It is clear that Auxiliary Verbs are used only with the Non-finite Forms of a Verb when they are used as a verb taking a Subject.
We now know that Verb has six Forms, namely:
1. Present form and 2. Past form are always used with Subject and never without Subject. These two forms are called Finite Forms of Verbs. They never take any help from any Auxiliary Verb. That means a Finite form never takes any Auxiliary Verb.
3. Present Participle, 4. Past Participle, 5. Infinitive and 6. Gerund are used as other Parts of Speech without any Subject. These four forms are called Non-finite Forms of Verbs.
Among the Non-finite Forms, Present Participle, Past Participle and Infinitive can be used as verb being helped by an Auxiliary Verb. Without the help of any Auxiliary Verb the Non-finite Forms cannot be used as a verb and never can take a Subject.
But Gerund is never used as a verb. It is always a Noun and so, it never takes any help from any Auxiliary verb.
So we can say that out of the 6 forms of Verbs Present form, Past form and Gerund do not take any auxiliary Verb.
And the rest 3 forms i.e. Present Participle, Past Participle and Infinitive take Auxiliaries only when they are used as verbs with a Subject.
Now we should know which verbs are Auxiliaries. There are three verbs which called Principal Auxiliaries. They are: Verb ?to be?, verb ?to have? and verb ?to do?.
They have their own meaning. So they are used as Principal verb and keeping their meaning silent they can also be used with the Non-finite Forms as Auxiliary verb.
He is honest. Here, ?is? Principal Verb and helped none.
He is going. Here, ?is? expressed no meaning of its own but helped the Non-finite Form.
I have a pen. Here, ?have? Principal Verb and helped none.
I have eaten rice. Here, ?have? expressed no meaning but helped the Non-finite Form.
I do the sum. Here, ?do? Principal Verb and helped none.
I do not go. Here, ?do? expressed no meaning but helped the Non-finite Form (bare Infinitive).
These three Auxiliaries are called Principal Auxiliaries.
Besides, there are six Modal Auxiliaries or Modals. Modals only help the Infinitives to change the mode of action. The Modals are: shall, will, can, may, must, and ought.
Note: modals have only two forms: Present form and Past form i.e. they have only Finite Forms. They have no Non-finite From. ?Must? and ?Ought? have only Present Form.
I shall/will go. ? Simple Futurity
I ought to go. -- Obligation
After Modal Auxiliaries the form of the Principal Verb must be Bare Infinitive. But only after ?Ought? the Verb must be in Full Infinitive.
So the total number of the Auxiliary Verbs is 9. Auxiliary Verbs are of two kinds: 1. Principal Auxiliaries (3 in number) and 2. Modal Auxiliaries or Modals (6 in number)
Finite Forms (Present form and Past form) never take any Auxiliary Verb.
When Non-finite Forms are used as Principal Verb, they must take an Auxiliary Verb.
As Gerund is never used as a Principal Verb, it never takes any Auxiliary Verb.
So the Auxiliary Verbs help only the Present Participle, Past Participle and Infinitive.
In the above figure we see, BE can help all the 3 Non-finite Forms: Present Participle, Past Participle and Infinitive.
i) BE + Present Participle = Continuous Tense
I am going Present Continuous Tense
I was going Past Continuous Tense
I shall be going Future Continuous Tense
ii) BE + Past Participle = Passive Voice
A bird is seen by me = Passive
iii) BE + Full Infinitive = Arrangement
He was to go to America = Arrangement
HAVE helps either Past Participle or Full Infinitive
i) HAVE + Past Participle = Perfect Tense
I have eaten rice = Present Perfect Tense.
I had eaten rice = Past Perfect Tense.
I shall have eaten rice = Future Perfect Tense.
ii) HAVE + Full Infinitive = Compulsion
I have to go now = I must go now (Compulsion)
DO + bare Infinitive = Negative (with ?not?) / Interrogative / Emphasis
To make the Sentence Negative, ?do? is used with ?not? if there is no other Auxiliary Verb. (Only in Present Indefinite and Past Indefinite Tenses)
Do you know me? = Interrogative
To make the Sentence Interrogative, ?do? comes before the Subject, if there is no other Auxiliary Verb. (Only in Present Indefinite and Past Indefinite Tenses)
I do believe you. = Emphasis
To emphasis the Verb ?do? comes before the Verb.
When ?do is used as an Auxiliary Verb, it always takes the bare Infinitive form of the Principal Verb.
MODALS + bare Infinitive = Future Modes of action
But OUGHT + Full Infinitive = Future Obligation
Present Form and bare Infinitive are of the same appearance but they are different.
Now we see that Preset Participles take only verb ?To be? as an Auxiliary Verb
Past Participles take verb ?To be? and ?To have? as an Auxiliary Verb.
Infinitives take all the Auxiliary Verbs.
Again, verb ?To be? can help all the 3 Non-finite forms.
Verb ?To have? can help only Past Participle and Infinitives.
And verb ?To do? and 6 Modals can help only Infinitives.
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