What is the affirmative form of the following sentence?
He is neither poor nor honest.
I tried, but couldn't transform it from negative to affirmative.
There is no singular "affirmative form" of the sentence. The original sentence tells you something about what the person is not. To make it affirmative, you would have to say something about what the person is.
It says he is not poor, but is he rich? Middle class?
It says he is not honest, but is he a pathological liar? A harmless rake?
"He is a rich liar" would be a simple example of how one might be able to turn it into an affirmative sentence, but that could significantly change the meaning of the text.
For example: "It is red" tells you a great deal more about an object than "It is neither blue nor green."
Because the phrase poor but honest, well, isn't uncommon, I would read the above sentence as "He is a rich liar".
As Lynn notes, there are other, weaker, ways of reading it. And as Lynn further notes, it's usually more informative to tell someone what something is rather than what it isn't. However, there are usually pragmatic reasons for a speaker to choose a superficially less-informative phrasing.
To my ears, He is neither rich nor honest has a trace of irony to it. I'd attribute that to the use of scalar implicature. Compare, for example:
So how did your date go last night?
Well, she wasn't beautiful...
So what was she?
Implicatures are cancelable: That is, an utterance might suggest a meaning pretty strongly, but the speaker can always choose to say, No, that's not what I meant. For example, the litotic phrase isn't uncommon might mean is unavoidable; however, when I used it in my first sentence, I didn't mean anything beyond the surface meaning.
Not poor = lives above poverty = lives with material comforts
Not honest = dishonest
He lives above poverty dishonestly.
He lives with material comforts dishonestly.
He is both rich and dishonest.
or more simply
He is rich and dishonest.
He is rich, but dishonest.
Are the immediate affirmatives that comes to mind.
But the problem is that 'not poor' isn't necessarily the same as being rich. One can be rich, or one can have a modicum of wealth. There isn't a single word or phrase that expresses 'not poor' better than 'not poor'.
The affirmative for 'not honest' isn't so much of a problem, IMO. There are several words to express different levels of honesty.
Liberal with the truth. Known to tell lies. A liar. Dishonest
It really depends on the context, and what you're trying to say. If you're trying to say that the guy is rich, then you can use 'rich'. However if you're simply saying that he's not about to starve to death, then 'not poor' is what fits best.
For the honesty one, as I've mentioned, using an affirmative is a bit easier.
There's a poetic element to the phrase, but describing what this guy is not you paint quite a broad picture of what he could be. That might be more effective than trying to paint a specific picture, which might detract from what you're trying to say.
He was neither poor nor honest.
He was both rich and dishonest.
The latter explicitly paints the subject's wealth and honesty as a key part of his personality. Whereas the former more allows the subject to act in certain ways in certain situations, without it being a key part of who he is.
The affirmative of
He is neither poor nor honest.
He is poor and honest.
If you want to retain meaning,
He is wealthy and mendacious.