But 'Strong as John is, Bill can lift it easily' is definitely wrong because both clauses must talk about one and the same person, right? Without 'being as', the first clause requires the second one to talk about John too.
Hello again, Alexey:
Let me give this another shot. As you can see, I have deleted my previous answer to your follow-up question. After massaging this topic in my mind for the better part of the afternoon, I realized that it was inaccurate of me to represent "As John is strong" as a viable paraphrase of the circumstantial interpretation of "Strong as John is." I also disagree with my affirmative answer to your question about whether both clauses must share the same subject. Consider the following sentences:
(1) Wet as the floor is, people could very easily slip on it.
(2) Wet as the floor is, there is little risk of people slipping on it.
In neither of those sentences does the main clause have the same subject as the "as"-clause. In the "as"-clauses, the subject is "the floor"; in the main clauses, the subjects are "people" and "there," respectively. (The logical subject of [2] is "little risk of people slipping on it," which is also not the same as "the floor.") This shows that the subject need not be the same, on either interpretation of the "as"-clause. Also, note that if you preceded (1) with "Being (as)," the sentence would imply that
people are wet!
Here is how I would now paraphrase the two sentences. Each has an "as"-clause that receives a different interpretation from that of the other.
(1') With the floor being as wet as it is, people could very easily slip on it. (circumstantial interpretation)
(2') As wet as the floor is, there is little risk of people slipping on it. (concessive/contrastive interpretation)
The second paraphrase (the paraphrase of the concessive/contrastive type) is borrowed from Quirk
et al., who point out that clauses like "Wet as the floor is" or "Strong as John is" can be regarded as the "as . . . as . . ." construction with the first "as" omitted. They do not give a paraphrase of the circumstantial interpretation. Earlier, I conveniently supposed that it could be like an "as"-clause of reason ("As John is strong" / "As the floor is wet"), but now I see that that paraphrase doesn't work at all.
The circumstantial reading of such "as"-clauses, which I believe is properly matched in meaning by the "With [subject] being as [adjective] as [subject] is" construction, "as" indicates the extent or degree to which the subject has the property denoted by the adjective. In the sentence "Strong as John is, he can lift it easily," "as" can be replaced by "to the extent that" -- on the circumstantial reading of the "as"-clause. The sentence says that, because John is strong to the extent that he is, he can lift it easily.