Translate this sentence to predicate logic

Question

The question given asks to translate to predicate logic:

Every positive real number has a unique positive real root.

My solution to this problem is to separate it into the appropriate quantifiers.

C(x) = "Every positive real number x"

S(x) = "x has a positive real root"

the final logic form being: ∀x(C(x)→S(x))

It seems to easy for it to be correct. I need help in confirming my answer.

Answer 1

The formula you have translates to:

"For all $x$, if every positive real number $x$, then $x$ has a positive root".

This does not make sense, and does not correspond to the statement you wish to formalize.

What you want to say is

$$\forall x(\text{$x$ is a positive real number} \rightarrow \text{$x$ has a unique positive root})$$

i.e.,

"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.

which is equivalent to

"Every positive real number has a unique positive root."

Answer 2

Couple of things:

First, you don't want to define:

$C(x)$ = "Every positive real number $x$"

The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:

$S(x)$ = "$x$ has a positive real root"

The obvious thing to do is to define:

$C(x)$ = "$x$ is a positive real number"

Second, if we now look at

$$\forall x (C(x) \rightarrow S(x))$$

then we see that this means: "Every positive real number has a positive real root"

.. but you were supposed to translate "Every positive real number has a unique positive real root"

So, you're missing the "unique" part.

What to do? One thing you can do is to simply redefine:

$S(x)$ = "$x$ has a unique positive real root"

But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$

As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:

$H(x,y)$: "$x$ has $y$ as a positive real root"

OK, so now we can nicely re-express that "Every positive real number has a positive real root":

$$\forall x (C(x) \rightarrow \exists y \ H(x,y))$$

Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:

$$\forall x (C(x) \rightarrow \exists y (H(x,y) \land \neg \exists z (H(x,z) \land z \not = y)))$$

See how that works? There is a positive real root $y$, but there is not other possible real root $z$

Another way to think about this is that every real root will have to be $y$. So:

$$\forall x (C(x) \rightarrow \exists y (H(x,y) \land \forall z (H(x,z) \rightarrow z = y)))$$

That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.

Finally, though a little less intuitive, you can do:

$$\forall x (C(x) \rightarrow \exists y \ \forall z (H(x,z) \leftrightarrow z = y))$$

This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$