My Math Forum “No head injury is too trivial to ignore”

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 August 24th, 2015, 01:31 PM #1 Newbie   Joined: Aug 2015 From: Isengard Posts: 7 Thanks: 0 “No head injury is too trivial to ignore” Hi, I was studying logic and mathematical thinking and I came across this sentence that has been puzzling me for a while: “No head injury is too trivial to ignore”. Apparently the sentence has a unintended meaning which makes it a nonsensical expression. I am trying to figure out what the flaw is and I came up with the following example: “This couch is too big to fit in the house”. Which I believe it can be re-written as “the couch is so big that it does not fit in the house”. On the contrary if I were to say, "no couch is too big to fit in the house" I think is equivalent to "no couch is so big that it does not fit in the house", which essentially means "all the couches, even big ones, fit in the house". (Please correct me if I'm wrong). Based on this logic saying "no head injury is too trivial to ignore" which can be re-written as "no head injury is so trivial that it should not be ignored". I don't even know if I understand what this means. If this was the case can we do something like the situation with the couch in the house? I would really appreciate if someone could explain this in clear manner with details and using predicate logic if necessary.
August 24th, 2015, 02:14 PM   #2
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Quote:
 Originally Posted by mick17 re-written as "no head injury is so trivial that it should not be ignored"
I think you have an error. The second "not" is incorrect.

One possible reason for your error is that the original is an implicit "not less than" relation, while your example is an implicit "not greater than" relation.

Last edited by v8archie; August 24th, 2015 at 02:22 PM.

August 24th, 2015, 02:32 PM   #3
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Quote:
 Originally Posted by v8archie I think you have an error. The second "not" is incorrect. One possible reason for your error is that the original is an implicit "not less than" relation, while your example is an implicit "not greater than" relation.
Can you expand a a little bit on what you mean?

 August 24th, 2015, 03:14 PM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 As what is wrong with the sentence, is that it is saying that, because no head injury is trivial, they can all be ignored! "No head injury is too trivial to be ignored"
August 25th, 2015, 08:39 AM   #5
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 Originally Posted by Country Boy As what is wrong with the sentence, is that it is saying that, because no head injury is trivial, they can all be ignored! "No head injury is too trivial to be ignored"
Can you explain as if I were a 10 year old kid how you came to that conclusion? I mean in a more fundamental sense.

I don't get why you interpret "because no head injury is trivial" then "they can all be ignored!". What makes you think there is an implication there? What are the consequences of having that "too" in the sentence?

I know there is a flaw in the sentence but the reason is quite unclear to me.

August 25th, 2015, 10:12 AM   #6
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Quote:
 Originally Posted by mick17 Can you explain as if I were a 10 year old kid how you came to that conclusion? I mean in a more fundamental sense. I don't get why you interpret "because no head injury is trivial" then "they can all be ignored!". What makes you think there is an implication there? What are the consequences of having that "too" in the sentence? I know there is a flaw in the sentence but the reason is quite unclear to me.
Do you understand what "too trivial to be ignored" means? Can you explain your understanding of that portion of the sentence?

August 26th, 2015, 12:45 AM   #7
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Quote:
 Originally Posted by Country Boy Do you understand what "too trivial to be ignored" means? Can you explain your understanding of that portion of the sentence?
To me the kind of expression "Too x to y", where x is an adjective and y represents a verb (action) either in active or passive form, is the equivalent to "so x that it can't y".

So in this case I think "Too trivial to be ignored" means "so trivial that it can't be ignored".

August 26th, 2015, 04:09 PM   #8
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Quote:
 Originally Posted by mick17 To me the kind of expression "Too x to y", where x is an adjective and y represents a verb (action) either in active or passive form, is the equivalent to "so x that it can't y". So in this case I think "Too trivial to be ignored" means "so trivial that it can't be ignored".
Yes, which is what I said before, "because no head injury is trivial, they can all be ignored!"

August 27th, 2015, 11:58 AM   #9
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Quote:
 Originally Posted by Country Boy Yes, which is what I said before, "because no head injury is trivial, they can all be ignored!"
I don't think I understand the process behind your conclusion. But pondering about it I believe the confusion arises firstly when the negation is introduced in the sentence and secondly when we misunderstand the phrase "too trivial".

If

(1) "too trivial to be ignored"

means

(2) "so trivial that it cannot be ignored"

which per se is a semantic nonsense, then if we say

(3) "there are injuries that are too trivial to be ignored",

we would mean

(4) "There are head injuries that are so trivial that cannot be ignored".

If we introduce the negation

(5) "There does not exist any head injury too trivial to be ignored"

which based on (4) would be the same as

(6) "There does not exist any head injury which is so trivial that cannot be ignored"

extracting the negation of the quantifier of (6)

(7) "Every head injury, no matter how trivial, can be ignored"

This is because being $\displaystyle x$ a head injury and a predicate such that

$\displaystyle P(x) :=$ "$\displaystyle x$ is so trivial that cannot be ignored"

$\displaystyle \nexists x. P(x) \iff \forall x. \neg (P(x))$

It is easier to think of it in a more generic way. For example:

"No WUG is too DAX to be ZONGED"

WUG can always be ZONGED, whether WUG is DAX or not. If we fill the sentence with real words such as:

"No football team is too good to be defeated"

This can be re-formulated as:

"Every football team, no matter how good, can be defeated"

Last edited by mick17; August 27th, 2015 at 12:06 PM. Reason: something missing

August 30th, 2015, 06:55 AM   #10
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Quote:
 Originally Posted by mick17 "No football team is too good to be defeated" This can be re-formulated as: "Every football team, no matter how good, can be defeated"
Yes, and "No head injury is too trivial to be ignored" can be re-formulated, in exactly the same way, as:
"Every head injury, no matter how trivial, can be ignored".

It's basic English grammar.

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