Theories of Definition: Belnap
Belnap is less concerned than Kant with categories of definitions, than with the ”good” properties of definitions. For him, a definition tries to explain the meaning of a word or phrase.
”I consider [definitions] only in the sense of explanations of the meanings of words or other bits of language. (I use ’explanation’ as a word from common speech, with no philosophical encumbrances.) As a further limitation I consider definitions only in terms of well-understood forms of rigor. Prominent on the agenda will be the two standard ’criteria’ – eliminability and conservativeness – and the standard ’rules’.” 
A definition explains by relating the new word or phrase, that which it defines, with words and phrases you may know already. I can only explain X to you by using something which you already know. I cannot, of course, know what you know, but I can make assumptions, and adapt my definition through our communication. The idea that a definition explains by relating to prior knowledge is an important one.
Here is how Belnap clarifies how an explanation should be, when it defines:
”Under the concept of a definition as explanatory, (1) a definition of a word should explain all the meaning a word has, and (2) it should do only this and nothing more. That the definition should (1) explain all the meaning of a word leads to the criterion of eliminability. That a definition should (2) only explain the meaning of the word leads to the criterion of conservativeness.” 
A definition satisfies the criterion of eliminability, if you can replace the term it defines by the definition, anytime you use that term, and this would change nothing to your and others’ understanding of what you are or were trying to say. In other words, if some term T is defined by D, with D being a sentence, a few of them, or anything else that you consider being the definition of T, then T:D satisfies eliminability if you can use D anytime you used T, and it would change nothing at all.
Let’s illustrate this with a definition of ”giraffe”.
Giraffe: ”a large African mammal with a very long neck and forelegs, having a coat patterned with brown patches separated by lighter lines. It is the tallest living animal.” 
I need a convention to make writing easier. Above, ”giraffe” is called definiendum, the term being defined, and the sentences which define it are called definiens.
What eliminability requires, is that anytime you, I or anyone else uses the term ”giraffe”, we can replace that term, in that context in which she or he used it, with the definiens above, and this replacement would change nothing (relative to using the term ”giraffe”). Which is to say that eliminability requires definiendum and definiens to be interchangeable at all times, everywhere, without effect on what is being understood, in any context where either is used. Continuing the illustration, if you define ”giraffe” as above, then the following sentence, and the subsequent paragraphs need to have the exact same meaning, if the definition above satisfies eliminability.
”Okapis are very different in their ecology and behavior from giraffes.”
”Okapis are very different in their ecology and behavior from large African mammals with a very long neck and forelegs, having a coat patterned with brown patches separated by lighter lines, they are the tallest living animals.”
Even for simple examples, as above, it is a stretch to claim that the definiens and definiendum are such that nothing changes if you change one for the other. If anything, the mood of the statement changes – imagine we talk, and I tell you that okapis and giraffes are different in ecology and behavior; then try to imagine the same conversation, where I’m giving you the definition of giraffe at length. And the definition above is a simplistic one; what if I were instead using the following one:
“The giraffe (Giraffa) is a genus of African even-toed ungulate mammals, the tallest living terrestrial animals and the largest ruminants. The genus currently consists of one species, Giraffa camelopardalis, the type species. Seven other species are extinct, prehistoric species known from fossils. Taxonomic classifications of one to eight extant giraffe species have been described, based upon research into the mitochondrial and nuclear DNA, as well as morphological measurements of Giraffa, but the IUCN currently recognises only one species with nine subspecies.” 
You could argue I am caricaturing. Would anyone do this replacement, except for irony, sarcasm, or comedy?
But the ”why” does not matter much, the point is that the very notion of preserving meaning probably makes little sense if you take seriously that you and I cannot have the same ideas in mind, regardless of how same the things we say or write. So there is no perfect preservation, and eliminability can only be something to want, but which you cannot have. (This is probably fine to say for natural language, but is too pessimistic for formal languages. In a mathematical logic, eliminability makes perfect sense, since we can – provided it is computable – determine if there is preservation of meaning, when meaning means all logical consequences. So you can take eliminability seriously, but only if you put many constraints on the language you use to make definitions. But innovation languages are in the realm of natural language.)
In the context of Plastic Definitions, eliminability remains a useful idea. Remember that we want to be precise, clear, and accurate. So if you replace definiendum with its definiens, and this leads you to easily draw conclusions which make ideas less precise, clear, and accurate, then this needs to be looked into. Again, there is a limit to how useful this eliminability idea is, since it is defined with a counterfactual: should I have replaced it, we would have had something else; but if I do not, then I cannot know that it would. Despite flaws, it remains a sanity check.
What about conservativeness? Eliminability was about having the definition explain every meaning, or meaning in all contexts, of the defined term. In innovation languages, I suggested settling with eliminability as a reminder to check what might be concluded when definiendum replaces definiens, especially if conclusions go against precision, clarity, and accuracy. Conservativeness is about a definition doing not more than explaining meaning.
Consider the following example, where I have two definitions of soccer.
- Soccer: ”Association football, more commonly known as football or soccer, is a team sport played with a spherical ball between two teams of eleven players. [. . . ] The game is played on a rectangular field called a pitch with a goal at each end. The object of the game is to score by moving the ball beyond the goal line into the opposing goal.” 
- Soccer: ”Association football, more commonly known as football or soccer, is a team sport played with a spherical ball between two teams of eleven players. It is played by 250 million players in over 200 countries and dependencies, making it the world’s most popular sport. The game is played on a rectangular field called a pitch with a goal at each end. The object of the game is to score by moving the ball beyond the goal line into the opposing goal.” 
If I wanted a definition which remains neutral on how this sport is or is not popular, or widely played, then the first definition is conservative, and the second is not. I can conclude nothing about popularity from the first, but I can from the second.
It might seem that conservativeness has an obvious and hard problem. Namely, it is relative to the purpose of the definition, or intention of the maker of the definition. Or, it is relative to the meaning intended by the maker of the definition. If I was making a definition of ”soccer”, I might think it necessary to say something about popularity, while someone else would not. So this is not about what soccer is, in the context of, say, a foundational ontology. If it is my intent that sets meaning, then it is me who draws the line between success or failure to satisfy conservativeness. And this is a problem, because if it is up to intentions, it is up to something inaccessible. I cannot, as I emphasized several times by now, see or otherwise directly access your intentions, since they are ideas ”in your mind”. So again, just like for eliminability, conservativeness is easiest to precisely define if we are making definitions in a formal language, such as some mathematical logic. There, a definition is conservative if replacing definiendum with definiens leads to the same conclusions.
There is a way to approach this, without giving too much attention to either eliminability or conservativeness, and in the context of natural language, as I argue next.
- Nuel Belnap. “On rigorous definitions”. Philosophical studies 72.2 (1993), pp. 115–146.
- Giraffe | Definition of Giraffe in English by Oxford Dictionaries. 2019. url: http://en.oxforddictionaries.com/definition/giraffe
- Wikipedia – Giraffe. 2019. url: https://en.wikipedia.org/wiki/Giraffe
- Wikipedia – Association Football. 2019. url: https://en.wikipedia.org/wiki/Association_football