Open and closed implication formula

Question

So I am a newbie in mathematical logic and one of the problems I have faced throughout this one week of study is the one concerning open or closed statement. Let me know if I have used some terms such as "statement", "sentence", "formula", etc in this question the wrong way.

I posted probably a related question before here: Connection between universal Quantifier and implication, and have stumbled upon new terms such as "open", "closed", "free", and "bound". I still need some confirmation about this but I state a different question.

So, am I right to say that $\forall x[x\in \mathbb{R}\implies x^2\geq0]$ is a closed formula since I know it has a no free variable?

Does every closed formula have a truth value?

I know that $x\in \mathbb{R}\implies x^2\geq0$ is a formula and is true for every assignment of $x$, or IS it? Why can't I just conclude that $x\in \mathbb{R}\implies x^2\geq0$ is just true and why should I consider its hidden universal quantifier (which, of course, it does not sound the same using existential quantifier)? Is it just for the sake of the existing rule of making sentence in FOL?

Probably the same question: Why can't all quantifiers be bounded quantifiers, and be written that way, considering the existence of domain of discourse? Why don't we explicitly write that domain in the sentence? Meaning, instead of saying "In the domain of natural numbers, $\forall x[x\geq0]$", why not simply $"\forall x\in \mathbb{N}[x\geq0]"$?

Hope my confusion is understood as a newbie. Thanks for the help! :D

Answer 1

The "founding" idea is that logic must be formal.

This idea was firstly investigated by Aristotle : the modern way to investigate the concept "formal" is to define it through syntax :

a formula [a grammatically correct expression] is an expression built-up according to specific rules.

Then we have semantics :

the way to give meaning (and truth value) to an expression through an interpretation.

The interaction of syntax and semantics is the key-point of modern formal logic.

You can see : John MacFarlane, WHAT DOES IT MEAN TO SAY THAT LOGIC IS FORMAL (2000):

What does it mean, then, to say that logic is distinctively formal?

Three things: logic is said to be formal (or “topic-neutral”)

(1) in the sense that it provides constitutive norms for thought as such,

(2) in the sense that it is indifferent to the particular identities of objects, and

(3) in the sense that it abstracts entirely from the semantic content of thought.

Thus, "topic neutral" and "to abstract entirely from the semantic content" mean to separate semantics from syntax : the domain of interpretation is obviously semantical.

Answer 2

How I learnt about the difference is associativity of "truth value".

From wikipedia of open-formula

An open formula does not have a truth value assigned to it, in contrast with a closed formula which constitutes a proposition and thus can have a truth value like true or false.

To answer your question about: Does every closed formula have a truth value?

Yes, again, as the wikipedia says about closed-formula

A sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables.

So if we do some reverse engineering on

$$\forall{x} [x \in \mathbb{R}, x^2 \ge 0]$$

Collectively, if you were to assign the output of this expression to the variable, what choices do you have? That is correct, true or false. Since they are boolean, we can say it is a closed formula.

Also, when quantifiers $\forall$ or $\exists$ are used with any variable $x$ $y$ and so on..., they are no longer a free variable.

You can learn more about quantification here https://plato.stanford.edu/entries/quantification/