Learn English – use of ‘let’ in passive voice

The passive voice exists for a reason; as far as I can tell all languages have some way of creating a construction equivalent in semantics (and pragmatics) to the English passive.

At the semantic level, the passive voice operates on transitive verbs (=2 or more arguments; note that the subject is included as an argument) and "promotes" the transitive object to a subject. The meaning is equivalent semantically (barring idiomatic usages), as the subject of a passive construction is still the object of that transitive verb. Consider these two sentences below, where the event (=what happens to the object) is the same, in either active or passive voice.

1) John ate [the cake]

2) [The cake] was eaten (by John).

In either 1) or 2), this information is the same: there was a salient cake and it was eaten. However, note that the passive allows one to drop out the agent.

So, why would one use the passive voice over the active? Perhaps the writer/speaker wants to give attention to the object. In English the subject position is often where topics go, and so is considered the "important" element. Promoting the object to a subject via passive allows one to emphasize this. Another possible case: we simply don't know who the agent was or perhaps the agent doesn't really matter. The passive is also helpful for generalizing / avoiding assigning blame / being polite.

Consider these cases:

3) The trash can was blown over (by the wind).

4) John was really screwed.

5) That project got really messed up. (<= note how it doesn't blame anyone in particular)

In 3), it probably obvious what blew the trash can over (in general people don't go around blowing down trash cans, etc.). For 4), this begins to get idiomatic--we could try to attribute reason(s) why John is not in a good situation, but usually this is secondary to stressing that John is really screwed.

We can apply this reasoning to the example sentence you gave:

6) [Tonight's moon] can be seen from anywhere worldwide.

In this sentence, the fact that the moon is visible from anywhere in the world is being stressed, hence why the writer probably chose to use the passive. We really don't care who is seeing the moon.

You should use the passive when its the idiomatic way to talk about an event (e.g. the agent usually is implied or not important) or when you want to focus on the object of such an event, and therefore promote it to subject position via the passive construction.

The book is giving you some screwy examples. "Let his game be played by him" is correct but weird. You would only say something like that to make very unusual emphasis.

Here are a couple things to know.

How to make the subordinate clause

To make the object of let into the subject of its own clause, you need to put it into the objective case and put the verb into the infinitive. That's why you say:

Let him play his game.

rather than:

Let he plays his game.

Similarly, you would say Let him be helped by us, not Let he is helped by us. However, this sentence is equally as weird as Let his game be played by him.

Two (or three) senses of let

I think what the book is trying to do is teach two different senses of let at the same time that it's teaching you a tricky form of passive voice. Two of the main senses of the word let mean: (1) allow/permit the clause to happen; (2) suggesting or agreeing that "we" do the clause.

A classic example of the allow/permit sense: Let me go! is what a person who is being held against their will says to their captor.

A classic example of suggesting that "we" do something: Let's go! or Let's get started! is what you say when you want to start doing what you and your listener were just talking about doing together. This sense nearly always has us contracted to 's. (It has to be us rather than we because the subject of the clause has to be in the objective case, as above.)

Those are the most common and simplest examples to remember in order to learn the sound of the language. However, those examples can't be converted to a passive form.

Here's a more-realistic example in both active and passive form:

Let Dr. Kildare see the patient.

Let the patient be seen by Dr. Kildare.

Possibly your book has confused the suggesting/agreeing sense with a third sense of let, expressing a wish. A classic example of using let to express a wish is: Let peace prevail on Earth. Here's a realistic version of what I think your book is trying to demonstrate:

Let us beat our swords into plowshares.

Let all our swords be beaten into plowshares.

or, passively again, without using let:

May all swords be beaten into plowshares.

I can see why your book might have confused these senses. They really are a big, muddy mess. They are all variations on the basic sense of allow/permit, stretched to mean different things by repeated usage. The wishing sense can often be understood as the allowing sense and the suggesting sense simultaneously, where the request/suggestion is addressed to a deity, like Oh, God, please let there be peace on Earth.