IOE, Chapter Five
Definitions
by Svein Olav G. Nyberg
Date: 24 Feb 1992
Forum: Moderated Discussion of Objectivist Philosophy
Copyright: Svein Olav G. Nyberg
"A concept is a mental integration of two or more units which are isolated according to a specific characteristic(s) and united by a specific definition." (ItOE, 2nd expanded edition, p.10)
"A definition is a statement that identifies the nature of the units subsumed under a concept." (ItOE, p.40)
"The purpose of a definition is to distinguish a concept from all other concepts and thus to keep its units differentiated from all other existents." (ItOE, p.40)
1. FORM
The formula for a definition is 'Genus - differentia', and has been handed down to us over the centuries from Aristotle. This formula means that the definition of a concept should be given so that one part, the genus, ties it in with other concepts by inclusion, and the other part, the differentia, should specify how the units subsumed by the concept are distinguished from other units subsumed by the genus. This follows the pattern of human thought in general. First we integrate - then we differentiate.
Because of this form, the definition serves to integrate the concept into the totality of our knowledge. By the genus, it tells us what other concepts to associate with it. By the differentia, it tells us how it should be understood apart from them.
2. CONTENT
A definition is a way to retain and communicate a concept within the context of one's knowledge. It will therefore be natural to refer to the nature of the concepts themselves when discussing the nature of their definitions.
a. A definition is a condensation.
A concept is not an arbitrary collection of characteristics, but an integration of units. Only in mathematics can the two be equated. In reality, the units usually have more characteristics in common than the ones known to us at any given time, and we might encounter existents that share all the characteristics in the arbitrary collection but still are not subsumed under our concept. Just imagine if some rational animals arrived from Altair IV.
This should show that we cannot define a concept simply by listing its known characteristics.
Also, it would be quite counter to the cognitive process that concept formation is to represent the concept through a large list of characteristics. A definition should serve to condense the data of these characteristics just as the concept itself serves to condense its units.
But, although only a few characteristics are explicit in the definition, Ayn Rand says the remaining characteristics one knows are implicit in it, as they are what determines and validates the definition.
b. All definitions are contextual.
The definition is a way to relate a concept to the other concepts we have at a given time, while at the same time it serves to identify the nature of the units as seen given our present knowledge. Thus, definitions are doubly contextual - they are conditional both on our overall knowledge and on our knowledge of the units of the particular concept. If the aforementioned creatures from Altair IV should arrive, we could no longer define man as 'the rational animal'. Similarly, if we were learn of a new characteristic 'X' of man that explained more of his characteristics than 'rational' does, we should define man as 'the X animal'. (See under Objectivity / Truth of a definition.)
This is equivalent with the fact that definitions are not changelessly absolute. For, in order to be changelessly absolute, the definition would have to be some perfect description, or an (exhaustive) list of the characteristics common to the units subsumed under the concept. The definition is, however, absolute _contextually_, in that there is no way to escape the context.
It is important to note that if we choose to retain our old concept when our knowledge is increased, the old and the new definition do not contradict. The latter merely serves to expand the former. By this we mean that the new definition refers to a wider body of knowledge and that it implies (the characteristics given in) the old definition.
c. A definition is in terms of essentials.
Man is as far as we know the only present animal posessing a thumb, so why not choose the characteristic 'thumb-having' instead of 'rational' in our definition of 'man'? It differentiates just as well as does 'rational', and it is even easier to check.
Ayn Rand's answer is that this would be wrong, as such a 'definition' would not be in terms of essentials. After all, how much of man's nature can be explained by such a non-essential characteristic as the thumb?
Both the cognitive and the communicative aspect of the definition are lost if one tries to define by non-essentials. Who would understand which concept one was trying to point out when one talked about "the tailless thumb-haver"?
"The 'essential' characteristic(s) is the fundamental characteristic(s) which makes the units the kind of existents they are and differentiates them from all other existents" (OPAR, p.100)
d. Fundamentals
It is important that the essentials assigned to the concept through definition are fundamental, or in Peikoff's words in 'OPAR', that they are 'causally significant'. This is 'the rule of fundamentality'. A fundamental is, in Rand's words:
"Metaphysically, a fundamental characteristic is that distinctive characteristic which makes the greatest number of other possible; epistemologically, it is the one that explains the greatest number of others." (ItOE, ch.5)
As I see it, there are two reasons for the rule of fundamentality. The first is that of the contextuality of definitions. As your body of knowledge grows, you will repeatedly be in need of revising your definitions. If your definitions should happen to be in terms of non-fundamentals, you will have to change your definitions both more frequently and more radically than if your definitions had been made following the rule of fundamentality.
