a definition depends on context

[Richard H. McCullough]

I rarely see any discussion of this fact. Perhaps it's too obvious. Perhaps it's "impossible" to identify a definition.

I use the Aristotle definition: genus and differentia. When I see the typical ontology concept hierarchy, I say to myself:

this concept has many genera; it is a combination of different definitions from different contexts. I define the ambiguity

of a concept hierarchy as the sum of log(number of genera) over its concepts.

What is your view of definitions?

Richard H. McCullough

http://ContextKnowledgeSystems.org
What is your context?

[Ronald Stamper]

i’ll follow a lexicographer over several definitional bridges from word to words only so far,

indeed, up to the gulf between signs and reality.  

Then I want an ostensive definition, the kind of bridge for which metaphysicians built ontologies.

At that point, I grow nervous in Ontolog company, because I have yet to discover what kind of bridges

you build between signs and reality.  

Alternatively, can you direct me to a step-by-step demonstration of computer science ontology helping the brother 

and sister, in the opening scenario in the T B-L et al, 2001 Sci Am paper, helping their mother with her healthcare.

From time to time, I ask a version of this question.  Each time I feel I’m gatecrashing a party in the wrong clothes.

I do so because our research was always conducted under Popper’s Refutationist rules, so we sought challenges

of that kind and regarded them as substitutes for wine in the hands of a gatecrasher.

We arrived at a very different way of handling semantics, which we have rigorously tried to falsify.  At this stage,

it seems acceptably refutation-proof for practical purposes, where it has been hugely successful.

I would greatly appreciate having my curiosity satisfied.

Kind regards to all,

Ronald

 

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[Alex Shkotin]

Richard,

for me, the definition is an element of a particular theory. For example in geology, we have definitions of igneous rocks recommended by IUGS.

It's possible to have more than one definition of the same something in the same theory, but then we must prove equivalence.

So Theory first!:-)

Alex

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[Jon Awbrey]

Ontologgers,

In some early math course I learned a fourfold scheme
of Primitives (undefined terms), Definitions, Axioms,
and Inference Rules. But later excursions tended to run
the axioms and definitions together, speaking for example
of mathematical objects like geometries, graphs, groups,
topologies, etc. ad infinitum as defined by so many axioms.
And later still I learned correspondences between axioms
and inference rules that blurred even that line, making
the distinction appear more a matter of application and
interpretation than set in stone.

Be that as it may, the important theme running through
all these variations is always whether the formal system
inaugurated by whatever ritual is a system of consequence
or not, whether and how well it determines a category of
mathematical objects and, if you are of an applied mind,
whether these objects serve the end of understanding the
reality that does not cease to press on us.

Regards,

Jon

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[John F Sowa]

Richard, Ronald, Alex, and Jon,

I'll start with the point that Alex made:

AS

> for me, the definition is an element of a particular theory.

I agree that every definition from Aristotle to the present is
a statement within some theory T. But you would have to add
more detail about syntax and truth conditions.

RHM

> I use the Aristotle definition: genus and differentia.

Yes. That is a method of stating those conditions in a convenient
way, but there are other equivalent ways.

RHM

> it is a combination of different definitions from different contexts.
> I define the ambiguity of a concept hierarchy as the sum of log(number
> of genera) over its concepts.

In a system of logic, you might have many equivalent definitions
in the same theory. But if that theory had two non-equivalent
definitions for the same term, it would collapse in contradiction.

If you're talking about natural languages, you get into all
the complexities of lexicography. See, for example, the article
"I don't believe in word senses" by the lexicographer Adam Kilgarriff:
https://www.kilgarriff.co.uk/Publications/1997-K-CHum-believe.pdf

RS

> I grow nervous in Ontolog company, because I have yet to discover

> what kind of bridges you build between signs and reality...

> our research was always conducted under Popper’s Refutationist rules,

I very strongly agree with that observation. The bridges I have
been building are based on Peirce's philosophy of signs and science.

