2 ideas after our last meeting (2018.1.24)

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alex.shkotin

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Jan 25, 2018, 8:27:51 AM1/25/18
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Hi All!

1) Sometimes I think it's better not to use word "ontology" at all. And try to substitute "formal theory" or "finite model" instead. And only if some formal text keeps together elements of formal theory and finite model we should use word ontology;-)

2) Trying to apply Category theory to knowledge representation is as good as old. Let me refer to http://beniaminov.rsuh.ru/ (as my FB-friend pointed out me), but nowadays we ask CTh-enthusiasts what kind of tools do they give us? pencil and paper? :-)

Alex

Tom Tinsley

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Jan 25, 2018, 11:51:53 AM1/25/18
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Hi All,

 

The discussion on category theory has been excellent. My take away is that It has a strong mathematical base but an almost zero level of usage.

 

Some answers may be found at: https://otterserver.com/category/catalog-for-knowledge-documents/

 

Tom

 

Sent from Mail for Windows 10

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Cory Casanave

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Jan 25, 2018, 12:05:11 PM1/25/18
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The discussion and presentation on ontolog sounds a lot like what has been done for the “DOL” (Distributed Ontology, Model, and Specification Language Specification) standard at OMG. (Not that I understand all the math).

 

DOL: http://www.omg.org/spec/DOL

Ontolog: http://ontologforum.org/index.php/ConferenceCall_2018_01_24

 

 

-Cory

Pat Hayes

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Jan 27, 2018, 3:54:38 PM1/27/18
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> On Jan 25, 2018, at 7:27 AM, alex.shkotin <alex.s...@gmail.com> wrote:
>
> Hi All!
>
> 1) Sometimes I think it's better not to use word "ontology" at all. And try to substitute "formal theory" or "finite model" instead. And only if some formal text keeps together elements of formal theory and finite model we should use word ontology;-)

Finite model? In what sense of ‘model’? Many formal theories don’t have finite models in the sense of ‘model thoery', and shouldn’t have them. Arithmetic, for instance. But perhaps (?) you mean some other sense of ‘model’ ?

Pat



Alex Shkotin

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Jan 29, 2018, 8:15:12 AM1/29/18
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Pat,

We use a finite model of rational numbers - something like the algebra of numbers modulo 10^20.
Any DB may be seen as a finite model but not very math.
Some finite categories mentioned on the last meeting are finite models for us.
What is a model? We may ask it instead of reality and get the same answer. Math models are cheaper than physical and more robust than DBs.

Alex

Pat Hayes

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Jan 29, 2018, 1:49:32 PM1/29/18
to Alex Shkotin, ontolog-forum
Alex

I think we are talking at cross purposes. At any rate, you do not seem to be using “model” in the sense of model theory, ie an interpretation which makes a theory true, so I should not comment on your postings any further. 

Best wishes

Pat Hayes

Alex Shkotin

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Jan 30, 2018, 7:57:36 AM1/30/18
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Pat,

I count on finite models of Description Logics.

Alex

Jack Hodges

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Jan 30, 2018, 10:24:13 AM1/30/18
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The term ‘model’ has such a broad range of definitions that it seems almost absurd to seek consensus on it. Is model theory the baseline for discussions of semantic models? I would be interested in knowing your views on the relationship between conceptual models and model theory.

Jack

Sent from my iPad

On Jan 30, 2018, at 4:57 AM, Alex Shkotin <alex.s...@gmail.com> wrote:

Pat,

I count on finite models of Description Logics.

Alex


2018-01-29 21:49 GMT+03:00 Pat Hayes <pha...@ihmc.us>:
Alex

I think we are talking at cross purposes. At any rate, you do not seem to be using “model” in the sense of model theory, ie an interpretation which makes a theory true, so I should not comment on your postings any further. 

Best wishes

Pat Hayes


On Jan 29, 2018, at 7:15 AM, Alex Shkotin <alex.s...@gmail.com> wrote:

Pat,

We use a finite model of rational numbers - something like the algebra of numbers modulo 10^20.
Any DB may be seen as a finite model but not very math.
Some finite categories mentioned on the last meeting are finite models for us.
What is a model? We may ask it instead of reality and get the same answer. Math models are cheaper than physical and more robust than DBs.

Alex

2018-01-27 23:54 GMT+03:00 Pat Hayes <pha...@ihmc.us>:


> On Jan 25, 2018, at 7:27 AM, alex.shkotin <alex.s...@gmail.com> wrote:
>
> Hi All!
>
> 1) Sometimes I think it's better not to use word "ontology" at all. And try to substitute "formal theory" or "finite model" instead. And only if some formal text keeps together elements of formal theory and finite model we should use word ontology;-)

Finite model? In what sense of ‘model’? Many formal theories don’t have finite models in the sense of ‘model thoery', and shouldn’t have them. Arithmetic, for instance. But perhaps (?) you mean some other sense of ‘model’ ?

Pat






Alex Shkotin

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Jan 30, 2018, 11:57:11 AM1/30/18
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Jack,

Pat and I we keep the same approach in mind - Model Theory from Math-Logic. I'd like to point that we need to create formal theories for every science and technology (for ex. our group have made a project for Geology) and get a lot of finite models for them.
But the problem comes from numbers: we use numbers but do not axiomatize them. And as I understand from the previous discussion, Pat points that there is no finite model if we use numbers.  
It's a little bit subtle to show that in practice we use a finite model of Numbers.

Anyway, the main point is that ontology = formal theory + finite model.

Alex


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henson graves

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Jan 30, 2018, 2:41:07 PM1/30/18
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In my experience engineers and logicians use the term "model" very differently. Engineers develop models for systems under design or analysis, perhaps in OWL or UML. If formalized the model becomes an axiom set used to reason about the interpretations in the physical or a simulated world. Logicians speak of the interpretations of axiom sets as models. So when this is formalized one has

axiom set <-> engineer's model

logician's model <-> engineer's interpretation. Interpretations include simulations of engineer's models.


- Henson




From: ontolo...@googlegroups.com <ontolo...@googlegroups.com> on behalf of Jack Hodges <jhodg...@gmail.com>
Sent: Tuesday, January 30, 2018 9:24 AM
To: ontolo...@googlegroups.com
Subject: Re: [ontolog-forum] 2 ideas after our last meeting (2018.1.24)
 

Jon Awbrey

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Jan 30, 2018, 3:56:12 PM1/30/18
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Henson, List,

Different senses of “model” usually divide
into the “logical” and the “analogical”.
Here's some thoughts along those lines:

Objects, Model, Theories
1. https://inquiryintoinquiry.com/2013/09/10/objects-models-theories-1/
2. https://inquiryintoinquiry.com/2013/11/20/objects-models-theories-2/
3. https://inquiryintoinquiry.com/2013/11/21/objects-models-theories-3/
4. https://inquiryintoinquiry.com/2013/11/27/objects-models-theories-4/

Regards,

Jon

On 1/30/2018 2:41 PM, henson graves wrote:
> In my experience engineers and logicians use the term "model" very differently. Engineers develop models for systems under design or analysis, perhaps in OWL or UML. If formalized the model becomes an axiom set used to reason about the interpretations in the physical or a simulated world. Logicians speak of the interpretations of axiom sets as models. So when this is formalized one has
>
> axiom set <-> engineer's model
>
> logician's model <-> engineer's interpretation. Interpretations include simulations of engineer's models.
>
>
> - Henson
>

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Jack Hodges

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Jan 30, 2018, 7:14:23 PM1/30/18
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Our work is on the engineering side of things, in OWL/RDFS.

Jack

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John F Sowa

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Jan 30, 2018, 9:15:10 PM1/30/18
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Spencer, Cory, Alex, Pat H, and Jack H,

In last week's Ontology Summit, I grew impatient while listening
to a talk by Spencer Breiner about categories.

Cory
> The discussion and presentation on ontolog sounds a lot like what
> has been done for the DOL...

Yes, but there is a difference. Category theory and institutions
are great for mapping formal systems. See the attached dol1.jpg,
which shows the mappings among 24 formal logics used for ontology
and related applications.

But the thousands of independently defined ontologies are just a
miscellaneous selection. They don't form a coherent category with
the kinds of systematic mappings shown in dol1.jpg.

However, I agree with the statement on slide 4 of Spencer's talk:
> Build mixed contexts by inheriting from libraries

When people build ontologies by selecting predefined modules from
a library, they are using a lattice of theories with a lattice of
mappings. Those theories and mappings form a category. In his talk,
I wish that Spencer had emphasized the *libraries*, how to build them,
and how to use them. The fact that they form a category is worth
*one slide* for anyone who happens to know something about categories.

