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Can anyone explain the perfect particle in detail? what do these following sentence mean and rule to apply the perfect participile

1) he admitted having made a mistake.

2) having said that.

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First of all let's address what each of your two examples mean, then explore the perfect participle, and finally wrap up by converting your sentences to statements using the perfect participle.

1) He admitted having made a mistake.

The above sentence is straight-forward. Somebody (a male) 'he' , admitted (past tense) having (gerund form of have) made (past participle of make) a mistake.

This sentence has two of the three qualities you need to form a sentence using the perfect participle. It has the gerund form of 'have' and it has a past particple 'made'. Not bad!

2) Having said that.

'having' (gerund form of have) good! 'said' (past participle of say) good! 'that'

Although your two examples are not demonstrating the perfect participle in complete sentences, they do have two of the three qualities they need to be such examples.

(1)The gerund form of 'have' - 'having' ; and (2) a past participle 'made/said'.

Let us explore the perfect participle in some detail now, and finally return to your sentences, once you are clear about exactly what the perfect participle is, and how you might use it.

The perfect participle is a compound verb form.

Let us break down the the term to begin to better understand it.

Particple

The form of a verb that usually ends in "ed" or "ing" and is used as an adjective.
ref: Cambridge

So that's something kind of interesting, we are using a verb as an adjective. Let's dig a little deeper...

Wikipedia has the following to say about participles.

A participle is a form of a verb that is used in a sentence to modify a noun, noun phrase, verb, or verb phrase, and then plays a role similar to an adjective or adverb...
ref: Wikipedia

So it is clear, a participle will involve a verb and that verb will take on a role similar to an adjective.

Perfect Participle

What then is the 'perfect participle'? We already know that the perfect participle, will involve a verb that is being used as an adjective. But what does the word perfect mean here?

The perfect participle indicates completed action. You form the perfect participle by putting the present participle having in front of the past participle.
ref: Learn English

We have clarified the essence of the perfect participle now. We are always going to take the gerund form of have (having) and use it with a past particple.

Here are some examples: -

Perfect Participle: ACTIVE VOICE

Having washed the car, she was allowed to watch TV.
Having saved the cat, the firefighter felt happy.
Having heard the rain, Eddie decided to put his coat on.

Notice in all three of these examples, we have the gerund form of have, 'having' coupled with the past participles: washed, saved, and heard.

So we have two of the three qualities that constitute perfect participle usage. We having the gerund form of have, 'having', and we have a past participle. This forms our perfect particple clause.

The final crucial feature that makes each of our sentences work, is that in each sentence we two clauses, and the first (the perfect participle clause) influences, and is before, the second.

e.g.

Having washed the car, she was allowed to watch TV.

The perfect participle clause here is "Having washed the car" and the result is that "she was allowed to watch TV". This pattern of a prior action influencing a later action is a key feature of present participle use.

We can swap the order of our clauses if we like, that is completely fine.

e.g.

I got an A on my English Test, having spent every waking moment in ELU Stack Exchange ;)

We still have the three key features. (1) the gerund form of 'have'- 'having' (2) a past participle, 'spent'[1 and 2 form the perfect particple clause] and (3) a second clause that the perfect participle clause can influence.

The perfect participle can be used for active and passive voice. The above examples were all in the active voice. If you've followed this far, you have perfect particile use in the bag. It's a very minor additional detail to learn the passive form.

Perfect Participle: PASSIVE VOICE

Having been caught red handed, Tracey gave her friends t-shirt back.
Having been admonished in front of his peers, Jack felt embarrased.
Having been made late by her boyfriend, Sam had to run to work.

Do you notice the difference?

If you recongized the presence of 'been', well done. That is the hallmark of the perfect participle in the passive voice.


Let's return to your two sentences now and convert them into sentences which make use of the perfect participle.


he admitted having made a mistake --> having made a mistake, he admitted his wrongdoing.

Do you understand the change we made? We kept your participle clause 'having made a mistake' and we changed the action of him addmiting this, such that it happened after and was influenced by the participle clause.

having said that. --> Having said that, he stood up and left.

In the second sentence you supplied the perfect participle clause, I added the second clause such that the sentence now makes sense (a perfect participle clause must be prior to it's result).

By way of reminder...

active voice: having + past participle (Having cooked, he set the table.)
passive voice: having been + past participle (Having been cooked, the food looked delicious.)

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Perfect participles start with having -- the present participle form of the auxiliary verb for the Perfect -- and follow it with the past participle form of the verb -- made and said in this case -- which is also required for the Perfect construction. It's used to put a past reference into a participial clause, which is supposed to not be inflected for tense. Having said this, for instance, is a fixed phrase that means 'I have now said that which I needed to say. Now I change the subject.' It's perfect because it refers to to the relevant near past. – John Lawler Jul 30 '16 at 23:33

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