Popper is a special case of Peirce. Popper wrote his major books
before he discovered Peirce, but when he did, he admitted that
Peirce had expressed the same observations in somewhat different
terminology. For more about them, google "Peirce and Popper".

JA

> In some early math course I learned a fourfold scheme of Primitives
> (undefined terms), Definitions, Axioms, and Inference Rules. But

> later excursions tended to run the axioms and definitions together...

Yes. The choice of which terms to designate as primitive is
often arbitrary. It's more general to say that a theory is
the deductive closure of a set of axioms -- and treat the
definitions as axioms, perhaps in a specialized syntax.

JA

> And later still I learned correspondences between axioms and

> inference rules that blurred even that line...

That distinction can be drawn much more sharply. But it would take
a fair amount of time and effort to explain it in an email note.

JA
> the important theme running through all these variations is...

All the issues about what is a logic and how to use various
logics have been thoroughly analyzed. For example, look at the
attached diagram dol1.jpg. (This is from the OMG standard for DOL.)

But those issues about logic should be clearly distinguished
from the issues of ontology (what assumptions about existence are
we expressing in some logic), epistemology (what can we know, how
can we know it), and philosophy of science (epistemology applied
to experimental and theoretical methods in science).

John

[Alex Shkotin]

Jon,

Any definition may be written in a form of an axiom, but this is just a trick in mathematical logic. It does not eliminate the logic of definition when we build a theory and study its objects.

It's like with many-sorted logics: any set of sorts may be reduced to set of unary predicates with disjoint axiom. As Maltzev mentioned in his book "Algebraic systems".

But someone should have a very special mind to keep real numbers and vectors in one sort of vector spaces formal theory. See Kolmogorov and Dragalin book "Introduction to mathematical logic". 

Any axiom A1 may be expressed as an inference rule |-A1, but this is another trick in mathematical logic. 

Math objects are constructed or sometimes exist before any axioms and we investigate what kind of axioms they are satisfying.

And let me add that one math objects are finite (graphs, some algebraic systems) and others are infinite (another algebraic systems). Even categories (in math) some are finite, other - not.

I like finite:-) It's simpler to work with them especially if we put them into a computer.  

And having applied mind may I underline that nowadays we have the situation when all things you have mentioned must be implemented in a formal language with processor realized on a computer and available for public:-)

For example from math area:

- We have Coq as an example of a language with a public processor https://coq.inria.fr/

- And HoTT is an example of formal math http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations.html 

There are definitely some more. And maybe we just have a competition of applicability where it seems today OWL2/DL is a leader?

And what do you think about " ONTOLOGY=FORMAL_THEORY+FINITE_MODEL"? 

See more here http://ontologforum.org/index.php/Blog:Formal_theory,_finite_model_and_DL_reasoner

Alex

[Alex Shkotin]

" AS

for me, the definition is an element of a particular theory.

I agree that every definition from Aristotle to the present is
a statement within some theory T.  But you would have to add
more detail about syntax and truth conditions.
"

John, let's first agree on what particular theory we do develop: formal geology, formal chemistry, formal biology etc., ed est which one of an existing theory we are formalizing.

For example to get formal definitions of igneous rocks we need linear inequalities in syntax.

Alex

[Richard McCullough]

To all:

I definitely do approach this topic from the context of natural language and epistemology. 

I think mostly bottom-up -- concepts are formed by grouping perceptual measurements 

of things that I am sure do exist. 

I am more concerned with how people use mathematical tools,

than with how machines use them.

Because of Ontolog Forum, I am aware of articles like "I don't believe in word senses".

But I view a concept hierarchy as a characterization of human interpretations and intents,

not a pure mathematical object.

Dick

[John F Sowa]

On 4/21/2018 1:29 PM, Richard McCullough wrote:
> Because of Ontolog Forum, I am aware of articles like
> "I don't believe in word senses". But I view a concept
> hierarchy as a characterization of human interpretations
> and intents, not a pure mathematical object.

