Nuances between Modals

You seemed not accept CAN / COULD as an expression of possibility but what do you think of the following ? Are my statements and examples true ?

[Actually, "could" is quite usable as "It is possible that ...": William could be waiting for us right now. (It is possible that William is waiting for us right now.) "can" is not used in this context, however.]

CAN = POSSIBILITY

1) CAN in the PASSIVE expresses possibility:

a) Arsenal can beat (is able to beat) Nottingham Forest.
b) Nottingham Forest can be beaten by Arsenal. (it’s possible for Arsenal to beat N.F.) [Yes.]

2) CAN: expresses PHYSICAL, FACTUAL possibility. ("en fait")

a) If it is raining tomorrow, the sports CAN take place indoors. [Yes.]

b) Jane CAN / COULD / MAY / MIGHT BE at home. [No. All but "can" OK here. The version with "can" is highly unusual; however, "Jane can stay at home" is normal. It means "Jane has the ability to stay at home" or "Jane is permitted to stay at home" or "Jane has the option of staying at home (if she wishes)".]

but if I say:

c) "Jane CAN / COULD WORK at home." (= ABILITY?) would it be different from :
d) "Jane CAN /COULD BE WORKING at home." (= POSSIBILITY ?) [Yes, these are different. In the sense of possibility, "can" is not used. It's "may", "might", or "could".]
e) "Jane MAY / MIGHT WORK at home". [With "may" it means "It is possible that Jane works at home" or "Jane is permitted to work at home". With "might" it means "It is possible that Jane works at home".]

3) CAN + Bare infinitive: is used to express:

A/ a GENERAL POSSIBILITY. It means it is possible (? may) [Not sure what you mean here, since "may" = "it is possible that ..." in one reading of "may". The difference, in my opinion, is that "can" is 'subject-oriented'. Whereas "may" means "it is possible that" in one reading, "can" means "it is possible for 'the subject' to ... Also, "may" in this reading focuses on the logical properties of the proposition; "can" focuses on matters of fact in the real world, so to speak.]
a) You CAN ski on the hills. (there is enough snow) [Yes. Conditions in the real world make it possible.]
b) CAN you get to the top of the mountains in one day / by cable car ?
(= it’s possible for you to… ) [Yes.]

B/ an OCCASIONAL POSSIBILITY in the affirmative form.
a) He CAN be silly at times. [Yes.]
b) Even doctors CAN MAKE mistakes (it’s possible sometimes) [Yes.]
c) Measles / Driving at night CAN BE quite dangerous. [Yes.]

4) COULD + Bare infinitive: is used to express a DOUBTFUL possibility, a guess (?) in the PRESENT time.

ex: What kind of tree is this? It COULD BE a maple tree. [Yes. It is possible that it is a maple tree. / It would be possible that it is a maple tree.]
(I guess it is possible for this to be a maple tree)

5) COULD + Perfect infinitive: expresses

A/ an UNREALIZED PAST opportunity.

ex: She COULD HAVE finished today, but her typewriter broke down.
(It was quite possible for her to finish if her typewriter hadn’t broken down = she didn’t finish typing) [Exactly.]

B/ a past possibility where we don’t know whether the action was performed or not. It is similar to MAY and MIGHT.

a) The money has disappeared! Who COULD HAVE taken it ? [Yes. Who had an opportunity to take it?]
b) Tom COULD HAVE (taken it); he was here alone yesterday. [Yes. Tom had an opportunity to take it.]

c) He COULD HAVE have sent a message =
- He did not send it. / - We don’t know if he sent it. [Yes. Two different interpretations of the sentence. 1. He had an opportunity to send a message; we know he did not follow through on that opportunity; we know he did not send it. 2. It is possible that he sent a message; we don't know.]

6) CAN’T + Bare Infinitive: denotes an impossibility in the present or a near certainty.

a) He CAN’T do that to me. (negative deduction based on present evidence) [Besides the literal meaning, there is the idiomatic reading "I won't let him do that to me", "I won't tolerate him doing that to me", "Just let him try to do that and I'll get revenge".
b) He CAN’T be at home. [It is not possible that he is at home. Also expressed (less often, perhaps) as "He must not be at home" ("One can only conclude that he is not at home.")]

7) COULDN’T + Bare Inifinitive: is more tentative, less definite than CAN’T.

ex: He COULDN’T be at home. (the impossibility is less certain) [Perhaps less certain, yes.]

8) CANNOT / COULD NOT + Perfect infinitive: expresses an impossibility in :

a) He CAN’T HAVE done that. (negative deduction based on past evidence) [Yes. It is not possible that he did that / that he has done that.]
b) That crime CAN’T / COULDN'T HAVE BEEN committed by my brother; he was abroad with his wife at the time. (It’s impossible that he could have) [Yes. Same idea as in 8a).]

