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## Main Question or Discussion Point

(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

[tex]

\begin{equation}

\forall x\in\mathbb{Z} (f(x)=x^2),

\end{equation}

[/tex]

I make sense of this statement by invoking the "for loop" analogy from computer science: for the first element in [itex] \mathbb{Z} [/itex], [itex] x [/itex] is equated with it, and the defining function equation is then evaluated, which makes sense to do because [itex] x [/itex] has been prescribed particular meaning --- [itex] x [/itex] has temporarily been made into a determined constant for the duration of the iteration. After the evaluation, [itex] x [/itex] is equated with the next number in [itex] \mathbb{Z} [/itex], and the process continues until [itex] f [/itex] has been defined for all values in [itex] \mathbb{Z} [/itex].

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

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

[tex]

\begin{equation}

\forall x\in\mathbb{Z} (f(x)=x^2),

\end{equation}

[/tex]

I make sense of this statement by invoking the "for loop" analogy from computer science: for the first element in [itex] \mathbb{Z} [/itex], [itex] x [/itex] is equated with it, and the defining function equation is then evaluated, which makes sense to do because [itex] x [/itex] has been prescribed particular meaning --- [itex] x [/itex] has temporarily been made into a determined constant for the duration of the iteration. After the evaluation, [itex] x [/itex] is equated with the next number in [itex] \mathbb{Z} [/itex], and the process continues until [itex] f [/itex] has been defined for all values in [itex] \mathbb{Z} [/itex].

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