That article by Adam K. is about the hierarchies in dictionaries
written about the way humans use words when they're talking with
and writing to other humans.

Sue Atkins originally said "I don't believe in word senses."
She came to that conclusion during her career of writing
definitions for dictionaries, originally for Collins and
later for the OED.

Sue and Adam frequently worked together in organizing and
teaching a "master class" for people whose job is to define
words for dictionaries designed for humans to communicate
more effectively with other humans.

So if you want to know what kinds of hierarchies you need
to represent what people think, read that article.

For a book by a linguist and lexicographer who has some
good insights into the ways that people use words and
relate them to one another, I recommend

Hanks, Patrick (2013) Lexical Analysis: Norms and Exploitations,
Cambridge, MA: MIT Press.

John

[Alex Shkotin]

Dick,

I am more concerned with formal ontology:-)

Alex

[John F Sowa]

Alex,

> I am more concerned with formal ontology :-)

The more formal your ontology happens to be, the less likely
that it will be useful for any practical application.

Just look at all those so-called "formal" ontologies -- which the
designers "align" with the very informal WordNet. Even if you
don't use WordNet, your users will think and talk in ordinary
language -- so any data they enter will be hopelessly informal.

In any case, I'll mention two of your very formal points:

> Any definition may be written in a form of an axiom, but this
> is just a trick in mathematical logic. It does not eliminate the
> logic of definition when we build a theory and study its objects.

First, a definition is assumed to be true "by definition", but
an axiom is just assumed to be true. If an axiom is false about
something, there is no paradox. It just means that the thing
in question doesn't exist in any model of the theory.

But the claim that certain axioms are definitions gives them
a higher status. That creates the so-called "Russell" paradox
about the set of all sets that are not members of themselves.

Cantor noticed that so-called "set" long before Russell.
But he dismissed it by saying that it violates the axioms.
Therefore, it cannot exist in any model. End of paradox.

But Frege stated the critical axiom in his *definition* of sets.
That meant that the paradoxical thing must exist "by definition".
It could not be dismissed. Ergo, contradiction. And panic.

Common Logic avoids paradoxes by adopting Cantor's policy.
CL does not have any keyword spelled D-E-F-I-N-I-T-I-O-N.
In CL, you can write axioms, but you can't declare that any
of them are true "by definition".

> It's like with many-sorted logics: any set of sorts may be reduced
> to set of unary predicates with disjoint axiom. As Maltzev mentioned
> in his book "Algebraic systems".

Yes, but. And this is a very big **BUT**: The set of models of
the axioms with the reduced version is much, much bigger than the
set of models of the sorted logic. This is significant for theorem
proving. The proofs with sorted logic can be orders of magnitude
faster than the proofs with the reduced version.

There are also important proofs about sorted logic that are not
true about the reduced version. For quotations and citations,
see http://jfsowa.com/logic/sorts.pdf

John

[Jon Awbrey]

RM: What is your view of definitions?

A recurring question, always worth some thought,
so I added my earlier comment to a long-running
series on my blog concerned with Definition and
Determination.

Definition and Determination : 15
https://inquiryintoinquiry.com/2018/04/21/definition-and-determination-15/

Those two concepts are closely related, almost synonyms in their etymologies,
both of them having to do with setting bounds on variation. And that brings
to mind, a cybernetic mind at least, the overarching concept of “constraint”,
which figures heavily in information theory, systems theory, and engineering
applications of both.

As it happens, I have been working for as long as I can remember on a project
that eventually came to fly under the banner of “Inquiry Driven Systems” and
in the early 90s I returned to grad school in a systems engineering program
as a way of focusing more resolutely on the systems aspects of that project.

Here's a budget of excerpts on Definition and Determination I collected around
that time, mostly from C.S. Peirce, since his pragmatic paradigm for thinking
about information, inquiry, logic, and signs forms the platform for my efforts,
plus a few bits from sources before and after him.

