Question about contrapositives. Logic Experts needed! Forum
- Geetar Man
- Posts: 564
- Joined: Wed May 26, 2010 4:13 am
Question about contrapositives. Logic Experts needed!
*EDIT: Please see the fifth post down (by myself). I'm having trouble writing the contrapositive of a given statment.
Okay, so contra-positives of basic conditionals is easy.
P -> Q
Contrapositive: -Q -> -P
-P -> Q
Contrapositive: -Q -> P
But what about a contrapositive of a conditional statement with a conjunction as the consequent, or a disjunction as the consequent?
I'm not a logic master, but I believe it goes like this:
Conjunction:
P -> Q and R
Contrapositive: -Q or -R -> -P
OR
Disjunction:
P -> Q or R
Contrapositive: -Q and -R -> -P
For anyone who knows logic, does this seem right?
I know you're supposed to change "and" to "or" if you're trying to get the contrapositive, but I can't wrap my head around it in practice.
Please advise,
Thanks!
Okay, so contra-positives of basic conditionals is easy.
P -> Q
Contrapositive: -Q -> -P
-P -> Q
Contrapositive: -Q -> P
But what about a contrapositive of a conditional statement with a conjunction as the consequent, or a disjunction as the consequent?
I'm not a logic master, but I believe it goes like this:
Conjunction:
P -> Q and R
Contrapositive: -Q or -R -> -P
OR
Disjunction:
P -> Q or R
Contrapositive: -Q and -R -> -P
For anyone who knows logic, does this seem right?
I know you're supposed to change "and" to "or" if you're trying to get the contrapositive, but I can't wrap my head around it in practice.
Please advise,
Thanks!
Last edited by Geetar Man on Sun Mar 18, 2012 7:03 pm, edited 2 times in total.
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Re: Question about contrapositives. Logic Experts needed!
You have it totally right
- Br3v
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Re: Question about contrapositives. Logic Experts needed!
I believe you have it correct.
If I am outside when it rains, then I can get wet and struck by lightning.
If I can not get struck by lightning nor get wet, then i am not outside when it rains.
If I buy a $1 lotto ticket, then I can win the jackpot or just lose $1.
If I do not lose $1 and I do not win the jackpot, I did not buy a lotto ticket
Rushed with real world flaws, but hope it helps wrap your head around it
If I am outside when it rains, then I can get wet and struck by lightning.
If I can not get struck by lightning nor get wet, then i am not outside when it rains.
If I buy a $1 lotto ticket, then I can win the jackpot or just lose $1.
If I do not lose $1 and I do not win the jackpot, I did not buy a lotto ticket
Rushed with real world flaws, but hope it helps wrap your head around it
- Geetar Man
- Posts: 564
- Joined: Wed May 26, 2010 4:13 am
Re: Question about contrapositives. Logic Experts needed!
Hey guys! Thanks for the quick responses.
This was something that I've come across in a logic game, so I figured I'd ask to double check.
I tried searching for the answer, but couldn't find it. So thankful I have some members on TLS to help me out.
best,
GM
This was something that I've come across in a logic game, so I figured I'd ask to double check.
I tried searching for the answer, but couldn't find it. So thankful I have some members on TLS to help me out.
best,
GM
VasaVasori wrote:Yep, that's right! And it should make sense intuitively, too. Consider the following statement: If the sky is blue, then I will fly and I will be happy (B -> (F & H)). In this circumstance, if i'm not happy or I'm not flying, then it can't possibly be the case that the sky is blue ((~F v ~H) -> ~B).
Or, this statement: If the sky is blue, then I will fly or I will be happy (B -> (F v H)). If I'm either flying or I'm happy, it's possible that the sky might be blue; but, if I'm neither flying nor happy then it cannot possibly be the case that the sky is blue ((~F & ~H) -> ~B).
So, you're spot on!
kaiser wrote:You have it totally right
Br3v wrote:I believe you have it correct.
