The contrapositive of this - LR Forum

[TTX]

This comes from PT 62, LR 1, Q 19.

So the original conditional statement is IF A but not B, THEN C.

I take the contrapositive to be: IF ~C, THEN ~A or B...

Why is that wrong? Am I wrong to equate IF..BUT (If A but ~ B) with IF...AND (If A and ~B)?

Having a hard time wrapping my head around this. Please Help!

[gaud]

Hopefully this can help you

http://www.manhattanlsat.com/forums/q19 ... c3c5d9825d

[BlaqBella]

But = and. You've made no error.

[TTX]

BBella:
Thanks for the affirmation! I think the mistake I made was I forgot that [~A or B] includes three possibilities: [A & B], [~A & ~B], and [~A &B].

Please confirm if I am right on this.

gaud:
I read manhattan's explanation before I posted, and I found their explanations insufficient.
The one explanation that seems to prove me right made the mistake of concluding from [A --most-->B --most-->C] that A some C.
So I was skeptical of the rest of their explanations.

[BlaqBella]

BBella:
Thanks for the affirmation! I think the mistake I made was I forgot that [~A or B] includes three possibilities: [A & B], [~A & ~B], and [~A &B].

Please confirm if I am right on this.

Correct

gaud:
I read manhattan's explanation before I posted, and I found their explanations insufficient.
The one explanation that seems to prove me right made the mistake of concluding from [A --most-->B --most-->C] that A some C.
So I was skeptical of the rest of their explanations.

If I may, from the above you can actually conclude some As are Cs/some Cs are As.

Most + Most = Some

[TTX]

Here is how I think of it.

We have 3 sets: A, B, and C.
Set A includes #s [1,2,3]. Set B includes #s [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Set C includes # [6 - 100]

Even though most As are Bs, and most Bs are Cs. It is not true that some As are Cs. In this case, none of the As are Cs.

So [A --most--> B --most--> C] does not mean [A some C].

On the other hand, If it were [B --most-->C] and [B --most-->A], then it is correct to say [A some C].

Correct me if I'm wrong.

[BlaqBella]

MOST + MOST = SOME
ALL + MOST = MOST
ALL + SOME = SOME

You cannot infer anything from:

Some + Some
Most + Some

[Mik Ekim]

Hi -- just wanted to add a couple of thoughts on this problem --

1) When I run into a complicated conditional statement like this one, I prefer to reason through the contrapositive rather than rely on a notational system that I've memorized -- I know not everyone agrees with me on this, but for me, I feel much more comfortable solving problems this way under pressure --

So, to come up with the contrapositive for a statement like "Any store that sells fish but not birds sells gerbils," I think to myself "what constitutes the result "not happening" and what constitutes the trigger "not happening?"

In this case --

The consequence not happening is no gerbils (~G)

The trigger not happening is any situation other than F & ~B, namely (-F, -B), (F, B), (-F, B).

Thus the contrapositive is that ~G leads to one of these three possibilities.

So, if I were to right anything down for this question (and I may not write anything out for this q - more on that in just a sec) -- I would write out the original conditional, and the contrapositive with G crossed out (~G per your notations) leading to the three different possibilities. Having reasoned through it, I feel confident my understanding of the contrapositive is correct, and, with all three possibilities written out (as opposed to combined into one complex notation that represents all three scenarios), it's much easier for me personally to evaluate answer choices.

2) The key to getting these types of questions correct quickly and accurately is to focus on the answer choices, rather than the stimulus. More specifically, focus on finding reasons why four answers are not justifiable. If I saw this question on the exam, I would quickly read through the stimulus, see that there are some most statements that can be combined, and a complex conditional statement, and another conditional link --

and I would take a deep breath --

and I would hope I don't have to do too much work (there are TONS of very complex ways in which all of that information could be brought together into a must be true answer) --

then go into the answers, looking for absolute reasons why four of them are not provable --

I don't need to do a lot of work to see that (A), (B), and (C) are all clearly not provable --

For (A), we're not given any information about non-independent stores not selling exotic birds.

Checking (B) against the stimulus, I can see that we don't have any info about "If fish and birds -> gerbils" - I can see that (B) is a "messed up" version of the condition in the stimulus and I can eliminate it.

