Sorry Frank! I readily admit it was a hasty conclusion and I just misunderstood you. I really don’t want to dig into it.
Okay, no bother.
You say (5) is wrong because “When you say a second time, there is an omitted preposition for, so that a second time = for a second time, in the adverbial sense of 'again'."
Yes, you've understood me. What I'm saying here is just one way of explaining things. There are other ways to do it too.
I don’t understand why the same logic doesn’t work in the case of this is the second time? Isn’t for omitted there either?
No, it isn't. That's the difference. The phrase
the second time is not adverbial there. It's just a noun phrase. It isn't modifying
reading Hamlet. It's a subject complement.
Let's simplify the sentence a bit.
1)
This is the second time.
This sentence is equative. It's like saying
X is Y, where X and Y are identical things. There's no modification, only equation.
2)
I succeeded the second time.
Unlike sentence 1, the exact same phrase here is used adverbially, modifying
succeeded. The phrase looks the same but the grammar is very different. There's no omitted preposition
for that can help understand the adverbial use of the phrase, but you could understand there to be an omitted
on in some way (although it wouldn't be natural), because the meaning of the phrase is something like 'on my second attempt' or 'on the second occasion'.
3)
I'm reading Hamlet the second time.
Here, the same phrase is used adverbially again, in modification of
reading Hamlet, but the meaning is different again, and this time, you
can understand an omitted
for, because the phrase basically means 'again'. The sentence places this particular reading in sequence of possible readings: This one comes between the first and third reading.