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Re: [ontolog-forum] Thing and Class

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Tue, 09 Sep 2008 14:26:28 -0400
Message-id: <48C6BFD4.2060803@xxxxxxxxxxx>
Dick,    (01)

Aristotle's syllogisms are basically that subset of FOL
that uses just monadic predicates.    (02)

 > I am surprised that all of Aristotle's syllogisms
 > are still valid under multiple inheritance.    (03)

There is nothing magic about multiple inheritance.  It is
just a use of implication applied to monadic predicates.    (04)

> Multiple inheritance may not have contradictions,
> but it does introduce ambiguity.  For example,
>     A is genus-1 with differentia-1
> and
>     A is genus-2 with differentia-2
> In other words, A has two different definitions.    (05)

If the two definitions are equivalent, there is no ambiguity
whatever.  For example, you might define the type Dog as
Mammal with differentia A, and I might define Dog as Animal
with differentiae A & B.    (06)

But if (Mammal & A) is logically equivalent to (Animal & A & B),
then our definitions are identical.  With software such as FCA,
all such implications are automatically generated.    (07)

 > In today's jargon, A has two different word senses,
 > which apply in two different contexts.    (08)

If two word senses are provably identical, they are the
*same* word sense.  In mathematics, it is very common for
different authors to state different definitions that are
provably equivalent.  For some good analysis and examples,
I recommend the article cited below, from which I extracted
an example that illustrates the above point.    (09)

That article should be required reading for anyone who is
defining, using, or talking about ontologies.    (010)

________________________________________________________________    (011)

http://www.sfu.ca/philosophy/swartz/definitions.htm    (012)

Notes on Definitions by Norman Swartz    (013)

Excerpt:    (014)

Consider the case of "square".  What is 'the' intensional definition
of "square"? Here are five competing candidates, i.e. sets of logically 
necessary and jointly sufficient conditions.    (015)

    1. "square" =df "a plane closed figure that has exactly four sides 
all of which are straight and equal to one another and whose interior 
angles each measure 90 degrees"    (016)

    2. "square" =df "a plane closed figure having four straight sides 
and whose diagonals are both equal in length to one another and bisect 
one another at right angles"    (017)

    3. "square" =df "a straight-sided, plane, closed figure, every 
diagonal of which cuts the figure into two right isosceles triangles"    (018)

    4. "square" =df "an equilateral parallelogram containing no 
(interior) acute angles"    (019)

    5. "square" =df "an equilateral parallelogram containing four axes 
of symmetry"    (020)

The first of these may be the closest (fairest) representative of what 
most persons 'have in mind' when they use the term "square", but the 
other four 'definitions' also 'pick out' the identical extension. If the 
former is to be privileged as being the one to earn the accolade "'the' 
intensional definition", it cannot be on its logical features alone: it 
can be regarded as 'the' intensional definition only by invoking 
empirical data about language-users' tacit rules.    (021)

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