The sentence: There aren't any reasons to do it.

I'd like to say other form of this phrase with opposite meaning. Something like this: There aren't any reasons to don't do it.

How to say this form of the phrase with double-negations of affirmative meaning correctly? And are there some rules to make a sentence of such form?

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    Couldn’t you just say “There aren’t any reasons not to do it”? Either way, the true opposite would be “There is every reason to do it.” This form of double-negation conveys a distinct meaning from either the phrase or its opposite, and is a good example of a litote. Dec 4 '13 at 16:42
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    This is not, by the way, the "rule for using double-negations in sentences". It's a lot more complex than that, and there are a lot of different kinds of negative, not all of which work the same. Dec 4 '13 at 17:57

In the sentence ‘There aren't any reasons to do it’, to do is the infinitive form of the verb. You have to keep that form in the negative: you can’t change it to the finite don’t. So the negative version will be ‘There aren't any reasons not to do it.’

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    Or, There aren't any reasons to not do it. Putting the negative after the complementizer to instead of before it gets it closer to its focus (do it) and therefore more emphatic. Dec 4 '13 at 17:54

As Tyler James Young has suggested, using a double negative in order to say something affirmative, goes as follows:

"There aren't any reasons not to do it."

Moreover, as Barrie England has suggested, one mustn't change the infinitive when double-negating. Think of your sentence in the following way:

"To do it, there are no reasons not."

With completely different sentences, the same rule applies. For example, with to apply,

"It's not that he didn't refuse to apply himself, it's just that his heart wasn't in it." (i.e., "Not to apply himself, he did not refuse, it's just that his heart wasn't in it")

Or with the infinitive to boast,

"May it be not far me to boast not about him when he is in my presence." (i.e., "To boast not about him, may it not be far from me, when he is in my presence.")

That sort of thing. In some ways, double negation is a kissing cousin to litotes, the figure of speech which uses the negation of the positive to indicate something positive (There was no small stir when the unpopular decision was made. In other words, there was a big ruckus when the unpopular decision was made.)

Both litotes and double negation can be used artfully, but in using them you run the risk of being misunderstood!

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