At 05:26 PM 1/22/2006, you wrote:
>Various themes have emerged from this discussion, and I'll
>try to sort them out:
>
> 1. The fundamental distinction which many of us have
> emphasized for years is between _intension_ and
> _extension_. To avoid prejudging the choice of
> technical terms, I'll use the example of Rabbit vs.
> Peter or Thumper. The intension of 'Rabbit' is
> determined by some definition that is independent of
> any actual rabbits, real or fictional. The extension
> of "Rabbit" at a particular point in time consists
> of all the rabbits that happen to exist at that time. (01)
If 'rabbit' refers to a type, and if rabbit instances existed before
human beings, and if rabbits were able to recognize other instances
of like type (e.g. for dating purposes) then the type rabbit existed
before any definition was formulated. Talk of types should thus not
rely too much on talk of definitions/intensions. Indeed there are
many types in biomedicine for which definitions have not yet been
supplied, and many types in all domains for which incorrect
definitions have been (or were for many centuries) supplied. (02)
> 2. The word 'set', as used in all versions of set theory
> for over a century, refers to the extension. Since
> rabbits are notoriously prolific, the set of rabbits
> at the instant you're reading this sentence is very
> different from the set at the instant I wrote it.
> The fact that sets are constantly being replaced by
> new ones means that sets are not suitable for defining
> any "meaning" that is supposed to remain constant.
>
> 3. Aristotle introduced the word 'kategoria' as his term
> for the intension, (03)
not true, I'm afraid;
'kategoria' means: that which is said (in the agora, or market);
for Aristotle the terms was used to mean something like: 'highest type';
Aristotle did not have our modern notion of intension (04)
>and that word was translated to Latin
> as 'praedicatum'. (05)
> From those, we get the English words
> 'category' and 'predicate'. The latter term has become
> popular for the symbols used in _predicate calculus_, and
> I would say that the defining rule for a predicate is
> sufficiently stable to be used as the basis for drawing
> a clear distinction between intensions and extensions:
>
> - The intension of 'Rabbit' would be determined by the
> definition of a monadic predicate rabbit(x).
>
> - The extension of 'Rabbit' at time t would be the set
> of all x at time t for which rabbit(x) is true. (06)
Again, when biologists (e.g.) develop large ontologies representing
types, they do not do this by scanning their minds for corresponding
definitions, or intensions; they look at the entities in reality,
e.g. at patterns of DNA. (07)
> 4. If we wanted to be traditional, we could use the four-
> syllable word 'category'. Otherwise, we could adopt
> a one-syllable word, of which the two main contenders
> seem to be 'type' and 'class'. The word 'set' is not
> a contender because all technical uses are extensional. (08)
> 5. The word 'type' has a long history of use as a technical
> term whose meaning was very close to its use as a common
> English word. Unfortunately, Bertrand Russell gave it a
> very narrow sense in logic, which caused many logicians
> to introduce the word 'sort' with a slightly different
> technical sense. Meanwhile computer language designers
> adopted the word 'type' for the kinds of data structures
> used in their systems. All these uses fall within the
> scope of the informal senses of the word 'type', and it
> seems possible to adopt 'type' as a generalization that
> would include all the technical senses and most, if not
> all of the common senses.
>
> 6. In another unfortunate twist, the object-oriented group
> adopted the word 'class' to distinguish their thingies,
> which have associated procedures called 'methods', from
> the more static data types of the older languages. More
> recently the Semantic Webbers adopted 'class' for the
> thingies used in OWL, but as Chris, Michael, and others
> have pointed out, OWL also uses the word 'type', and
> all of its definitions are very confused and confusing.
>
> 7. Meanwhile, the word 'class' has long been used as a
> synonym for 'set' in some versions of set/class theory.
> In other versions of set theory, a class is something
> that is just as extensional as a set, but too "big"
> to be a proper set.
>
>The major argument for using the word 'class' is that it has
>been adopted by the Semantic Web movement, which currently
>has a great deal of momentum. But one could also say that
>the Semantic Web has also adopted the word 'type' in a way
>that seems to be very closely linked to the word 'class'.
>
>Summary, I strongly believe that the word 'class' is the
>*worst* possible choice to represent the intensional side
>of the distinction between intensions and extensions:
>
> 1. Logicians use the word 'class' in the same extensional
> sense as the word 'set'.
>
> 2. Object-oriented programmers use the word 'class' as
> something that has associated procedures, and that
> is definitely *not* a meaning we want to suggest.
>
> 3. The OWL group uses both 'class' and 'type' in ways that
> are related, but it seems safe to say that there is
> probably nobody in the OWL community who could give
> a clear argument for any distinction between them.
>
>In short, the distinction between intension and extension
>is fundamental to ontology, and it *must* be highlighted.
>The word 'class' would hopelessly confuse the issue. (09)
I prefer: the distinction between type and instance, and thus also
between type and extension (= class of instances) is fundamental to
ontology, and I hope it will be taken as seriously as it deserves to
be taken. The word 'class' would hopelessly confuse the issue.
BS (010)
>John Sowa
>
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