But note what Cory said: "Not that I understand all the math."
One slide would inform him that the ideas are related to DOL.
The details of that relationship would require a short course,
not a half-hour talk.

Alex
> Pat points that there is no finite model if we use numbers. It's a
> little bit subtle to show that in practice we use a finite model of
> Numbers.

Pat is absolutely correct. And Alex is correct that we can never
use or represent more than a finite subset of numbers. What Pat
meant is that there is no upper bound on that subset. As our
computers become bigger and faster, we use more of them. There
is no natural stopping point: You can always enlarge your subset.
That is the meaning of infinity: there is no end (last number).

Jack H
> The term ‘model’ has such a broad range of definitions that it
> seems almost absurd to seek consensus on it. Is model theory
> the baseline for discussions of semantic models?

For any logic used to specify an ontology, there is a consensus,
and Alfred Tarski stated it: A model M of a formal theory T
expressed in a logic L is a set theoretic construction for which
every axiom of T is true.

The attached mthworld.gif illustrates that consensus: On the right
is a theory T expressed by five axioms in a first-order logic L.
In the middle is a model M, which consists of a set of nodes
(represented by dots) and a set of relations (represented by
lines that connect the does). Each axiom has a denotation True
of False in terms of the model M.

That abstract model M can be used as an approximation to some aspect
of the world w, which is shown on the left of mthworld.gif. But that
mapping depends on methods of measurement that can never be absolutely
precise. Therefore, the mapping of M to w is only an approximation,
which may be judged as good, fair, or poor for some application.

John
dol1.jpg
mthworld.gif

Pat Hayes

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Jan 30, 2018, 9:49:15 PM1/30/18
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> On Jan 30, 2018, at 8:15 PM, John F Sowa <so...@bestweb.net> wrote:
>
> Spencer, Cory, Alex, Pat H, and Jack H,
>
>
….
> Alex
>> Pat points that there is no finite model if we use numbers. It's a
>> little bit subtle to show that in practice we use a finite model of
>> Numbers.
>
> Pat is absolutely correct. And Alex is correct that we can never
> use or represent more than a finite subset of numbers. What Pat
> meant is that there is no upper bound on that subset.

Not exactly, though that is indeed true. What I meant was that pretty much any axiomatization of arithmetic does not have finite models. That is the whole point of arithmetic: it's the theory of *the natural numbers*. Obviously, being finite creatures, we can never use more than a finite number of numerals; that is a trivial observation. But what we might call the machinery of arithmetic, which we also use, itself is based on there being infinitely many numbers.

Alex says that he is using a finite model of Numbers. I am not sure what this means, but if he has an axiomatisation of Arithmetic which has finite models, I would love to see it. Just for a start, if 0 is a Number, and if every Number has a successor which is different from it, and if no Number’s successor is 0, then there are infinitely many Numbers. Putting addition and multiplication into the mix just confirms the infinity.

Pat



John F Sowa

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Jan 31, 2018, 1:26:08 AM1/31/18
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On 1/30/2018 9:49 PM, Pat Hayes wrote:
> Alex says that he is using a finite model of Numbers. I am
> not sure what this means, but if he has an axiomatisation
> of Arithmetic which has finite models, I would love to see it.

There are, of course, no finite models of the integers or the
real numbers. But IEEE floating-point arithmetic, which is
usually used as an approximation to the real numbers, is finite.

On the other hand, all the theoretical work and theorem provers
are based on the real numbers or the integers. Mathematicians,
physicists, and engineers happily use systems such as Mathematica
to do the theorem proving and symbolic computation.

For symbolic computation, infinite models are actually *simpler*
than theorem proving with IEEE arithmetic -- primarily because
you never bump up against the upper bounds.

John

Alex Shkotin

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Jan 31, 2018, 8:02:43 AM1/31/18
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Henson,

interpretation for logicians as I know is a function (usually recursive one) how to calculate some logical formulas on some math system (first of all - algebraic, second - categoric). If all axioms give True, the system is a model of this axioms.
In Description Logics, they call an axiom anything from a theory statement (like "any human is mortal") to system statement (like "Socrates is a human."). 
Keeping theory and system statements separately should be very useful IMHO.

Alex

2018-01-30 22:41 GMT+03:00 henson graves <henson...@hotmail.com>:

In my experience engineers and logicians use the term "model" very differently. Engineers develop models for systems under design or analysis, perhaps in OWL or UML. If formalized the model becomes an axiom set used to reason about the interpretations in the physical or a simulated world. Logicians speak of the interpretations of axiom sets as models. So when this is formalized one has

axiom set <-> engineer's model

logician's model <-> engineer's interpretation. Interpretations include simulations of engineer's models.


- Henson



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henson graves

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Jan 31, 2018, 10:21:51 AM1/31/18
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Alex,

I agree with what you are saying. One needs to distinguish between a theory and interpreted statements about the system that the theory is intended to describe.

A typical engineering problem is to modify a product to have some new capability.  The typical scenario is to build an engineering model of the product and its operating environment in a modeling language such as UML. One then attempts to determine consequences from the combined product-operation context model and construct simulation code from the model and execute it to better determine the modified system behavior. One also may operate some modified product and collect data for the same purpose.  Generally this results in revision and refinement of the engineering model. For an example see https://scholar.google.com/scholar?oi=bibs&cluster=8615220581478398249&btnI=1&hl=en

If you view this engineering activity within a standard logic formalism paradigm, the model is an axiom set, the conclusions derived from the model are statements in the theory of the axiom set, and the simulations, as well as the product operation scenarios are interpretations of the theory.

I am suggesting that the logic paradigm is a good description of the engineering paradigm with of course the change in terminology, e.g., engineering model = axiom set. There are a lot of consequences from adoption of the paradigm. Here are three.

1. The kind of logic used is not given a prior. The kind of logic and form of the theories are determined by the test and evaluation methods accepted in the domain.

2. Physicists and philosophers often think that they are trying to build an axiom set which has a single unique (categorical) interpretation. For engineering the axiom sets that they build to describe systems generally have more valid interpretations than intended. This simply means that part of the engineer’s job is to determine what assumptions need to be added to an axiom set to constrain the possible interpretations to correspond with the physical world.

3. At some stage the engineer may need a meta theory in which he can represent object theories and their interpretations. This is likely the consequence of your statement.

Henson





From: ontolo...@googlegroups.com <ontolo...@googlegroups.com> on behalf of Alex Shkotin <alex.s...@gmail.com>
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Alex Shkotin

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Jan 31, 2018, 10:25:36 AM1/31/18
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​​
Pat,

in fact, if ontology uses numbers for any particular value (as we do not have just natural or rational numbers but always number with a unit of measure) it should be axioms for upper, low boundaries and accuracy for any value. For example, percentage value must be in 0..100, but accuracy depends on particular science and technology. Or if I say that I have $10^80 in my bank account nobody put it in his/her finite system (sorry, ontology).
But of course, as John mentioned, it's easier to be infinite, i.e. ignore boundaries and accuracy matters.

Alex

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Alex Shkotin

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Jan 31, 2018, 10:58:06 AM1/31/18
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Henson,

for me "the model is an axiom set" is not a good point of view as this should be a very special kind of axioms. It's better for me to think (as Tarski:-) that the model is math-structure satisfying some axioms.
W need a special language to build math-structures by the way.
What kind of math-structures engineers use to model their production it's another matter, but this structures (for example labeled graphs) are finite with numbers:-)

Alex

2018-01-31 18:21 GMT+03:00 henson graves <henson...@hotmail.com>:

Alex,

I agree with what you are saying. One needs to distinguish between a theory and interpreted statements about the system that the theory is intended to describe.

A typical engineering problem is to modify a product to have some new capability.  The typical scenario is to build an engineering model of the product and its operating environment in a modeling language such as UML. One then attempts to determine consequences from the combined product-operation context model and construct simulation code from the model and execute it to better determine the modified system behavior. One also may operate some modified product and collect data for the same purpose.  Generally this results in revision and refinement of the engineering model. For an example see https://scholar.google.com/scholar?oi=bibs&cluster=8615220581478398249&btnI=1&hl=en

If you view this engineering activity within a standard logic formalism paradigm, the model is an axiom set, the conclusions derived from the model are statements in the theory of the axiom set, and the simulations, as well as the product operation scenarios are interpretations of the theory.

I am suggesting that the logic paradigm is a good description of the engineering paradigm with of course the change in terminology, e.g., engineering model = axiom set. There are a lot of consequences from adoption of the paradigm. Here are three.

