One example that I'm reminded of recently pertains to the issue of theft. For instance, say a watch is stolen from someone, and that thief later sells that watch to another person. That person later finds out (by interaction with the owner) that this watch was stolen. Is he then obligated to give the watch back to the original owner upon request. If you've developed a more foundational understanding of property, the answer is a clear, "Yes!" But even some fervent defenders of property often get tangled within their own response. After all, how can you justify taking something from someone that they have paid for?
This becomes even more confusing if you move to ask such a person if title to such property has exchanged hands at any point in this, to which they will likely reply, "No." So, on the one hand, you have a customer of a thief who you believe now rightfully owns the watch. And yet you believe that the original owner still holds title to it. This is an obviously inconsistent view. For if the original owner still holds title to this property, and yet the property is now rightfully in the hands of some other person, then what does property imply at all? Is property not the rightful ownership or possession of an item? If you hold property that rightly belongs to someone else, it would seem like this is not your property at all.
One can certainly sympathize with the third party here. After all, they have likely spent hard-earned money on an item, and it would seem almost as a tort to pry it from them. But what such an observer would be failing to see is that the criminal has committed fraud against them, and has essentially stolen their money - much in the same way that if I had claimed I owned the house across the street, and "sold" it to you, on the spot, for $1,000 and left the premise, I would have just robbed you of $1,000. I'd expect that we'd certainly feel sorry for you. But what we probably wouldn't do is tell the rightful owner that his property has now effectively changed hands, and that he will now have to take it up with the criminal. No - the title of ownership was never transferred to the criminal to begin with. He was, in effect, fraudulently selling a house. In other words, he was selling you ownership that he never had...he was selling you, quite literally, nothing. The original owner has still retained the title of ownership.
It's examples like this, I think, that show how important it is for us to think about the implications of terms before we move forward with employing them. Someone who contests the right of the original owner to reclaim his property in the above story, and yet still claims that title had never been transferred, has not resolved the implications of the term "property" correctly - however fervent they may be about it.
Ayn Rand less famously used to make a similar rhetorical point regarding broader economic terminology. She argued that fascism and socialism were not really inseparable on the account of what property actually implies. In one system (socialism) the state owns the property. In the other (fascism) the state directly controls it but private ownership remains. But Rand recognized that this dichotomy was false - based purely on the implications of property. Socialism is clearly state-ownership. But how is fascism fundamentally different? What is the difference between someone owning your car and someone having ultimate control over whatever is done with that car? We could simply say, on a piece of paper, that you still "own" the car in the second example. But what does property imply if it does not imply rightful control over something? At that point, to Rand (and I for that matter), it sounds more like verbal masking than it does an accurate description of reality.
When we move back to talking about core social concepts, it's important that we're not only coherent, but consistent. If something is the property of someone, we want to recognize the consequences of that reality, and not merely pay lip-service to it and substitute for it our own sympathies. If you find yourself arguing that property isn't really property, theft isn't really theft, or that violence isn't really violence, then it's probably worth retracing your steps.