## Question about contrapositives. Logic Experts needed!

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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.

Thanks!
Last edited by Geetar Man on Sun Mar 18, 2012 7:03 pm, edited 2 times in total.

VasaVasori

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Joined: Sun Dec 11, 2011 2:36 pm

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Last edited by VasaVasori on Sat May 02, 2015 10:58 pm, edited 1 time in total.

kaiser

Posts: 2878
Joined: Mon May 09, 2011 11:34 pm

### Re: Question about contrapositives. Logic Experts needed!

You have it totally right

Br3v

Posts: 4272
Joined: Mon Jun 13, 2011 7:18 pm

### 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

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

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

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

jR -> kS

halp!

kaiser

Posts: 2878
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

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!

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

jR -> kS

halp!

They are the same thing since its a binary grouping game

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!

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

jR -> kS

halp!

1. Use Parentheses to avoid any confusion and mixture of variables between Group(S,R) and Elements(j,k, etc)
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: 4272
Joined: Mon Jun 13, 2011 7:18 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

jR -> kS

halp!

Yeah you messed up the beginning
If jS then kR

jS > kR