1. The kind of logic used is not given a prior. The kind of logic and form of the theories are determined by the test and evaluation methods accepted in the domain.

2. Physicists and philosophers often think that they are trying to build an axiom set which has a single unique (categorical) interpretation. For engineering the axiom sets that they build to describe systems generally have more valid interpretations than intended. This simply means that part of the engineer’s job is to determine what assumptions need to be added to an axiom set to constrain the possible interpretations to correspond with the physical world.

3. At some stage the engineer may need a meta theory in which he can represent object theories and their interpretations. This is likely the consequence of your statement.

Henson





Sent: Wednesday, January 31, 2018 7:02 AM
To: ontolog-forum

Subject: Re: [ontolog-forum] 2 ideas after our last meeting (2018.1.24)
Henson,

interpretation for logicians as I know is a function (usually recursive one) how to calculate some logical formulas on some math system (first of all - algebraic, second - categoric). If all axioms give True, the system is a model of this axioms.
In Description Logics, they call an axiom anything from a theory statement (like "any human is mortal") to system statement (like "Socrates is a human."). 
Keeping theory and system statements separately should be very useful IMHO.

Alex
2018-01-30 22:41 GMT+03:00 henson graves <henson...@hotmail.com>:

In my experience engineers and logicians use the term "model" very differently. Engineers develop models for systems under design or analysis, perhaps in OWL or UML. If formalized the model becomes an axiom set used to reason about the interpretations in the physical or a simulated world. Logicians speak of the interpretations of axiom sets as models. So when this is formalized one has

axiom set <-> engineer's model

logician's model <-> engineer's interpretation. Interpretations include simulations of engineer's models.


- Henson



Cory Casanave

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Jan 31, 2018, 11:07:51 AM1/31/18
to ontolo...@googlegroups.com

Henson,

A good analysis. Also consider the “forward engineering” scenario. For physical items this could be 3D printing a design. For a software system this could be producing software artifacts (A.K.A. Model Driven Architecture – MDA).

 

In both cases there is a “source model” (set of axioms) and a set of “production rules”, which can be thought of as “production axioms”.

 

There is an interesting difference between physical and software production – the 3D printed item is the final “real thing” in the world. Produced software is, of course, a real thing in the world but is also, essentially, a set of axioms describing the data and processes the software will process (this assumes software can be accepted as a set of axioms). So we have the “source model” (set of axioms) transformed by a transformation model/rule (set of axioms) producing software (set of axioms) that act on statements about the world. Those “statements about the world” are what logicians typically call models! Perhaps they are all models.

 

Of course the source model (set of axioms) can also be processed by a different set of axioms for the simulation paradigm you describe. That the same model can be interpreted with different axioms for different purposes points to the need for unifying semantics.

 

-Cory

 

From: ontolo...@googlegroups.com [mailto:ontolo...@googlegroups.com] On Behalf Of henson graves
Sent: Wednesday, January 31, 2018 10:22 AM
To: ontolog-forum <ontolo...@googlegroups.com>
Subject: Re: [ontolog-forum] 2 ideas after our last meeting (2018.1.24)

 

Alex,

John F Sowa

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Jan 31, 2018, 11:55:55 AM1/31/18
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Alex and Henson,

Alex
> if ontology uses numbers for any particular value (as we do not have
> just natural or rational numbers but always number with a unit of
> measure) it should be axioms for upper, low boundaries and accuracy
> for any value.

As Pat and I have been repeating, the word 'finite' does not
simplify anything -- and more often than not, it's false.

Re axioms for ontology: If you want your ontology to be general,
you don't want it to include upper & lower bounds, accuracy, etc.
That info may differ from one application to another.

Pat said that he would not reply to anything more on this topic.
This is also my last note on the subject. We agree that your use
of the word 'finite' is not helpful and usually wrong. But that's
your problem, not ours.

Henson
> the model is an axiom set, the conclusions derived from the model
> are statements in the theory of the axiom set, and the simulations,
> as well as the product operation scenarios are interpretations of
> the theory.

That first sentence is the most confusing. Axioms are always statements
in some theory. The word 'model' has different meanings in different
fields, but no field ever talks about the axioms of a model.

Besides logic, there are two other uses of the word 'model' that
may be used in applications:

1. Engineering models: They may be physical things, such as a scale
model that is smaller than the final product. Or they may be
computer simulations of the physical things.

2. Data models: That term is used for database systems to distinguish
different representations, such as a relational model, a network
model, a hierarchical model, or an object-oriented model.

Suggestion: I believe that the three-way distinction shown in the
attached mthworld.gif could be generalized to cover all these ways
of using the words 'theory' and 'model':

1. I first drew that diagram to illustrate a Tarski-style model of
a theory stated in logic and the relationship between a model
and the world.

2. For engineering models, the theory could be stated by axioms in
logic, but it could also be stated in mathematics (which could
also be mapped to logic). And it could even be stated in ordinary
language supplemented with mathematics.

3. The engineering model could be a simplified or scaled down
physical system or it could be a computer simulation. In either
case, it would conform to the theory as exactly as possible,
and the final product or physical implementation would conform
to the model as exactly as possible.

4. For data models, the tables, networks, or hierarchies are
different ways of representing the same abstract theory (AKA
conceptual schema). In fact, mthworld.gif shows the model
as a network. But a set of tables could also be true of
exactly the same axioms.

In summary, I believe that it would be possible to redraw mhtworld.gif
in ways that would be acceptable for logic, engineering, and databases.
In each version, the thing on the left would be physical, the thing
in the middle would be called a model, and the thing on the right
would be called a theory, ontology, or specification.

John
mthworld.gif

henson graves

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Jan 31, 2018, 11:58:26 AM1/31/18
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Alex,

The fact that different communities use the same terminology for very different things should not be the show stopper that it seems to be in this forum. Logicians have one set of terminology, engineers have another. They use the word model for different things. Since I occasionally talk to people of both communities, I often use the words engineering model or descriptive model for what engineers talk about and interpretation of a model for what logicians mean when they say model.


I don't see that Cory needs to change his terminology e.g., MDA.  One doesn't have to conflate or confuse the concepts once one understands what is going on.  What I have outlined is completely consistent with the Tarskian view point, and with what one finds in books on model theory.

I agree that engineer's models (aka axiom sets) are often somewhat different from the ones that logicians typically deal with and the interpretation theory (logician's model theory) has some interesting aspects that Cory mentions.  But that only raises interesting issues for ontology discussion.

Henson


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henson graves

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Jan 31, 2018, 12:03:12 PM1/31/18
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John,

Of course it doesn't make sense to talk about the axioms of a model in the Tarskian sense, but the way that engineers use the word model such as the design model for an aircraft, the design is an axiom set, and the intepretations are; model in the Tarskian sense of the word.


Henson


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henson graves

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Jan 31, 2018, 12:19:44 PM1/31/18
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I have spent considerable building design and analysis models in various UML languages for both aircraft and molecules. I have also spent some time using OWL to build the same kind of "models" of aircraft and molecules. Of course these models are called axiom sets in OWL. My purpose was to see if OWL could be used in engineering model development as OWL of course provides reasoning and the UML languages do not. I was doing the same kind of activity in both cases. Same activity and artifacts, but different names.


The important thing to me seems to realize that the same activity is going own in both domains with different names. It seems not a point of real interest that the same names are used for different concepts.


I know pretty much what one finds in a book on model theory. It doesn't change what I am saying.


Henson




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Pat Hayes

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Jan 31, 2018, 3:53:21 PM1/31/18
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On Jan 31, 2018, at 11:19 AM, henson graves <henson...@hotmail.com> wrote:

... It seems not a point of real interest that the same names are used for different concepts.

It only of interest, or better of importance, when people use such a wildly ambiguous technical word in a multidisciplinary forum like this, without clarifying which sense of the word they mean. If one took all the emails to just this forum which have been devoted to clearing up the confusion caused by one person’s use of “model” being mis-read by others as meaning something different from what the writer intended, it would probably fill a fairly large book. The point is not that anyone is right or wrong, only that communication sometimes requires brief amounts of pedantry, to avoid mutual misunderstanding.

Best wishes

Pat Hayes

Pat Hayes

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Jan 31, 2018, 8:19:45 PM1/31/18
to ontolog-forum, John F. Sowa
Just for the record:

> On Jan 31, 2018, at 10:55 AM, John F Sowa <so...@bestweb.net> wrote:
>
> ...
> Suggestion: I believe that the three-way distinction shown in the
> attached mthworld.gif could be generalized to cover all these ways
> of using the words 'theory' and 'model':

I disagree, and think that this diagram is profoundly misleading. But John and I have had this argument in public now at least three times, and we should probably not have it again.

