Question on conditional logic Forum

[afa_brandon]

So i get from the LG Bible and elsewhere that "if and only if" begets a double arrow: <-->

but i don't understand why :/


ex: I can buy a pencil if and only if I spend a dollar

symbolically, P <--> Spend dollar

or

there is no way either could occur without the other occurring also.

that's the part i don't get. While, if i have a pencil, i must have spent a dollar, i can spend a dollar without buying a pencil. so why is it a double arrow?

[tamlyric]

1. A if B = B arrow A

2. A only if B = A arrow B

Hence, A if and only if B = A double-arrow B

I imagine your issue is with 2? If so, here's the story. 2 means that B is necessary for A. But if B is necessary for A, then what do you know if A happens? You know that A couldn't have happened if B hadn't happened. Therefore, A only if B entails B if A.

Re: your example. It just happens to be false that A if and only if B.

[tartugas]

there is no way either could occur without the other occurring also.

that's the part i don't get. While, if i have a pencil, i must have spent a dollar, i can spend a dollar without buying a pencil. so why is it a double arrow?

What you have to remember here is that you are being given a very specific circumstance. What you're being told is not just that if X occurs Y must occur (normal conditional) but that in this case, the only circumstance under which X can occur is for Y to occur, thus making X a necessary condition of Y also (If Y then X).

If and Only if turns a normal conditional into a special set of circumstances that create a singular relationship between two elements.

Normal Conditional:

If your dog itches, then he has fleas. (x--->y)

If your dog has fleas... that doesn't necessarily mean anything.

However if you switch the wording to if and only if

If and Only if:

Your dog itches IF AND ONLY IF he has fleas. (x<--->y)

Now you can draw an inference in the case of y occurring

If your dog itches, then he must have fleas.

Try to work with form more than content when breaking down logical statements. The LSAT is full of contextual scenarios that are designed to get you to ignore the formal logical rules. It's easy to get tripped up when you think of all the great things you could buy with that dollar in your pocket. But in this circumstance, you've been told the only thing you can do with the dollar is buy that pencil.

[afa_brandon]

Hence, A if and only if B = A double-arrow B

okay, i get it, so "if, and only if" are two separate-yet-interrelated phrases. this makes perfect sense

Try to work with form more than content when breaking down logical statements.

this is exactly my problem, i am realizing--spot-on advice.


thank you both!