Contrapositive of 'most' statements

[nbayar1212]

Can you take a contrapositive of a most statement?

That is, if you know that most As, are Bs, can you then conclude that if you are not a B you are not within the category that most As fall into?

[joshringu]

No, you cannot. That goes for most statements and also some/many statements.

Since we are on the topic, can we take the contrapositive of a double arrow? Such as:

A <----> B

Which is a combination of A ----> B and B -----> A

Thanks

[nbayar1212]

Yes, you can take the contrapositive of a double arrow. It just means that if you don't have one, you will necessarily not have the other - irrespective of which variable is absent.


BUT, on the point about "most statements": I have yet to get an explanation of why we can't take the contrapositive of a most statement..... it makes perfect sense to me.

Also, it seems like taking the contrapositive of a most statement is how we get to the correct answer on question 19 of S4 in PT20....

Please help.

[christine.defenbaugh]

Great question nbayar1212!

No, you cannot take the contrapositive of 'most' statements, but let's spend a moment delving into why that is.

Mary Adkins recently posted a great blog post outlining the meaning of various quantity terms on the LSAT. http://www.manhattanlsat.com/blog/2013/ ... ity-terms/ As you can see, the only thing that 'most' means on the LSAT is 'more than half'.

So, in your example, most A are B. That means that more than half of A are B, but we actually don't know a thing about the whole world of 'not A'. This could be laid out a number of ways. Take a look at the diagrams below, and note that these are only 2 possibilities - there are many, many more!

We cannot conclude anything about any of the other groups other than A. The "not A" group (everyone outside the box) could be mostly Bs or mostly "not B"s, depending on the diagram. Similarly, the Bs could be mostly As or mostly "not A"s, and the "not B"s could be mostly As or mostly "not A"s. We just don't know about any of them! It's even possible that ALL As are Bs! (That's not on either of the diagrams!)

In fact, the only things we can know from the statement "Most A are B" is the "some A are B" and "some B are A". That's it! It's extremely limited.

Now, your statement attempts to avoid all of these classic invalid inference attempts. You would likely never see a statement quite like that on the LSAT, nor does it enable you to conclude anything else. But even that statement "if you are not a B you are not within the category that most As fall into" is still problematic. If you are not a B, it is of course true that you are not in the B category, and the B category does have most of the As. But is that the only category that contains most of the As? Perhaps there's another category we can use!

What if most of the As are roses, and ALL the "not B"s are roses too. Then, being a "not B" would still leave you in a category that most of the As are in - the roses category!

It's great that you being sensitive to the idea of the contrapositive, but it's important to only apply it to conditional statements. The critical thing about 'most' statements is to be mindful of all the things you're NOT allowed to infer!

As for PT20, S4, Q19, a "contrapositive" isn't necessary to get to the correct answer, and in fact would potentially lead to incorrect inferences. There's a great discussion of this specific question here: post407.html

Please let me know if this answers your question completely!

[nbayar1212]

Christine,

Thank you for the response. I have been thinking about it all day but I am still unclear (and unconvinced) that we can't infer the contrapositive of a "most statement."

The problem I see with your response to my example is that you are considering a statement that I didn't make i.e. I said Most A's are Bs and you changed it to Most As are Roses (which made me think B is now being defined as a rose) but then you go on to negate B and ask me to think about a world where "not Bs are roses too".

I don't think such a world can exist. If we are defining B to be roses, the "not B" would simply be no roses.... we can't have a world where Bs are defined as a rose and so are "not Bs".

Let me give you an example that I think support my point:
IF we have a statement that says "Most Americans are against a strike in Syria, and then we have a statement that say Obama is not against a strike in Syria, then I would say that we can safely conclude that Obama's views are not the same as most americans....

If this doesn't make sense, where is the flaw?

Thank you!

[christine.defenbaugh]

Thank you for clarifying your question!

First, let's clear up a significant point from the examples above: in the example where I introduced roses, I did not say that roses were the same thing as "B". It's an entirely separate category, one that partially overlaps A and B. Check out the diagram that illustrates this.



As you can see, most A are B. As a completely separate matter, it just so happens that most A are also roses. But there are some roses that aren't Bs, and some Bs that aren't roses. Take Joe, for instance. He's an A, and he's a rose, but he's no B!

So why do we care? Well, the original statement that you wanted to infer was "If you are not a B, then you are not in the category that most As are in." Okay, so Joe is not a B. Is it true that he is not in "the category that most As are in"? Well, he's not in the B category, which most As are in. But he could be in the 'rose category' (as he is in our diagram), and most As are in that!

We cannot, therefore, infer a blanket statement such as "Joe is different from most As" - different how? That statement would suggest he's different in *every* way! You could say something very narrow and specific, such as "with respect to the question of B-ness, Joe differs from most As". But that's all.

Let's take your later example and adjust it. Let's say most Americans like ice cream. As a separate matter, most Americans like pizza. But the pizza lovers and the ice cream fiends are not the exact same group.



Now, Joe likes ice cream, just like most Americans. But he abhors pizza. So, with respect to the question of liking pizza, he is not in the category that most Americans are in. But with respect to the question of liking ice cream, he is! So, if all we knew about Joe was that he didn't like pizza, we can't conclude much about his relation to most Americans - there are millions of other potential categories where he might be like, or not like, most Americans.

While the statement "with respect to the question of liking pizza, Joe is not in the category that most Americans are in" is arguably a technically valid statement, it would not be called a 'contrapositive'. But more importantly, it's not that useful of a statement. We cannot use that statement to lead us anywhere new, and we are unlikely to be able to link it to, or combine it with other statements to understand something interesting about our players the way we can with the contrapositive of a conditional statement.

All we've really done is torture a 'most' statement into a very narrow grammatical pretzel. It is a rather interesting linguistic exercise, but it's not likely to help you score points on the LSAT.

The key points to remember about most statements are:
If most A are B
1) Some A are B
2) Some B are A
3) It's possible that all A are B (but you don't know!)
4) You cannot say that most B are anything
5) You cannot say that most -A are anything
6) You cannot say that most -B are anything

I hope that this helps clarify the issue! Please let me know if you have additional questions!

[nbayar1212]

It all makes perfect sense now.

Thank you!