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Dictionary.com defines bear as:

Any of the plantigrade, carnivorous or omnivorous mammals of the family Ursidae, having massive bodies, coarse heavy fur, relatively short limbs, and almost rudimentary tails.

(Just focusing on the definition of the animal bear).

I would just start by defining a bear as "An animal". Or, "An animal that looks like this ". I might say "it is a big furry animal that sometimes walks on two legs, occasionally catches fish out of the river, can climb trees, and hibernates in the ground".

At what point can we say we have a definition of a bear that is satisfactory?

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We need first to distinguish between two kinds of definition, nominal and real.

Nominal

The Aristotelian tradition distinguishes 'nominal' and 'real' or 'essential' definitions. To bring out this distinction one must consider answers to questions which begin: 'What is ... ?' 'What does it mean to ... ?' For example, 'What is "positive health"?' This question is asking for the definition of a term, a word or words; as such it is asking for a nominal definition. One way of identifying the request for a nominal definition is to consider whether what we are asking about (as in the example given) should be enclosed in inverted commas. If however, like jesting Pilate, we ask without quotes 'What is truth?' we are not asking about the meaning of a word, for we all know what the word 'truth' means, but about the nature of what the word refers to. And, of course, that nature is very complex, which is why Pilate did not wait for the answer. This second sort of inquiry is an inquiry into 'realist' or 'essentialist' definitions. There are many methods of nominal definition. One method - beloved of writers of essays - is to look the word up in a dictionary. This method of definition is called 'lexical' because it is concerned only with words, or 'reportive' since it reports current usage of words in terms of their approximate synonyms. At some point, however, we must get outside the enclosure of words or we shall not really know what any word means; lexical, reportive or 'word-word' definitions presuppose the existence of 'word-thing' definitions. (R. S. Downie, 'Definition', Journal of Medical Ethics, Vol. 20, No. 3 (Sep., 1994), pp. 181-184: 181.)

Real

'Real' has no honorific status : it indicates merely a definition of the thing (Latin re) rather than just the word.

Those who were first concerned with definition were concerned with the nature of things and not mainly of words. For example, Plato's dialogues are typically concerned with establishing a definition - of justice in the Republic, of courage in the Laches, of love in the Symposium, and of knowledge in the Theaetetus. But these definitions are of the things and not of the words. Similarly, Aristotle offers a range of definitions, of motion, happiness, virtue and so on, but again he is concerned with what these things are and not simply with the words. Indeed, Aristotle defines 'definition' as the statement which gives the essence, and he is clearly thinking of the essence of the thing or the type of thing, and not mainly, or not at all, of the word. And it is not just the ancient Greeks who take this line on definition, nor indeed is it just philosophers. Linnaeus provided real definitions of plants through his system of botanical classification, and the World Health Organisation (WHO) in 1946 offered a much criticised definition: Health is a state of complete physical, mental and social well-being, and not merely the absence of disease or infirmity. Clearly, the WHO were not offering a definition of the word 'health', but of the very essence of health itself. What is it that Linnaeus and the WHO were trying to do?

Linnaeus was attempting to classify. In the traditional logic the classification is in terms of 'genus' and 'species', where the class that is to be divided is called the 'genus' and the sub-classes are called the 'species'. Classification involves devising a process of dividing such that all the sub-classes are mutually exclusive. This process of classification employs some technical terms which are widely agreed: they are 'difference', 'property', and 'accident'. A quality is said to be a 'difference' if it serves to distinguish the class of entities of which it is a quality from other species of the same genus, or in other words, if it is used in the definition of the class. A quality is said to be a 'property' if it is a quality possessed by every member of the class, yet is not used to distinguish the class from other species of the same genus. Finally, a quality is said to be an accident if it may indifferently belong or not to all or any members of the class. This terminology may be archaic but the ideas are perfectly familiar and are perhaps expressible as 'defining characteristics' and 'accompanying characteristics'. For example, we might wonder whether certain symptoms are defining characteristics of a disease or just accompanying ones, sometimes or always. (R. S. Downie, 'Definition', Journal of Medical Ethics, Vol. 20, No. 3 (Sep., 1994), pp. 181-184: 182-3.)

Comments and examples

Not everyone accepts the objective existence of genus and species. Are these classifications which we project on to the world or do they exist 'out there', independently of human conceptual schemes? Too big and complex a question to deal with here - at least for me.

I can, however, illustrate the distinction between the two kinds of definition:

Nominal: 'Bear: Any of several large, heavily built mammals constituting the family Ursidea (order Carnivoral), with thick fur and a plantigrade gait' (OED).

This is satisfactory for a range of descriptive and explanatory purposes.

Real: A real definition might, by contrast, specify Ursidea in terms of its mitochondrial DNA. An organism possessing such DNA belongs to the bear family.

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