Why can't we use "doesn't talk" instead of "don't talk"? when referring to he/she why is "doesn't talk" wrong?
e.g. When you are talking in class and your teacher said: "don't talk please." Why didn't she use "doesn't talk" in that case?
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When someone gives a command like this, it is called an "imperative sentence". The subject of the sentence is an implied "you", and so you use the form of the verb that goes with "you". In this case, "[You] do not talk." "Doesn't" is used with the third person singular, such as "he". But you can't give a command to a third person, that is, you can't give orders to someone other than the person you are talking to. So we don't use the third person for imperatives, always the second.
The phrase you are using is called an 'imperative'. We use the imperative when we command other people to do something. For example,
Be quiet, Lisa!
Eat your breakfast, Timmy!
In English, the imperative is mostly the same as the bare infinitive form, but can be contracted. Since the infinitive in question is 'to do', you should use "don't talk" and not "doesn't talk". This because 'don't' is the contraction of 'do not', and 'doesn't' is the contraction of 'does not'.
Ice Girl, according to situation given in your question the teacher is telling the students to do something or NOT do something. This type os sentence is called a command or an order - it is also described as an imperative sentence form (as pointed out by sanchises and others).
This type of sentence (command/order/imperative) requires the verb to be in the BARE infinitive form (INFINITIVE FORM: to talk; BARE INFINITIVE: talk). So the teacher says:
When the teacher tells you NOT to do something that is called the NEGATIVE form of the sentence. The negative form needs an auxiliary (helping) verb. The helping verb is the PRESENT INDICATIVE form of the verb to do. As correctly pointed out by oerkelens this must be the second person singular/plural - that is, DO.
So for the negative forms of commands the teacher says:
Don't talk, please! - or - Do not talk, please!
Don't be quiet!
Hope that helps!
Simply put. Words like "Doesn't, Don't, Can't, Won't" are all contractions. The reason they are used is to shorten the sentence speaking/writing length. Therefore when wondering why a specific contraction is preferred over the other I'd suggest you look at the full form and then ask. For example.
Don't talk Thomas! = Do not talk Thomas!
Doesn't talk Lisa! = Does not talk Lisa!
So it means that the second contraction is actually incorrect as saying "Does not talk Lisa" is not an Imperative form which is used to command others to do something.
Answers here tend to mix up two different aspects: one being imperative/indicative, another being 2nd person/3rd person. I'm not versed well enough in Early Modern English to know whether it sported a 3rd person imperative like Kaiserzeit German, but if it did, it would have been somewhat like "Talk he not to me, commoner!" in analogy to the German "Spreche Er mich nicht an, Gemeiner!".
Of course, neither modern English nor German use anything but the second person in their imperative nowadays, but the difference in ending between "doesn't" and "don't" cannot be mainly attributed to 2nd/3rd person difference since "does" never worked as an imperative even for 2nd person.