Appealing to emotion works great, because beliefs tend to stem from what "feels" right. But if you don't
want to be a manipulative asshole, and if you don't want to indoctrinate people, stick to using patience and logic.
Hence the question, "what is your goal?"
Think of Ebeneezer Scrooge. Both his nephew and the ghost of his former business partner tried to
reason with him. Yet he was so dismissive that he was able to look a ghost in the eye and seriously claim that it was a hallucination.
But Christmas Past showed him how happy he could be if he changed, and Christmas Future showed him pain, loneliness and death if he didn't.
BAM! Changed man.
People are not so different.
thinks overpopulation is the worlds biggest issue
How would you go about challenging anyone of these beliefs
If you really want to change their mind, try visiting them with the ghosts of christmas past and future.
First...tell them that they're absolute right and that you agree with them. This identifies you as a member of their personal "in" group, and therefore not a threat that needs to be argued with as you would appear to be if you try to argue with them. It also identifies you in their mind as someone who is smart and well informed, which obviously you must be because you agree with them.
Next, encourage them to tell you about why they belief what they do. Not like you're a new convert asking questions. Conversationally, like you already agree with them and know what they're going to say, but enjoy the process of two like minds sharing the same thoughts in agreement. Whatever they say,
agree with it. Agree with their reasoning. Tell them they're absolutely right and you agree completely.
...therefore...Since they're right and you agree with them and the population must obviously be reduced, how shall we go about it? (Say "we." Not "you." This reaffirms that you share group affiliation and encourages them to think with you rather than against you.) Would it be better to do it slowly via forced sterilizations, or more quickly by euthanizing the elderly? Of course it would be most efficient to do both. Yeah...you have grandparents too, but they're kind of old and it would really be for the greater good if they were euthanized. It's a sacrifice you'd be willing to make. Then look at your "friend" and ask them if they'd be willing to do that too, since it's obviously what would have to happen in order to reduce world population.
Don't worry about what they say. Just let them think about that. Makes them feel bad for believing what they do by associating things they obviously don't like, don't agree with and would strongly and personally object to...with the beliefs you want them to no longer have.
If you really want to go for broke, act annoyed that the last hippy-wannabe you tried to convince of this very sensible approach went off on some stupid tangent about how Hitler was a supporter of these very same ideas: euthanasia, forced sterilizations and population control. Because of course we don't want to be like Hitler. He was a very bad man, and it's a totally irrelevant that he happened to engage in these very sensible actions that we should definitely be doing ourselves.
Don't say this sarcastically or humorously. Say it genuinely. The goal is to
associate things that your listener strongly disapproves of, doesn't like and would feel bad having anything to with...with the belief you want them to give up. They're not listening to your words. The brain works by association. So associate population control with murder and death and Hitler and pain and suffering and guilt.
Then...after they've made that association, give them an alternative. Mention how ridiculous it is that people who don't believe in population control actually believe that technology can solve all the problems and that we can all live in some kind of futuristic utopia where everything is all happy and nice. Point out that they've obviously been watching too much star trek if they really believe that technology can allow us to accommodate even more people just because it's allowed us to accommodate the billions we have now . By doing this, yes, you're subtlely giving them solutions, but most people are not listening to the words. The real goal is to simply associate happy nice imagery with the beliefs you want them to adopt. Of course, you my as well do both.
If you want to go full on NLP mode, and with a little practice...you can make statements like these and deliver them via subtle changes in tone and inflection to make stealth points like:
"...how ridiculous it is that people who DON'T BELIEVE IN POPULATION CONTROL actually believe that TECHNOLOGY CAN SOLVE ALL THE PROBLEMS."
You see what's going on? The trick is to deliver these subconscious commands with enough subtlety that they'll go unnoticed by the conscious mind of the listener, but picked up on by the subconscious.
This is the sort of thing that changes beliefs.
It's also how you make compelling advertising. For example, show regular guys drinking beer, being happy, having friends, and with beautiful girls, and the subconscious may tend to associate being happy, having friends and beautiful girls with drinking beer. Nobody would buy beer if the commercials listed rational, logical reasons for drinking it. Want to sell insurance? Make people feel fear by associating not having insurance with how awful it would be if they were to die and leave their loved ones with nothing. Want to sell your college? Make them feel fear of not having a job and associate feelings of wealth and security with having a degree.
...if you want to sell
belief, then use mechanisms that instill belief.