CJ

1) Do May / Might NOT (like CANNOT) express an interdiction = must not ?

ex: ?
["may not" and "cannot" and "must not" can be used to forbid, but not "might not" and "could not".

You [may not / cannot / must not / *might not / *could not] begin the exam before you are given the signal.

"could not", and less convincingly, "might not", can be used to report an act of forbidding in a subordinate clause. I would avoid "might not" in this context. It is too ambiguous in modern English, but you might find it in older literature with this meaning.

We were told that we [could not / ?might not] begin the exam before we were given the signal.]

2) Mrs James is in hospital and HASN’T BEEN ALLOWED TO have (and NOT “could have”) a cigarette after her meals. (correct ?) [Yes. Or "is not allowed to have a ...]

3) What's the difference between possibility and probability ?

a) probability = near-certainty, action likely to happen;
b) possibility = doubtful situation, action less likely to happen ?
[If an action is possible, it just means that nothing is stopping it from happening, nothing physical and nothing logical. It says nothing about how likely (probable) it is that it will happen. It may happen once every 5000 years or it may happen once an hour. It may be an event that is very likely to happen or very unlikely to happen. The point is that it has some liklihood (big or small) of happening. It's possible.
If an action is probable, in my opinion it has better than a 50-50 chance of happening. We estimate that it is more likely than not to occur.]

4) In the following examples is the verb in the –ing form considered a present continuous tense or a gerund? [I don't know. I suspect that it will depend on which reference you consult. I would consider it a present continuous tense.]

a) “Where is John?”
“I don't know. He may BE READING in his room.
He may BE WAITING at the station.”

b) Where is John? I don't know. He may BE COMING tonight.

4) What does ‘COULD + Bare infinitive’ express ? Is it the same as “would be able to” (= present conditional) ? Should an if-clause be always be implied ?

ex: a) I’m sure you COULD GET into university. = unrealized past action ? [No.] i.e. you were capable of / it was possible for you to enter university but you didn’t ?? [No.] [It means, "I'm sure that you would be able to get into ... (if you tried / if you wanted to / etc.)"]

b) I’m sure you COULD HAVE GOT into university. (correct sentence ? meaning ) [Correct. I'm sure that you would have been able to get into ... (if you had tried / etc.) (but we know that you didn't)]

c) I’m sure you COULD get / WOULD BE ABLE TO get into university, if you applied. = improbable action in the present or future ? [Not improbable. Not to me, anyway. Especially with "I'm sure", which indicates certainty, of course.]

d) I’m sure you COULD HAVE GOT / WOULD HAVE BEEN ABLE TO get into university, if you had applied. = unreal past action ? [Yes. "Counterfactual".]

- What’s the difference between a) and c) then between b) and d)? [The difference is whether the condition expressed in the "if"-clause is implicit or explicit.]

5)a) A: That crime can’t/could have been committed by my brother; he was abroad with his
wife at the time. [Either is OK.]

B: You MAY believe me. / You CAN believe me. (difference ?)

[With "may" there are two readings: 1. "It is possible that you [do/will] believe me", AND 2. literally, "You are permitted to believe me" -- but with the idiomatic meaning "I assure you that I am speaking the truth" / "Rest assured that you would not be wrong to believe/trust me". Of the two, I find the second reading a bit strained. The second reading belongs more properly, in my opinion, to "You can believe me."
With "can" there are also two readings that come to mind. 1. "It's up to you to decide whether you want to believe me (or not)". 2. "I assure you that I am speaking the truth" and all the others mentioned above for "may".
A lot of the interpretation will depend on context and even tone of voice and phrasing.]

You MAY be right (perhaps you are right, perhaps not) / You CAN be right (possible / correct ?)
[You [may, might, could] be right. I find it extremely difficult to contextualize "You can be right" in any but the most strained readings. I'd say it's just not used in the meaning you're thinking of. Still, there's " Everything is in flux. Circumstances are constantly changing. You can be right in the morning and wrong in the afternoon.]

b) He begins to wonder if the man might be some sort of a lunatic. (correct sentence?) [No, but not because of the modal. It should be, "He's beginning to wonder if the man might be ..."]
c) He began to wonder if the man might be some sort of a lunatic. [Correct.](difference with b) ?) [Just the obvious change of tense in the main clause.]

[The paraphrases are:

He is beginning to wonder if it is possible that the man is some sort of lunatic.
He began to wonder if it was possible that the man was some sort of lunatic. ]

CJ