1. Concerning the Mohists and their competition:

The critic says that the Mohists agree with "normal people" on the propositions in Section II. All agree that the superset is more relevant than the subset. They disagree with normal people on the propositions in Section III because they want to affirm that the subset cannot be more relevant than the superset. In the propositions in Section IV, everybody else originally maintains that in these cases a subset is indeed a subset, i.e., that we have a situation like this:
{cats (Fluffy)}
rather than like this
(pets {Rover) working dogs}
where we are really considering two intersecting sets. But these other people are not consistent. For when we talk about life spans and fate, we find {total life cycle (death)} being believed by the Mohists, whereas "everybody else" believe this picture:
{normal life cycle ( ** } deaths).
Most deaths fit in the joint region ( ** } but some are found outside the normal life cycle, according to "the world."

2. On Logic in Early China:

I think that upon examination of this chapter it is very clear that the author(s) were well aware of the need for clarity in talking about sets and subsets, and were clearly aware that the Mohists and other thinkers could come to opposite conclusions and that they could not both be right. So it seems to me that there was a substantial basis for logic here some three hundred years B.C. If the Chinese language did not make it impossible to begin this enterprise, and if, indeed, the enterprise was taken forward a considerable distance, then it must be that there were other, non-linguistic, forces that inhibited the development and use of logic in China.

3. Regarding the "是而然 shì ér rán) characterizations: (I replace S and H, B and H, and A and F from the above formulations with A for antecedent and C for consequent in order to make the comparisons of these cases easier.)
1. shì ér rán  A → C, paradigmatic: A = 1, C = 1; counter: A = 1, C = 0
2. shì ér bù rán ¬(A → C), paradigmatic: A = 1, C = 0; counter: A = 1, C = 1
3. bú shì ér rán  ¬A → ¬C, paradigmatic, A = 0, C = 0; counter: A = 0, C = 1

The three cases indicate that shi` and bú shì  may refer to the truth status of the antecedent and consequent, while rán and bù rán refer to whether the conditional is affirmed or denied.

If we try to state things in terms of sets and subsets, then:

1. shì ér rán: both superset and subset characteristics apply; superset is essential, subset is accidental. If we lost members of the superset we would not care that we might incidentally lose that member as the member of a particular subset. Everyone affirms (say "rán") applicability of the predicate involved in the proposition to all members of the superset. (Subset membership is accidental.)

2. shì ér bù rán: superset membership (e.g., being human) is irrelevant to non-Mohists to what is being predicated of the subset member (e.g., being a bandit); i.e., superset is not relevant, subset is essential. Non-mohists deny (say "bù rán") to the assertion of statements such as, "If Mary is a robber, then Mary is a human being." 

3. bú shì ér rán: Cases where, e.g., somebody approaches the door and enters the door, are subsets of the basic situation wherein, e.g., somebody approaches the door. It may be true that somebody approaches the door but does not enter the door.  It is never the case that someone enters the door immediately and yet from some distance away.  The important connection that the Mohists want people to see is that it is a correct interpretation of the way the world works to maintain that if one does not do the anticipatory things, the beginning of the process, e.g., move all the way to the door, then the culmination of the process will never occur, e.g., one will never enter the door. Moving all the way to the door does not mean that one will necessarily go through the door. There is nothing remarkable about this statement. Not moving toward the door and somehow still entering the door is an impossibility. Denial (say "bú shì") of superset membership and of subset membership analytically guarantees that the causal sequence will be true. For instance, saying that it is not true that somebody approached too close to the well, and also saying that they did not fall into the well will always, in this view, be true. Similarly, saying that if you do not have any laying hens then you will not have any eggs laid in your henhouse is always true. 

Thus, oddly, the formal characterization of these kinds of sentences by translating them into implications seems more appropriate to the "shì ér rán" kind of terminology than is describing them in terms of sets -- even though the main body of the discussion appears to be about sets. It almost seems that the issue of implication is a subtext of the discussion that did not get carried over into the actual writing of this chapter.

4. The fifth section of this chapter of the Mo Zi has to do with the distinction between "some" and "all." It also has to do with the strong requirement for scientific proof of something that asserts that it is possible to gain more and more confirmation of some hypothesis, but that the next case examined may disprove that hypothesis. One case examined in this chapter involve the question of whether someone truly "loves people," meaning "loves all people without exception."  It is impossible to make this assertion confidently until the candidate has been tested against all human beings. Until that point, one might always find at least one person that the candidate could not live. The requirement suggested for riding horses is much weaker. The authors of this chapter of the Mo Zi are willing to credit someone with being able to ride horses if that individual can ride some one horse.  They are, to the contrary, unwilling to assert that the person cannot ride horses until the candidate has failed with all the horses in the world. The text says, "These are situations in which something either is or is not predicated universally." 

The authors clearly are aware that factors such as social context affect the meaning of utterances. So this section seems to me to have more to do with how people use language than with set content or logical connectives per se. The deeper implications, however, is that if people wanted to then they could use terms such as "all" and "not all" to clarify their meaning.