If I am outside when it rains, then I can get wet and struck by lightning.
If I can not get struck by lightning nor get wet, then i am not outside when it rains.
If I buy a $1 lotto ticket, then I can win the jackpot or just lose $1.
If I do not lose $1 and I do not win the jackpot, I did not buy a lotto ticket
Rushed with real world flaws, but hope it helps wrap your head around it
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- Geetar Man
- Posts: 564
- Joined: Wed May 26, 2010 4:13 am
Re: Question about contrapositives. Logic Experts needed!
This comes from PT 34, Game 4. Doctors; binary grouping.
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
-
- Posts: 3019
- Joined: Mon May 09, 2011 11:34 pm
Re: Question about contrapositives. Logic Experts needed!
Geetar Man wrote:This comes from PT 34, Game 4. Doctors; binary grouping.
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
You mixed it up. "If" indicates the sufficient condition, regardless of whether that sufficient condition is given first or second.
"K is at R if J is at S" can simply be rephrased as "If J is at S, then K is at R"
jS --> kR
The contrapositive would be "If K is NOT at R, then J is NOT at S". But we of course know that this game has 2 mutually exclusive groups (the hospitals). So we can rephrase once again since saying "K is NOT at R" is the same as saying K must of course be at hospital S. And on the other side, saying J is NOT at S is the same as saying that he is at hospital R. Thus, the contrapositive can be rephrased in situations where there are only 2 mutually exclusive groups, and it would look like this:
If K is at S, then J is at R
- pizzabrosauce
- Posts: 74
- Joined: Wed Oct 26, 2011 6:06 pm
Re: Question about contrapositives. Logic Experts needed!
They are the same thing since its a binary grouping gameGeetar Man wrote:This comes from PT 34, Game 4. Doctors; binary grouping.
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
- Easy-E
- Posts: 6487
- Joined: Fri Feb 18, 2011 1:46 pm
Re: Question about contrapositives. Logic Experts needed!
Oooh that game sucked. That is all.
- lovejopd
- Posts: 544
- Joined: Tue Dec 01, 2009 1:00 pm
Re: Question about contrapositives. Logic Experts needed!
1. Use Parentheses to avoid any confusion and mixture of variables between Group(S,R) and Elements(j,k, etc)Geetar Man wrote:This comes from PT 34, Game 4. Doctors; binary grouping.
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
2. Pay attention to the word "if" which is a sufficient condition. A if B is NOT A --> B BUT B -->A. Thus, the location of if in a sentence does not matter.
3. Make some deduction in this rule
K is at R if j is at S
: j(S) --> k(R)
Deduction: J and K CANNOT go to "S" group TOGETHER. However, It does not MEAN J or K SHOULD go to "S" group as both j and k can go R group together.
There are three scenarios you can think of in terms of this rule
1) S:j R:k
2) S:k R:j
3) S: nothing R: j, k
I simply write "~(J=K)" ABOVE "S" group set-up.
Contrapositive
-K is NOT at R if j is NOT at S
: k(~R=S) --> j(~S=R)
~S=R/~R=S only applies to "Binary Grouping" as the elements has only two groups to go
Hope this helps
Last edited by lovejopd on Mon Mar 19, 2012 10:42 am, edited 1 time in total.
- Br3v
- Posts: 4290
- Joined: Mon Jun 13, 2011 7:18 pm
Re: Question about contrapositives. Logic Experts needed!
Yeah you messed up the beginningGeetar Man wrote:This comes from PT 34, Game 4. Doctors; binary grouping.
How would you write the contrapositive to this statement:
K is at R if J is at S.
This games asks you to put 6 doctors at two hospitals, S and R.
I understand that it should be:
kR -> jS
The contrapositive to this (intuitively) seems to be:
-jS -> -kR
But I saw that someone else said the contrapositive was this:
jR -> kS
halp!
If jS then kR
jS > kR
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