Can someone please help me write a contrapositive for this string of information.
W>S>O and M>~R
Thanks!
Help with Contrapositive!

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Re: Help with Contrapositive!
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Last edited by 03152016 on Tue Mar 15, 2016 3:03 am, edited 1 time in total.

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Re: Help with Contrapositive!
Thanks vas!
Jose
Jose
 homestyle28
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Re: Help with Contrapositive!
VasaVasori wrote:josemnz83 wrote:Can someone please help me write a contrapositive for this string of information.
W>S>O and M>~R
Thanks!
R > ~O v ~M > ~S > ~W
Not to stir stuff up but the above is logical nonsense and needs some parentheses.
If the original sentence is:
[W>(S>O)] & (M>~R) , then it's a conjunction and strictly speaking doesn't have a contrapositive.

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Re: Help with Contrapositive!
I’ll take a shot at it.
I’m going to assume that the series of conditionals looks like this:
W > S > (O & M) > ~R
This is the contrapositive:
R > (~O or ~M) > ~S > ~W
Or, it might be more useful to put the contrapositive in either of these ways:
R > ~O > ~S > ~W
or
R > ~M > ~S > ~W
I’ll explain how I worked this out below.
As you know, a string of any length of conditionals has a contrapositive. For instance, G > H > M > L has the contrapositive ~L > ~M > ~H > ~ G. So we can do the same for your conditional:
Conditional: W > S > (O & M) > ~R
Contrapositive: R > (~O or ~M) > ~S > ~W
Now, you will probably want to simplify ~(O & M.)
So, to put it the way logicians do, how do you distribute the negation over (O & M)?
Well, it’s easy. You negate each statement separately, so you get ~ O & ~ M, and then you change the “&” to an “or”. So you get the following: ~O or ~M.
The logical laws that allow you to do this are called DeMorgan’s Laws. If you google for that name, a huge number of hits will come up that will explain them. (I forgot to mention, if you have an “or” in the expression instead of an “&”, then the rule is to change the “or” to “&” and negate the statements as before.)
So the whole contrapositive will look like this:
R > (~O or ~M) > ~S > ~W
Now, because you have an expression with an “or” in it, or what logicians call a “disjunct”, you have the option in a Logic Game of using ~O alone and without ~M, the option of using ~M alone and without not ~O, and the option of using both ~M and ~O together. The last option is already written above. But you can use your contrapositive as in the two options. Like this:
R > ~O > ~S > ~W
or
R > ~M > ~S > ~W
To make my meaning clearer, suppose your conditional (and contrapositive) is one of the rules of the logic game, and one of the questions gives you the premise ~O. From ~O, ~S and ~W follow. Or suppose a question gives you the premise ~M. From ~M, ~S and ~W follow as well.
I hope that explanation was clear.
I’m going to assume that the series of conditionals looks like this:
W > S > (O & M) > ~R
This is the contrapositive:
R > (~O or ~M) > ~S > ~W
Or, it might be more useful to put the contrapositive in either of these ways:
R > ~O > ~S > ~W
or
R > ~M > ~S > ~W
I’ll explain how I worked this out below.
As you know, a string of any length of conditionals has a contrapositive. For instance, G > H > M > L has the contrapositive ~L > ~M > ~H > ~ G. So we can do the same for your conditional:
Conditional: W > S > (O & M) > ~R
Contrapositive: R > (~O or ~M) > ~S > ~W
Now, you will probably want to simplify ~(O & M.)
So, to put it the way logicians do, how do you distribute the negation over (O & M)?
Well, it’s easy. You negate each statement separately, so you get ~ O & ~ M, and then you change the “&” to an “or”. So you get the following: ~O or ~M.
The logical laws that allow you to do this are called DeMorgan’s Laws. If you google for that name, a huge number of hits will come up that will explain them. (I forgot to mention, if you have an “or” in the expression instead of an “&”, then the rule is to change the “or” to “&” and negate the statements as before.)
So the whole contrapositive will look like this:
R > (~O or ~M) > ~S > ~W
Now, because you have an expression with an “or” in it, or what logicians call a “disjunct”, you have the option in a Logic Game of using ~O alone and without ~M, the option of using ~M alone and without not ~O, and the option of using both ~M and ~O together. The last option is already written above. But you can use your contrapositive as in the two options. Like this:
R > ~O > ~S > ~W
or
R > ~M > ~S > ~W
To make my meaning clearer, suppose your conditional (and contrapositive) is one of the rules of the logic game, and one of the questions gives you the premise ~O. From ~O, ~S and ~W follow. Or suppose a question gives you the premise ~M. From ~M, ~S and ~W follow as well.
I hope that explanation was clear.
 ben4847
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Re: Help with Contrapositive!
Church is mad about contrapositives.

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Re: Help with Contrapositive!
VasaVasori wrote:josemnz83 wrote:Can someone please help me write a contrapositive for this string of information.
W>S>O and M>~R
Thanks!
R > ~O v ~M > ~S > ~W
Gotta be careful with taking the contrapositive of conjunctions/disjunctions in conditional chains.
Most likely, on the LSAT, you'll get a rule that says M>R to generate the string above. If that's the case, when you take the contrapositive, you can't just switch that conjunction to a disjunction, because O shouldn't end up in that chain.
So I'd do it as (assuming the rule is M>R and not O>R or M and O>R; and S>M and O)
R>~M>~S>~W
^
~O
Hmm, not showing up well on the page. That '~O' arrow should be going to ~S. It's a second leg to the chain that reflects there is no relationship between R and O or M and O, but a relationship between O and S.
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