What do you call it when a person asks somebody a question when they know they'll criticise any answer regardless? For instance, a man asks you something like "If you were recruiting staff would you employ an ex-convict?". If you answer yes he says "Well that shows you don't care about security". But if you'd said no then the reply would have been "So you wouldn't give a person a second chance then?"
I upvoted David's loaded question because it's a very common usage, but on reflection I realised that's not quite right for OP's context.
A loaded question is nearly always one that's asked in such a way as to force or encourage a particular answer (that the answerer might not give if the question were presented "fairly").
But a trick question is one where the questioner usually doesn't care what you answer - you'll be wrong no matter what you say. I don't normally cite Urban Dictionary, but here's their definition...
An inquiry having no correct answer, or one asked for the sole purpose of starting controversy or eliciting certain responses. Basically, a no-win situation.
Girlfriend asks: "Do I look fat in this?" (trick question)
If you tell her she does, she'll throw a fit and tell you how insensitive you are. If you tell her she doesn't she'll call you a liar and go off on a tangent about how "all men are the same" or some nonsense like that.
My favourite trick question is "Have you stopped beating your wife yet?", but David's "When polar bears eat penguins, do they get indigestion?" is neat too. But they're both slightly different to the example cited above (that has two wrong answers, ours don't really have "answers" at all).
I would call that a "loaded question."
A loaded question is one where the person asking it has an agenda behind it. While there are other cases where a loaded question is the appropriate term, I believe this to be one type.
Of course, one can say that traditionally a loaded question has some information that forces the other person to agree to unsavory terms to answer the question. See here
But, I would still characterize this as a subtype of loaded question.
Essentially the asker of this type of question is only asking it as a means of embarrassing the other person. Hence, my characterization.
It does not give any name to this type of question, but there is an excellent discussion of the usage in James Joyce's Portrait of the Artist as a Young Man.
Then he went away from the door and Wells came over to Stephen and said:
— Tell us, Dedalus, do you kiss your mother before you go to bed?
Stephen answered: — I do.
Wells turned to the other fellows and said:
— O, I say, here’s a fellow says he kisses his mother every night before he goes to bed. The other fellows stopped their game and turned round, laughing. Stephen blushed under their eyes and said: — I do not.
Wells said: — O, I say, here’s a fellow says he doesn’t kiss his mother before he goes to bed.
They all laughed again. Stephen tried to laugh with them. He felt his whole body hot and confused in a moment. What was the right answer to the question? He had given two and still Wells laughed. But Wells must know the right answer for he was in third of grammar.
So now you have a choice.
You can strike a blow for evolving language and the insights new and vigorous young commentators offer, and show that you are not mired in stodgy tradition by checking David M's loaded question.
Or you can recognize the subtle distinctions offered by a seasoned veteran and not fall for the easy titillation of a flashy newcomer, by acknowledging FumbleFingers erudite trick question.
Sounds like a no win situation to me.
There is a term: a trap question.
As I've been answering people's comments, I've noticed myself using a phrase over and over again.
I do not claim this to be current or common parlance. But, in fact, I think I will sit here and begin the coining process!
I acknowledge the similarity to the term trap question, but I've chosen this word order to reflect that it is a trap that takes the guise of a question.
It is adequately descriptive of the intent and process going on here. And, best of all as the coiner, I can set whatever terms I choose for its usage. Hence, I state it to be an all-inclusive term that comprises, but is not limited to, the subset of both Trick and Loaded Questions.
And, for all who question my motivation as a fame whore, I will set this answer to community wiki status!
In casual usage, you could call it a gotcha question, although that can encompass both trick questions (there is no correct answer because the question makes incorrect assumptions), obscure questions (What is the name of the Lieutenant Governor of the tenth-largest US state?), and the kind of question you describe.
This was made famous by Sarah Palin a few years ago and many politicians since have complained about them. This article gives a good run-down: http://dailycaller.com/2011/08/19/politicians-complain-about-gotcha-questions-but-what-exactly-are-they/
The most prominent example is a very close fit to what you have described. The question to Palin "What do you read?" could only serve to harm her: no one would be persuaded to vote for her ticket however she answered, and whatever the answer there would be someone who could find fault with it.
It sounds like the scenario you are describing is the common phrase "Catch 22", aka "damned if you do, damned if you don't". This is popularized by the novel of the same name ( http://en.wikipedia.org/wiki/Catch-22). Essentially there is no right answer.
"If you were recruiting staff would you employ an ex-convict?" could also be considered a Rhetorical question, that is, a question with no intended answer. The questioner could then continue with something along the lines of "Of course not, because ...etc." As the question was not intended to be answered any answer given could possibly be criticized.
I think the term describing a question which elicits an answer that provokes an argumentive statement from the questioner is 'argumentative'. This is like the cross-examiner at a trial whose question is objected to by the opposing counsel: 'Your honor, the question is argumentative', i.e., inducing the witness to engage in argument with the cross-examiner rather than stating facts.
I always thought this type of question was, 'Begging the Question'.
The answer is included in the premise, so he can criticize it no matter what you say.