Perplexing claim in Chang and Overby's "General Chemistry" I'm reading the chapter on gases in Chang and Overby's book &quot;General Chemistry: The Essential Concepts&quot; [1, p. 145], and the following passage is causing me some confusion:

The work of the Italian scientist Amedeo Avogadro complemented the studies of Boyle, Charles, and Gay-Lussac. In 1811 he published a hypothesis stating that at the same temperature and pressure, equal volumes of different gases contain the same number of molecules (or atoms if the gas is monatomic). It follows that the volume of any given gas must be proportional to the number of moles of molecules present; that is,
$$V \propto n$$
$$V = k_4 n \tag{5.7}$$

I don't quite understand the emphasized sentence; in particular, I don't understand how the emphasized claim &quot;follows&quot; from the preceding statement. In fact, I don't actually think it does.
For example, suppose instead that the volume of any gas at a given temperature and pressure was inversely proportional to the number of moles of the gas present; that is, suppose the relationship between volume and amount was instead
$$V \stackrel{?!}{=} \frac{k}{n},$$
for some constant $k$. Then it would certainly still be true that equal volumes of different gases contain the same amount of gas. For example, if I had $\pu{5 L}$ of $\ce{H2}$, then, according to the above &quot;false law&quot;, I would have $k/5$ moles of $\ce{H2}$. Similarly, if I had $\pu{5 L}$ of $\ce{N2}$, I would have $k/5$ moles of $\ce{N2}$ — the same amount!
So the claim that the proportionality follows from the fact that equal volumes of different gases contain the same number of molecules seems illogical. In fact, it seems to me that the only thing one could logically deduce from the first hypothesis is that there is some relationship between volume and amount which holds for all gases (that is, the relationship doesn't depend on the gas itself).
Am I missing something, or is the statement in the book an error?

Added much later:
To clarify what I'm asking here (though I thought it was pretty clear), consider the following two statements:
Statement A: at a given temperature and pressure, equal volumes of different gases contain the same number of molecules.
Statement B: at a given temperature and pressure, the volume $V$ of a gas is related to the number of molecules by $V = kN$ (where $k$ depends on temperature and pressure).
My point is simply that these two statements are not logically equivalent. B implies A, but not the other way around.
So why is it that the two statements are (seemingly quite often) presented as logically equivalent?
References

Chang, R.; Overby, J. General Chemistry: The Essential Concepts, 6th ed.; McGraw-Hill: New York, NY, 2011. ISBN 978-0-07-337563-2.
The  other answer is, I think excellent.

Your question is basically "but what if there isn't a direct proportion between number of gas molecules and volume?" Instead, that was Avogadro's hypothesis in a nutshell: the volume of the gas is determined exactly proportionate to the number of molecules in it.

You are right in principal that you could have such a "false law" where the number of molecules is inversely proportionate to the volume and still have equal number of molecules for $\ce{H2}$ and $\ce{N2}$.

Let's follow your course of logic. That would mean if I get 10 L of $\ce{H2}$, I would then have $k/10$ molecules. Let's imagine I add more and more volume, to get 100 L then 1000 L of gas. At some point, I must necessarily divide the molecules in fractions in your false law. Certainly a molecule must occupy some indivisible volume, since Avogadro believed in the atomistic theory.

Instead, the obvious solution is that the number of molecules must be a measure of the amount of "stuff." So if we have more "stuff," we need more volume for it to occupy. The expression must be directly proportional between the volume and the number of molecules.