On: Theirself or Themself?

[Symmetry]

What are you consulting to find that the correct replacement for 'them' in formal writing is 'him' here? There isn't any College of the English Language to make authoritative prescriptive statements, and the Chicago Manual of Style at least has been against the use of 'him' here for several editions.

[gr3yh47 → Symmetry]

I am not claiming that 'him' is a replacement for 'them'

'them' is plural and flatly incorrect when used as a singular pronoun.

edit: Again, 'them' is formally incorrect but informally acceptable in the singular usage.

[70forty → gr3yh47]

'them' is plural and flatly incorrect when used as a singular pronoun.

Why? How do you know this?