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I'm Italian and live in Italy, where books on English are often odd, grammatically speaking....
For instance, in those books you can find a rule about the negative forms of "may" and "can" saying that "cannot" indicates something sure (you cannot swim = you are not able or not allowed to swim at all), while "may not" just indicates something negative as possible (you may not swim = it's not necessary you swim).

Yesterday I found another book saying the opposite, however. "Use "may not"/"must not"/"cannot" for prohibition. Examples: a) Unauthorized personnel may not enter. b) He must not forget. c) You cannot go yet".

Now, I'd understand the sentence a) as meaning that perhaps unauthorized personnel doesn't enter but may enter too....

For prohibition, I'd put: "Unauthorized personnel is not allowed/is not permitted/must not/cannot enter".

Am I wrong?
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Hello Fortiter, welcome to English Forums! Good screen name.

Your example A is a 'polite but forceful' use of 'may'. There is no doubt that unauthorized personnel are not allowed to enter. The use of 'may' is simply to soften the prohibition.

Of your other versions, 'are not allowed' and 'cannot' would sound quite mild. 'not permitted' would be fine. 'must not' is a little stronger. But strangely, 'may' seems most forceful of all. It's rather like a policeman calling you 'sir'.

The distinction between 'can' and 'may' is difficult. As a general rule, it's probably true that 'can' has a more pragmatic, solid aspect, while 'may' has more of an air of 'permission' or 'formality'. But both words have several uses, and each use probably has to be learnt separately. Here are some examples:

1. can = know how to do something (sapere)

'I can't swim.'
'I couldn't swim in those days.'

2. can = be able to do something (potere, essere capace di)

'I can't swim any further.'
'I couldn't swim any further, so I drowned.'

3. can/may = be permitted to do something (permettere)

'You can't swim here – it's not allowed.'
'I couldn't swim there – it wasn't allowed.'

'You may not swim in my swimming pool!' (quite formal or authoritative)
'She said I might not swim in her pool.'

4. can/may: when asking permission (potrei parlarti? etc)

'Can I go swimming?' (direct)
'Could I go swimming?' (less direct)

'May I go swimming?' (slightly more tentative, or formal)
'Might I go swimming?' (very tentative)

5. can/may = have the possibility of doing something (potere in conditional?)

'We can go swimming tonight.' (direct)
'We could go swimming tonight.' (more tentative)

'We may go swimming tonight.' (possibility still quite uncertain)
'We might go swimming tonight' (tentative and uncertain)

6. can: used to express disbelief (non è possibile che...)

'She can't be going swimming at this time of night!'

'Could' is also used to:
7. talk about actions that were possible but didn't happen, in the past (potere in conditional)

'I could have gone swimming, if I'd wanted to.'

8. express possibility (può darsi che)

'He could be in the swimming pool.'

9. discuss recent actions (come hai potuto...?)

'How could you have gone swimming without me?'

'Might' can be used in 7 and 8, but not 9. Again, it seems slightly more tentative.

(There may be other uses of can/may, but these are all I can think of for the moment!)

MrP
Very helpful and clear, Mr. Pedantic! Thanks a lot!

Yet, I'd ask you something more about your kind reply:
1) how could I just express uncertainty (not a prohibition) in such a sentence as: "You may not swim in my swimming pool"? How can I say that - after a certain time, for example - "you can swim in my s.p. if you wish (but you're not obliged to, of course)"? In Italian we have a simple "trick": we always use "potere" (=can/may) but we change the negative word place. So "tu NON puoi (a form of "potere") nuotare (=to swim)" means you absolutely cannot/are not allowed to swim, while "tu puoi NON nuotare" indicates mere possibility (in negative) "you may not swim" "perhaps you don't swim but you could". It means possibility at the same "level" of a sentence like "workers MAY experience fatigue" (but also they may not, depending on their personal strength and/or the type of their job and/or specific day and/or...). So, how can I convey that uncertainty in sentences like:
- You may not swim in my swimming pool
- She said I might not swim in her pool (simply meaning that I needn't - so to say - or I didn't have to swim this time)
without expressing clear inability neither any clearly prohibitive intention, nor formal nor authoritative (referred to your "bocks" 2 and 3)?

I hope I made me understood
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In other words, how can I render the same mood, same meaning same nuance of such a sentence as "he MAY NOT arrive on time" in the following sentence:
- You MAY NOT swim in my swimming pool
- She said I MIGHT NOT swim in her pool?
You can't make a direct translation in this way as in English, we do not use the negative to imply uncertainty. The word 'may' does this for you. 'You may swim in this pool' means they can if they want, certainly no one is going to push them in to force them to. To say that it is compulsory to swim in the pool you would say 'You must swim in this pool'. 'You can swim in this pool' means that it is possible to swim in this pool - alright if it is a verbal permission but it would look ridiculous on a sign as it is rather stating the obvious - what else could you do in it instead! It is okay in reported speech though so, 'She said I could swim in her pool'.
Thanks Nona for your help.

May I tell you that what you wrote - "we do not use the negative to imply uncertainty" - sounds a little odd to me? I thought "He may not arrive on time" did imply uncertainty.... Or what?

I'm waiting for your feedback.

Thanks in advance
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Hello Fortiter

Both forms are ambiguous:

1. 'He may not arrive on time.'
a) It is possible that he will not arrive on time.
b) He is not allowed to arrive on time.

2. 'He may not use the swimming pool.'
a) It is possible that he won't use the swimming pool.
b) He is not allowed to use the swimming pool.

We have to judge which one is correct by context. 1b would be very rare, for instance, because we rarely prohibit someone from arriving on time. But 2a and 2b are both quite possible.

Another aspect of 'can' and 'may' occurs to me. Take the phrase 'to have a word with someone'. As a direct question, in BrE at least, this sometimes gives a clue to the relative status of the speakers:

The two parties know each other quite well:
1. Can I have a word? (I have higher status than you)
2. Could I have a word? (I have lower status than you)

The two parties don't know each other very well:
3. May I have a word? (I have lower status than you, or I have higher status than you )
4. Might I have a word? (I have much lower status than you; or I have much higher status than you)

5. I want a word with you/I'd like a word with you (I have higher status than you and we both know it; or, I am angry and so am assuming higher status out of rudeness)

The ambiguity of #3 and #4 may seem surprising; but I suppose 'assumed deference' is often a form of 'politeness' (or of 'ironic politeness', 'insolent politeness').

MrP
Edit: The two preceding posts weren't there when I started answering, so apologies to both of you for taking the conversation back a step.
1) 'You may not swim in my swimming pool!' (quite formal or authoritative)

2) 'She said I might not swim in her pool.'

>>>>>>>>>>>>>>>>>>>>>>>>>

JT: Are you offering 2) as an example of reported speech for 1), Mr P?
I think that what Fortiter was asking about is: can we use the phrase “you may not swim in this pool” as a permission “not to swim” (if one doesn’t feel like it), i.e., a permission to refrain from swimming, and if we can, then how to distinguish such a permission from a prohibition?
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