Pat

John F Sowa

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Feb 1, 2018, 3:12:04 AM2/1/18
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On 1/31/2018 8:19 PM, Pat Hayes wrote:
>> I believe that the three-way distinction shown in the attached
>> mthworld.gif could be generalized to cover all these ways
>> of using the words 'theory' and 'model'
>
> I disagree, and think that this diagram is profoundly misleading.
> But John and I have had this argument in public now at least three
> times, and we should probably not have it again.

I agree. But I'll summarize the two positions. Anyone who has
heard the arguments before may stop reading *here*.

I thought of Pat when I wrote that note. I have shown
that diagram (copy attached) to many people who have strong
backgrounds in logic and philosophy. Some of them agree with
Pat's position and sometimes object even more violently than Pat.
But others look at the diagram and say it's obvious.

Pat's position, as I understand it, is that the domain of discourse
of a statement in logic may be some abstract set (such as integers
or other mathematical constructs). But it just as well could be
a set of things in the physical world.

An argument for Pat's position is that words of ordinary language
can and do refer to things in the physical world. When you translate
a sentence from some NL to some version of logic, the referents of
the variables in the logical sentence should be the same as the
referents of the corresponding words in the original NL sentence.

One reason for a distinction between the model and the physical world
is that it's more flexible. It enables us to distinguish the way
people think and talk about the world from the way it actually is.
Different people may have had different experiences and ways of
thinking. Some might be more accurate than others. For some, the
model might be a plan for a future that doesn't exist now or ever.

But mthworld.gif does not rule out the possibility that the model
in the diagram has an *exact* mapping to the world. In that case,
you could, if you wish, identify the model with part of the world.

John
mthworld.gif

Alex Shkotin

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Feb 1, 2018, 10:44:54 AM2/1/18
to ontolog-forum
Henson,

I put the core question of mine in ontologforum blog http://ontologforum.org/index.php/Blog:Formal_theory,_finite_model_and_DL_reasoner
Let's continue there in comments and perhaps we get something useful for all.

Alex 

2018-01-31 19:58 GMT+03:00 henson graves <henson...@hotmail.com>:

Alex,

The fact that different communities use the same terminology for very different things should not be the show stopper that it seems to be in this forum. Logicians have one set of terminology, engineers have another. They use the word model for different things. Since I occasionally talk to people of both communities, I often use the words engineering model or descriptive model for what engineers talk about and interpretation of a model for what logicians mean when they say model.


I don't see that Cory needs to change his terminology e.g., MDA.  One doesn't have to conflate or confuse the concepts once one understands what is going on.  What I have outlined is completely consistent with the Tarskian view point, and with what one finds in books on model theory.

I agree that engineer's models (aka axiom sets) are often somewhat different from the ones that logicians typically deal with and the interpretation theory (logician's model theory) has some interesting aspects that Cory mentions.  But that only raises interesting issues for ontology discussion.

Henson

From: ontolo...@googlegroups.com <ontolog-forum@googlegroups.com> on behalf of Alex Shkotin <alex.s...@gmail.com>
Sent: Wednesday, January 31, 2018 9:58 AM

To: ontolog-forum
Subject: Re: [ontolog-forum] 2 ideas after our last meeting (2018.1.24)
 
Henson,

for me "the model is an axiom set" is not a good point of view as this should be a very special kind of axioms. It's better for me to think (as Tarski:-) that the model is math-structure satisfying some axioms.
W need a special language to build math-structures by the way.
What kind of math-structures engineers use to model their production it's another matter, but this structures (for example labeled graphs) are finite with numbers:-)

Alex
2018-01-31 18:21 GMT+03:00 henson graves <henson...@hotmail.com>:

Alex,

I agree with what you are saying. One needs to distinguish between a theory and interpreted statements about the system that the theory is intended to describe.

A typical engineering problem is to modify a product to have some new capability.  The typical scenario is to build an engineering model of the product and its operating environment in a modeling language such as UML. One then attempts to determine consequences from the combined product-operation context model and construct simulation code from the model and execute it to better determine the modified system behavior. One also may operate some modified product and collect data for the same purpose.  Generally this results in revision and refinement of the engineering model. For an example see https://scholar.google.com/scholar?oi=bibs&cluster=8615220581478398249&btnI=1&hl=en

If you view this engineering activity within a standard logic formalism paradigm, the model is an axiom set, the conclusions derived from the model are statements in the theory of the axiom set, and the simulations, as well as the product operation scenarios are interpretations of the theory.

I am suggesting that the logic paradigm is a good description of the engineering paradigm with of course the change in terminology, e.g., engineering model = axiom set. There are a lot of consequences from adoption of the paradigm. Here are three.

1. The kind of logic used is not given a prior. The kind of logic and form of the theories are determined by the test and evaluation methods accepted in the domain.

2. Physicists and philosophers often think that they are trying to build an axiom set which has a single unique (categorical) interpretation. For engineering the axiom sets that they build to describe systems generally have more valid interpretations than intended. This simply means that part of the engineer’s job is to determine what assumptions need to be added to an axiom set to constrain the possible interpretations to correspond with the physical world.

3. At some stage the engineer may need a meta theory in which he can represent object theories and their interpretations. This is likely the consequence of your statement.

Henson




henson graves

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Feb 1, 2018, 11:17:37 AM2/1/18
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 John

My view of your diagram is very close to yours. But the inability to recognize how engineering terminology has evolved in the last 50 years, perhaps improperly from your point of view, is causing an immense amount of unnecessary confusion. One wonders if propagating confusion is the point of this thread.

To attempt to explain the difference in terminology, how it arose, and see how engineers are beginning to employ a standard logic paradigm as depicted in your diagram consider the following engineering creation myth.

People in ancient times, before say 1985 proceeded pretty much like you describe. People use test and verbal language to describe things they want to build or analyze. They then often built prototypes often on reduced scale which most everybody called models. So far everything is exactly as you say. Then something happened over a course of the next several years. The engineers started replacing their verbal and text description with artifacts in languages such as UML and OWL. This didn’t happen overnight and was not successful overnight. The reasons should be of great interest to logicians and KRR folks. But after year 2000 things changed. Now these artifacts are becoming the authoritative source of information.

Reasoning from the artifacts  are used to make design decisions;  they are used to generate simulations which help understand properties of systems under design or analysis. Engineers called these new artifacts “models” which is certainly not in keeping with the terminology of the Tarskian tradition. Some engineers have realized that these artifacts, which they call models, are axiom sets or can be embedded as axiom sets in logic. The result of doing so gives some well-known tools from logic to apply to questions of correctness of reasoning and validity of simulations (interpretations in synthetic worlds). This development promises to fundamentally change the way engineering is done on a daily basis.

 Unfortunately some people with a logic background get confused about the different use of terminiology, e.g., engineering model as axiom set and logical model as interpretation. Perhaps engineers should be chastised for calling their artifacts models, but they do. Perhaps “ model driven analysis”, “model based system engineering”, and “Unified Modeling Language” should be forced to change their names.  Historically the engineering artifacts were as you describe. But now the term model is generally used in the engineering community to mean the diagrams which translate to axioms or are directly axiom sets.  As I have said the interpretations of the engineering models are Tarskian models for the appropriate logic.  

So while your diagram is ok, the gloss is incomplete in my opinion. The real unfortunate aspect is it seems to preclude logicians from understanding how engineering is beginning to almost be applied logic. I would like for logicians to understand this scenario and contribute to the developments in logic, model theory needed to obtain specific benefits from using this well-known paradigm as expressed in your diagram, as opposed to chasing one’s tail regarding the use of the word “model”.

Henson





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Pat Hayes

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Feb 1, 2018, 3:09:23 PM2/1/18
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> On Feb 1, 2018, at 2:12 AM, John F Sowa <so...@bestweb.net> wrote:
>
> On 1/31/2018 8:19 PM, Pat Hayes wrote:
>>> I believe that the three-way distinction shown in the attached
>>> mthworld.gif could be generalized to cover all these ways
>>> of using the words 'theory' and 'model'
>> I disagree, and think that this diagram is profoundly misleading.
>> But John and I have had this argument in public now at least three
>> times, and we should probably not have it again.
>
> I agree. But I'll summarize the two positions.

Sigh. I had hoped you would not do this, but here goes.

> Anyone who has
> heard the arguments before may stop reading *here*.
>
> ..
> Pat's position, as I understand it, is that the domain of discourse
> of a statement in logic may be some abstract set (such as integers
> or other mathematical constructs). But it just as well could be
> a set of things in the physical world.

