I'm trying to understand the difference between the following two sentences:
We must stay at home.
We have to stay at home.
Is the meaning of these two sentences completely equivalent? In what cases are #1 and #2 used?
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must is a modal verb never followed by to. That says it all. Other such verbs include can, could, may, might etc.
On the other hand, have can have to following it.
Kindly note that you may come across ...must to.. in some sentence but then there, must must be serving as a noun.
Passport is an important document. Yes, it is a must to bring (document)
He is a must to succeed. = He'll certainly succeed.
Note that I had to add an article to make must a noun.
Must as a noun means an essential or necessary thing.
"seat belts are an absolute must"
Your two sentences are semantically identical. At one time have to was regarded as strictly informal, but it is now generally acceptable in all but the stuffiest registers.
The most important difference between must and have to is that must is a modal verb and therefore limited to the simple present and past constructions‡, while have to is formally a lexical verb and can be employed in the entire range of active verb constructions: progressive, perfect and modal, and their combinations.
If she hadn't had the surgery she might still have been having to wear a neck brace at her wedding.
Have to and parallel phrases like be able to, be supposed to and be going to have other distinct syntactic properties. As snailplane points out, these phrases do not act as auxiliary verbs, as modals do: they require DO-support in questions and negations.
Must you go? ... You must not go.
Do you have to go? ... You do not have to go.
On the other hand, these phrases do act like modals in other ways. Like modals, but unlike catenative verbs like want or intend, they strongly resist separation from their dependent infinitives. In constructions which call for such separation, these phrases call for a different sort of DO-support, just as modals do:
ok What I want is to go to London.
∗ What I must is go to London. ... ok What I must do is go to London.
∗ What I have to is go to London. ... ok What I have to do is go to London.
They also strongly resist separation from the to component:
∗ What I have is to go to London.
This resistance is so strong that the to is phonologically integrated with the preceding word, effecting a distinct change in pronunciation. Many linguists prefer to use the spellings hafta, hasta, gonna, sposta in discussing these phrases.
For all these reasons, these phrases are usually not classified as modals but as semi-modals or periphrastic modals.
‡And in fact the simple past use is so rare today that many unsophisticated native speakers and even English teachers will tell you that must cannot be used in the past.
∗ marks a usage as unacceptable
Your two sentences are similar but not identical.There is a little difference between 'must' and 'have to'.The sentence with 'must' means the speaker feels the obligation whereas the sentence with 'have to' means someone else is the reason behind obligation.
e.g.If you want to buy a gift for your mother's birthday.You can say:
I must buy a gift for my mother's birthday .
here you are feeling the obligation.
And,if your wife is forcing you to buy a gift for her mother's birthday.You can say:
I have to buy a gift for my mother-in-law's birthday .