# Meaning of "no two"

Is the following sentence ambiguous?

The eight queens puzzle is the problem of placing eight queens on an 8x8 chess board so that no two queens attack each other.

Do we need to add "or more" as follows?

The eight queens puzzle is the problem of placing eight queens on an 8x8 chess board so that no two or more queens attack each other.

I am sorry, I am still confused, so let me give another example as follows:

There are 4 red balls and 5 white balls in a box. Five balls are randomly taken from the box.

Find the probability that no 2 balls are red.

In my understanding the following configuration is allowed.

• 1 red and 4 white
• 3 red and 2 white
• 4 red and 1 white

but only 2 red and 3 white balls is not allowed.

Short answer: no two queens attack each other is unambiguous; no two queens or more attack each other is confusing.

Long answer: I have a slightly different interpretation than the other answerers. A common issue in mathematical communication arises over ambiguous statements like

Two queens are red.

To a layperson, or in some mathematical contexts, this means "there are exactly two red queens". But in logical, it gets unpacked as "there exist two queens which are red" which is true as long as there are at least two red queens. In cases like these, it's usually a good idea to add at least or exactly to avoid any possible confusion.

However, things are very different when we add a negation in there. Notice that

No queen is red,

unlike one queen is red, is unambiguous to both logicians and laypeople. The quantifier here is universal, rather than existential; the equivalent statement "every queen is non-red" has no numbers attached to it. On the other hand,

No two queens are red

barely even makes sense. So how do we interpret your sentence? The key is that "attacking each other" is a binary relation; it's a property not of queens but of pairs of queens. We could rewrite your sentence as

The eight queens puzzle is the problem of placing eight queens on an 8x8 chess board so that no pair of queens attack each other.

Now it's clear that the statement is of the type no (object) is (property) --- the object is "pair of queens". Just like no queen is red, there is only one possible interpretation. If you want a better wording, this one should be unambiguously unambiguous! But I should stress that your original sentence is just fine.

In any case, I'd recommend against changing it to no three queens or more attack each other. Since "attacking each other" is a binary relation, this actually adds to the confusion. If Queen A attacks Queen B attacks Queen C, but Queen A and C are non-attacking, does that count as three queens attacking each other? Leaving out the or more would be the right thing to do.

Edit to respond to your second example

Contrary to the first example, I have a lot of trouble parsing the sentence

Find the probability that no 2 balls are red.

I think this sentence is ambiguous at best. (In fact, I would argue that it's semantically incorrect; in any case, it would be best changed to something else.) I mentioned a similar example above and rejected it as meaningless, but I didn't properly explain myself. So here goes:

"No" and "not" can sometimes be confusing in mathematics. "No" is what the logician calls a quantifier. It says that something is false for all objects. So no ball is red means for every ball, that ball is not red. And in general, no (object) has (property) means for every (object), that (object) does not have (property). (Logicians write this in symbols as ∀x ¬p(x).)

So let's try to analyze the sentence no two balls are red. Here we get a bit stuck. What does the "no" quantify over? The most sensible interpretation is probably to unpack

No (two balls) are (red)

as

For every (two balls), those (two balls) are not (red)

and then further assuming that a pair of balls are red if and only if they are both red. In other words, you get a logically equivalent statement to

At most one ball is red

which is really what you should write in the first place, since as you point out "No two balls are red" is confusing.

No two balls are red

as

It is not the case that (exactly) two balls are red.

This is an understandable attempt to tackle the sentence. It's not the way that "no" is usually interpreted, but I wouldn't say it's wrong under the circumstances. It's really the phrase no two balls are red which is meaningless; the reader needs to resolve this by either changing "no" to a quantifier that makes sense (as you did) or by interpreting (two balls) as a single object and extending the definition of (red) to pairs of objects.

TL;DR

"No" is a logical quantifier. You can only quantify over singular objects. So

No two balls are red

doesn't make sense. The confusion here arises because you can quantify over a group of things if you treat the group as the singular object:

No two queens are attacking

works because "two queens" (or better, the singular "pair of queens) are treated as a singular object which can have the property "attacking".

The TL;DR was TL;DR

All this confusion is the fault of the word "no".

No, I don't think "or more" is necessary. The meaning is clear.

If you want to get pedantic about it: If more than two queens can attack each other, then two queens can attack each other.

So, you really don't need to add "or more."

• I'd even go further to say that there is not really a need for the word `two`. Simply stating so that no queens may would also suffice (however personally I think no two sounds much cleaner).
– Phil
Aug 1, 2011 at 16:28

The usage is a bit of a mathematical idiom coming from a literal interpretation of the number 'two'. Follow this reasoning:

If you have -exactly- two items in your hand, how many do you have in your hand? Two of course. Can you give me two? Yes, of course!

If you have -exactly- seven items in your hand, how many do you have in your hand? Let me not answer that. Can you give me two? Yes, of course!

For a mathematician it is correct to answer that with anything from 0 to 7. You can answer 'two' and be correct, because it is true that you have two items in your hand (you also have 5 others). For normal humans, it sounds strange and can be tantamount to lying to answering that way; asking 'do you have two in your hand?" normally means you're really asking "do you have -exactly- two in your hand?".

In your example, if seven queens are mutually attacking each other (all on a row?), then it is certainly already the case that two of those seven are attacking each other, so for brevity one leaves out '... or more'.

• By the same logic, a literally-minded mathematician could answer your first statement with any number greater than one, since we can always assume "two" or "seven" are shorthand for "[number] or more". Aug 1, 2011 at 17:18
• @FumbleFingers: Actually, no, it doesn't work in the reverse direction. Suppose you have exactly 2 in your hand. To answer 'Do you have 7? and answer 'Yes' would be an outright lie. Aug 1, 2011 at 17:53
• Except your original statements don't say "If you have exactly [number]". But I fear we're in danger of becoming as literal-minded as our hypothetical mathematician here! :) Aug 1, 2011 at 17:56
• @FumbleFingers: that's the problem with language, in that we're using it to talk about it at the same time. So it is not apparent if we're talking about or using, or which context. I will modify my answer to say 'exactly' and clarify (but it is really not necessary) Aug 1, 2011 at 18:13