Help with Contrapositive! Forum

[josemnz83]

Can someone please help me write a contrapositive for this string of information.

W----->S----->O and M----->~R

Thanks!

[VasaVasori]

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[03152016]

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[VasaVasori]

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[josemnz83]

Thanks vas!

Jose

[homestyle28]

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.

[VasaVasori]

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[TylerJonesMPLS]

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.

[ben4847]

Church is mad about contrapositives.

[bp shinners]

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.