“if and only if”

[jgray]

I’m working on the LGB workbook, and having a hard time understanding “if and only if” answers. How should I interpret “if and only if” statements and questions? How should I draw them?
thanks

[Dave Killoran]

Hi J,

Thanks for the question!

"If and only if" statements indicate that each condition is both sufficient for, and necessary for, the other condition, and thus they produce a double arrow:

Under a double arrow, only two things can happen: both occur, or both don't occur. So, if you had "A if and only if B," the only possible results would be:

• 1. A and B both occur (A and B)
  
  or
  
  2. Neither A nor B occurs (A and B)

Thus, if you had a rule like "A if and only if B" and you found out B had occurred, then you'd know that a would have occurred as well. Or, if you found out that B did not occur, then you'd know that A did not occur as well.

Please let me know if that helps. Thanks!

[akanshalsat]

Hello! I'm having some issues understanding the below statements

A [not] B

I just need help in understanding how to diagram this, and what a possible scenario/rule on a logic game would look like in order for me to come to the conclusion of diagramming this

I know that the inference for this is that one of either A or B are in and one of either B or A is out

So if I made an "in and out" chart,

under in I would write: (A/B) and under out I would write (B/A)

THanks!

[akanshalsat]

P.S

I rather like diagramming it like: A

rather than A B bc that is so much more confusing to me and I don't think its wise to retrain my thought processes this late in the game, so if you all could help me understand the first way and all of its inferences in a logic game scenario and what rules would lead to me diagramming it alittle better that would be great

[Dave Killoran]

Ok, these aren't the same statements, so you need to be careful! Consider the following:

• 1. A B
  
  • Possible:
    
    A, B
    A, B
    A, B
    
    
    Impossible:
    
    A, B
  
  
  
  2. A B
  
  • Possible:
    
    A, B
    A, B
    
    
    Impossible:
    
    A, B
    A, B

There is a different set of possibilities/impossibilities under each representation, so you can't represent those ideas in the same way

Thanks!

[akanshalsat]

oops I think I made a mistake

1. A B is similar to A [not] B am I correct??

Question2, Is there a way to write

2. A [not] B in a way that does not incorporate the double arrow??

Also what could a possible LG rule look like that would make me diagram

A [not] B / A B

[Dave Killoran]

No problem, here you go:

• 1. Yes, that is indeed the case! The double-not arrow there is a "super" statement that encapsulates the implications of A B as fully as possible.
  
  2. No, not in one diagram just using a single arrow.
  
  For the diagram question, there are tons of examples from real LSATs, and here are a few: "L and R cannot both stay open;" "Forest and turquoise are not used together in a rug;" "If N is selected, V cannot be selected."

Please let me know if that helps. Thanks!

[akanshalsat]

No problem, here you go:

• 1. Yes, that is indeed the case! The double-not arrow there is a "super" statement that encapsulates the implications of A B as fully as possible.
  
  2. No, not in one diagram just using a single arrow.
  
  For the diagram question, there are tons of examples from real LSATs, and here are a few: "L and R cannot both stay open;" "Forest and turquoise are not used together in a rug;" "If N is selected, V cannot be selected."

Please let me know if that helps. Thanks!

THank you for your responses, Dave!!

If I can, just wanted to clear a few more things up.

I know that for a logic game, the A [not]B / A B has inferences that say that either, A will be in and B will be out; B will be in and A will be out, or BOTH A AND B will be out

I'm wondering what the diagram/rule would look like for something that would force A/B to be IN for sure and one of B/A to be OUT for sure! (either one has to be in, and either one has to be out)

(sorry for so many questions!!)

[akanshalsat]

Actually looking back at your explanation from before, I now see that

A [not] B is what will FORCE one of A/B to be in and one of A/B out, since both can't be in and both can't be out together!!

But just for reference - what would a possible game scenario/rule sound like in order for me to diagram it?

I'm so sorry for all these posts, I just want to clearly differentiate these two in my brain so I can make the correct diagrams + inferences when doing LG's

[Adam Tyson]

"A is included if and only if B is not" would give us that double arrow, akanshalsat. So would "B is not included if and only if A is included."

That would be a rare rule, but an interesting one! It's much more common to get "Either A or B is included, but not both," which gives you the same set of possibilities (exactly one is in and exactly one is out). You could also show that as two different rules, if that makes it clearer (although for some folks it makes it worse - it's a question of style and comfort):

A B

and

A B

The first rule is "not both" and the second is "at least one," so together they tell you that one is in and one is out. Exactly the same as the very creative A B

I hope that helps!