All the Fallacies, Baby

People make all kinds of mistakes. One of the more popular songs the kids are listening to is “Rich Girl” by Gwen Stefani. Its refrain goes like this:

If I was a rich girl (Na, na, na, na, na, na, na, na, na, na, na, na)
See, I'd have all the money in the world, if I was a wealthy girl
No man could test me, impress me, my cash flow would never ever end
Cause I'd have all the money in the world, if I was a wealthy girl

On two accounts Gwen mixes up her cause and effect, committing the Casual Fallacy of Wrong Direction.

-Just because you are rich does not imply that you have all the money in the world. I can think of lots of rich people that do not fulfill the criteria (like, all of them).

-Just because you have all the money in the world does not imply that quantity is infinite. In fact, the amount of money is a set comprised of a limited number of elements, thus it is actually finite.

Most people would agree that it’s simply a matter of direction and the reverse (if I had all the money in the world, I would be a wealthy girl) is logically valid—assuming "I" is a young female). Casually speaking, I would agree with them.

But let’s be anal retentive economists and see if there’s a way to make it invalid. Money is often quantified into four broad categories: M0, M1, M2 and M3 (economists aren’t very creative). Each one of these definitions can be applied to different currencies. Suppose Gwen is talking about M0—all cash and coin in circulation—of, say, the Soviet ruble. Since the Soviet ruble is no longer in circulation, that total is zero. Having all the money in the world can result in no change to your bank account. If you have no other holdings, then you certainly are not rich even with the inclusion of these rubles.

She also made the Inductive Fallacy of Hasty Generalization when she said “No man could test me, impress me” as a result of her theoretical limitless funds. Perhaps in her experience all the men she’s encounter have been intimidated by wealth but I doubt she’s been tested by a large enough portion of the population to logically make that conclusion (especially since all we need is one case that makes the assertion invalid). However, it’s reasonable to conclude a proportion of men could not test nor impress her if she was wealthy, which I think she is.

To make her song logically valid to our anal retentive selves, it would have to go a little something like this:

Suppose I was a rich girl (Na, na, na, na, na, na, na, na, na, na, na, na)
Now suppose I have all the legal tender in the world (assuming that quantity of legal tender represents a significant proportion of economic activity and the economy in question is widely defined), thus I would be a wealthy girl
A larger portion of men couldn’t test me, nor could they impress me, my cash flow would be larger than possibly most
Because I'd have all the legal tender in the world (assuming that quantity of legal tender represents a significant proportion of economic activity and the economy in question is widely defined), I would be a wealthy girl

Logical errors may make result in poor understanding, but they are certainly useful in music.

Note: I'm aware that "widely defined" and "a significant proportion" are not terms appropriate to logical agruments but the vagueness of the word "wealthy" barred me from constructing a more precise logical statement.

Bonus Question: There is an economic argument that questions the validity of my claim that even after correcting for Wrong Direction and Hasty Generalization, the lyrics still don't logically hold. What's the argument?