for you to figure out what's possible in a standard FL situation.

(TL;DR The results of a statement and its contrapositive can always happen together, and the triggers can never happen together)d
Take the statement: A ---> ~B, and it's contrapositive B ---> ~A:

A ---> ~B
B ---> ~A

Every statement and its contrapositive allow for three possibilities. Two of them are obvious, and one is still visually marked for you. In this case, you could have:

1) Only A (original statement: A --> ~B)
2) Only B (contra- B --> ~A)
3) Neither A nor B.

You CANNOT have both A and B.

I wish I had drawing tools here, but if you want to see the third possibility- circle the results of a statement and its contrapositive.

A ---> ~B
B---> ~A

The results can always happen together.

In this case, the only triggers are A and B. Formal Logic statements only ever do anything when a trigger occurs. These statements *ONLY* matter if A or B happens. In any situation where neither trigger happens, the statements are inert/irrelevant. So in a case where there's no B and/or there's no A, nothing occurs.

And if you want to see what ISN'T allowed- look at the triggers. In this case:

A ----> ~B
B ----> ~A

You can never have both A and B. In this case, having an A knocks out B, and having a B knocks out A. The quick visual trick is just to look at the triggers, and you'll instantly know they can't happen together.

[Walrus]

Hi Pepperdine2014

Either A or B
Allowed:
1) A and ~B
2) B and ~A
3) A and B
Not allowed:
4) ~A and ~B

Either A or B but not both
Allowed:
1) A and ~B
2) B and ~B
Not allowed:
3) ~A and ~B
4) A and B

Bi-conditionals
A<-->B
Allowed:
1) A and B
2) ~A and ~B
Not allowed:
3)A and ~B
4)B and ~A

A<-->~B
Allowed:
1) A and ~B
2) B and ~A
Not allowed:
3) A and B
4) ~A and ~B

Does anyone know where I can find materials that test on either or, either or not both and bi-conditionals? I would really like to practice these.

Logic games. Just practice tons of LG and you will have a lot of practice.
Bi-conditionals are more common in recent games.

If i made mistakes feel free to correct me

[Pepperdine2014]

Thank you very much, KDLMaj and Walrus, for the responses! I actually understand the different possibilities now!!! Thank you both for the explanations! *feeling victorious right now*

Logic games. Just practice tons of LG and you will have a lot of practice.
Bi-conditionals are more common in recent games.

Got it! Thank you!!