Judge says "on a weekday," and that it will be a surprise to the person being executed. But the judge has not specified which weekdays are actually in play.
Now, let's attempt to specify it:
"Judge will schedule the execution for one of the following days in the upcoming week: Monday, Tuesday, Wednesday, Thursday, or Friday. Judge will pick one of these days at random. This random process will determine the day of the execution, such that the person being executed will not be able to use a process of elimination to predict the day of the execution."
Now we can see that there is no paradox, only a self-contradicting specification.
So your real point is the tension between random process and process such that the person being executed will not be able to use a process of elimination to predict the day of the execution?
If we don't care about randomness, can't the judge just use the strategy of Always pick Wednesday?
If we do care about randomness, the job is to explore whether we're guaranteed to surprise the convict, and of course we're not: if the random process selects Friday, the convict won't be surprised.
> If we don't care about randomness, can't the judge just use the strategy of Always pick Wednesday?
The point is that because the process is underspecified the prisoner is using one interpretation while the judge is using another. (Otherwise the judge would have had to conclude that no choice could be made.) The reader switches between the two specs in an apparent paradox without considering that they are two different specs.
For example, the judge could simply be breaking the rule by considering Friday as a possible choice. That is a different spec from the prisoner, and if such a judge ended up settling on Wednesday then the prisoner would be surprised.
But here's another angle. If the prisoner logically concluded that the judge cannot pick any weekday, the judge can pick Friday and still satisfy the requirement that the prisoner be surprised.
I think that's cheating, though, and the important part is the underspecification.
The surprise is a surprise in relation to the week. If we use the relation of truth value of surprise to week day, as the prisoner uses to reason, there's a chance it won't be a surprise. That's because the prisoner is treating the statement value as though the judge stated each day, that the hanging will be a surprise. I agree it's not a paradox.
Judge says "on a weekday," and that it will be a surprise to the person being executed. But the judge has not specified which weekdays are actually in play.
Now, let's attempt to specify it:
"Judge will schedule the execution for one of the following days in the upcoming week: Monday, Tuesday, Wednesday, Thursday, or Friday. Judge will pick one of these days at random. This random process will determine the day of the execution, such that the person being executed will not be able to use a process of elimination to predict the day of the execution."
Now we can see that there is no paradox, only a self-contradicting specification.