Yes, that is the first observation, which is not a ‘position’, but simply a fact. But it goes beyond this, since of course the domain of discourse *could* also be made of mathematical entities. Your diagram is seriously misleading because it takes the matter of how a ‘model’ (in any sense) can be approximation to a reality – issues of degrees of precision, tolerance, approximation, accuracy and so forth – outside the semantic framework of formal ontologies and their semantics altogether. If your diagram were accurate, the relationship on the RHS of the diagram would have to be something that cannot *in principle* be described by any ontology on the far left. But (1) such matters *can* be described in formal ontologies; and more seriously (2) if they were outside this scope, as the diagram claims, what theory or framework do you suggest we could use to talk about them? I have never seen anyone, including your good self, explain how we can even begin to talk about the proposed relationship between ‘formal models’ and reality, if our semantic theories – that is, model theories – stop before these matters can even be brought into their scope.

>
> An argument for Pat's position is that words of ordinary language
> can and do refer to things in the physical world. When you translate
> a sentence from some NL to some version of logic, the referents of
> the variables in the logical sentence should be the same as the
> referents of the corresponding words in the original NL sentence.
>
> One reason for a distinction between the model and the physical world
> is that it's more flexible. It enables us to distinguish the way
> people think and talk about the world from the way it actually is.

It makes a distinction, but it does not provide any way to talk about it. In fact, it makes it *impossible* to talk about it, because all talk can only be about what the semantics of that talk says are the referents of the talk, and that only gets us to the models. So how can ANY talk be about the real world?

> Different people may have had different experiences and ways of
> thinking. Some might be more accurate than others. For some, the
> model might be a plan for a future that doesn't exist now or ever.

Of course, but that has nothing to do with the debate here. These are just observations about semantics generally.

>
> But mthworld.gif does not rule out the possibility that the model
> in the diagram has an *exact* mapping to the world. In that case,
> you could, if you wish, identify the model with part of the world.

But you still have this strange distinction between the real things on the right and their intermediary doppelgangers in the middle, even when they are in exact 1:1 correspondence. And this is just plain wrong: that is not how model theory works, nor how the original designers of it were thinking. Tarski’s running example of a sentence was “Snow is white”, and the truth conditions for this are that snow - actual, real, snow - is in fact white - the actual color, white. The language is related to the world; names refer to referents. Semantics is not a three-way relationship, it is a direct mapping between names the things they denote. That is how model theory works, and how it always has worked. If you want to talk about approximations and so forth, by all means do so, but the Tarskian semantics applies to that talk just like it applies to all other talk, which is why we can have ontologies about things like approximation and measurement tolerances and the difference between a quantity and a measurement of that quantity and so on.

But, as I say, let us not have this argument yet again :-)

Pat


>
> John
>
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> <mthworld.gif>


Edward Barkmeyer

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Feb 2, 2018, 12:28:07 AM2/2/18
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Cory,

 

I have a real problem with this:

> In both cases there is a “source model” (set of axioms) and a set of “production rules”, which can be thought of as “production axioms”.

 

“Production rules” are transformation rules.  The significant question for transformations is:  What properties of the source do they preserve in the image?  And, to some extent, their preservation capability may be limited by the differences in nature between the source milieu and the target milieu.  A precise mathematical vector maps to an imperfect graphical display and to an even more imperfect physical cut line or deposition line.  And as Pat pointed out, a mapping from an n-ary fact to RDF does not maintain the integral sense when viewed as a set of triples; that sense must be imposed on the RDF graph, but it is not present in the RDF milieu per se.

 

So, in what sense are “production”/”transformation” rules “axioms”?  What is the nature of the logic in which they are “true”?  The truth seems to be only that the target image is a representation of the source.  And those are the kind of axioms we often call “simple facts”.  The interesting axioms are those that enable one to reason about behaviors of the target entities in the target milieu from the facts and axioms that describe the behaviors of the source entities in the source milieu.  That is: the preservation axioms and the mutation axioms (what is predictably different).

 

I have spent a large part of my life developing rule-based transformations in software (which is the nature of 90% of all software) and in machine control.  They are all “algorithmic”, but it is not clear that any of them is “axiomatic”.  And it is really important not to confuse those concepts.

 

-Ed

John F Sowa

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Feb 2, 2018, 9:49:37 AM2/2/18
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On 2/2/2018 12:28 AM, Edward Barkmeyer wrote:
> I have spent a large part of my life developing rule-based
> transformations in software (which is the nature of 90% of all
> software) and in machine control. They are all “algorithmic”,
> but it is not clear that any of them is “axiomatic”. And it is
> really important not to confuse those concepts.

Every algorithm can be specified by a set of axioms in FOL,
and those axioms can be automatically translated to algorithms.
Production rules can also be translated to statements in logic,
which can be executed directly by logic-programming systems
or be translated to algorithms.

Those issues were hashed out and concluded in debates between Gödel,
Church, and Turing at Princeton in the 1930s. Their conclusion was
that recursive functions defined by axioms (Gödel's preferred form),
lambda calculus (by Church), and computers with an infinite tape
(Turing) had exactly the same computational power. They proved
that claim by defining translations from one to the other.

A major part of computer science and practice has been devoted
to implementing those ideas from the 1930s in the latest and
greatest programming languages and systems. The modern debates
are ways of saying "My version of the 1930s is better than yours."

John

Jack Hodges

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Feb 2, 2018, 10:16:09 AM2/2/18
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I would be interested in a moderated forum devoted to semantics and engineering. Perhaps a companion forum to ontolog so that these interesting philosophical, theoretical, and historical discussions can take place, here, and topics which are engineering specific can take place there, trying to keep them independent as much as practical.

Jack

Sent from my iPad

John F Sowa

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Feb 2, 2018, 11:10:30 AM2/2/18
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Pat and Mary-Anne,

I changed the subject line to include notes by both of you.

PJH
> ["Pat's position"], which is not a ‘position’, but simply a fact.

Whoa! A fact about what? The physical world? Some publication?
If the latter, please cite the source. It was certainly not Tarski.

Note the title of Tarski's original paper (1933): "The concept of
truth in formalized languages." For example, "Schnee ist weiß"
if and only if snow is white. But then he said that the issues
about natural languages and the world are too vague and complex.
That paper only addressed the right-hand side (RHS) of mthworld.gif.
For the LHS, see the informal philosophical paper by Tarski (1944):
http://jfsowa.com/logic/tarski.pdf

PJH
> Your diagram is seriously misleading because it takes the matter
> of how a ‘model’ (in any sense) can be approximation to a reality
> – issues of degrees of precision, tolerance, approximation, accuracy
> and so forth – outside the semantic framework of formal ontologies
> and their semantics altogether.

I believe that the correct term is 'metalevel', not 'outside'.

Some excerpts from Tarski (1944), sections 20, 21, and 22:
> The most natural and promising domain for the applications of
> theoretical semantics is clearly linguistics — the empirical study
> of natural languages...
>
> The relation between theoretical and descriptive semantics is analogous
> to that between pure and applied mathematics, or perhaps to that between
> theoretical and empirical physics... another important domain for
> possible applications of semantics is the methodology of science; this
> term is used here in a broad sense so as to embrace the theory of
> science in general... The semantics of scientific language should be
> simply included as a part in the methodology of science... One of the
> main problems of the methodology of empirical science consists in
> establishing conditions under which an empirical theory or hypothesis
> should be regarded as acceptable...
>
> As regards the applicability of semantics to mathematical sciences and
> their methodology, i.e., to metamathematics, we are in a much more
> favorable position than in the case of empirical sciences.

Formal mathematics is the only field for which Tarski (1944) claimed
that his definition of truth was directly applicable. He didn't deny
that it could be extended, but his discussion implied that extensions
would be in the "methodology" -- i.e., metalevel.

PJH
> I have never seen anyone, including your good self, explain how
> we can even begin to talk about the proposed relationship between
> ‘formal models’ and reality, if our semantic theories – that is,
> model theories – stop before these matters can even be brought
> into their scope.

It's a two-step mapping: In 1933, Tarski specified the RHS of
mthworld.gif. In 1944, he discussed the then current methodologies
for the LHS and admitted that they weren't formal. I doubt that he
would approve of making them formal by magic: Waving your hand and
declaring "Presto-zingo, my domain consists of things in the world."
That statement is OK as a hypothesis, but it's not an observation.

