Formal logc of " If X then A or B but not both"?

User avatar
MS415
Posts: 48
Joined: Mon Dec 03, 2012 3:02 pm

Formal logc of " If X then A or B but not both"?

Postby MS415 » Sat Dec 15, 2012 3:12 pm

Hey Guys. I just wanted to see if someone can tell me how to denote "A or B but not both" in formal logic with both sides of the contra-positive. Thanks in advance.

Edit: I meant to write If X -> then A or B but not both. ( sorry).

User avatar
jitsubruin
Posts: 23
Joined: Sun May 08, 2011 1:56 pm

Re: Formal logc of " If X then A or B but not both"?

Postby jitsubruin » Sat Dec 15, 2012 3:18 pm

A +B ---> not x

X ----> A or B

User avatar
dingbat
Posts: 4976
Joined: Wed Jan 11, 2012 9:12 pm

Re: Formal logc of " If X then A or B but not both"?

Postby dingbat » Sat Dec 15, 2012 3:30 pm

jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X

User avatar
Cerebro
Posts: 239
Joined: Thu Aug 09, 2012 9:22 pm

Re: Formal logc of " If X then A or B but not both"?

Postby Cerebro » Sat Dec 15, 2012 3:33 pm

dingbat wrote:
jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X



X --- > [(A or B) & ~(A & B)]

User avatar
Cobretti
Posts: 2560
Joined: Tue Aug 21, 2012 12:45 am

Re: Formal logc of " If X then A or B but not both"?

Postby Cobretti » Sat Dec 15, 2012 4:43 pm

Cerebro wrote:
dingbat wrote:
jitsubruin wrote:A +B ---> not x

X ----> A or B

not (A or B) ----> not X



X --- > [(A or B) & ~(A & B)]


Dingbat's response = contrapositive to jitsu's second statement.

Cerebro's X -> ~(A & B) = contrapositive to jitsu's first statement.

Everyone agrees!

bp shinners
Posts: 3091
Joined: Wed Mar 16, 2011 7:05 pm

Re: Formal logc of " If X then A or B but not both"?

Postby bp shinners » Mon Dec 17, 2012 1:33 pm

X -> (A or B) and (~A or ~B)

There are a few ways to write it (as seen above), but I find this to be the most straight-forward, the easiest to remember, and the easiest to work with.

TylerJonesMPLS
Posts: 74
Joined: Wed Jun 20, 2012 11:20 pm

Re: Formal logc of " If X then A or B but not both"?

Postby TylerJonesMPLS » Thu Dec 20, 2012 4:08 am

Celebro is right, X ---> [ (A OR B) & ~(A & B) ] is the correct logical formulation.

I’m not sure I understood the suggestions for the contrapositive above. So I’ll just suggest my own.

X ---> [ (A OR B) & ~(A & B) ]

So reverse and negate both sides .
~ [ (A OR B) and ~(A & B) ] ---> ~X

Then put the negation inside the brackets, and throw away the brackets.
~(A OR B) OR ~( A & B) ----> ~X

Then put the negations inside the parentheses.
(~A & ~B ) OR (~A OR ~B) ---> X

so (~A & ~B ) OR (~A OR ~B) ---> X is the contrapositive.

bp shinners
Posts: 3091
Joined: Wed Mar 16, 2011 7:05 pm

Re: Formal logc of " If X then A or B but not both"?

Postby bp shinners » Thu Dec 20, 2012 11:04 am

TylerJonesMPLS wrote:so (~A & ~B ) OR (~A OR ~B) ---> X is the contrapositive.


I don't have time to see where you went wrong, but that's not the correct contrapositive. Especially since it's redundant - the first parenthetical is included in the second parenthetical. And you forgot the negation of X.

I'm going back to my formulation because there's no need to mess with moving negations around:
X -> (A OR B) AND (~A OR ~B) (which is the same as Celebro, if you move the negation inside the second parenthetical instead of keeping it outside - it's more symmetrical that way)
(A OR B) is saying at least one of A or B; (~A OR ~B) is the same as saying not both A and B. Put them together and you get at least one, but not both.
Contrapositive:
(A AND B) OR (~A AND ~B) -> ~X
If I have both, or if I have neither, I don't have X.

TylerJonesMPLS
Posts: 74
Joined: Wed Jun 20, 2012 11:20 pm

Re: Formal logc of " If X then A or B but not both"?

Postby TylerJonesMPLS » Wed Sep 02, 2015 9:59 pm

The question is how to translate the sentence "If X then A or B but not both" into propositional logic, and to show the contrapositive.
Celebro's answer to the question of how to put the sentence in propositional logic is obviously correct: X ---> [(A OR B) AND ~(A AND B)]
(One can use DeMorgan's laws to derive (~A OR ~B) from ~(A AND B), but that requires a second step in the logic.)
I apologize for the typos and miscounting the negations in my earlier post; I wrote it out too quickly and carelessly. I will will begin with Celebro's formulation, and derive the contrapositive in two ways, because one may be easier to see than the other.

1. X > [(A or B) and ~(A and B)]
2. ~ [(A or B) and ~(A and B)] > ~X
3. [~(A or B) or ~~(A and B)] > ~X
4. [~(A or B) or (A and B)] > ~X
5. [(~A and ~B) or (A and B)] > ~X

1. X > [(A or B) and ~(A and B)]
2. X > [~(~A and ~B) and ~(A and B)]
3. ~[~(~A and ~B) and ~(A and B)] > ~X
4. [~~(~A and ~B) or ~~(A and B)] > ~X
5. (~A and ~B) or (A and B)] > ~X




Return to “LSAT Prep and Discussion Forum”

Who is online

Users browsing this forum: 34iplaw, Anon.y.mousse., bcapace, cianchetta0, goldenbear2020, lnsl123, mrgstephe, MZaf, Pozzo, sspeckk, The_Pluviophile and 13 guests