I disagree, John. The point of using mathematical (or logical) notation is to
make your statement precise and unambiguous.
(01)
You can't have it both ways: 1) avoiding terminology wars by using precise
logical statements (witness the recent discussion here on "individual"), and 2)
avoiding logical and mathematical notation by using supposedly clearer natural
language.
(02)
One reason for the success of ontological engineering in general, as for formal
ontology in philosophy, is that one can write logical axioms, thereby
curtailing somewhat the endless argumentation in English and other natural
languages that has gone on in philosophy for generations (and often goes on
here).
(03)
Logic and mathematics are formal tools that ontologists and computer scientists
need to use, so they can express their ontological analysis in logical
expressions that they can then share/compare. Ontological analysis does not
reduce to logical analysis, but uses logic as a way to formalize what it wants
to say.
(04)
I do think that you need to use simple but precise English for neophytes, but
at some point ontologists (and ontological engineers) need to learn logic and
at least some mathematics. One can use good English and formal definition
together.
(05)
Thanks,
Leo
(06)
>-----Original Message-----
>From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
>bounces@xxxxxxxxxxxxxxxx] On Behalf Of John F Sowa
>Sent: Thursday, January 03, 2013 6:36 AM
>To: ontolog-forum@xxxxxxxxxxxxxxxx
>Subject: Re: [ontolog-forum] Intensional relation
>
>On 1/3/2013 12:19 AM, Hassan Aït-Kaci wrote:
>> The notation "2^S" for a set S denotes the set of all subsets of S -
>> i.e., it powerset (also written P(S) sometimes). It is because its
>> cardinality |S| is equal to 2^|S| that this notation has been used.
>
>I agree.
>
>But that is no excuse for writing statements like:
>
>> "An intensional relation (or conceptual relation) ρ^n of arity n
>> on <D,W> is a total function ρn : W → 2^D^n from the set W into
>> the set of all n-ary (extensional) relations on D"
>
>Mathematicians don't think like that. They only use such language
>when they are deliberately trying to frighten the unwashed.
>
>Following is a quotation by Paul Halmos, whose books were used to
>teach the mathematicians who talk that way. But thankfully, he
>never wrote that way. If he had, nobody would have ever read
>his books.
>
>As Halmos said, the intuition is fundamental. After you understand
>the fundamental ideas, writing the formalism is trivial: "it is more
>the draftsman’s work not the architect’s."
>
>Teaching ontology by burying the fundamental insights under
>the trivial notation is pedagogical malpractice.
>
>John
>___________________________________________________________
>_____________
>
>Paul Halmos:
>
>“Mathematics — this may surprise or shock some — is never deductive
>in its creation. The mathematician at work makes vague guesses,
>visualizes broad generalizations, and jumps to unwarranted conclusions.
>He arranges and rearranges his ideas, and becomes convinced of their
>truth long before he can write down a logical proof... the deductive
>stage, writing the results down, and writing its rigorous proof are
>relatively trivial once the real insight arrives; it is more the
>draftsman’s work not the architect’s.”
>
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(07)
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