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

  • $\begingroup$ Correct; the formula $∀x[x∈R ⟹ x^2≥0]$ is a closed formula (or sentence) because it has no free occurrences of variables. $\endgroup$ Jun 9 '18 at 9:16
  • $\begingroup$ Yes; a closed formula has a definite truth value in an interpretation. $\forall x (x=0)$ is FALSE in $\mathbb N$, while an open one, like e.g. $(x=0)$ may change truth value (for a specific interpretation) according to the value assigned to the free variable $x$. $\endgroup$ Jun 9 '18 at 9:18
  • $\begingroup$ You can see Classical Logic and Model Theory and Logical Truth and Logical Constants for an introduction. $\endgroup$ Jun 9 '18 at 9:43
  • $\begingroup$ The last point is that usually the "pure" presentation of first-order logic has no predicate constant, like $\in$. Thus the way to formalize $∀x \in \mathbb N [x≥0]$ is $∀x[N(x) \to x≥0]$ where $N(x)$ is a unary predicate. Now again, the truth-value of the sentence depends on the way to interpret $N(x)$. $\endgroup$ Jun 9 '18 at 10:04
  • $\begingroup$ Thus, from the "pedagogical" point of view, we have reduced restricted quantification to quantification of a conditional: thus, in any case, we have to first learn how to manage quantifiers. $\endgroup$ Jun 9 '18 at 10:11

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.

  • $\begingroup$ I get it now. The idea is to formalize that sentence formed with bounded quantifier. So, my last point, how do you think about introducing $\endgroup$
    – bms
    Jun 9 '18 at 13:17
  • $\begingroup$ get it now. The idea is to formalize that sentence formed with bounded quantifier. So, my last point, what is a reasonably good way to introduce my student the way of proving an implication? Do you think it is fine to assume (with careful context reading) that there are always hidden universal quantifiers before the implication statement that bound the variables in the hypothesis, and tell them that? Hence, they will always formally start with "Take an arbitrary x. If p(x) is true, then.... Thus q(x)". I ask this since my usual understanding is $\endgroup$
    – bms
    Jun 9 '18 at 13:24
  • $\begingroup$ that "for all" sentence is equivalent to "If, then" sentence because the way we prove them is essentially the same. $\endgroup$
    – bms
    Jun 9 '18 at 13:26

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