Negating using "not"

Hamaseh_S

Can anyone help me with this negation?

The sentence is: “The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km”

I negated it like this: “The highest point of any volcanic ash cloud will not eventually exceed an altitude of 5km”

I ran into trouble with this question bc I negated the answer choice wrong and it should have been negated as: “It is not the case that the highest point of any volcanic ash loud will eventually exceed an altitude of 5km”

I am having trouble seeing how these two negated sentences differ. Will using the word “not” get me into trouble? I reviewed the negation lesson and “not” was a word used to negate. I thought the meaning of negation (and from what I got from the lesson) was to say the logical opposite of something. But now I am wondering if the logical opposite means the thing you're negating can sometimes happen?

My sentence is saying that no volcanic ash cloud will exceed 5km. But the recommended negated sentence seems to mean that some volcanic ash clouds will exceed and some volcanic ash clouds won’t exceed 5km. Maybe I put the word “not” in the wrong place. If my negated sentence read “The highest point of not any volcanic ash cloud will eventually exceed an altitude of 5km” would that be right? Why?

This is from PT 83, S1, Q17.

Thanks in advance, sorry if this is super confusing but I am right there with you haha

goingfor99th

“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km.”

Translation :
All [volcanic ash cloud high points] will [eventually exceed an altitude of 5 km].
All [A] are [B].
If [A], then [B].

Negation:
Some [volcanic ash cloud high points] will [not eventually exceed an altitude of 5 km].
Some [A] are [not B].
If [A], then [not necessarily B].

Hope this helps. This is the concept of 'logical opposites.'

samantha.ashley92

As stated above, the opposite of "all A then B" is "some A not B". Essentially, the opposite of "all" is not "none"-- it's "some".

goingfor99th

I think your confusion is rooted in this translation:
Some [A] are [not B].
If [A], then [not necessarily B].

These two statements are equivalent, but when you go from a "Some..." statement to an "If, Then..." statement (conditional statement), you have to account for the "some" in the conditional statement by acknowledging it's "not necessarily" the case that if we have A then we must also have B. When I think of "negation," I think of it as a negation of the relationship between the two terms (the antecedent and the consequent), not necessarily a negation of either term on its own.

J.CHRIS.ALST

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Hamaseh_S

Ohhhhh this makes a lot more sense. So it seems like I've been "negating" for opposites, not true negations. If the logical opposite/negation of all is some then in context, using the word "not necessarily" seems to make more sense then sticking a "not" in the sentence.

I could fix my sentence by saying “The highest point of any volcanic ash cloud will not necessarily eventually exceed an altitude of 5km”

@goingfor99th I'm confused about negating for meaning. Here is how I would negate the sentence -

“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km”
Volcanic ash cloud ---> exceed altitude 5km

Negated
exceed altitude 5km --> volcanic ash cloud

is this what you're doing? it made sense when I read your comments but now I am confused as to where the "some" statement came in

goingfor99th

@"J.CHRIS.ALST" said:
Functionally, they seem to say the same thing. From what I can tell, your version and the other version are both correct. From what I remember from an intro. logic class is that a lot of the time when you are trying to negate a very complex idea, one way to do it is to simply say "it is not the case that said complex idea occurs."

This makes it easier because sometimes it can be difficult to know where to put the "not," or any other negating word, when doing it the way that you did it.

For instance, one could potentially misplace the "not" by saying:

"The highest point of any volcanic ash cloud will eventually **not **exceed an altitude of 5km."
This would leave the possibility that the smoke is already above the 5km open for interpretation. But that's a different discussion. but, the point is that would obviously completely change the meaning of the idea, and would be wrong.

As far as a correct interpretation of the two, your answer and the suggested answer, they both imply the the smoke will never exceed 5km. Both are saying no possible worlds, universes exist in which the highest point of the smoke exceeds 5km. Notice how I am basically saying the same thing over and over, just using a different way to express the idea, both of which would be acceptable answers.

If you are able to determine the antecedent and consequent terms in a conditional statement, which admittedly isn't always easy on the LSAT, then to go from 'All A are B' to 'Some A are not B,' you only need to place a 'not' in front of the consequent and change the "all" to "some."

goingfor99th

@Hamaseh_S said:
“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km”
Volcanic ash cloud ---> exceed altitude 5km

Negated
exceed altitude 5km --> volcanic ash cloud

is this what you're doing? it made sense when I read your comments but now I am confused as to where the "some" statement came in

That is the contrapositive. That's another important function but it's not the same as negation. To take the contrapositive, you switch the antecedent and consequent terms and then do an absolute negation of both (i.e. you change the places of the terms, and cross them out like you did in the example, A goes where B went, B goes where A went, 'A' becomes 'not A,' 'B' becomes 'not B').

Hamaseh_S

@goingfor99th said:
“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km.”

Translation :
All [volcanic ash cloud high points] will [eventually exceed an altitude of 5 km].
All [A] are [B].
If [A], then [B].

Negation:
Some [volcanic ash cloud high points] will [not eventually exceed an altitude of 5 km].
Some [A] are [not B].
If [A], then [not necessarily B].

Hope this helps. This is the concept of 'logical opposites.'

A --> B negated /B--> /A
Im missing something... when does negating become "some"

Hamaseh_S

@goingfor99th said:

@Hamaseh_S said:
“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km”
Volcanic ash cloud ---> exceed altitude 5km

Negated
exceed altitude 5km --> volcanic ash cloud

is this what you're doing? it made sense when I read your comments but now I am confused as to where the "some" statement came in

That is the contrapositive. That's another important function but it's not the same as negation.

OHHHHHHHHHHHHHHHH my god and it clicked!!! thank you, I can't believe I confused the two

goingfor99th

@Hamaseh_S said:

@goingfor99th said:

@Hamaseh_S said:
“The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km”
Volcanic ash cloud ---> exceed altitude 5km

Negated
exceed altitude 5km --> volcanic ash cloud

is this what you're doing? it made sense when I read your comments but now I am confused as to where the "some" statement came in

That is the contrapositive. That's another important function but it's not the same as negation.

OHHHHHHHHHHHHHHHH my god and it clicked!!! thank you, I can't believe I confused the two

It's alright. Negation is tricky. I can tell you'll never forget that one again, though.

Hamaseh_S

@goingfor99th said:

It's alright. Negation is tricky. I can tell you'll never forget that one again, though.

hahha never!!

Hamaseh_S

@"J.CHRIS.ALST" said:
.

This has nothing to do with you but why do ppl put these dots in threads. I've always been curious lol

_oshun1_

The highest point of any volcanic ash cloud will eventually exceed an altitude of 5 km

Negation: It is not the case that the highest point of any volcanic ash cloud will eventually exceed an altitude of 5km.
The highest point of any volcanic ash cloud will sometimes not eventually exceed an altitude of 5km.

That isnt really a conditional statement, but if you take it to be a conditional statement, you just throw in "some not" into it to negate it.

A-->B
Negation: Not (A-->B)
A some not B

As opposed to the contrapositive which is /B -> /A

Negation becomes useful when you run into "not all" statements. I.e. Not all As are Bs. It is not the case that All As are Bs. Some As are not Bs. A some /B

J.CHRIS.ALST

Sorry, I originally tried to delete the comment because I immediately realized it was wrong! You can’t delete a comment entirely. Only edit it.

After I had typed out my response I re read it and notice my mistake and accidentally hit post.