A: It was half past five. Dad would have finished work.
Does that sentence mean there was an expectation that Dad was going to have finished work by half past five?
I found this sentence under the use of future in the past on some websites.
You're right about the expectation.
All modals in English have two sets of meanings: the more familiar set is the deontic meaning, which is about the objective world, and about obligations and necessities. Deontic meanings of would are generally about futurity, or intention (including when it serves as the past of will).
Epistemic meanings are about our knowledge or assumptions. Epistemic meanings of would and will are about our expectations. So
Dad will have finished work.
in most contexts means something like "I expect or conclude that Dad has finished work"; and
Dad would have finished work.
is the backshifted version of that.
These uses of will and would are absolutely not "future" (or "future in the past"). But I'm afraid that people have been referring to will a "future tense" for a long time now.
Yes, sentence A means that the speaker expects dad to have finished work by 5:30. Since it is 5:30 at the time, I would not call this future in the past, but the expectation is clear.
Unlike Jeff Morrow, I don't see the OP's sentence as being a type of elliptical conditional. Despite the appearance of would, it is not a conditional sentence because there is no "if-clause" (protasi) present in the sentence and none is implied.
There is no uncertainty or probability when the speaker mentions the time. It was half past five, so that meant their father was not at work.
Compare that sentence with this one
Dad would have finished work (but he hadn't) if it had been 5.30 (but it wasn't).
and
If it had been 5.30 (but it wasn't), Dad would not have been working (but he was still working).
It was the speaker's assumption (a knowledge based on verified experience) that their father could not have been working at 5.30 PM because he always finished working before that hour.
You have the idea. I am not sure I would explain it as a past future. I might prefer to explain it as a past conditional with ellipsis.
It was half past five. [If things were progressing normally,] Dad would have left work.
It was half past five. [Unless something quite unusual had happened,] Dad should have left work.
It is absolutely true that “would” and “should” are past forms of “will” and “shall” (just as “could” and “might” are past forms of “can” and “may”), but modal verbs such as those have special rules. The main use of those modals in modern English is expressing conditionals that are often implied rather than explicit.
It means the dad died at work or otherwise did not go to work. Would have means the condition never happened. (coulda, woulda, shoulda). So he either did not go to work or was not present at work at 5, or he died at work and was unable to meet the condition of finishing work.