Because as we all know, cats and their owners share a telepathic link. Why should something have to die because you have a beef with your neighbor.
Because the beef with my neighbour would be the cat in this circumstance, and since I couldn't feasibly erect a type of barrier tall enough to keep their cat out (the fact that I would pay for this myself is beyond the type of compromise I should consider, but I did say I'd exhaust other options first), it's up to the neighbours that own their cat(s) to keep them off my property, especially considering the health effect cats have on my wife. The
cat's health to me is completely irrelevant, even more irrelevant is the lack of telepathic link an owner shares with their pet, I value my property more than someone's pet, aka an animal, and I value my wife's health way beyond that animal and it's health. If it sounds bad or inhumane that's unfortunate but I don't really care, health and property are pretty serious and contentious issues, and I don't bend over for anyone else to be completely disregarding of others in this respect. I wouldn't allow any pet of mine into another property like that, I expect the same treatment all things considered.