Must not versus cannot

Question

It is stated in my grammar textbook that in American English people use “must not” when they want to say that something is improbable or it is improbable that somebody has done something:

American English:

It must not be broken, because I checked it before delivery.

British English:

It cannot be broken, because I checked it before delivery.

Later in the textbook, it is stated that we use “must not” to order someone that it is necessary that they do not do something:

This machine is very dangerous, you must not even touch it.

Now, my question is: can’t we use “can not” instead of “must not” for the first case? Because I think it’s clearer and it is even more logical.

Answer 1

The first examples are what is called the epistemic use of the modals "can" and "must": they're saying things about the speaker's knowledge and expectations, not about possibilities or powers in the real world. They both say "I am sure that" or "I conclude that" it isn't broken. Certainly can't is more likely there in BrE: I don't know about AmE.

In BrE, epistemic "mustn't" is more commonly used to mean something like "I conclude that", often with a hint of surprise, eg he mustn't have found the box means something like "I thought he would find the box, but judging from what happened afterwards, he didn't find it". It musn't be broken is certainly possible, but has a hint of surprise which is not there in It can't be broken.

The other use is deontic: it's not about the speaker's knowledge, but about potentialities and obligations in the real world. In that context, "must" and "can" have very different meanings: you must not even touch it means "you are not permitted to touch it", or "you risk some awful consequence if you touch it", whereas "you cannot even touch it (note that "cannot" is one word, not two) means "it is not possible for you to touch it". This can sometimes be intepreted as social or institutional possibility, and so means the same as "must not", but it could also refer to some other kind or possibility, such as "it is not physically possible for you to touch it".

The relationship between epistemic can and must has little in common with the relationship between deontic can and must.

Answer 2

Can refers to ability. If you can do something, you're able to do it--it doesn't mean you will do it, or have to do it, but you are able to do it.

Must means not having a choice to do something. If you must do X, you don't have a choice not do X.

Keep in mind modals in English are often "misused" for politeness and authority-deference purposes. For example, instead of directly asking Help me with this you might say Can you help me with this to sound more polite.