If you want a one-step mapping, look at fuzzy logic. Since I had
made some sympathetic comments about fuzzy systems, I was invited
to contribute to the Festschrift for Lotfi Zadeh. I wrote a 7-page
article on the question "What is the source of fuzziness?" and
included mthworld.gif as Figure 3: http://jfsowa.com/pubs/fuzzy.pdf

Since I didn't want to offend Lotfi, I didn't criticize fuzzy logic
directly. But I implied that it would be better to use a two-stage
mapping with classical logic on the RHS and fuzzy sets (or something
related) on the LHS. I don't believe that the LHS can ever be
completely formal, because no system of measurement can be perfect.

MAW
> a cool formalisation of symbol grounding
> http://www.benjaminjohnston.com.au/papers/formal.pdf

Symbol grounding addresses the LHS of mthworld.gif. Page 5
of the article discusses the informal issues:
> Representation units may or may not have any particular semantic
> interpretation, and may be manipulated by rules (such as interaction
> with the environment or hyper-computational systems) that are beyond
> formal definition.

Yes. For humans, symbols are grounded by what Peirce called
"the gates" of perception and purposive action. Methods of
pattern recognition and robotics address those two gates, but
none of them can be completely formal at the points of contact.

John
mthworld.gif

Rich Cooper

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Feb 2, 2018, 11:44:08 AM2/2/18
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Jack,

If you start one, sign me up!

Sincerely,

Rich Cooper,

Rich Cooper,

Chief Technology Officer,

MetaSemantics Corporation

MetaSemantics AT EnglishLogicKernel DOT com

( 9 4 9 ) 5 2 5-5 7 1 2

http://www.EnglishLogicKernel.com

From: ontolo...@googlegroups.com [mailto:ontolo...@googlegroups.com] On Behalf Of Jack Hodges
Sent: Friday, February 02, 2018 7:16 AM
To: ontolo...@googlegroups.com
Subject: Re: [ontolog-forum] 2 ideas after our last meeting (2018.1.24)

I would be interested in a moderated forum devoted to semantics and engineering. Perhaps a companion forum to ontolog so that these interesting philosophical, theoretical, and historical discussions can take place, here, and topics which are engineering specific can take place there, trying to keep them independent as much as practical.

Jack

Sent from my iPad


On Jan 30, 2018, at 11:41 AM, henson graves <
henson...@hotmail.com> wrote:

In my experience engineers and logicians use the term "model" very differently. Engineers develop models for systems under design or analysis, perhaps in OWL or UML. If formalized the model becomes an axiom set used to reason about the interpretations in the physical or a simulated world. Logicians speak of the interpretations of axiom sets as models. So when this is formalized one has

axiom set <-> engineer's model

logician's model <-> engineer's interpretation. Interpretations include simulations of engineer's models.

- Henson

Pat Hayes

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Feb 2, 2018, 12:30:44 PM2/2/18
to ontolog-forum, John F. Sowa
Hi John

The various formal models of computation all have the same power in the sense that they all define equivalent notions of computable function. But that is not to say that all computational architectures define the same notion of computation, or that they can all perform the same computations as all the others. For just one thing, there is lot more to computation than which function gets computed. The computer in my pocket can, for example, respond to a phone call within a couple of seconds, an ability which is not captured by talking about what function it computes.

I agree with Ed that ‘axiomatic’ and ‘algorithmic’ are distinct ideas which we should try to keep distinct. Even when talking about something like Prolog which has both aspects, it makes sense to distinguish these aspects of its design. For example, the model theory of Prolog really has nothing to do with the algorithmic machinery of a Prolog engine, and the use of, say, a linear-time unifier has got nothing much to do with the axiomatics.

Pat

henson graves

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Feb 2, 2018, 12:45:01 PM2/2/18
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Ed,


Cory’s comments about axioms and production rules can be interpreted in the following way.

Engineering is transitioning from constructing artifacts in natural language to artifacts in UML and OWL. However, as axiom sets these artifacts are very weak as they generally have a lot more models (interpretations) than their constructors intended. Only gradually have engineers understood the implicit assumptions needed to constrain the valid models to what is intended. It takes a lot of work to identify and formalize this implicit knowledge. Some of us view these assumptions as the context of the axiom set. This usage of context is consistent with its use in logic and lambda calculus and other places.

What Cory refers to as production rules are rules used to construct simulation models for engineering and to generate code from “incomplete” artifacts used to generate software. These rules add information to the artifacts. However, this info should be and can be formalized and made part of the axiom set. This is necessary if one wants to reason correctly from the axiom sets to their interpretations.  This is now a concern in engineering.

 I don't think this view is really inconsistent with what you say.

Henson





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Sent: Thursday, February 1, 2018 11:28 PM

Pat Hayes

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Feb 2, 2018, 6:43:18 PM2/2/18
to ontolog-forum, John F. Sowa

On Feb 2, 2018, at 10:10 AM, John F Sowa <so...@bestweb.net> wrote:

Pat and Mary-Anne,

I changed the subject line to include notes by both of you.

PJH
["Pat's position"], which is not a ‘position’, but simply a fact.

Whoa!  A fact about what?  The physical world?  Some publication?
If the latter, please cite the source.  It was certainly not Tarski.

Tarki’s account of truth conditions, now usually referred to as ‘model theory’, characterizes the domain of an interpretation as being a non-empty set containing the referents of symbols. The *only* condition imposed by the theory on this set is that it be non-empty. In particular, the theory imposes no conditions whatever on the nature of the entities in this set. Ergo, it applies when the set contains real-world entites. QED.

If we need to argue about whether sets can contain real-world entities (as I recall we once did), I will refer you to Russell, Zermelo, Quine, and a host of other authorities – indeed almost everyone, including writers of textbooks, who has written extensively on the topic – but I hope we won’t need to go there again. 


Note the title of Tarski's original paper (1933):  "The concept of
truth in formalized languages."  For example, "Schnee ist weiß"
if and only if snow is white.  But then he said that the issues
about natural languages and the world are too vague and complex.

No, he said that natural languages were. (And he was right, in spite of valiant subsequent efforts by linguists.) Tarski never said that formal languages could not refer to reality. As we both know, his own running example was ‘snow is white’. 

That paper only addressed the right-hand side (RHS) of mthworld.gif.
For the LHS, see the informal philosophical paper by Tarski (1944):
http://jfsowa.com/logic/tarski.pdf

There is nothing, in any of Tarski’s writings, to suggest that his conception was like your diagram. I challenge you to give an exact citation, if you believe otherwise. That he talked about formal languages and truth in one place, and about measurement and approximation in another place, is not evidence for this claim. 

But in any case, subsequent authors have most certainly allowed formal logics to refer to real-world things, while assuming the Tarskian model theory (or some technical variant of it) without any qualms or hesitation or qualifications. The most obvious being Carnap, but also all the work on axiomatic mereologies, and of course just about all of modern formal ontology-building. 


PJH
Your diagram is seriously misleading because it takes the matter
of how a ‘model’ (in any sense) can be approximation to a reality
– issues of degrees of precision, tolerance, approximation, accuracy
and so forth – outside the semantic framework of formal ontologies
and their semantics altogether.

I believe that the correct term is 'metalevel', not 'outside’.

No, it takes it outside. The metalevel would be the (hypothetical, mysterious, unrealized) semantic theory of the RHS model-to-real-world mapping. But the image makes clear that this would not be part of model theory, the semantic theory of the LHS formal ontology: that is the LHS mapping, the Tarskian interpretation. So the ontology itself *cannot possibly* talk about the RHS of the diagram. It cannot talk about the real world, because whatever it says can only be interpreted through the middle domain of dry abstractions; so it cannot talk about the relationship between that domain and something else, such as the real-world RHS. 

What is so frustrating to me in these discussions is that you seem to keep missing this obvious point, that your diagram shoots itself in the foot in this way, because we both know that ontologies *can* be created for talking about such matters as the distinction between a real-world quantity and an observable measurement of that quantity, the bounds on possible errors of measurements and so forth: the very stuff of the RHS of your diagram, in fact. (I know this in part because I have done it, as part of an effort to creating an ontology of quantities and measures.) But on your account, any such ontology has to be some kind of illusion, since all its terms are obliged to refer to things in the central desert of mere formality, and can never reach across to the real world on the right. 

Some excerpts from Tarski (1944), sections 20, 21, and 22:
The most natural and promising domain for the applications of
theoretical semantics is clearly linguistics — the empirical study
of natural languages...
The relation between theoretical and descriptive semantics is analogous
to that between pure and applied mathematics, or perhaps to that between
theoretical and empirical physics... another important domain for
possible applications of semantics is the methodology of science; this
term is used here in a broad sense so as to embrace the theory of
science in general...  The semantics of scientific language should be
simply included as a part in the methodology of science... One of the
main problems of the methodology of empirical science consists in
establishing conditions under which an empirical theory or hypothesis
should be regarded as acceptable...
As regards the applicability of semantics to mathematical sciences and
their methodology, i.e., to metamathematics, we are in a much more
favorable position than in the case of empirical sciences.

