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Re: [ontolog-forum] Simplifying the interface for teaching and using ont

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: sowa@xxxxxxxxxxx
Date: Sun, 6 Jan 2013 12:58:52 -0500 (EST)
Message-id: <3f2c3fe85f1ded223556a75ae726659d.squirrel@xxxxxxxxxxxxxxxxxxxx>

I'm still traveling, and this email handler is an abomination.  Following is the note I've tried to send for the past two days.

------------------------------------------------------------------------------

Leo,

We have no disagreements about the need for logic:

 1. As Peirce said, "The very first lesson that we have a right to
    demand that logic shall teach us is, how to make our ideas clear."

 2. And formal notations of any kind (logic, mathematics, programming
    languages, etc) are designed to represent highly specialized topics
    more clearly, precisely, and unambiguously than ordinary NLs.

But I also want to emphasize two additional principles:

 3. Anything that can be stated in any artificial notation can also be
    expressed in any natural language -- but usually in a longer sentence
    that is not as convenient for specialists in the field.

 4. Unnecessary complications in either notation -- natural or artificial
    -- are reflected in unnecessary complications in the other.  They are,
    in Peirce's terms "a failure to make our ideas clear."

As an example of these principles, I'd like to cite Aristotle's four
sentence types in controlled English and their mapping to logic:

   "Every A is B."  =>  subtypeOf(A,B).

   "Some A is B."  =>  x is some A; instanceOf(x,B).

   "No A is B."  =>  disjoint(A,B).

   "Some A is not B."  =>  x is some A; ~instanceOf(x,B).

Note that there is no need for the word 'class'.  That word is a composite
term that can be defined in terms of a monadic relation (AKA type) and
some notion of set or collection:

   Definition:  A class A is a pair (S,R), where S is the set of all x
   for which the relation R(x) is true.

Aristotle's four sentence types can be expressed in terms of classes,
but the English sentences are more awkward and far less readable:

   "The class of all A is a subclass of the class of all B."

   "Some member of the class A is a member of the class B."

   "The class A and the class B are disjoint."

   "Some member of the class A is not a member of the class B."

It's possible to choose symbols for an artificial notation that can
express these sentences more concisely.  But each of them would require
a definition in terms of sets and relations:  another sign that the
word 'class' is not an ideal choice "to make our ideas clear."

The other mathematical _expression_ that was overly complex was based
on ideas derived from Cantor's set theory.

In mathematical English, one can say

   "The function f maps a set D to a set R."

In the usual notation for functions, one would write

   f: D -> R.

This is a good, clear notation that is easy to teach and learn.

But Cantor showed that the cardinality of the set of all functions
from D to R is the cardinality of the set R raised to the power of
the cardinality of D:  card(f) = card(R)^card(D).

Because of that fact, mathematicians who are talking about set theory,
sometimes use the abbreviation R^D for the set of functions from D to R.

Instead of writing "f: D -> R", they write "f is a member of R^D".

The first notation has a simple mapping to English.  The second can
also be mapped to an English sentence, but it would only be readable
by people who have been heavily indoctrinated in Cantor's set theory.

When we're trying to teach ontology to ordinary university graduates,
we cannot assume that they are professional mathematicians.  Therefore,
it is essential to use the barest minimum of mathematical jargon.

For a popular summary of the basic notations for math & logic, see

   http://www.jfsowa.com/logic/math.htm

This 30-page summary has received over 176,000 hits.  In presenting and
and talking about ontology, there is no need to go beyond the level of
that summary -- except, possibly, for some extremely specialized topics.

John


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