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Person A: How was your hunting today.

Person B: I shot one dead.

Person A: You only shot a single dear.

As I understand it, in the sentence ''You only shot a single dear'', a is a quantifier which means one and single is also a quantifier which also means one. Is my understanding correct?
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Single emphasizes that there in only one of something: not two, not more, just one.
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deer.

Singular countable nouns in English must have a determiner.
single is an adjective, not a determiner.
It has nothing to do with meaning. That is, just because an adjective means the same thing or something similar to the determiner does not mean you can leave out the determiner.

CJ
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Comments  
Grammar GeekSingle emphasizes that there in only one of something: not two, not more, just one.

Then what is the function of a in that sentence? Doesn't a also tell that there is only one dear?
 CalifJim's reply was promoted to an answer.
Students: We have free audio pronunciation exercises.
Once my English teacher told me that in sentences like ''this is a book'', a means one. It just represents quantity. Does it really represent quantity? Well, as I now know, my teacher told many incorrect things.
Jackson6612Once my English teacher told me that in sentences like ''this is a book'', a means one. It just represents quantity. Does it really represent quantity? Well, as I now know, my teacher told many incorrect things.

'One' is used for emphasising. For example, "I've caught one fish."