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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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