I really don't understand this. You give a counterexample to a theorem; you don't give one to a definition. If someone disagrees with the definition and wants to use a different one, then they are talking about a different thing (even if they want to use the same English word to refer to each of them) and their conclusions can not be compared in any meaningful way.
Imo, you can't "proceed carefully" with a definition that is open to interpretation. If you could do so safely, then you know enough about how people could interpret your definition to form a more rigorous definition.
>> that somebody won't be able to
>> disagree with by counter example
> I really don't understand this.
> You give a counterexample to a
> theorem; you don't give one to
> a definition.
The classic example is "connected" from topology. Speaking very loosely, a set is "disconnected" is there is a "disconnection", which is a separation of the set into two pieces which are contained in disjoint open sets. Basically, the set is disconnected if it's made up of two (or more) pieces that can be divided by a "surface". A set is "connected" if there is no disconnection.
The problem is that there are sets we want to think of as not connected, and yet which satisfy the definition of connected as given above. Here's an example:
X = { (x,sin(1/x)) : x in R, x>0 } u { (0,y) : -1 < y < 1 }
If that example had been thought of earlier, it's possible that the definition of "connected" might have been fixed earlier. So in a sense, X is a counter-example to the definition of connected.
Further reading:
http://planning.cs.uiuc.edu/node140.html
https://en.wikipedia.org/wiki/Connected_space
https://en.wikipedia.org/wiki/Connected_space#Path_connected...
What you sometimes see among philosophers is that even definitions that are the result of many iterations are still treated more like rules of thumb than precise definitions. Or they at least acknowledge that their definition is probably flawed, and proceed carefully.
Some proceed as if they are doing math.