Following is an excerpt from Chapter 6 of the book _Knowledge Representation_,
which is in the production stages now. This refers to a classic debate
between William Whewell and J. S. Mill, which states the issues more succinctly
and clearly than the more recent discussions of the topic, which are mostly in
the literature on cognitive psychology and connectionism (subjects that are
not noted for either brevity or clarity).
John Sowa
_____________________________________________________________________________
PROTOTYPES VS. DEFINITIONS. Aristotle introduced the method of formal
definition in terms of genus and differentiae, and he also began the
systematic description and classification of animal and plant species.
But as biology developed, the classification of species came to be based
on _prototypes_ rather than formal definitions. The historian of
science William Whewell (1858) gave a classic description of biological
practice:
Natural groups are given by Type, not by Definition. And this
consideration accounts for that indefiniteness and indecision
which we frequently find in the descriptions of such groups, and
which must appear so strange and inconsistent to anyone who does
not suppose these descriptions to assume any deeper ground of
connection than an arbitrary choice of the botanist. Thus in the
family of the rose-tree, we are told that the _ovules_ are _very
rarely_ erect, the _stigmata usually_ simple. Of what use, it
might be asked, can such loose accounts be? To which the answer
is, that they are not inserted in order to distinguish the
species, but in order to describe the family, and the total
relations of the ovules and the stigmata of the family are better
known by this general statement....
The type-species of every genus, the type-genus of every
family, is then one which possesses all the characters and
properties of the genus in a marked and prominent manner. The
type of the Rose family has alternate stipulate leaves, wants the
albumen, has the ovules not erect, has the stigmata simple, and
besides these features, which distinguish it from the exceptions
or varieties of its class, it has the features which make it
prominent in its class. It is one of those which possess clearly
several leading attributes; and thus, though we cannot say of any
one genus that it _must_ be the type of the family, or of any one
species that it _must_ be the type of the genus, we are still not
wholly to seek; the type must be connected by many affinities with
most of the others of its group; it must be near the center of the
crowd, and not one of the stragglers.
In responding to Whewell, the philosopher John Stuart Mill (1865) argued
that the practice of defining a category by prototype does not exclude
the possibility of finding a formal definition in terms of necessary and
sufficient conditions:
The truth is, on the contrary, that every genus or family is
framed with distinct reference to certain characters, and is
composed, first and principally, of species which agree in
possessing all those characters. To these are added, as a sort of
appendix, such other species, generally in small number, as
possess _nearly_ all the properties selected; wanting some of them
one property, some another, and which, while they agree with the
rest _almost_ as much as these agree with one another, do not
resemble in an equal degree any other group. Our conception of
the class continues to be grounded on the characters; and the
class might be defined, those things which _either_ possess that
set of characters, _or_ resemble the things that do so, more than
they resemble anything else.
And this resemblance itself is not, like resemblance between
simple sensations, an ultimate fact, unsusceptible of analysis.
Even the inferior degree of resemblance is created by the
possession of common characters. Whatever resembles the genus
Rose more than it resembles any other genus, does so because it
possesses a greater number of the characters of that genus, than
of the characters of any other genus. Nor can there be the
smallest difficulty in representing, by an enumeration of
characters, the nature and degree of resemblance which is strictly
sufficient to include any object in the class. There are always
some properties common to all things which are included. Others
there often are, to which some things, which are nevertheless
included, are exceptions. But the objects which are exceptions to
one character are not exceptions to another: the resemblance
which fails in some particulars must be made up for in others.
The class, therefore, is constituted by the possession of _all_
the characters which are universal, and _most_ of those which
admit of exceptions.
The arguments by Whewell and Mill are compatible for closed classes
whose members have long been known and thoroughly analyzed. They
disagree, however, on open-ended classes for which many members have
never been discovered or analyzed. Whewell's argument would have been
stronger if he had not chosen such a common example as the Rosaceae.
Critical examples arise with prehistoric animals such as dinosaurs and
early hominids, for which the fossil record is fragmentary and many
species are known only by a single jaw bone or just a tooth. That is
the lesson of Waismann's principle of open texture: it is not possible
to state the necessary and sufficient conditions for class membership
when many or even most of the members have never been discovered.
The nineteenth-century debate between Whewell and Mill was continued
by Wittgenstein with his discussion of family resemblance. More
recently, it was revived in AI debates over the use of logic or
prototypes for definitions. In general, formal definitions are possible
in any subject where the person who states a definition has the
authority to make it true by _legislation_. Government legislatures are
the prototype for legislating, but anyone who has the power to make
rules also has the power to legislate definitions: schools, businesses,
clubs, parents, computer programmers, participants in a discussion,
engineers who invent a new device, and authors who write a book or
article on a new subject. In empirical subjects, formal definitions are
possible, but only when all the evidence has been gathered. When the
evidence is incomplete, definitions by prototype are not only useful,
but necessary. Since all children and most adults are constantly
learning about things that were previously unknown (at least to them),
the human brain is designed to learn better and faster through
prototypes than through formal definitions. Even mathematicians and
logicians need examples to sharpen their insights into the meaning of a
new definition. Prototypes and other examples are difficult to use in
deductive reasoning, but they are the raw material for inductive
reasoning.
REFERENCES
Mill, John Stuart (1865) _A System of Logic_, Longmans, London.
Whewell, William (1858) _The History of Scientific Ideas_, 2 vols.,
J. W. Parker & Son, London.