I wouldn't prefer to or I would prefer not to
 

I was asked by a student, who is also a teacher, which is correct and why. I wouldn't prefer to do this or I would prefer not to do this. I know on 'usage' the second is correct, but I'd like to give her a more complete answer if one exists. I've checked all my usual reference texts and can't find a satisfying answer.

Any advice would be appreciated.

Thanks

Re: I wouldn't prefer to or I would prefer not to
 

I would prefer not to do it.

Preferences in English are always expressed positively. It is the action which is negated. In other words, if you say eg I (would) prefer not to work on Saturdays, the thing which you don't want is " to work on Saturdays". You have a positive preference for not doing something. Therefore, it's the verb work which has to be negated. After prefer the verb is in the infinitive with to and this is negated simply by placing not in front. So the structure of the sentence is :
I (would) prefer / not to work on Saturdays.
Positive preference + negative action

Expressing preferences therefore differs from expressing likes or wants, where it is negative liking/wanting which is usually expressed :
I don't want to work on Saturdays
I don't like working on Saturdays.
but when you're expressing preferences the preference is always positive and it is the action which is negated. Some more examples:
I'm not a complete vegetarian but I prefer not to eat red meat
I'd prefer not to fly - let's take the train.
I preferred not to talk to him, because I knew it would just make me angry.

Notice that it is the semantic concept of preference which is important here, not the verb prefer itself. If you express preference in any other way, the rule is the same : the preference remains positive and the action is negated :
I'd rather / not talk about it
I'd sooner / not go

Re: I wouldn't prefer to or I would prefer not to
 

Hi. Thank you.

How about "believe" as in the following?

A) I strongly believe that Japan was wrong at that time.
B) I strongly believe that Japan wasn't right at that time.
C) I don't strongly believe that Japan was right at that time.

Which of B) or C) is the equivalent expression of A) ?

Re: I wouldn't prefer to or I would prefer not to
 

B is OK grammatically and means the same as A, but English doesn't like negating an opinion - usually the "opinion verb" is negated : Eg the opposite of I believe/think that Japan was wrong is I don't believe/think that Japan was wrong
or I believe think that Japan was wrong. With believe it's just possible to negate the opinion : I believe that Japan wasn't right though I suspect it would be less common than the others, but with think it's not possible: * I think that Japan wasn't right ( The asterisk indicates an incorrect sentence.)

The real problem with your sentences is the presence of the word strongly. C changes the meaning. Your sentence is : I don't strongly believe that Japan was right here, not no longer negates believe but strongly . So the meaning becomes : I believe that Japan was right but not strongly.

So, in answer to your question, the answer to "What's the opposite of I strongly believe that Japan was wrong..."
is most likely to be I strongly believe that Japan was right.... But if you want an equivalent, B is possible but less likely than A, and C changes the meaning completely

Re: I wouldn't prefer to or I would prefer not to
 

Thank you, susan53, for your detailed explanation.
I think I got it.
I agree that "strongly" is the problem.
So I create a new sentence D).
How about D) as the synonym of A)?

From your lesson, I think D) is better than B) as the synonym of A).

Re: I wouldn't prefer to or I would prefer not to
 

That works. Or put an adverb in front of the verb phrase :
I really don't believe that Japan ...
I honestly don't believe that Japan...

Interestingly strongly can't be used in this way - ie in front of the verb phrase.