What does the existential quantifier imply about the variable it comes before?

Question

For example, what does the statement

imply about x?

Is x a free variable (if so, isn't the set of real numbers imposing a restriction on x?)?

or is it a bound variable with the domain of discourse being the set of real numbers? (if so, would it be appropriate to claim that x is a defined variable?)

Answer 1

Without the quantifier, the variable is free.

But with the quantifier, the variable becomes bound. And yes, we are also claiming that the variable is an element of $\mathbb{R}$

Answer 2

If you take the set of real numbers, and look at which satisfy the condition, this statement asserts that that set is non-empty. So this can be seen as a statement about that set, rather than about "x".

Whether a variable is free or bound depends on what scope you are considering. For instance, consider the expression $y= \int_0^1 \sin(x) dx$. Outside the integral, $x$ is a bound variable. For instance, as far as the equal sign is concerned, $x$ is a bound variable. But inside the integral, $x$ is a free variable.

If you say "There exists a real number $x$ such that $x^2$ is rational", then you can imagine someone going through all the real numbers, and for each one, squaring it and seeing whether the result is rational (obviously, you can't actually perform the calculation for all the real numbers, but you can model checking the statement as this). While you're checking the real numbers, $x$ is a free variable. But once you've checked all the real numbers, and determined that the statement is true or false, $x$ is bound.

You can have multiple levels of quantifiers. For instance, consider $\forall y\exists x:x^2=y$. You can think of this as:

value1 = True
for y in R:
     value2 = False
     for x in R:
           if (x^2 == y):
                value2 = True
     if not value2:
            value1 = False

That is, you go through each $y$. For each $y$, you initialize a boolean to True. You then go through each $x$, and for each $x$, you set a boolean to False. But then if any $x$ work, then you set the inner boolean to True. Then, going back to the outer for-loop, if any $y$ didn't have any $x$ that worked, then you set the outer boolean to False.

In the outer for-loop, $y$ is free but $x$ is bound. In the inner for-loop, both $x$ and $y$ are free. Outside the outer for-loop, both are bound.

That is, in the expression $\exists x: x^2=y$, $x$ is bound, while $y$ is free. Once you put that expression inside the expression $\forall y\exists x:x^2=y$, both $x$ and $y$ are bound outside the entire expression, while $y$ is free inside the inner expression.