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English reflexive pronouns are usually used in a binding domain. There are, however, some restrictions about its distribution when we consider some dimension of meaning and other considerations about grammatical properties.

1. He looked about him.

2. She had her fiance beside her.

3. John ignored the oil on himself.

4. John has no covering over himself.

5. John has no control over him.

6. John has many books about him.

Those are extracted and contained in my book. It says #1, 2 and 6 are acceptable.

Q1) If I change him into himself in #1 and 6, isn't the meaning the same?

Q2) It looks like #3,4 and 5 are similar to #1 and 6 but why they are not acceptable?
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I cannot really interpret your questions. Only #5 is ambiguous without changing it to 'himself'. #3 and #4 are both fine.
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The only thing I can suggest is that "near him, besides him, about him" are all showing the location of things that are near but not on that person. The emphasis is more on the location of thing A and less about person B.

Person B had thing A near/about/beside her. It's again more about the location of thing A and less about person B.

In the example with the oil, it's more about Person B. Person B had thing A on himself.

This is very vague. I can't give a better answer than Mr. M's answer.

Sorry.

(As you requested, I reviewed this post, but I don't think I can contribute much.)
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Comments  
Thank you for both answersEmotion: smile

I didn't know 'about' can refer to location; all I guessed about #6 was that 'about him' means his 'book'.

I thought #3 and 4 are acceptable too, but I don't still know why my book says they are not acceptable.
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