Does American English allow the use of "must not" instead of "can't" to say that something is believed to be logically impossible?

Please consider the following examples:

It must not be true!

How can you say such a thing? You must not be serious! (-you gotta be kidding!)

A: Joe wants something to eat. B: But he just had lunch. He must not be hungry already. (-it's impossible that he is hungry because he just had lunch)

I just bought a box of cereal yesterday. It must not be empty already.

You just started filling out your tax forms 10 minutes ago. You must not be finished with them already?

If A is bigger than B, and B is bigger than C, then C must not be bigger than A.

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    I can't say the usage is wrong, per se ... but I would not have phrased any of those statements that way, and I would notice if someone else did. – Dan Bron Sep 23 '15 at 17:10
  • Elian... welcome back!!! :)) – user66974 Sep 23 '15 at 17:18
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    "Cannot be true", taken literally, means that "it" (the item in question) is provably false. "Must not be true", on the other hand, implies that either the speaker does not want "it" to be true, or that there is some prohibition (perhaps in the instructions for a form) of "it" being true. But these terms are often used "loosely". – Hot Licks Sep 23 '15 at 17:18
  • (And the sense of the terms changes somewhat when you slip into subjunctive mood.) – Hot Licks Sep 23 '15 at 17:23
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    I would add that in maths, the final "If A is bigger than B, and B is bigger than C, then C must not be bigger than A." is completely correct usage. In this limited context, for this one sentence, "cannot" and "must not" are interchangeable. – Patrick Stevens Sep 23 '15 at 21:33

What you're asking about is called the Epistemic sense of a modal.

Every modal auxiliary verb that's still used in English has at least two senses:

  1. The Epistemic sense, which has to do with logic and perceived probability
    This may/might be the place ~ This must/should be the place ~ This could be the place.

  2. The Deontic sense, which has to do with permission and obligation
    You may/might kiss his ring now ~ You must/should kiss his ring now.

In addition, there is an Alethic sense, which has to do with personal ability, and shows up only with can. E.g, He can bench-press 150 kg means 'He is able to bench-press 150 kg', which obviously entails that it is logically possible but attributes this possibility to the subject rather than to logic.

That's general. Specifically, modals are irregular as hell, and must and can are more so than usual. In addition, all modals interact with negatives, but not in the same ways. Again, they're different.

In American English (UK Englishes vary), epistemic can occurs only in a negative environment.
This can't be the place ~ This can't be true, but not *This can be the place ~ *This can be true.
I.e, epistemic can is a Negative Polarity Item (NPI). In American English, at least.

Can, outside negative environments, usually takes an Alethic sense instead of an Epistemic one,
but it's equally common as equivalent to Epistemic may 'permitted to'.
You can/may leave 'You are permitted to leave'.

Epistemic must is not an NPI, but it has a specific meaning with not.
You must not go to the ball means 'You are obliged not to go to the ball'.
It does not mean 'You are not obliged to go to the ball'.
Logically, that's Not (Possible (Go (You, Ball))).

On the other hand, the periphrastic modal have to, which means must in the affirmative,
(You have to leave now = You must leave now 'You are obliged to leave now'),
means 'You are not obliged to leave now' in the negative,
and not 'You are obliged not to leave now'.
Logically, that's Possible (Not (Go (You, Ball))).

So, to answer the original question, Yes. I.e, since they're both negative, can't and must not both negate possibility (i.e, Not (Possible (P)) instead of Possible (Not (P))) in their epistemic senses, so they're equivalent.

Of course, the differences among logical possibility, personal ability, and official permissions can be very complex and hard to distinguish, so there are -- Surprise! -- exceptions. Just like everything else about modals, negatives, and quantifiers (these are logically Operators, which have logical superpowers of binding foci).

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The interchangeable use of "must not" for "can't" in colloquial American English seems most natural (and appropriate) to me when, for example, both "[the person] must not" and "[the person] can't" stand for a longer idea amounting to "it must be the case that [the person] does [or did] not." For example, a person seeing that someone has smashed a glass pane in the front door of a house—presumably in order to unlock the door from the inside—might say

The burglar must not have noticed that the key was in the keyhole of the door.

or equivalently

The burglar can't have noticed that the key was in the keyhole of the door.

because both wordings convey the logical reasoning:

It must be the case that the burglar did not notice that the key was in the outside keyhole.

In other situations, however, must not and can't carry different implications. For example,

It can't be true!

when viewed in isolation, may be read as a distressed denial of something that may very well be true but is most unwelcome (although under other circumstances it may not). In contrast,

It must not be true!

carries a triumphant note of certainty appropriate to stating a logical conclusion. Though both expressions are applicable to an array of situations, their implications differ in many settings.

A further complication involves wordings that function as set phrases. For example, the expression

You can't be serious!

is common as a dismissive exclamation meaning "I don't believe it." But

You must not be serious!

is not widely used in that sense, so it is more likely to be understood as the conclusion reached at the end of a logical appraisal of something said than as an out-of-hand rejection of the thing said.

And finally, some expressions show very little overlap in meaning between the "must not" form and the "can't" form. For example,

Such freedom of choice can't be permitted.

strongly implies that the freedom of choice in question poses an existential threat to whatever authority opposes it. To the contrary,

Such freedom of choice must not be permitted.

expresses categorical opposition, but merely as a matter of policy, to permitting a particular freedom of choice that may already exist.

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