negating quantifying statements

glitterfyme
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negating quantifying statements

Postby glitterfyme » Tue Aug 28, 2012 11:04 am

when negating quantifying statements, do you negate the verb, the quantifier, or both??

Which is the correct logical opposition????
1. [quantifier change] All A's are B's <--> NOT ALL A's are B's = some A's are not B's
2. [verb change] All A's are B's <--> All A's are NOT B's = A's can never be B's
3. [both change] All A's are B's <--> NOT ALL A's are NOT B's = some A's are B's

Is it #1?

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Jeffort
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Re: negating quantifying statements

Postby Jeffort » Tue Aug 28, 2012 11:19 am

Of the options you gave, #1 is the only valid negation.

Never negate both the quantifier and the verb/verb phrase. When a quantifier referring to the subject of the sentence is present, negate the quantifier to its logical opposite.

TylerJonesMPLS
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Re: negating quantifying statements

Postby TylerJonesMPLS » Tue Aug 28, 2012 3:37 pm

I agree with Jeffort that it is #1- so you are right. Change the quantifier in the subject from All to Some, and negate the predicate (verb phrase) to get the logical opposite (contradictory).

1) All A’s are B’s.

2) Opposite (contradictory): Some A’s are non-B’s.

You can think about it as a Venn diagram. In 1) Circle A has to be entirely inside circle B. (Or the part of A that is outside B has to be marked out.) In 2) at least one A is outside the B circle.
Obviously, 1) and 2) are mutually exclusive. You can’t have Circle A entirely inside B and also partly outside B. So they are the strongest kind of logical opposition, called contradictories in logic textbooks.

Your #1 is an affirmative universal statement. Unfortunately, you can’t use the same method with a negative universal statement, like your second part of #2.

Since it is a negative universal statement - No A is B -to get its opposite (contradictory) you change the quantifier of the subject from No to Some, and you leave the predicate (verb phrase) unchanged, so it’s Some A’s are B’s. It’s a pain that you can’t use the same method.

But you can still diagram it as a Venn diagram. #2 has no intersection between Circle A and Circle B. (Or, the intersection is marked out.) But the opposite (contradictory) of #2 does have an intersection between A and B. Again, these are mutually exclusive, so these are contradictories.
Last edited by TylerJonesMPLS on Wed Aug 29, 2012 12:25 am, edited 1 time in total.

tamlyric
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Re: negating quantifying statements

Postby tamlyric » Tue Aug 28, 2012 3:59 pm

glitterfyme wrote:when negating quantifying statements, do you negate the verb, the quantifier, or both??

Which is the correct logical opposition????
1. [quantifier change] All A's are B's <--> NOT ALL A's are B's = some A's are not B's
2. [verb change] All A's are B's <--> All A's are NOT B's = A's can never be B's
3. [both change] All A's are B's <--> NOT ALL A's are NOT B's = some A's are B's

Is it #1?


Haven't looked closely, but I think I agree with the previous replies.

Here's the thing. Don't get caught up in gimmicky rules about changing symbols. Just use your head and be careful.

The negation of any sentence is just a way of saying that the sentence isn't true. -A = "A is false." The same goes for sentences that have quantifiers. If I say, "All Republicans are liars," and you say, "That's not true!", then you have said that there are at least some honest Republicans. That is why (1) is correct. (2) is not the negation because it is too strong -- just because some Republicans are honest (i.e., not all of them are liars) doesn't mean that all Republicans are honest. And (3) is not the negation because it is too weak -- it would be odd, indeed, for you to say defend Republicans against my aspersion by saying, "Bologna! Some Republicans are liars."

TylerJonesMPLS
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Re: negating quantifying statements

Postby TylerJonesMPLS » Tue Aug 28, 2012 8:28 pm

I agree with the poster above.

But I also think you should do it the easy way. Memorize the stuff that will help you get through sitting the LSAT while expending as little energy as possible.

Of course you should think through the logic yourself- but do it before sitting the LSAT as much as possible. For instance, if you use your head you can think through stuff like negations of conditionals. What is the negation of P > Q? Well, you can just think of what P > Q means, and what it would mean for it not to be true. P > Q means that when you have a P, then you have a Q as well. What would it mean for that to be false? Just that there is a P and no Q. So the negation of P > Q has to be P and ~Q.

But the logic on the LSAT gets complicated. There are lots of double and triple negations. Of course, you can think through triple negations while you are sitting the LSAT, but it is tiring and take lots of time.