Formal mathematics is the only field for which Tarski (1944) claimed
that his definition of truth was directly applicable.

The first paragraph you cite, above, says the exact opposite.

 He didn't deny
that it could be extended, but his discussion implied that extensions
would be in the "methodology" -- i.e., metalevel.

Tarski was indeed much more optimistic about applying logic to mathematics than to more empirical fields. In this he was of course not alone. But nothing you cite here, or indeed you can cite, I believe, suggests that he thought it was impossible in principle, or that his semantic picture needed to be “extended” along the lines of your diagram. Montague, following Tarski, used his semantics directly on natural language without significant modification to the notion of interpretation. Kripke extended the interpretation structures to cover modal logics, but did not interpose any new mappings between the content of interpretations and reality, and all the extensive literature on the nature of possibilia or counterparts in modal logic has clearly assumed that the things in Kripkean universes are parts of the actual world, or of possible worlds. 


PJH
I have never seen anyone, including your good self, explain how
we can even begin to talk about the proposed relationship between
‘formal models’ and reality, if our semantic theories – that is,
model theories – stop before these matters can even be brought
into their scope.

It's a two-step mapping:  In 1933, Tarski specified the RHS of
mthworld.gif.  In 1944, he discussed the then current methodologies
for the LHS and admitted that they weren't formal.

He did no such thing. As Tarskian scholarship, this is pure fantasy. You have retrofitted Tarski’s ideas onto your misleading diagram.

 I doubt that he
would approve of making them formal by magic:  Waving your hand and
declaring "Presto-zingo, my domain consists of things in the world.”

Why do you think that to claim to be talking about reality is to claim some kind of magic power? We all talk about reality much of the time. I daresay that some of the emails in this very discussion group have referred to reality on occassion. If I say that my ontology is about, say, family relationships among gerbils, I am not saying anything fundamentally different from someone who says his ontology is about homeomorphisms of finite groups. It’s just a different subject matter.

[**](Note, I do not need to have an axiomatic *definition* of ‘gerbil’ in order to talk about gerbils. Do you feel that any ontological claim has to be justified mathematically, by providing such definitions? Because that misapprehension could account for your strange views on this topic.)

That statement is OK as a hypothesis, but it's not an observation.

It is neither of these. It is true by fiat. If I am the author of the ontology, it is about whatever I say it is about. Now, it might of course be wrong, or confused, etc. – I am not omniscient when it comes to gerbils, no doubt –- but what it is *referring to* is my decision to make. 

If you feel this is hubris, ask yourself: what makes, say, the EPISTLE framework be about fluids and pipes and so on? Is it because Matthew West managed to reduce the oil and gas industries to a mathematical theory, a kind of Principia Processia?  Or is it about that because the authors said it was, and the users find it useful to use it in that way? 


If you want a one-step mapping, look at fuzzy logic.

I really would rather not, particularly as it has nothing whatever to do with what we are talking about. 

... I don't believe that the LHS can ever be

completely formal, because no system of measurement can be perfect.

You have said things like this previously, and I really don’t understand why you think this is even remotely relevant. Measurement has got nothing whatever to do with reference. I can refer to Julius Caesar – I just did – without measuring him, indeed without measuring anything. If we could only refer to things that were defined by measurement, we would all still be living on Laputa. 

Pat



MAW
a cool formalisation of symbol grounding
http://www.benjaminjohnston.com.au/papers/formal.pdf

Symbol grounding addresses the LHS of mthworld.gif.  Page 5
of the article discusses the informal issues:
Representation units may or may not have any particular semantic
interpretation, and may be manipulated by rules (such as interaction
with the environment or hyper-computational systems) that are beyond
formal definition.

Yes.  For humans, symbols are grounded by what Peirce called
"the gates" of perception and purposive action.  Methods of
pattern recognition and robotics address those two gates, but
none of them can be completely formal at the points of contact.

What do you mean by “completely formal”? (See my aside comment, [**] above.)

John F Sowa

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Feb 4, 2018, 10:51:58 AM2/4/18
to Pat Hayes, ontolog-forum
On 2/2/2018 6:43 PM, Pat Hayes wrote:
> Tarki’s account of truth conditions, now usually referred to
> as ‘model theory’, characterizes the domain of an interpretation
> as being a non-empty set containing the referents of symbols.
> The *only* condition imposed by the theory on this set is that
> it be non-empty... Ergo, it applies when the set contains real-
> world entites. QED.

Nothing prevents you from having a set that consists of physical
entities. I'm just saying that such a set could not be the domain
of a Tarski-style model, but it might be isomorphic to the domain.

If you derive a model *for* a set of axioms, that model will consist
of mathematical objects. If you derive it *from* observations of
the world, you get data that you can store in a computer. It might
resemble, represent, or be analogous to something in the world,
but a set of data is not physical.

As a simple example, let's take an axiom for which we don't
need a computer or even pencil and paper:

Axiom: There are three people in Pat Hayes' living room.

If you are one of them, I'm sure that you could verify that the
axiom is true of the current state just by looking. Testing that
axiom would be trivial.

But there is something between that axiom and the physical situation:
the occipital lobes in the back of your head, where a "mental image"
of the room and things in it is formed. There are also the temporal
lobes that recognize some things called people, the parietal lobes
for matching patterns, and the frontal lobes for reasoning, counting,
and saying "Yes!"

As you said above, "the domain of an interpretation [is] a non-empty
set containing the referents of symbols." For that example, the set
of referents are the people and the room. But the image of those
referents and the process of verifying the axiom are performed on
patterns in your head. Psychologists such as Johnson-Laird would
call those patterns a "mental model".

For a more complex theory and situation, you would need a more
elaborate specification, aided by pencil and paper or a computer.
That spec would be more specific than the axioms. It would contain
"data", such as diagrams, measurements, lists of parts and subparts...

Whether you call it a mental model, an engineering model, or a
Tarski-style model, there is always something data-like between
your axioms and the physical things or situation. QED.

John

Pat Hayes

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Feb 4, 2018, 12:26:31 PM2/4/18
to John F. Sowa, ontolog-forum

On Feb 4, 2018, at 9:51 AM, John F Sowa <so...@bestweb.net> wrote:

On 2/2/2018 6:43 PM, Pat Hayes wrote:
Tarki’s account of truth conditions, now usually referred to
as ‘model theory’, characterizes the domain of an interpretation
as being a non-empty set containing the referents of symbols.
The *only* condition imposed by the theory on this set is that
it be non-empty... Ergo, it applies when the set contains real-
world entites. QED.

Nothing prevents you from having a set that consists of physical
entities.  I'm just saying that such a set could not be the domain
of a Tarski-style model, but it might be isomorphic to the domain.

I know you are saying that, but you are wrong. There is no such restriction anywhere in any account of model theory. In fact, if one structure is isomorphic to another, and one of them is a Tarskian interpretation, then so is the other, by definition of “Tarskian interpretation”. 

(What do you think the universes of a Tarski-style model are restricted to contain? Not real things, so… unreal things? Spots in a diagram? Nodes of a graph? What? )


If you derive a model *for* a set of axioms, that model will consist
of mathematical objects.

That would be false if it made sense. What is a “mathematical object”? Mathematics can also describe real things, which is why it is so useful for people who deal with reality, like physicists and engineers. 

 If you derive it *from* observations of
the world, you get data that you can store in a computer.  It might
resemble, represent, or be analogous to something in the world,
but a set of data is not physical.

The observations are not physical, but they are (parts of) descriptions of things. The things they refer to, that they are descriptions of, can be physical. That is often the very point of making the measurements in the first place. When I note, with regret, that I now weigh 15lb more than I once did, it is the increase in girth of my all too, too physical, waist that I am concerned about. 

As a simple example, let's take an axiom for which we don't
need a computer or even pencil and paper:

Axiom:  There are three people in Pat Hayes' living room.

If you are one of them, I'm sure that you could verify that the
axiom is true of the current state just by looking.  Testing that
axiom would be trivial.

Indeed. A good example. But I note in passing that we aren’t talking about *testing* anything, in this argument we are having. 

But there is something between that axiom and the physical situation:
the occipital lobes in the back of your head, where a "mental image"
of the room and things in it is formed.  There are also the temporal
lobes that recognize some things called people, the parietal lobes
for matching patterns, and the frontal lobes for reasoning, counting,
and saying "Yes!”

