# “Exactly Two” or “Exactly k” from English into (Quantificational) Logic

The inspiration is Example 2.2.3 #2(d) on P71 of How to Prove It by Daniel Velleman.

Analyze the logical forms of the following statement: The number $x$ has exactly $k$ $n$th roots.

Answer : $\color{#FF4F00}{\exists \, r_1 \cdots \exists \; r_k} {\huge{[}}\, \color{#007FFF}{r_1 \;\& \, \cdots \& \,r_k \text{ are$n$th roots of$x$}} \; \text{ and } \; \color{green}{ r_1 \neq \cdots \neq r_k} \quad \text{ and }\color{#960018}{\text{ nothing else is a$n$th root of$x \;$}}{\huge{]}}$ $= \color{#FF4F00}{\exists \, r_1 \cdots \exists \; r_k} {\huge{[}}\, \color{#007FFF}{r_1^n = x \;\& \, \cdots \& \, r_k^n = x \text{ are$n$th roots of$x$}} \; \wedge \; \color{green}{r_1 \neq \cdots \neq r_k} \quad \wedge \;\color{#960018}{ \lnot \, \exists y \, {\Large{[}} \,y^n = x \, \wedge \, y \neq r_1 \, \wedge \cdots \wedge \, y \neq r_k \, {\Large{]}}} \;{\huge{]}}$.

I understand the necessity of the blue and green statements.
However, why is the (carmine) red necessary? At the beginning of each sentence, in orange, I declared the existence of only $k$ variables (ie $\color{#FF4F00}{r_1, ..., r_k}$), so there are simply no more variables that could serve as the $(k + 1), (k + 2), ...$ variables. Thus, how and why does the orange NOT imply the red tacitly and wordlessly?

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The red statements relate to the work EXACTLY in the original statement. –  Nick R Aug 28 '13 at 17:31
@NickR: I sadly don't understand your comment. Would you please elaborate? –  LePressentiment Aug 29 '13 at 7:03
I shall elaborate below in the form of an answer. –  Nick R Aug 29 '13 at 17:22
@NickR: Thank you very much. So did you intend to write "...relate to the wor$\Large{D}$ EXACTLY"? You wrote "work." –  LePressentiment Aug 30 '13 at 13:10

Your orange, blue and green parts declare that there are at least $k$ n'th roots. However it is perfectly possible that there are more than $k$ \emph{objects} which the variables are currently standing in for. That is what your red sentence excludes.

For instance $\exists x (x^2=1)$ says that $1$ as a square root (e.g. 1) however $-1$ could also stand in here.

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Is there a mathematical symbol equivalent to "there exists only"? Or would I have to invent such a symbol for my own purposes? –  Ryan Aug 28 '13 at 18:13
Sometimes we use $!$ for unique i.e. $\exists !x(\ldots)$ but this is only an abbreviation. First-order logic doesn't have a short-hand for this, you have to do it the long way if you want it to be formally correct. –  James Aug 28 '13 at 18:17

Let's take a simpler case. Suppose you want to render 'There is exactly one F' into standard logical notation.

$\exists x(Fx \land \forall y(Fy \to y = x))$

or some equivalent like

$\exists x(Fx \land \neg\exists y(Fy \land y \neq x))$.

Now imagine someone asking what the clause after the conjunction was doing. "Why is it necessary? At the beginning of the sentence I declared the existence of only one variables (i.e. $x$), so and why doesn't the first conjunct imply the second one tacitly and wordlessly?". Well it should be obvious in this case what's gone wrong. If we write down only the first conjunct, i.e. just write

$\exists xFx$

this tells us that at least one thing (in the relevant domain) is F. That's what the existential quantifier means. It doesn't declare that there is only one $F$ (though of course it only uses one variable). That is to say, it doesn't rule out there being more than one F; we need the extra clause precisely to rule that out.

That should give a clue, then, why you need the "red" clause in the more complex case.

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Thank you. Could you please demystify the notation: $Fx$ and $Fy$? Do you mean $F(x)$ so that $F(x)$ is a statement which depends on the variable $x$? –  LePressentiment Aug 29 '13 at 7:07
You don't need to insert redundant brackets here -- but if it helps you, be my guest! –  Peter Smith Aug 29 '13 at 11:19
Many thanks! Since MSE doesn't admit of multiple votes, I upvoted. –  LePressentiment Aug 30 '13 at 13:18

The importance of the red part may be more clear with the equivalent phrasing

If $y$ is an $n$-th root of $x$, then $y=r_1$ or $y=r_2$ or .. or $y = r_k$.

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The orange, blue, and greed section states the existence of $k$ distinct roots. The red statement says that there are no others. Thus there are EXACTLY $k$ roots. The red statement formalizes the meaning of the work exactly in the statement "The number $x$ has exactly $k$, $n$-th roots.