Collection Of Source Materials on Definition and Determination
http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey/EXCERPTS

Regards,

Jon

[John F Sowa]

On 4/22/2018 12:34 PM, Jon Awbrey wrote:
> Collection Of Source Materials on Definition and Determination
> http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey/EXCERPTS

That's a very good collection of quotations about definitions,
most of them by Peirce and some by other notables. It reminded me
to download page 1504 of the _Century Dictionary_, which contains
Peirce's definitions for the words 'definite', 'definition',
'definitional', 'definitive', and 'definitum':
http://jfsowa.com/peirce/defs/definition.jpg

I also recommend the article "Definitions, dictionaries, and meanings"
by Norman Swartz: https://www.sfu.ca/~swartz/definitions.htm

Swartz earned a BA in physics and a PhD in the philosophy of science.
He developed this article for the 30+ years of students that he taught
at Simon Fraser U. I recommend it for anybody who is defining or using
an ontology.

On a related theme, I also recommend an article on "Laws of Nature",
which Swartz wrote for the International Encyclopedia of Philosophy:
http://www.iep.utm.edu/lawofnat/

The following excerpt states a critical controversy for ontology:
> Within metaphysics, there are two competing theories of Laws of Nature.
> On one account, the Regularity Theory, Laws of Nature are statements
> of the uniformities or regularities in the world; they are mere
> descriptions of the way the world is. On the other account, the
> Necessitarian Theory, Laws of Nature are the "principles" which govern
> the natural phenomena of the world. That is, the natural world "obeys"
> the Laws of Nature. This seemingly innocuous difference marks one of
> the most profound gulfs within contemporary philosophy, and has quite
> unexpected, and wide-ranging, implications.

Section 3, "Shared Elements in the Competing Theories", states the
issues on which both sides agree. A general-purpose ontology must
agree on these points. To avoid endless arguments, it should avoid
a commitment to issues on which they disagree.

Swartz also wrote an article about theories of truth for the IEP.
It's not bad, but it shows that he had never studied Peirce.
The following excerpt would apply to James, but it is absolutely
and egregiously false about Peirce. In fact, this was the issue
that caused Peirce to reject the term 'pragmatism' and replace it
with the word 'pragmaticism' which, as he said, "is ugly enough
to be safe from kidnappers." (CP 5.414)

> 6. Pragmatic Theories
> A Pragmatic Theory of Truth holds (roughly) that a proposition is
> true if it is useful to believe. Peirce and James were its principal
> advocates. Utility is the essential mark of truth. Beliefs that lead
> to the best "payoff", that are the best justification of our actions,
> that promote success, are truths, according to the pragmatists.

If you delete this section, the rest of the article is OK.

For my summary of Peirce's definition of truth and a criticism
of Quine's failure to understand Peirce, see pp. 31 to 38 of
http://jfsowa.com/pubs/signproc.pdf

John

[Richard McCullough]

John and Jon,

We clearly have some differences in the "definition" of "definition".

In my epistemology context, a definition is a 

statement/proposition which can be true or false.

I recommend the discussion in Chapter 7 of "How We Know"

by Harry Binswanger. I find the 21st century English

much easier to understand.

Dick 

[Jon Awbrey]

Dick,

I suppose it all depends on the sorts of things one wants to define.
We could call that the “context of application” I guess. I am not
as much focused on an ontology as a “Large Online Lexicon” (LOL) as
I am on the task of “Acquiring Scientific Knowledge” (ASK), and so
the sorts of things I need to define are systems of relationships,
formal or mathematical models that we use as intermediate objects
to deal with phenomena and the realities behind those phenomena.

These sorts of objects, intermediate and ultimate, typically have
high levels of complexity that we are forced to approach in stages,
often beginning with “toy worlds” in the time-honored AI fashion.
Those are the sorts of definitions I am after. We could call them
“specifications” if it helps to use another word.

Regards,

Jon