The second, and most important reason is that definition through non - fundamentals does not function as an aid in thinking, but leads, as Peikoff puts it, "to cognitive stultification". As an example of what you might get by not using the rule of fundamentality, Peikoff mentions in Objectivism: the Philosophy of Ayn Rand an example of a schizophrenic in New York City's Bellevue Hospital. That man had tried to make a concept by grouping sex, cigars and (sic!) Jesus Christ. They all had the attribute "encirclement" in common. In sex, man 'encircles' the woman; cigars are 'encircled' by tax bands; Jesus Christ is 'encircled' by a halo. A definition of this pseudo - concept "encirclist" would have as an essential - 'encirclement'. But this essential could in no way be a fundamental to the units subsumed under "encirclist". If one were to define a valid concept by non-fundamentals one would actually give that concept the same status as "encirclist" in one's cognitive grasp of the world. (OtPoAR, 'definitions')
e. Objectivity/Truth of a definition "An objective definition, valid for all men, is that designates the _essential_ distinguishing characteristic(s) and genus of the existents subsumed under a given concept - according to all the relevant knowledge available at that stage of mankind's development." (ItOE, ch.5)
To make a definition objective, one has to regard _all_ ( relevant ) knowledge available, not ignoring any evidence, and not the least, one should not make any pretentions of knowing more than one actually knows.
"The truth or falsehood of all of man's conclusions, inferences, thought and knowledge rests on the truth or falsehood of his definitions." (ItOE, ch.5)
"Truth is the product of recognition (i.e., identification) of the facts of reality." (ItOE, ch.5)
It might seem strange to evaluate a definition as 'true' or 'false', but - observing what Rand states about truth above - we can fit it into the scheme. The proper identification of a concept through definition, as we have seen in the above sections, lies in assigning the right essential characteristics to the concept. A true definition is thus one whose designation of essential characteristics is true, i.e. proper according to the rules above.
According to Ayn Rand, every concept stands for a number of propositions. A 1s.t level concept stands for implicit propositions, whereas 2nd level concepts stands for a whole chain of explicit propositions. Further on, a definition is a condensation of these. Because of this, a definition's truth rests on the truth of these propositions. Compare the two definitions "Man is the rational animal" and "Man is the religious animal".
Symmetrically, all propositions consist of words, and so refer to concepts. In this, their truth necessarily rests on the truth of the concepts, and thereby on the truth of the definitions, as every concept requires some definition; 2nd order non - axiomatic concepts require a verbal definition, and every 1st level concept requires ostensive definition.
f. Ostensive definitions
It is now widely recognized that you cannot provide all concepts through definition by other concepts. If you tried to rely on (explicit) definition only, your hierarchy of concepts would either become an air castle, or you would end up in an infinite recursion, which is even worse. We need some concepts to be basic, concepts that we do not define by other concepts.
Still, we have to communicate these concepts in some way. This is done through so - called 'ostensive definition'. In ostensive definition, one isolates perceptual concretes and displays them: THIS is what I mean by 3. MATHEMATICAL DEFINITIONS
Mathematical definitions belong to a very peculiar class of definitions, and reflect the deductive structure of the subject. The mathematical definition has to imply all the characteristics of the concept one wants to define - through deduction. Thus, the mathematical definition has to supply enough characteristics that the remaining characteristics of the concept are deducible from them. This is to be contrasted with the ordinary definition, where many if not most of the concept's characteristics are only implicit in the definition.
Let me take an example: Defining "fractal", a concept whose units are mathematical objects, we would get "an irregular geometric shape". But, even though this is a definition of a concept whose units are mathematical all right, it is not a mathematical definition. For a definition to be a mathematical definition it must be useful within mathematical proof. To be so, it has to give the concept's characteristics explicitly. And though it might seem cruel to cut a concept's wings like that, we just need to equate the concept with its definition, and no more. So, in order for us to be able to study a fractal mathematically, we will have to define it by stating its characters explicitly. What is then stated, is what will be a fractal in our mathematical study. No more - no less. The most popular definition of "fractal" is "A geometric object whose topological and Hausdorff dimension are different". From this definition we can make calculations and theorems. From the former definition, we cannot. Regarding some Cantor sets, which are the canonical fractals, we see that the mathematical definition cannot fill out the former definition.
It is important to understand that the mathematical definition cannot leave itself vulnerable to potential new knowledge. That would make the core of mathematics, mathematical proof, invalid. A mathematical definition has to be changelessly absolute, and thus has to stand for purely deductive propositions. The context of a mathematical definition is not only the (mathematical) existents known at the moment, but absolutely any conceivable (mathematical) existents whatsoever. So even though we might find irregular geometric shapes whose Hausdorff and topological dimensions are equal, and let them be subsumed under our concept of fractal as "irregular geometric shape", we can never let them in under the mathematically defined concept of 'fractal'. To subsume them under the same concept as those fractals given in the mathematical definition, the only possibility is to create a new mathematical concept and call that one "fractal" instead.
This should semonstrate why one cannot expect ordinary definitions to follow the form of mathematical definition. Our knowledge about the world is not deductive, but inductive. We _can_ allow rational animals to arrive from Altair IV without doubting our own humanity. For, our characteristics qua characteristics do not usually depend upon one another by necessity. Non- mathematical concepts are open-ended also in regard to their characteristics.
To conclude - mathematical definitions are definitions, albeit belonging to a very peculiar class. But conversely, most non-mathematical definitions can not and should not follow the form of mathematical definition.
References Ayn Rand: Introduction to Objectivist Epistemology, Second expanded edition, Meridian 1990. Paperback.
Leonard Peikoff: Objectivism: The Philosophy of Ayn Rand, Dutton 1991.
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