4 yr old niece still uses pacifier & throws tantrums!?

As sickening as it sounds, my Mother always gives it to her just to shut her up. The kid has a way with tantrums and my Mom just gives into them. My sister lived with my Mom and she doesn't want her to have them but my Mom makes excuses and says since she was a 26 wk baby that she needs it or that she grinds her teeth if she can't sleep with it. Okay, so my sister is in the hospital right now and when she gets out she's going to be moving in here. I already have my niece and OMG! She can throw a tantrum for HOURS ON END! She is screaming to the top of her lungs now as I type because I will not give in to her. I've been dealing with it the past 2 hours. I've had her a week and she does it every single night and then she'll wake up at 3:30am missing it. What do I do? I know there has to be a point where she realizes she isn't getting a paci and I threw them away so she won't be getting one again but I almost feel like going to the store and buying a binky... I won't, but I'm 7 months pregnant & these tantrums have to stop!