Can I use ": ," instead of ", implies" for this example?

Question

I've to write this statement in a formal manner:

if $x>1$ then $x^{2}>1$.

Writing the result of the exercise I face this problem, I wonder if these two statements are equivalent: $$\forall~x \in \mathbb{R}, x >1 \implies x^{2}>1 \ $$ $$\forall~x \in \mathbb{R}: x >1, ~~ x^{2}>1 \ $$

I read the first as:

for all the x in $\mathbb{R}$ if x is greater than 1 then x squared is greater than 1

For the second I read it so:

for all the x in $\mathbb{R}$ such that x is greater then 1, x squared is greater then 1

Where is my fault? Is there a better alternative?

---

Exercise 1.14-14 p.23 from Analisi matematica vol.1 3th ed. (2002) by Enrico Giusti

Answer 1

These statemens aren't equivalent, at least the way I read them. I read the first as $$ \forall x\,\left[\left((x \in \mathbb{R}) \land (x > 1)\right) \rightarrow x^2 > 1 \right] \text{,} $$ which of course is true. The second, however, I read as $$ \forall x \left[(x \in \mathbb{R}) \rightarrow \left((x > 1) \land (x^2 > 1)\right)\right] \text{,} $$ which is wrong, since $-1 \in \mathbb{R}$ but $x < 1$.

I'd say the general rule for reading such first-order formulas with extended syntax is that $$ \forall x\,\varphi_1,\varphi_2,\ldots,\varphi_n\,:\, \theta $$ means $$ \forall x \left[\left(\varphi_1 \land \cdots \land \varphi_n\right) \rightarrow \theta\right] \text{,} $$ i.e. in plain english that "For all $x$ such that $\varphi_1$ and $\varphi_2$ and ... $\varphi_n$, we have $\theta$".

Your first formula doesn't include a colon (:), and therefore doesn't really fit into that schema. But since the last comma-separated term is an implication, i.e. since it has the form $$ \forall x\,\varphi_1,\ldots,\varphi_n \rightarrow \theta $$ I'd say most people will read it as $$ \forall x \left[\left(\varphi_1 \land \cdots \land \varphi_n\right) \rightarrow \theta\right] \text{.} $$ But it really is a bit ambiguous, since another possible reading would be $$ \forall x \left[\varphi_1 \land \cdots \land \varphi_{n-1} \land (\varphi_n \rightarrow \theta)\right] \text{.} $$

Answer 2

This is only a "long" comment ...

We start form the condition you are working with :

if $x>1$ then $x^2>1$

where it is implicit that we are restricting the domain of discourse to $\mathbb R$.

We can easily symbolize it in an unambiguos manner as :

(a) --- $\forall x(x \in \mathbb R \rightarrow (x>1 \rightarrow x^2>1))$.

A "standard" way to abbreviate this kind of fomulae is through the so-called "restricted quantifiers".

We may have :

($a_1$) --- $(\forall x \in \mathbb R)(x>1 \rightarrow x^2>1)$

or :

($a_2$) --- $(\forall x \in \mathbb R : x>1 \rightarrow x^2>1)$.

Both are exploiting the usual set-theoretic symbolism : $\{ x \in \mathbb R : \varphi(x) \}$.

When I read your first formula, I assume that you are using the comma (",") as a "separator", like the colon (":") in my formula ($a_2$), and it's all right for me.

When I read your second formula :

$∀x∈R : x>1, x^2>1$

I assume that now you are using the colon (":") as "separator". But what is the role of the comma (",") ? It is also a "separator" ? of what ? or it takes the place of a connective...

Thus, we have too many possibilities :

($b_1$) --- $(∀x∈R)(x>1, x^2>1)$

and we may suppose that the comma is $\land$;

or we can read it as :

($b_2$) --- $(∀x∈R : x>1)(x^2>1)$.

This the source of my "disappointment" with the "double restricted" quantifier; we have to read it as ($a_2$) above :

($b_3$) --- $∀x(x∈R \land x>1 : x^2>1)$

i.e. as

($b_4$) --- $∀x((x∈R \land x>1) \rightarrow x^2>1)$ ?

But this one is different from ($b_1$) which, written in extenso, is :

$∀x(x∈R \rightarrow (x>1 \land x^2>1))$.

But if we read the comma in ($b_1$) as $\rightarrow$, we have :

($b_5$) --- $∀x(x∈R \rightarrow (x>1 \rightarrow x^2>1))$

and this is equivalent to ($b_4$).

Thus my conclusion is :

• stay with standard symbolism; commas and colons are not connectives but "separators"

• and we have already "standard" separators : the parentheses.