I was asked on another website how we should treat definitions that aren’t strictly convertible with the concepts that they define. In other words, in mathematics and logic, you can generally just substitute a concept whenever its definition appears, because there is nothing more to the concept than its definition states. You can’t do that with a lot of definitions elsewhere. For example, a horse without hooves is still a horse even though it doesn’t strictly meet the definition of a horse.
However, even when we define a term in mathematics, we don’t generally do so as an end in itself, and if we did our definition would quickly be forgotten or discarded. We formulate our definitions in order to prove things from them. The definition names a useful starting point in our reasoning for seeking out and identifying logical connections that exist objectively.
So, on the view I’m defending, our main goal when we define something isn’t necessarily to come up with a definition in which the subject is strictly convertible with the predicate, it is to identify the essence of the subject. This is a true proposition, in genus and differentia, from which the most of the subject’s known attributes follow, whether causally or logically. If the definition is convertible with the concept it defines, great, but if not, that’s fine too.
So, when we’re defining, say, a horse, the goal shouldn’t be to come up with a proposition that’s true of every horse, although it would be great if we could. We want to find the essence, a proposition which identifies the genus and differentia from which the most of the attributes of horses follow. We want the definition to do this because it will enable us to draw the most true generalizations about horses in our subsequent research, although we may have to take particular exceptions like the occasional horse without hooves into account when we are reasoning about them specifically.