OK. You can design a language that allows you to specify everything that you can express with a natural language.
The problem you get is that no computer can execute that thing anymore. Where I see a big problem are quantities. If you get from the classic logic to the predicate logic, we have already problems. We have problems with the all and exists quantifier.. and it
is getting better, if you extend that thing even more...
It is that we could express everything, but it's not executable or interpretable for a computer anymore.
I find programming languages, just to simple to have them classified with the same name as spoken languages. The idea to talk of programming formal languages, sound just better to me.