What it means to negate other people's ability to carry within an eruv chatzerot - how that happens, when one potential participant in the cooperative doesn't join, whether out of spite or simply not being Jewish, for example, and the range in between. Note that these cases are indeed practical - "real world halakhah," more so than the theoretical expansion of policy. The goal becomes circumventing the person who is inherently present in the cooperative, but can't actually participate in it, and therefore nullifies everyone's ability to make an eruv and carry. Some practical cases, which also being to the fire the teacher/student relationship between R. Yosef and Abaye (again). Also, a person can negate or nullify his own property rights for carrying, which can make all the difference in these cases - where the possibility of establishing an eruv exists. Similarly, if you could prohibit others from using the property... that's where the ability to negate your ownership rights kick in.