Understanding the Universal Quantifier and Variables

[cra18]

(I hope this question is in the proper place.)

I am confused about what effect the universal quantifier has on a variable. My understanding of variables is very simplistic: I view a variable as simply a placeholder for any of a set of possible values, where that set is the universe of discourse for the variable. If I consider the function definition
$$\begin{equation} \forall x\in\mathbb{Z} (f(x)=x^2), \end{equation}$$
I make sense of this statement by invoking the "for loop" analogy from computer science: for the first element in $\mathbb{Z}$, $x$ is equated with it, and the defining function equation is then evaluated, which makes sense to do because $x$ has been prescribed particular meaning --- $x$ has temporarily been made into a determined constant for the duration of the iteration. After the evaluation, $x$ is equated with the next number in $\mathbb{Z}$, and the process continues until $f$ has been defined for all values in $\mathbb{Z}$.

Given the above definition of a variable, the $x$ in the above process doesn't seem to function as a variable at all. There is never a case where $x$ is a placeholder for any more or any less than one particular number. There is never a case where $x$ acts as anything other than a constant. (I suppose I am viewing the $x$ as being a new variable upon each iteration, since it isn't as though the past history of the values $x$ has taken is relevant.)

Is my understanding of the universal quantifier incorrect? Is there a better way to understand the universal quantifier? My confusion stems from the fact that I encounter such strong emphasis on a variable as being a placeholder for any of a SET of possible numbers, but am having a hard time coming up with an example of such a thing that actually has any meaning (i.e., a truth value).

 

[Stephen Tashi]

Your understanding is essentially correct. Since a universal quantifier can apply to more than a countable number of objects, it isn't completely accurate to think of a loop in a computer program, which necessarily loops over a countable set.

The saying that a variable"is a placeholder" isn't a precise mathematical statement. If you heard someone that that a variable is a "placeholder for a set of values" , that was not a formal definition. Of course, It is possible for a single variable (like "x") to represent a set, but whoever said that a variable was a placeholder for a set of numbers probably meant to express the same idea as your concept.

In mathematical literature, the universal quantifier is often "understood" instead of be written explicitly. For example if someone declares "x + y = y + x", the intended meaning usually is "for each number x, for each number y ...".

There is also the mysterious form of logical proof called "universal generalization". For example, a proof might begin "let epsilon > 0 be given.". The variable epsilon is then treated as a specific number. If something is shown to be true for epsilon then the conclusion is "We have show that for each epsilon > 0 that ..." because the epsilon, although treated as a specific number had no special properties except that it was a number > 0.

Interpreting variables in computer code is possible because there are formal ways to define the "scope" of variables. In mathematical literature, sometimes "x" means something in one sentence and something different in the next sentence, or next page, or next chapter. If mathematics was written with a strict syntax then quantification like "for each x " and "there exists an x" would be used with syntax that showed the scope of the statement about x. In symbolic logic, this is done. But most mathematics is written more informally.

 

[cra18]

The saying that a variable"is a placeholder" isn't a precise mathematical statement. If you heard someone that that a variable is a "placeholder for a set of values" , that was not a formal definition. Of course, It is possible for a single variable (like "x") to represent a set, but whoever said that a variable was a placeholder for a set of numbers probably meant to express the same idea as your concept.

I never said that a variable is a placeholder for a set. I said that a variable is a placeholder for any of a set of numbers --- for any element of some set of numbers. This is a definition I decided upon because of its simplicity, and because the alternative concept of a "general number" makes no sense to me, but frankly I have no idea how to define what a variable is. This problem is aggravated by the fact that, when I look at equations like
$$\begin{equation} \forall x\in\mathbb{R}\left(g(x)=\sum_{i=1}^x \varphi(i)\right) \end{equation}$$
or
$$\begin{equation} \forall x\in\mathbb{R}\left( f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}\right) \end{equation}$$
or
$$\begin{equation} \forall x\in\mathbb{R}\left(F(x) = \int f(x)\,dx\right) \end{equation}$$
none of the variables seem to vary at all in any meaningful way, if each element of $\mathbb{R}$ is equated with $x$ in turn and the expression evaluated. And yes, I am aware that the universal quantifiers are rarely if ever written out like this, but they are most certainly implicitly there.

 

[Stephen Tashi]

I never said that a variable is a placeholder for a set. I said that a variable is a placeholder for any of a set of numbers

Hmm...

--- for any element of some set of numbers.

Ok. There is a difference between "for any set" and "for any element of some set".

This is a definition I decided upon because of its simplicity

I thought you preferred the idea of comparing "for each" to a loop statement. I think that's a better idea than using the terminology "placeholder".

, and because the alternative concept of a "general number" makes no sense to me, but frankly I have no idea how to define what a variable is.

Trying to answer the question "what is a variable?" can get into the same bull session type discussions as threads on "what is a differential? "! A rigorous definition of a variable is only possible if you establish exact rules for syntax and specify what a "well formed formula" is. It would be rather like defining what a (legal) variable is in a computer language and it wouldn't necessarily help with everyday math.

I don't know what you mean by a "general number" and I don't see that as the only alternative definition for a variable. I suggest you take the pragmatic approach. If you are comparing two alternative definitions for a variable, ask "What practical difference would it make if I used one definition instead of the other?". Try to give an example where the truth or falsity of a statement would change depending on which definition you used. If you can't find such an example then the two definitions you are using don't make any practical difference. They might amount to the same thing or neither might be precise enough to have any practical interpretation.

This problem is aggravated by the fact that, when I look at equations like
$$\begin{equation} \forall x\in\mathbb{R}\left(g(x)=\sum_{i=1}^x \varphi(i)\right) \end{equation}$$

(Technically you should call those "statements". An equation states an equality that is not necessarily true for each possible value of its variables. The values of the variables that make the equation a true statement are its solutions. )

 

[verty]

One way to look at it is that $\forall x \; P(x)$ means
$P(x_1) \land P(x_2) \land P(x_3) \land ...$, while $\exists x \; P(x)$ means
$P(x_1) \lor P(x_2) \lor P(x_3) \lor ...$. One can't say x is a constant, x ranges over all its values. The correct term is that x is a bound variable.

This is my preferred explanation.