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Is it now considered correct to use "them" or "their" after a singular noun such as "someone", or "person", in order to avoid the awkward phrase "him or her?"
For example, are these sentences correct?
"What would indicate that someone is not saying what is on their mind?"
"How can you show a listener you are interested in them?"
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Yes, it is considered correct by some (majority/minority I don't know). It's a great way to get around "he/she" "his/her" etc. Personally, I use it.

Look up "they/their (singular)" on this page:

http://www.wsu.edu/~brians/errors/errors.html

for more information.
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Well, 'Collins Cobuild English Dictionary for Advanced Learners' uses 'them', 'they', 'their', etc to refer back to 'someone', 'something', 'anyone', etc when giving definitions.

'Oxford Advanced Learner's Dictionary' says that you can use 'their' to refer to someone who sex is not mentioned or not known. The example sentence given is: If anyone calls, ask for their number so I can call them back.

(I believe these two dictionaries are published by careful people.)

The 'GrammarNow' newsletter I received says that 'the singular they is now standard practice in Canadian government documents'. Is there any Canadian on the forum who can verify this?

Ryan