I think it's better to think through the formal logic before you sit the LSAT, and also to memorize how to manipulate the logical symbols before you sit the LSAT. That way, you will be able to handle triple negatives quickly when they come up without having to think about them. And you will have the mental energy left to think through the really difficult questions on the LSAT that can't be solved just by manipulating logical symbols.

bp shinners
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Re: negating quantifying statements

Postby bp shinners » Tue Aug 28, 2012 9:58 pm

If it's any quantifier except 'most', negate the quantifier:
some<->none
all<->some don't

If the quantifier is 'most', change the verb ("most don't")

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sdwarrior403
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Re: negating quantifying statements

Postby sdwarrior403 » Tue Aug 28, 2012 10:16 pm

bp shinners wrote:If the quantifier is 'most', change the verb ("most don't")

That does not appear to be correct.

Most people buy shoes.

Half or less of people buy shoes.

The second statement is the correct way to negate the first statement.

Your post above would have this be the negation..

Most people do not buy shoes.

That statement neglects to realize that there could be an even 50/50 split of those buying/not buying shoes.

Could you perhaps explain why i am wrong if i am wrong?

TylerJonesMPLS
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Re: negating quantifying statements

Postby TylerJonesMPLS » Wed Aug 29, 2012 12:47 am

I think you’re right sdwarrior403.

The negation of Most People Buy Shoes can be put as:
It isn’t the case that Most People Buy Shoes.

And there are two cases that fit this statement.

1) If exactly half of all people buy shoes, then it isn’t the case that most people buy shoes.
2) If fewer than half of all people buy shoes, then it isn’t the case that most people buy shoes.

So your formulation is right:
If half or fewer than half of all people don’t buy shoes, then it isn’t the case that most people buy shoes.

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05062014
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Re: negating quantifying statements

Postby 05062014 » Wed Aug 29, 2012 12:55 am

With a parallel flaw question today I came across a question that said: Some people who like turnips don't like potatoes. T some (not P).
What is the contrapositive of this statement?

Some people who don't (don't like Potatoes) don't like Turnips?

tamlyric
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Re: negating quantifying statements

Postby tamlyric » Wed Aug 29, 2012 12:59 am

abdistotle wrote:With a parallel flaw question today I came across a question that said: Some people who like turnips don't like potatoes. T some (not P).
What is the contrapositive of this statement?

Some people who don't (don't like Potatoes) don't like Turnips?


What do you mean by contrapositive?

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05062014
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Re: negating quantifying statements

Postby 05062014 » Wed Aug 29, 2012 1:02 am

The equivalent. Dude, I actually have no clue lol. sigh

One statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa -- (Source: http://en.wikipedia.org/wiki/Contraposition)

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RCinDNA
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Re: negating quantifying statements

Postby RCinDNA » Wed Aug 29, 2012 5:18 am

Does anyone have a test, question and section number that I could look at as an example of a question where this would come into play?

tamlyric
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Re: negating quantifying statements

Postby tamlyric » Wed Aug 29, 2012 8:46 am

abdistotle wrote:The equivalent. Dude, I actually have no clue lol. sigh

One statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa -- (Source: http://en.wikipedia.org/wiki/Contraposition)


OK, right. This is the right idea. But it applies to conditional (If A, then B) statements, not quantified statements. Roughly the same kind of conversion will work with universal statements (All As are Bs), but not with existential statements (Some As are Bs). Your statement is of the latter kind, so, really, it doesn't have a contrapositive. Your statement "Some Ps are not Ts" is logically equivalent to "Some not Ts are Ps". Both statements just say that there is at least one thing in the world that is both P and not T. It might be worth noting, though, that this statement is the negation of the conditional statement "All Ps are Ts", which is logically equivalent to the negation of "All not Ts are not Ps".

bp shinners
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Re: negating quantifying statements

Postby bp shinners » Thu Aug 30, 2012 1:38 pm

sdwarrior403 wrote:Could you perhaps explain why i am wrong if i am wrong?


You're not wrong.

I went for simplicity over being 100% logically accurate. Mainly because so many of my students have a terrible time negating answers that I try to keep it simple. I also can't think of an example on the LSAT where changing it to 'most don't' would result in getting it wrong. So while 'half or less of people buy shoes' is the negation, it's also (in my experience) much harder for people to get a grasp on than 'most people don't buy shoes' (which is a negation that misses a single, outlier case - 50/50 split), with almost no benefit to them on the test.

So, in short, you're correct, but I think the simplicity of my Rule of Thumb trumps the negligible chance of it mattering on the test. YMMV.




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