I think that "I don't want to allow him to go to the Vietnam war." is better than "I want to not allow him to go to the Vietnam war."
Can I use both, or not?
I think that "I don't want to allow him to go to the Vietnam war." is better than "I want to not allow him to go to the Vietnam war."
Can I use both, or not?
“I want X” expresses desire for X. From a logical perspective, the negation “I don't want X” should express a lack of desire for X, which could be a desire for the negation of X, but could also be indifference towards X — “*I want not-X”. However, in English, “I don't want X” is the idiomatic and unambiguous way of expressing “*I want not-X”.
To express the lack of desire without implying a desire for the negation, possibilities include “I don't particularly want X” (hints at mild desire for X), or “I don't really want X” (can range between mild desire for X and mild desire for not-X, not to be confused with “I really don't want X” which conveys a strong desire for not-X).
“I don't want to allow him to go to the Vietnam war” is idiomatic English and means that the speaker wishes to not grant that person the permission to go to the Vietnam war. This effectively means the same thing as “I want to forbid him to go to the Vietnam war”, unless some distinction is made between not allowing (not saying yes) and forbidding (expressly saying no).
“I want to not allow him to go to the Vietnam war” would be understood to mean the same thing, but it sounds awkward. “I want to not …” is rarely idiomatic. It can sometimes be used to emphasize the lack of desire, but the very reason it's perceived as emphasis is its non-idiomatic nature.
The first is the usual form. You would be saying that the other person has asked for your permission to go to war and you would like to deny it. That's the best way I can see to explain it. The second is saying something different and is quite clumsy in the way it says it. It's as though the person is asking you for an exemption from going to war and you want to grant that exception.