# Negation of a statement in two different ways. Which one is correct?

What is the negation of the following statement?

All triangles are not equilateral.

Should it be : There exists a triangle which is equilateral?

Or is it

All triangles are equilateral?

I think the former is correct. But I have seen the latter as the correct answer in two different books.

• @KaviRamaMurthy I think you might have misread; the negation is "There exists a triangle which is equilateral". (switch all the quantifiers) Commented Jun 16, 2021 at 8:49
• the latter "all triangles are equilateral" cannot be the negation as both this and the original statement are clearly false and it is not possible for a statement and its negation to both be false. Commented Jun 16, 2021 at 8:55
• @KaviRamaMurthy I don't think this is the case. But even if the statement was "not all triangles are equilateral" then the negation would trivially be "all triangles are equilateral" Commented Jun 16, 2021 at 8:58
• @Kavi Rama Murthy I apologise if the question was not up to your standards. But like I said, I have seen this exact question in a number of books and simply wanted to clear my doubt. Commented Jun 16, 2021 at 9:01
• I think that the sentence is ambiguous. Can be read as "All triangles are not-equilateral", meaning that no triangle is equilateral. But also "Not all triangles are equilateral". Before the question can be answered, this must be clarified.
– user65203
Commented Jun 16, 2021 at 9:23

I'm not a native speaker, so a comment from one (better with some degree in English language studies) would come handy, but I think that the original statement has only one reading.

Being equilateral is a property of a single triangle. All X are not P means that for all $$x\in X$$ it holds that $$x$$ does not have property $$P$$. That is, if $$\mathcal P$$ is a collection of all equilateral objects and $$\mathcal T$$ is a collection of triangles, the original statement formalizes as $$\forall t\in \mathcal T:t\notin \mathcal P$$. I think from that point it should be clear that its only negation is $$\exists t\in\mathcal T:t\in \mathcal P$$, as you thought yourself.

• Native speakers (and I am one) do say things like "All A are not B", but they shouldn't, because it is ambiguous. Often the meaning is clear from the context, but not always. Commented Jun 16, 2021 at 9:50

It is impossible to say, because the statement "All triangles are not equilateral" is ambiguous. It can mean either "No triangles are equilateral" or "Not all triangles are equilateral". You should never say in English "All A are not B".

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– Pedro
Commented Jun 17, 2021 at 13:01

What does the original statement mean?

Does it mean "For every triangle, it is true that it is not equilateral" or does it mean "It is not true that every triangle is equilateral"?

The negation of the first statement is "There exists a triangle which is equilateral".

The negation of the second statement is "Every triangle is equilateral".

Because the original statement is unclear, an exact answer to your question is impossible.