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zxcvbnm

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"A unless B" question
« on: July 19, 2004, 02:13:21 PM »
All of the LSAT study materials I've seen want to translate "A unless B" as ~B -> A. My instincts tell me that A -> ~B is probably more correct though. If B is a condition that prevents A, then you certainly know that given A, B must be false. John will go to the party unless Susan goes; JP -> ~SP, and conversely, SP -> ~JP. This seems like the most we can confidently assert from the statement. If John is at the party, then Susam definitely isn't going to be there, and if Susan goes to the party, John won't. But doing it the prescribed way, we get ~SP -> JP, or "If Susan does not go to the party, then John will go." This seems to claim too much though, saying that Susan going is the only thing that would stop John from going. In addition, it's consistent with John and Susan both going to the party; if the first half of the conditional is false (it is not the case that Susan doesn't go"), and the second half is true, then the whole thing will be true.

So where am I going astray here? It's been a while since I'd had logic, so I may be overlooking something. Is there something peculiar about the example I'm using that I haven't caught?

Kwertee

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Re: "A unless B" question
« Reply #1 on: July 19, 2004, 02:36:56 PM »
"A unless B" only states that A is true any time B is not, or ~B -> A.  If B is true, we don't know anything about A.  B is not a condition that prevents A.  You could add another condition to "A unless B" such as "If B, then A iff C".  Using your party example we could have the following:

John will go to the party unless Susan goes.
If Susan goes to the party, then John will go if and only if Charles goes.

Now, if Susan does not go then John will go no matter what.  However, if Susan does go, John's going coincides with whether or not Charles goes, which is independant of Susan.

There are times that "unless" can be treated as biconditional, but they are rare and probably out of the scope of the LSAT.

trinityx90

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Re: "A unless B" question
« Reply #2 on: July 19, 2004, 08:45:55 PM »
I figured I'd complicate this one step further.  "A unless B" can also be symbolized as "A or B" which can be easily proved by using a truth table.

assume A-->B   symbolized as "~B-->A"   "OR AS"  "A or B"

truth table:  (sub A for variable x, and B for y,  "v" symbolizes the "or")
 
           (~Y-->X)        (X v Y)       
  x   y

  t   t     f T  t         t T  t 
  t   f     t T  t         t T  f
  f   t     f T  f         f T  t
  f   f     t F  f         f F  f

If you look at the middle column under each stmt. (the cap letters) you can see that they form a tautology.  (a fancy word for logically equivilant).  You will end up with the exact same result using either symbolization for every possibility. 

Another way to see this, possibly a bit easier, is to note that:

   "~B-->A"   is the same as "B v A"  (or "A v B" however you look at it b/c)  This is evident from the rule of conditional exchange.

For anyone who has been through logic this is quite trivial, however if this dosent' make sense to you, or if it's been a while since you've seen logic, just take my word for it. 

"A unless B", can also be stated as "A or B"                 (or if not B than A)

 

Kwertee

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Re: "A unless B" question
« Reply #3 on: July 19, 2004, 08:47:41 PM »
"A unless B" is also equivalent to "B unless A".

America's Next Top Lawyer

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Re: "A unless B" question
« Reply #4 on: July 20, 2004, 11:31:26 PM »
This is good stuff. Thanks for posting it. I forgot it long ago from logic class. I think that it really simplifies things when you are translating the rules and you run into the tricky unless word.

I figured I'd complicate this one step further.  "A unless B" can also be symbolized as "A or B" which can be easily proved by using a truth table.

assume A-->B   symbolized as "~B-->A"   "OR AS"  "A or B"

truth table:  (sub A for variable x, and B for y,  "v" symbolizes the "or")
 
           (~Y-->X)        (X v Y)       
  x   y

  t   t     f T  t         t T  t 
  t   f     t T  t         t T  f
  f   t     f T  f         f T  t
  f   f     t F  f         f F  f

If you look at the middle column under each stmt. (the cap letters) you can see that they form a tautology.  (a fancy word for logically equivilant).  You will end up with the exact same result using either symbolization for every possibility. 

Another way to see this, possibly a bit easier, is to note that:

   "~B-->A"   is the same as "B v A"  (or "A v B" however you look at it b/c)  This is evident from the rule of conditional exchange.

For anyone who has been through logic this is quite trivial, however if this dosent' make sense to you, or if it's been a while since you've seen logic, just take my word for it. 

"A unless B", can also be stated as "A or B"                 (or if not B than A)

 

kingoftorts

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Re: "A unless B" question
« Reply #5 on: July 21, 2004, 07:48:17 PM »
A unless B is certainly not the same as B unless A


Kwertee

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Re: "A unless B" question
« Reply #6 on: July 21, 2004, 10:53:02 PM »
A unless B is certainly not the same as B unless A

It certainly is.

dta

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Re: "A unless B" question
« Reply #7 on: July 21, 2004, 11:08:10 PM »
Kwertee is right.

A unless B = ~B -> A   (this is the classic indisputable interpretation)
B unless A = ~A -> B

~B -> A and ~A -> B are the contraposative of one another, and therefore equivalent.

Kwertee

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Re: "A unless B" question
« Reply #8 on: July 21, 2004, 11:11:23 PM »
Kwertee is right.

A unless B = ~B -> A   (this is the classic indisputable interpretation)
B unless A = ~A -> B

~B -> A and ~A -> B are the contraposative of one another, and therefore equivalent.

Yup.  Also, "A unless B" is the same as "A or B" as proven above, and "or" is a commutative operation.  Therefore "unless" is also commutative.

QED

kingoftorts

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Re: "A unless B" question
« Reply #9 on: July 21, 2004, 11:16:15 PM »
wow.  it makes sense when I diagram it, but in terms of semantics, the fact that those 2 are the same just creeps me out.  Wow.  Ok, they indeed are the same.