In Old English, the pupil of the eye (the round, dark center) was called the 'apple'. It was thought that the pupil was a round object much like an apple. When you look at someone, 1)their reflection appears in your pupil. So if someone is the 'apple of your eye', he or she is someone that you look at a lot and enjoy seeing.

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1)their reflection.....

Is "their" grammatical? What about "his or her"? Or what about "his" (reflection)?

Since the antecedent is "someone" (obviously singular with "one" in it!), the pronoun "their" is incorrect. See below.

In Old English, the pupil of the eye (the round, dark center) was called the 'apple'. It was thought that the pupil was a round object much like an apple. When you look at someone, his or her reflection appears in your pupil. So if someone is the 'apple of your eye', he or she is someone you look at a lot and enjoy seeing.

Thank you....

Can we use 'their' instead of 'his or her' in a casual speech (in everyday conversation)?

Lots of people do that. It's incorrect, but they do it anyway!!

=)

Yes, the use of "their" in this context is grammatical and commonly used in modern English when the gender of the person is unknown or when referring to a group of people. In the phrase "their reflection," it implies that the reflection of any person or multiple people appears in your pupil.

The use of "his or her" is also grammatically correct and can be used as an alternative to "their" when referring to an individual whose gender is unknown. For example, you could say, "his or her reflection appears in your pupil."

Using only "his" would be grammatically correct if you were specifically referring to reflections of males. For example, if you were talking about someone who has a strong liking for men, you could say, "his reflection appears in your pupil."

However, when gender inclusivity is desired or when it is unknown, the use of "their" or "his or her" would be more appropriate.