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 genus1 with differentia1
and
A is genus2 with differentia2 (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: "[ontologforum]" <ontologforum@xxxxxxxxxxxxxxxx>
Sent: Tuesday, September 09, 2008 6:54 AM
Subject: Re: [ontologforum] 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, contradictionfree multiple inheritance.
>
> John Sowa
>
>
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> (06)
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