In Constructions of the type КАК БЫ НЕ..., what exactly does the "не" add to the phrase. I'm looking for a literal interpretation from those of you that know both English and Russian at a very high level.
For example—
Я боюсь, как бы он не забыл зайти в аптеку.
Правительство боится, как бы этот закон не привел к увлечению бюджетных мест.
What is a literal translation of these sentences, and why do we need "не" at all?
I think the sentence in question is a clearly pessimistic transformation of this optimistic one (where не directly negates the verb):
Надеюсь, он не забудет зайти в аптеку. (I hope he won't forget to visit the drugstore.)
The original version expresses uncertainty based on some possibility of things going wrong:
Боюсь, как бы он не забыл зайти в аптеку.
I'm afraid he might forget to visit the drugstore.
I'm afraid that maybe he'll have forgotten to visit the drugstore.
Typical English translation doesn't use direct negation, present in the Russian expression. But what is the negation не aimed at? Despite its position right before the verb, it does not directly negate its meaning but refers to unwanted (or unexpected - in similar cases of doubtful speculation) possibility of its action. It seems possible to give a close-in-sense isolated expression where не negates the action of an additional verb, while the existing verb stays in the affirmative (considering не originally didn't directly negate it):
как бы... не... ~ (мне) не хотелось бы, чтобы...
It works here independently of the emphasizing addition Я боюсь (I'm afraid):
(Я боюсь вот чего:) не хотелось бы, чтобы он забыл зайти в аптеку.
Correspondingly, an English version of the above can be given with a similar direct negation:
(I'm afraid,) I would not want him to somehow forget to visit the drugstore.
The presence of the particle бы suggests it is the conditional mood. This mood has several meanings, and since here there is no actual condition (like if or when), it has desiderative meaning, that is it expresses our desire, what we wish. Keeping this in mind, it becomes much more clear why не is used in such constructions: the word wish is implied but not expressed openly, other verbs are used instead.
So in Я боюсь, как бы он не забыл зайти в аптеку, the verb боюсь does not mean actual fear or awe, but it is the hope that he will not forget to visit the drug store. That is, the implied meaning of this sentence is actually Я надеюсь, что он не забудет зайти в аптеку.
In other words, in such sentences the second part expresses the desirable state of things in which the unwanted situation does not occur.
"Как бы не" is a traditional construct that is often utilized to convey negative meaning.
Let's look into the details of your example. "Правительство боится, как бы этот закон не привел к увлечению бюджетных мест." - "The government is afraid that this law can lead to the increase of subsidized enrollment(?)"
Now let's see if we can drop the "не":
"Правительство боится, как бы этот закон привел к увлечению бюджетных мест." - this phrase is clumsy and its meaning is unclear. Why? Because the word "боится" implies negative meaning - but the particle "не" is missing, so we are not sure what exactly the government is afraid of.
What if we change the negative verb into a neutral one?
"Правительство думает, как бы этот закон не привел к увлечению бюджетных мест." - "The government is thinking that this law can lead to the undesired increase of subsidized enrollment". This new phrase is pretty close in meaning to the original one.
"Правительство думает, как бы этот закон привел к увлечению бюджетных мест." - "The government is thinking how to make this law to increase the subsidized enrollment".
You see, now dropping the "не" completely changed the meaning!
And more examples:
"Как бы так" - "That's the way it is"
"Как бы не так" - "No way!"
