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The Modals  
Making Sense of The Modals:   
More clarification:
We saw that modals are used to indicate degrees of certainty, and ability.  Now let us look at this again.
"Degree of certainty" refers to how sure we are-what we think the chances are-that something is  true.  If we are sure that something is true in the present, we do not need to use a modal.  For example, if I say, "Martha is sick,"  I am sure; I am stating a fact that I am sure.  My degree of certainty here is 100%.  If I am asked "Why isn't Martha in class?" and I am not sure of the reason, I can respond in the following ways:
1- She must be sick.
    Here, I am 95% sure that she is (not 100%)
2- She may be sick.
    Here, I am 50% sure that she is.
3- She might be sick.
    Here, I am less than 50% sure that she is.
4- She could be sick.
    Here, I am stating a mere guess.  This is a very weak degree of certainty.

  Try it yourself:
 My grammar book is not on my desk. Where is it?  
  Answer with 100% degree of certainty:
  with about 90% degree of certainty:
  with about 50% degree of certainty:
  with less than 50% degree of certainty:
  with only making a guess.
The Negative:
   Forming such sentences in the negative can be confusing. However, here is my attempt to help  
   you eliminate any confusion.
   Read these sentences:
1- Maria is not hungry.
       I am 100% sure that she is not hungry.  
2- I don't know why Maria is not eating with us. She may
       not (or might not) be hungry.
       I am 50% or less certain that she is not hungry.
3- Maria cannot be hungry. She has just had diner.
       Here, I believe that there is no possibility that
       Maria is hungry, but I am not 100% sure.
4- Maria is not eating.  She must not be hungry.
     Here, I am expressing a logical conclusion, a best guess.
So far we have discussed modals in the present time.  Using modals in the past requires more  understanding of the perfect
 tense.  It is safe to say that a modal in the past is really a more perfect.  What is the modal perfect?
  To answer this question, we need to refresh our memory of the perfect tense form.  Do you remember?  I am sure you do.
     Perfect = verb to have + the past participle of a verb
     Example: I have studied English for two years.
                      have + verb to study in the past participle = perfect tense.
So, a modal perfect = a modal + have + past participle.    
            must have eaten
            may have been
                             should have gone  degrees of certainty
                             might have been
                             could have passed
          must not have eaten
          may not have been
         should not have gone
          might not have been
          could not have passed  
 Let us look at the degree of certainty in these negative forms.
If I have to respond to the question:
Why didn't Martha eat?
I can respond in the following ways:
   1- Martha was not hungry.
                                  (I am certain she is not)
   2- Martha could not have been hungry.
    (I believe it is impossible for her to have been hungry)
   3- Martha must not have been hungry.
          (I am making a logical conclusion)
   4- Martha might not have been hungry.
          (I am mentioning one possibility)
   One more example:
            Question: Juan was absent yesterday. Where was he?
Answer:  I don't know. He may have been at home.  He might have gone to a movie.  He could have decided to go to the zoo because
the weather
Was so nice.

You try it
. What if you overhear Juan say, "My sister's plane was late yesterday.  I had to wait almost three hours." Now what do you think?
Well done.  I am sure you have done very well.
Degrees of certainty: Future Time:  
We have examined the use of modals in the present and past times.  Now let us examine their use in future time. Well, should and ought to can be used to express expectations about future events.
For example: Maria has been studying hard. 
She should do well on the test tomorrow.
She ought to do well on the test tomorrow.
In these two possible sentences, the speaker is saying,
"Maria will probably do well on the test.  I expect her to do well.  That is what I think will happen."
 Your textbook, Understanding and Using English Grammar provides a wealth of examples and many exercises that would be very helpful here.
Progressive forms of modals
The progressive (or continuous as some texts call it) form of modals can be in present or past.  Certainly you remember the formula for the progressive, don't you?  Well, just to refresh your memory, it is verb to be + the verb in the -ing form.
So present progressive will be something like this:
                                I am studying grammar now.
And past progressive:
       I was studying grammar last night.
Again, progressive is verb to be + a verb + -ing.
So, logically, a modal in the present progressive will be something like this:
                Tom may be sleeping.
                             (modal) may (verb to be) be (verb) sleep with ing.
 Another example: "All of the lights in Ann's room are turned off.  She must be sleeping. "
 Here is a tricky situation.  Your remember we said that modals in the past are modals + perfect; example:
                             He must have been tired.
Then modals in the past progressive will have to be:
Modal + perfect progressive : Modal + have + been + verb + ing
She must have been sleeping
He might have been studying
She could have been playing
These mean that the action in progress at a time in the past.
Well, let us recapitulate. We discussed the modals, their meaning and their various uses.
 Modals are helping words that are used to express:

(1) possibility, (2) ability, (3) degree of certainty, and (4) level of authority.

And we also stressed that :
the degree of possibility decreases as we use the past for of the modal:
I may visit you tonight.
                (50% chance that I will)
I might visit you tonight.
(less than 50% chance I will)
I can run fast.
                           (I am positively sure I can)
I could run fast.     
    (I am not 100% sure -   possibility may depend on something else) 
Will you shut the door?
   (you are close to the door and the possibility of you shutting it is high)
Would you shut the door?
                (you are far from the door and you might not want to go shut it)
You should study harder. 
(I am advising you without authority)
You must study harder. 
   (I am commanding you with authority)    
Would and could in the past:
One more matter to tackle with is the use of would to express a repeated action in the past and the use of could to express ability in the past.  I will try to simplify this as much as possible.  Here it is:
When would is used in the past, it means "used to," for example:
When I was younger, I would run two hours every day.
This means that I used to run two hours every day and now I don't. 
Another example:
When I was in Paris, I would spend hours walking up and down the Champs-Elysees.
This means that I used to do that when I was in Paris.
When could is used in the past, it means was able to.
For example: My girlfriend could lift the desk, but I could not.

This means that my girlfriend was able to lift the desk, but   I was unable to.
Please note that your textbook has an excellent table on pp.199-200 "Summary Chart of Modals and Similar Expressions." I would certainly look at it carefully.
If you would like to do more exercises and practice using modals, you may try exercise 34 on page 207 in your Azar textbook.  These questions will provide you with the opportunity to practice the different forms and uses of modals. 
Lesson Summary
In this lesson, we examined the English modals and we focused on three points:
1- what modals mean,
2- how they are used, and
3- when and under what conditions we can use them. 
There are nine (9) main modals in English.  They are:
shall, should
will, would,
can, could
may, might

 Modals are used to help convey a specific meaning. They indicate degree of possibility, probability, authority, or ability.

Remember that would, could and might are used when the degree of probability is 50% less. Should is used to offer advice and must is used to express an order or a command; so, the degree of authority is different between the two.  Also notice that shall is no longer used in American English except in stating a question.  For example, "Shall we go to the theater?" "Shall we call Mary?