Re: [ontolog-forum] Types of Formal (logical) Definitions in ontology

[Alex Shkotin <alex.shkotin@xxxxxxxxx>]

John,

you wrote:

"AS
...
For a useful survey of the many issues about definitions, see
http://faculty.cbu.ca/rkeshen/Logic/Definitions,%20Dictionaries,%20and%20Meanings,%20by%20Norman%20Swartz,%20Dept_%20of%20Philosophy,%20Simon%20Fraser%20University.htm

"

and have moved me back to 06.11.10 as follows

"Dear Prof. Swartz,

it is my pleasure to read your "Definitions, Dictionaries, and Meanings".

And reading section 8.4 "The Possibility of an Overabundance of Necessary and Sufficient Conditions"

I'd like to share one formal idea:

definition is part of a theory.

It is hard to give a good definition. It is unique to build a system of useful definitions.

And following Euclid, Book I, Definitions item 22, we have

“square” =df “a right-angled equilateral quadrilateral figure”

Formally we may substitute definitions of "right-angled", "equilateral", "quadrilateral" and paraphrase, then we get 8.4 #1.

All other are necessary and sufficient conditions and need a proof.

To assign a rank of "definition" to one of equivalent conditions is a privilege of a theory author:-)

Sincerely yours,

Alex Shkotin"

thank you,

Alex

2014-06-24 20:28 GMT+04:00 John F Sowa <sowa@xxxxxxxxxxx>:

Robert, Alex, et al.,

RR

> what do you mean by "closed-form definitions"?

In mathematics and other fields, it's common to distinguish
an explicit definition from an implicit definition.

For example, following is an implicit definition of a function f:

    x^2 + 3*f(x) - 4 = 0

And following is an explicit or *closed-form* definition:

    f(x) = (4 - x^2) / 3

But many implicit definitions cannot be converted to closed-form:
See http://www-math.mit.edu/~tchow/closedform.pdf

RR

> I understand Edward Barkmeyer's reservations as well, but perhaps

> i'm not getting the gist of the whole context you have in mind.

In another note, I clarified the point I was trying to make:  the
proliferation of metalevel terminology is more confusing that helpful.
Whenever possible, reduce the number of metalevel terms by replacing
them or defining them in terms of some version of logic.

RR

> Based on my studies in philosophy FOL is essentially presented as

> a modern form or translation of [Aristotle's syllogisms].

A's syllogisms are a subset of FOL.  They are also the most widely
used subset of description logics such as OWL.

AS

> For example OWL 2 Full is not FOL.

RDF and OWL full allow quantifiers to range over relations.
Classical FOL does not allow that.  Common Logic is an extension
of FOL that was designed to support that option.

RR

> Now, my concern with this is that since syllogistic logic is not

> how the mind reasons...

First, nobody knows the details of how the mind reasons.  But one
point is certain:  the human mind supports *every* version of reasoning
that anybody has ever used.  That includes all of mathematics and logic
as well as every kind of intuition, creativity, meditation, or whatever.

Second, every digital computer is a logic machine.  Anything that cannot
be defined in some version of logic cannot be programmed
in a digital computer.

And third, every notation for any version of reasoning, formal or
informal, can be explained in and translated to natural languages.

RR

> why isn't a non-syllogistic-based logic used for ontologies?
> Why is FOL used?

The basis for syllogisms is a statement that term A is or is not more
general (applies to more instances) than term B.  That's an integral
part of every natural language.  You use it whenever you speak.

The reason for FOL is that it's a subset of every NL.  Whenever you
use the words 'and', 'or', 'not', 'some', and 'every', you're
speaking FOL.  If you just take something as simple as RDF triples
and add negation, you get the expressive power of FOL.

Basic point:  if you try to extend RDF, it's much easier to define
full FOL than it is to avoid FOL.  One of the major reasons for the
complexity of OWL is that they tried to *avoid* the full expressive
power of FOL.  That imposes many constraints on what you can say.

RR

> Aside from that, please continue mentioning any other logics
> that are used.

All other logics are subsets, supersets, or variations of FOL.
Fuzzy logic, for example, assumes a continuous range of fuzzy
levels from absolutely certain (true) to absolutely not certain
(false).  For a discussion of the relationships, see

   http://www.jfsowa.com/pubs/rolelog.pdf
   The role of language and logic in reasoning

AS
> non-FOL def:  VPC(p)(ob) ≝ 100*Volume_in(p,ob)/Volume(ob).

Any mathematical function or operator can be used in an FOL statement.

For a useful survey of the many issues about definitions, see
http://faculty.cbu.ca/rkeshen/Logic/Definitions,%20Dictionaries,%20and%20Meanings,%20by%20Norman%20Swartz,%20Dept_%20of%20Philosophy,%20Simon%20Fraser%20University.htm

John

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