Checking (C) against the stimulus, the only relevant part is "...not birds -> gerbils" -- no way we get gerbils -> birds (note that we don't even have to worry about thinking about some/most etc yet -- the incorrect use of the conditional is enough proof these answers are wrong).

Checking (D) against the stimulus, we know -
1) no indep owned store sells gerbils
2) if a pet store sold fish but not birds, it would sell gerbils!
Seems that we can infer that no index owned stores sell fish but not birds (on the exam I would leave it for now) --

Checking (E) against the stimulus, I don't see anything that forces stores to either sell one or the other.

Having eliminated A, B, C, and E, I would then check (D) more carefully against the stimulus, writing out the conditionals if need be to make sure (D) works out --

Again, not for everyone, but just wanted to offer an alternative way to go -- hope that helps.

[TTX]

BBella:
If you have the Powerscore LR bible, it says in Chapter 11 (formal logic section) on pg. 330 that [A--most-->B--most-->C] does not yield any inference.

I like to use hypotheticals in situations like this. Using the example you give that Most Cats (C) are Dogs (D) and that most Dogs are Fat (F). If we have 10 Cs, 100 Ds, and 1000 Fs. If 9 Cs are Ds, and 51 Ds are Fs, then there is still a good chance that the Cs and the Fs do not overlap. In that case, it is not true that some Cs must be Fs, since the example I give above leaves open the possibility that no Cs are Fs.

Mik:
I like your approach. I was afraid that I might have a huge blind spot in my conditional logic

[BlaqBella]

BBella:
If you have the Powerscore LR bible, it says in Chapter 11 (formal logic section) on pg. 330 that [A--most-->B--most-->C] does not yield any inference.

...and on the same page it goes on to say this:

"the fact that most Bs are both As and Cs allows us to conclude that some As are Cs"

[TTX]

BBella:
If you have the Powerscore LR bible, it says in Chapter 11 (formal logic section) on pg. 330 that [A--most-->B--most-->C] does not yield any inference.

...and on the same page it goes on to say this:

"the fact that most Bs are both As and Cs allows us to conclude that some As are Cs"

The direction of the arrows makes a huge difference. In the first instance where [A--most-->B--most-->C] all the arrows point in one direction and for that reason no inference an be drawn.

In the quote you give, [A<--most-- B --most-->C], the direction of the arrows are not all pointing in the same direction and as such, it is the only instance where combining a "most" and a "most" can yield an inference, which is A some C.

[KFV]

BBella:
If you have the Powerscore LR bible, it says in Chapter 11 (formal logic section) on pg. 330 that [A--most-->B--most-->C] does not yield any inference.

...and on the same page it goes on to say this:

"the fact that most Bs are both As and Cs allows us to conclude that some As are Cs"

No, just no. Your claim that "Most As are Bs + Most Bs are Cs = Some As are Cs" is just plain 100% wrong. Additionally, the guy's already given you an example of it not applying with those three number sets. Did you look at his example? How can you still be disagreeing with him?

Here's a less abstract example if that helps:

Most of my favourite rappers are dead people.
Most dead people were born before 1900.

Still going to claim that some of my favourite rappers were born before 1900?

[Power Clean]

The direction of the arrows makes a huge difference. In the first instance where [A--most-->B--most-->C] all the arrows point in one direction and for that reason no inference an be drawn.

In the quote you give, [A<--most-- B --most-->C], the direction of the arrows are not all pointing in the same direction and as such, it is the only instance where combining a "most" and a "most" can yield an inference, which is A some C.

Beating a dead horse here, but maybe thinking visually will hammer this home?

[BlaqBella]

The direction of the arrows makes a huge difference. In the first instance where [A--most-->B--most-->C] all the arrows point in one direction and for that reason no inference an be drawn.

In the quote you give, [A<--most-- B --most-->C], the direction of the arrows are not all pointing in the same direction and as such, it is the only instance where combining a "most" and a "most" can yield an inference, which is A some C.

Beating a dead horse here, but maybe thinking visually will hammer this home?

Thanks guys! I see my error.