Theories of Definition: Kant
For Kant, to define is to identify all primitive properties of that which you are trying to define, whereby that set of properties allows you, me, others, to unambiguously distinguish the thing from others. It is important that all these properties in the set, i.e., properties which together make up the definition, are primitive.
Primitive means that none of them is derived from others. Each is necessary; none can be removed from the set, and together they are sufficient for you to make the distinction [1,2,3].
Kant made the distinction between analytic and synthetic definitions.
- A definition is analytic if it defines something that is already there, independently from its definition, and its definition identifies only the necessary and sufficient primitive properties needed to distinguish that given thing from everything else. A definition of, say, the window you’re looking at.
- A definition is synthetic, if it creates the thing it defines. The thing that is being defined is not there, and starts being there by virtue of us producing its definition. The definition contains the necessary and sufficient primitive properties for us to introduce the thing and make it unambiguously distinct from all else. For example, I define
Then, there is a distinction between nominal and real definitions.
- A real definition will be such that, at least some properties it identifies are grounded in experience of reality.
- A nominal definition will identify properties which do not have to be grounded in such experience, for you to accept that the thing being defined exists.
Peano axioms, for example, seem to be a nominal definition of natural numbers if you take them to exist as abstractions in the framework of mathematics. They do not define what a natural number refers to, but define the relationships which we must stand between instances of natural numbers – it does not matter if you or I have ever had experiences of actual things which we would reasonably call natural numbers.
In more technical terms,
”Kant is saying that in a real definition we do not merely equate a word with a logical product of arbitrarily chosen logical predicates, but we make at least a problematical existential judgment and state the conditions under which this judgment could be verified so that the definiendum [i.e., name of that which is being defined] will be seen to have ’objective reference.’ There must be, in the definiens [the statement of properties, defining the definiendum], some determination or compound of determinations that can be ’cashed’ in possible sensible (intuitive) experience.” 
Analytic/synthetic and nominal/real are orthogonal, independent distinctions, combining into four kinds of definitions: analytic nominal, analytic real, synthetic nominal, and synthetic real.
How does this relate to Plastic Definitions? I said earlier that a Plastic Definition is open to change, and it needs to be precise, clear, and accurate, because that makes it easier to change.
The analytic/synthetic distinction has nothing to do with either of these – there is no reason why an analytic or a synthetic definition cannot change. If you do take that distinction seriously, then note that the plasticity of an analytic definition reflects the acquisition of more information about the properties of the given thing being defined; the plasticity of a synthetic definition means going from defining and making one thing, and when the definition changes, the thing being made changes too, as when, for example, one software specification, a synthetic definition of the software product to make, leads to one release of that product, and then a change in the specification leads to a next release, of a different software product.
Plasticity is also independent from the nominal/real distinction. Nominal definitions can change, if the abstraction they define changes – this could be because you know more than you did when you defined it initially, you thought it through more, you discussed it with others, and so on. These changes will amount to adding or removing of, the properties appearing in the definition. Same applies to real definitions, in which case that change reflects again the change in what we know or assume about the actual thing being defined.
If a Plastic Definition stops changing, if it graduated to a stable definition, then it may be analytic or synthetic, and nominal or real.
That independence from Kant’s dimensions of definitions is the first useful observation so far in this text. It consequently does not matter if you do, or don’t take Kant’s distinctions relevant.
A second observation has to do with the economy of definition, specifically that a good definition only identifies primitive properties. If a property can somehow be derived from primitive properties, then it shouldn’t be mentioned in the definition.
But how do you derive these other properties? What does it mean, in practice, to derive properties?
These questions matter, because what you answer decides if you care at all about the distinction between primitive and derived properties. If you don’t, it is one less complication to think of, when making and changing Plastic Definition.
A property is derived, if you can take the definition, and by applying some procedure, find that derived property of the thing being defined. So you do not need to have those derived properties written in the definition, as long as you know the primitive ones, and the procedure.
We need to distinguish two cases: Is the procedure well defined, or not?
Mathematical logic has well-defined procedures. Primitive properties need to be stated by grounded well-formed formulas of your logic of choice, and the logical closure over these formulas will include all derived properties. Same goes for statistics. If you have a dataset that describes, thus defines some sample of a population, a mean of a variable in there is a derived property, to be derived by computing the trivial, but well defined formula.
What if the reasoning procedure is not well defined? This is the realm of personal, subjective, biased, ordinary, everyday reasoning procedures, those which might be partly accounted for by Walton’s informal logic , behavioral economics , starting with Tversky and Kahneman , or other, but certainly not well defined procedures, procedures which you and I, if we applied them, would consistently lead us to the same conclusions. In those cases, if there is no guarantee that you and I will derive the same conclusions from the same definition, is the primitive and derived distinction useful?
It remains useful, but in a restricted sense: there is no need for synonyms in a definition, for example.
For Plastic Definition in particular, the primitive/derived distinction loses its appeal. Plastic Definition are used when working with new ideas, when trying to make them precise, clear, and accurate, so as to prepare them for others to change. It is more important to do that, than to worry about a Plastic Definition stating the obvious, which is the worry that the economy of derivation promotes. Better for the content of a Plastic Definition to be obvious to all, than missed by some.
Does this argument apply even when the reasoning procedure is well defined? I cannot compute the mean of a categorical variable, we all know that. I also cannot deduct a formula with a predicate which did not occur somewhere in the premises. (Unless I am drawing conclusions from an inconsistent set of formulas, and doing so under a procedure which is explosive (as in classical propositional or classical first-order logic, for example). But in that case, I can only conclude anything, same as you, and we may disagree, then. In other words, since you and I can conclude anything from inconsistency, how likely is it that we will come to the same conclusions? So my point holds for paraconsistent logics too.)
A well defined reasoning procedure ensures that both you and I, and anyone else, starting from the same explicit, written down premises or inputs, will compute the same conclusions. We will compute the same mean of the same variable, if we both have the same dataset. We will compute the same grounded formulas from the same premises, if we apply the same reasoning procedure. But while we will get the same formulas, or the same numbers, what we read from this, the references we make, can be different, and there is nothing the well defined reasoning procedure can do about that. So even if the procedure is well defined, it is you and I who read the conclusions, and when we do it, we end up applying our individual procedures which are not well defined, and are inevitably different.
Ultimately, then, there’s no need to worry if your Plastic Definitions only talk of primitive properties, or if they fail the economy of derivation. In practice, this translates into Plastic Definition which repeat the same things, in different ways, trying to clarify the important ideas they are being designed and changed to convey.
- Immanuel Kant. Critique of pure reason. Cambridge university press, 1999.
- Immanuel Kant. Lectures on logic. Cambridge University Press, 2004.
- Lewis White Beck. “Kant’s theory of definition”. In: The Philosophical Review 65.2 (1956), pp. 179–191.
- Douglas N Walton and David N Walton. Informal logic: A handbook for critical argument. Cambridge University Press, 1989.
- Sendhil Mullainathan and Richard H Thaler. Behavioral economics. Tech. rep. National Bureau of Economic Research, 2000.
- Daniel Kahneman and Amos Tversky. “Prospect theory: An analysis of decision under risk”. In: Handbook of the fundamentals of financial decision making: Part I. World Scientific, 2013, pp. 99–127.