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Re: [ontolog-forum] Thing and Class

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Richard H. McCullough" <rhm@xxxxxxxxxxxxx>
Date: Tue, 9 Sep 2008 07:31:49 -0700
Message-id: <074777F3D3024947A1F9ACE0FAAC3E7F@rhm8200>
John
Thank you.
I was not aware of the Leibniz method.
I am surprised that all of Aristotle's syllogisms
are still valid under multiple inheritance.    (01)

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    (02)

In other words, A has two different definitions.
In today's jargon, A has two different word senses,
which apply in two different contexts.    (03)

Dick McCullough
Ayn Rand do speak od mKR done;
mKE do enhance od Real Intelligence done;
knowledge := man do identify od existent done;
knowledge haspart proposition list;
http://mKRmKE.org/    (04)

----- Original Message ----- 
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
Sent: Tuesday, September 09, 2008 6:54 AM
Subject: Re: [ontolog-forum] Thing and Class    (05)


>I received an offline note with some questions that may have occurred
> to others.  Therefore, I'm sending my response to this forum.
>
> > I have never seen any discussion of multiple inheritance and
> > Aristotle's syllogisms.  Do you have any references?
>
> Multiple inheritance is implicit in Aristotle's theory of categories.
> The fundamental principles are
>
>  1. A definition of a new category (or type) by specifying its
>     genus or supertype and the differentiae that distinguish
>     the new type from any siblings of the same supertype.
>
>  2. The four types of statements (A, I, E, O), which relate
>     categories with four combinations of negation, existential
>     quantifier, and universal quantifier:
>
>     A:  Every A is a B.       (Universal affirmative)
>     I:  Some A is a B.        (Particular affirmative)
>     E:  No A is a B.          (Universal negative)
>     O:  Some A is not a B.    (Particular negative)
>
>  3. The valid forms of syllogisms, which the medieval Scholastics
>     named by the combinations of vowels in the sentence types.
>     The first pattern, for example, is named Barbara:
>
>        Every A is a B.
>        Every B is a C.
>        Therefore, every A is a C.
>
> Aristotle did not present his categories as a tree, and he did not
> talk about multiple inheritance.  In the 3rd century AD, Porphyry
> wrote an introduction to the categories, in which he drew a tree
> to illustrate the relationships in one of Aristotle discussions.
>
> In the 17th century, Leibniz developed a numerical method for
> reasoning about the categories, which defined a lattice with
> multiple inheritance:
>
>  1. Assign a unique prime number to each differentia.
>
>  2. Represent each category A by the product of the primes for
>     all its differentiae.
>
>  3. The number for any category can be computed by a recursive
>     definition:
>
>        Let 1 represent the Supreme Genus.
>
>        If any category C is defined by genus G with differentia D,
>        let the number for C be the product of the numbers for G and D.
>
>  4. Then you can test whether every A is B by checking whether the
>     number for A divides the number for B.
>
> Leibniz did not invent terms for lattice or multiple inheritance,
> but his system and Aristotle's supported them.  He didn't use the
> term 'Gödel numbering' for his method, but the similarity was not
> a coincidence, since Leibniz was Kurt Gödel's favorite philosopher.
>
> Many people think that multiple inheritance can create contradictions.
> But that is the wrong way to view the lattice.  Neither A. nor L.
> began by drawing a tree or lattice.  Instead, they began by stating
> *definitions* that *generate* the lattice.  If you generate the
> lattice from the definitions (which the FCA software does), you are
> guaranteed to have correct, contradiction-free multiple inheritance.
>
> John Sowa
>
>
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>
>     (06)



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