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*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!

.

You have it totally right

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

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

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!

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

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!

They are the same thing since its a binary grouping game

Oooh that game sucked. That is all.

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!

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

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!

Yeah you messed up the beginning
If jS then kR

jS > kR