Negation test revisited

[wj097]

Hello,

How would you negate the following sentences.

1) Students are more bored by history than math
--> Students are not more bored (equal or less) by history than math???

2) All students are more bored by history than math
--> Some students are not more bored (equal or less) by history than math???

Q: are statements 1 and 2 different from each other?? or am I mistaking on the negation of 1.

Thx

[ohthatpatrick]

You nailed it. That was all accurate.

First of all, negating "more than" means "less than or equals to", as you noted.

Secondly, when a sentence's truth value is based on a quantity term, you must negate the quantity term.

(I just re-read that sentence and want to punch myself in the face ... "when a sentence's truth value is based on a quantity term" .... sigh)

Here is such a statement:

Some of my friends are Libertarians.

If we negated the verb, we would get:

Some of my friends are not Libertarians.

Notice that this second statement does NOT contradict the first. So the truth value of the original claim was not "are" vs. "are not".

The correct negation would attack the quantity word "some".

Negating this:
Some of my friends are Libertarians.
we get this:
None of my friends are Libertarians.

That's a contradiction. So the truth value of the original claim hinged on "some" vs. "none".

So, long story short, everything you wrote was correct. Hopefully you understand why it was correct. Let me know if not.

[wj097]

Patrick, one follow up question.
How would you negate the following statement??

People express themselves more cautiously when something important is at stake.

Some people do not express themselves more cautiously even when something important is at stake.

or

People do not express themselves more cautiously even when something important is at stake.

and is inserting "even" before "when" necessary??

Thx

[ohthatpatrick]

I think the "even" is unnecessary. To me it adds some subtle emphasis that I don't think we need.

It's definitely not the easiest sentence to negate, but remember, you can ALWAYS negate something by simply prefacing it with "it is not true that ____"

So:
It is NOT true that People express themselves more cautiously when something important is at stake.

If I were trying to paraphrase what that means, I might just say "People don't necessarily express themselves more cautiously when something important is at stake".

Maybe they do, maybe they don't.

The original statement is technically a conditional, because "when" is a sufficient trigger.

So you COULD diagram the original statement as:
Something important at stake --> ppl express themselves more cautiously

When you want to negate a conditional, it's normally easiest to just use the "it is not true that" prefix.

You do NOT want to try negating either the sufficient or the necessary condition.

Instead, you're simply saying that no such conditional relationship exists. So when the sufficient idea happens, the necessary idea may or may not happen.

When something important is at stake, people may or may not express themselves more cautiously.

Let me know if that didn't make sense.

[brandonhsi]

I think the "even" is unnecessary. To me it adds some subtle emphasis that I don't think we need.

It's definitely not the easiest sentence to negate, but remember, you can ALWAYS negate something by simply prefacing it with "it is not true that ____"

So:
It is NOT true that People express themselves more cautiously when something important is at stake.

If I were trying to paraphrase what that means, I might just say "People don't necessarily express themselves more cautiously when something important is at stake".

Maybe they do, maybe they don't.

The original statement is technically a conditional, because "when" is a sufficient trigger.

So you COULD diagram the original statement as:
Something important at stake --> ppl express themselves more cautiously

When you want to negate a conditional, it's normally easiest to just use the "it is not true that" prefix.

You do NOT want to try negating either the sufficient or the necessary condition.

Instead, you're simply saying that no such conditional relationship exists. So when the sufficient idea happens, the necessary idea may or may not happen.

When something important is at stake, people may or may not express themselves more cautiously.

Let me know if that didn't make sense.

I have a question regarding your example of putting "it is not true that ___," when you negate a conditional statement.

Your example:
It is NOT true that People express themselves more cautiously when something important is at stake.

If I were trying to paraphrase what that means, I might just say "People don't necessarily express themselves more cautiously when something important is at stake".

Maybe they do, maybe they don't.

So, when the sufficient idea happens, the necessary idea may or may not happen. However, for the case that the necessary idea happens, "It is NOT true that People express themselves more cautiously when something important is at stake" does not make sense? It is like saying "It is NOT true that People express themselves more cautiously when something important is at stake," but "People express themselves more cautiously when something important is at stake" is also possible.

My take on the negation of "People express themselves more cautiously when something important is at stake" is all scenarios but "People express themselves more cautiously when something important is at stake."

[endless_sekai]

So, this is an interesting question. I can see how the problem arises the way that you are interpreting the negation of the condition, perhaps this may help answer your question.

So, more or less, a conditional relationship represents a guarantee. That is, if X happens, it is GUARANTEED that Y happens. Hence why X is referred to as the sufficient condition and Y as the necessary condition, because X is sufficient to guarantee the occurrence of Y. Now, I think the confusion on the negation of the conditional is arising because your missing this part. When, I negate a conditional, what is happening is that I am negating the guarantee. For instance, take this conditional

If Tommy is married to a movie star, then he is famous.

Essentially the conditional is saying....

It is guaranteed that if Tommy is married to a movie star, then he is famous.

Now when you negate the conditional essentially you are putting the negation infront of the guaranteed, hence...

Negated conditional:

It is not guaranteed that if Tommy is married to a move star, then he is famous.

Now the import part of understanding this implication is that well it may be the case that Tommy is married to a movie star and is famous, it is in no way guaranteed that Tommy will be famous by Tommy being married to a movie star, instead you can think of the being famous part as a chance occurrence (it may or may not of happened), but just because it so happened that he was married to a movie star and he was famous doesn't imply a contradiction or inconsistency in the negation.

So in the case of you example, (I am going to put it into standard if, then form)

It is guaranteed that if something important is at stake, then people express themselves more cautiously

When I negate it, it becomes

It is not guaranteed that if something is important is at sake, then people express themselves more cautiously

What this essentially means is that when you negate a conditional you essentially neuter the sufficient condition. That is the sufficient condition has no longer has an absolute ability to guarantee the necessary condition. It could make the necessary condition more or less likely, but the point is the sufficient condition no longer forces the occurrence of the necessary condition.