In an either or statement in logic, is "either TRUE STATEMENT or NONSENSE STATEMENT" a true or nonsense statement?

Question

I apologise for having no education in logic, so I am clueless on where to start with this issue. I will explain my predicament in full:

I recently loosely adapted the barber's paradox into a logic puzzle of sorts. This is the formulated puzzle:

In a town, everybody must shave their beards. All the people living there either shave their own beard or have the barber shave their beard. There is a person in this town named Ed. Does Ed shave his beard or have his beard shaved by the barber?

The answer which I considered to be correct is the following:

Consider the barber. It is given that everyone in the town either shaves their own beard or has the barber shave their beard. Were the barber to shave his own beard, he would shave his own beard and have the barber shave his beard. This is a contradiction of the previous 'either or' statement, and thus the barber cannot exist. Therefore everyone in the town shaves their own beard. So Ed shaves his own beard.

When I posed this problem to a friend, they responded by saying that the statement "All the people living there either ... or have the barber shave their beard" implies the existence of one barber. If no barber exists, the statement is nonsense and no implications can be made from it.

So, my question is essentially: In a town with no barber, is it true to say that "everyone either shaves their own beard or has the barber shave their beard" or is this just nonsense, and the puzzle solution thus invalid? Furthermore, could this puzzle be improved by writing "any barber" as opposed to "the barber", as this wording does not necessitate the existance of a barber?

Answer 1

If there is no barber, the statement "everyone either shaves their own beard or has the barber shave their beard" is equivalent to the statement "everyone shaves their own beard". It is true unless somebody has a non-barber (like their friend) shave their beard or somebody doesn't shave their beard at all. It is certainly not nonsense.

When I have seen the problem it was specified that there was exactly one barber in the town. It is important to get the details right on problems like this. Even with that added on, we can't know about Ed. It is important to have the puzzles ask about the barber, not about a different person. Then the statement should be that the barber shaves all those who do not shave themselves. Now you have a contradiction, but only with these changes.