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In his chapter on modal auxiliaries in The English Verb, Michael Lewis lists what he thinks are the basic semantic meanings for each. I cite those meanings below and ask if you agree with them.

Please remember, we are talking about the basic semantic meaning. In context, these auxiliaries take on wider meanings, but it is the basic meaning I want to discuss.

Paraphrased.

Can = I assert that it is possible that ...
Could = I assert that it is "remotely" possible that ...

May = If I have anything to do with it, it is possible that ...
Might = If I have anything to do with it, it is "remotely" possible that ...

Must = I assert that it is necessary that ...

Will = Given my percepton of the immediate situation, it is inevitable that ...

Would = Given the (hypothetical) situation which I perceive at the moment of speaking, the action described is also inevitably true.

Shall = According to my perception of the present situation, it is, if it's anything to do with me, inevitable that ...

From The English Verb by M Lewis. LTP 1986.

Should is dealt with seperately as it is a far more complex auxiliary and has many meanings.

Thanks.
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I don't find these paraphrases the best I've ever seen. Here are a few of my observations / opinions:

1. The basic meaning of "can" and "could" is ability.
2. I don't see why "if I have anything to do with it" is included.
3. I don't see why "I assert" is included.
4. I find "inevitability" a good paraphrase for "will" / "shall".

CJ
Hello, Milky and CalifJim. I haven't read Michael Lewis yet. But this question seems interesting.
Will = Given my perception of the immediate situation, it is inevitable that ...

Does this mean we should exclude will of ?
And what's the difference between and ?

I agree with you, Milky (and M.Lewis) in this respect:
Should is dealt with seperately as it is a far more complex auxiliary and has many meanings.

I've heard that we should distinguish should from should, and ... as far as I remember ... there's some problematic point in the treatment of the latter.
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One test might be: what happens if we take Lewis's 'basic' meaning, and present it as a sentence? Would a native speaker be likely to infer any hint of the 'modal' equivalent?

1. Given my perception of the immediate situation, it is inevitable that you go.

My immediate reaction is to translate this back into English as 'I think you have no choice but to go', or 'in my opinion, you have to go'; or even 'I think you should go'. But not 'you will go'.

How do others read it?

MrP
1. The dynamic meaning of "can" and "could" is certainly "ability", which comes under the banner of possibility in its dictinary meaning, but what about the epistemic and deontic meanings of those two modals?

2. Because the basic use of "may" is to give permission. Authority, on behalf of the speaker, whether real or imagined, allows him/her to say "if I have anything to do with it".

3. assert = to insist on having one's opinions, rights accepted. It also means the same as "to declare".

I insist that you...
You must...
I declare that you must...

4. Good.
Does this mean we should exclude will of ?

Why should it?

Given my perception of the immediate situation, it is inevitable that I will go to the party.

And what's the difference between and ?

It is necessary that you go to the doctor, but you may choose to avoid doing so.

It is an inevitable, unavoidable thing that you will eventually see a doctor because you are extremely ill.

I agree on "should":

Should uses:

a)Epistemic = Recommended
b) Deontic = Expected or Advice

There are more uses.
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1. Given my perception of the immediate situation, it is inevitable that you go.

My immediate reaction is to translate this back into English as 'I think you have no choice but to go', or 'in my opinion, you have to go'; or even 'I think you should go'. But not 'you will go'.

How do others read it?

MrP

Inevitabilty is related to the unavoidable future occurence.

Your above should be:

1. Given my perception of the immediate situation, it is inevitable that you will go. It will take place, in the opinion (modal) of the speaker.

"Will" connects two moments:that pertaining at the moment of speaking and a second one that is most commonly in the future.
Since you usually so much dislike finding the core meaning of "will" to be simple future, Milky, why do you prefer to find it in something that is given in our beloved textbooks as of one of the more obscure uses of "going to"!?
Hello, milky. Thank you for your answering the questions. I still feel uncertain about the definition of .
It's too strong, isn't it (as MrP also pointed out)...? For example, according to your interpretation:

(1) I will go to see a doctor today.
(= It is an inevitable, unavoidable thing that you will eventually see a doctor because you are extremely ill.)

(2) I must go to see a doctor today.
(= It is necessary that you go to the doctor, but you may choose to avoid doing so.)

Right ...? And as to my previous question, intention?>, I'm not certain yet .... I had in mind the following pair of sentences:

(3) She will be a good doctor.
(4) I will be a doctor.
(We should exclude the , tense meaning of will here, of course.)

Can we apply M.Lewis's definition of will also to (4)?

.... it's for the first time in my life(?) to tread in the field of modality. Don't be too hard on me, please, everyone..!
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