All true, and all completely irrelevant to what we are talking about. Model theory is not a theory of perception or neurology. It simply talks about the relation between symbols – in this case, parts of English sentences – and whatever those symbols refer to – in this case, three people in my living room. Real people in a real room. 

Now, a more elaborate system of symbols might indeed describe not only my living room but also the goings-on inside my head that result in my visual perceptions, etc.. That would be a different (and far more elaborate) sentence, and the model theory would apply to it just as well. But as I say, it would be a DIFFERENT sentence, so it is not relevant to our discussion of your simple sentence and its interpretation.

As you said above, "the domain of an interpretation [is] a non-empty
set containing the referents of symbols."  For that example, the set
of referents are the people and the room.

Exactly. This is all I have been saying: the referents are parts of reality.

 But the image of those
referents and the process of verifying the axiom are performed on
patterns in your head.  Psychologists such as Johnson-Laird would
call those patterns a "mental model”.

That might be true – I don’t think Johnson-Laird has this right, myself, but whatever – but it has nothing to do with what we are talking about. Even Johnson-Laird would not claim that when we describe what we see, that we are *referring to* the images in our heads. And reference is not a *process* of verifying: it is the mapping that is thereby verified (or not, as the case may be.)

For a more complex theory and situation, you would need a more
elaborate specification, aided by pencil and paper or a computer.
That spec would be more specific than the axioms.  It would contain
"data", such as diagrams, measurements, lists of parts and subparts...

Whether you call it a mental model, an engineering model, or a
Tarski-style model, there is always something data-like between
your axioms and the physical things or situation.

You make a conceptual mistake here by putting Tarskian models – let me write Tmodels – into the same category as engineering models – Emodels. This is just a (bad) pun on the word “model”. A Tmodel of some sentences is an interpretation of them that makes them true. It is not a model in the sense of being a simplified or scaled-down simalcrum of something, like a model of a bridge. And if we use the word “model” to encompass an Emodel then things get even more teminologically confused, since an Emodel comprises *symbolic descriptions* – as you say, data – of the reality being modelled, so in this case the 

Emodel <—> reality being modeled

relationship is exactly like the 

Description <—> Tmodel 

mapping, and the meaning of the word “model” has almost completely inverted. To say that the (actual) bridge is correctly modeled by the Emodel description – that is, that it is correct when understood as referring to that bridge –  is *exactly the same claim* as saying that the actual bridge is a Tmodel of the assertions which comprise the Emodel data; that is, that those assertions are true under that interpretation. 

When you say ‘between’, you might meant several things. If you mean, there is some kind of measurement process which supports any claim of reference, then I might agree, at least provisionally.[**] Or at any rate, that is something we might discuss in greater depth. But if you mean that any reference mapping must (therefore?) be decomposable into a functional composition of what we might call a reference-1 map to some abstract domain, and a reference-2 map from that abstract domain to the real world, and model theory can talk only about reference-1 – which is what your diagram clearly asserts – then I sharply and firmly disagree. And nothing in these or any other emails has actually made a coherent argument for that second decomposition-of-reference claim. So:

 QED.

Nope. Non demonstratum quod erat demonstrandum.

Pat

[**] Actually on more thought I won’t agree, but that is another discussion. Might be more productive than this one, though :-)


John



Nicola Guarino

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Feb 5, 2018, 6:02:45 AM2/5/18
to ontolo...@googlegroups.com
Folks,

I rarerely can afford the luxury of partecipating to Ontolog discussions, since they often tend to explode an my time is scarce… However, this time I can’t resist offering my two cents to this debate.

I think that, in a sense, John and Pat are both correct.

Pat is absolutely right while saying that the *domain* of interpretation of Tarski-style models can include real things. No question about that.

On the other hand, John is right in insisting on the distinction between a Tarskian *model* and the physical world.

These two claims concern two different things: a domain and a model. A Tarskian model includes a domain plus an interpretation function. In my view, it is this interpretation function which makes the difference between a model and the physical world.

Consider the example described in the attached slide, which I presented a number of times in formal ontology courses. The simple theory shown in the example is intended to axiomatize the relation “on” holding among blocks. It just says that the relation is asymmetric and anti-transitive, which is correct, so that all the models of this theory are actually intended. However, the theory can’t distinguish between the actual real-world situations shown to the right. This means that the interpretation function constrained by the theory is very different from the way people actually interpret two physical blocks as belonging to the extension of the ‘on’ relation.

As a result, the small ontology shown in the example is precise but not very accurate, since its intended models collapse intended and non-intended real-world situations. There are two ways to make it more accurate:

a) extending the domain of discourse in order to include other entities besides blocks (say, regions of space that may or may not be occupied by blocks).
b) extending the signature of the language in order to be able to talk of other primitive relations (say, topological connection among blocks, or geometrical arrangement of blocks)

In both cases, if we do things properly, the models of the resulting theory will be closer to the physical world, in the sense that the interpretation function constrained by the theory will be closer to the one actually used by competent English speakers using the ‘on’ preposition.

I hope this helps…

Cheers,

Nicola


Blocks.pdf

John F Sowa

unread,
Feb 5, 2018, 10:12:17 AM2/5/18
to Pat Hayes, ontolog-forum, Nicola Guarino'
Nicola and Pat,

Nicola
> I think that, in a sense, John and Pat are both correct...
> These two claims concern two different things: a domain and a model.
> A Tarskian model includes a domain plus an interpretation function.
> In my view, it is this interpretation function which makes the
> difference between a model and the physical world.

Thanks. That point is consistent with Tarski (excerpts below)
and with the following statements by Pat and me:

>> [John] Nothing prevents you from having a set that consists
>> of physical entities. I'm just saying that such a set could
>> not be the domain of a Tarski-style model, but it might be
>> isomorphic to the domain.
>
> [Pat] I know you are saying that, but you are wrong. There is
> no such restriction anywhere in any account of model theory.

I agree with Pat's last sentence: Nothing in Tarski's definition
restricts the domain of an interpretation. In fact, if a formal
ontology is about the real world, the variables in the logic must
refer to the world.

But the interpretation function, which maps sentences in the theory
to truth values, must have some formal method for accessing the
elements of the domain, their properties, and their relationships
to other elements. (I'm using the word 'relationship' to mean
"one instance of a relation" -- for example, one RDF triple or
one row of a relational DB.)

If those elements, properties, and relationships are represented
by names or other symbols, they can be stored in a database and be
indexed by character strings. But no logician would trudge through
a swamp to evaluate a relationship.

Tarski (1944) mentioned "mathematics and theoretical physics" as
suitable fields for applying formal semantics. (Section 6 below)

In Section 20, he discussed the difference between "empirical research"
and "theoretical semantics". The difference is that the empirical
(experimental) research is "concerned only with natural languages
and that theoretical semantics applies to these languages only with
certain approximation" -- as my mthworld.gif diagram shows.

In fact, experimental physicists have a rule: Never allow a
theoretician to walk into your laboratory. As soon as they do,
everything breaks. Neils Bohr was a very great theoretician.
As proof, when he took a train from Copenhagen to Zurich,
the minute his train passed through Göttingen, an experiment
at the university blew up.

In short, if you don't want your interpretation function to blow up,
restrict the elements of the domain to symbols, for which some
surveyor or experimenter determines the properties and relationships.

John
_______________________________________________________________________

Excerpts from Tarski (1944) http//jfsowa.com/logic/tarski.htm

Section 6

If in specifying the structure of a language we refer exclusively
to the form of the expressions involved, the language is said to be
formalized... the field of application of these languages is rather
comprehensive; we are able, theoretically, to develop in them various
branches of science, for instance, mathematics and theoretical physics.

Section 20

The fact that in empirical research we are concerned only with natural
languages and that theoretical semantics applies to these languages only
with certain approximation, does not affect the problem essentially.
However, it has undoubtedly this effect that progress in semantics will
have but a delayed and somewhat limited influence in this field. The
situation with which we are confronted here does not differ essentially
from that which arises when we apply laws of logic to arguments in
everyday life — or, generally, when we attempt to apply a theoretical
science to empirical problems...

The most natural and promising domain for the applications of
theoretical semantics is clearly linguistics — the empirical study of
natural languages. Certain parts of this science are even referred to as
"semantics," sometimes with an additional qualification...

It is perhaps unnecessary to say that semantics cannot find any direct
applications in natural sciences such as physics, biology, etc.; for in
none of these sciences are we concerned with linguistic phenomena, and
even less with semantic relations between linguistic expressions and
objects to which these expressions refer. We shall see, however, in the
next section that semantics may have a kind of indirect influence even
on those sciences in which semantic notions are not directly involved.

Section 21

Besides linguistics, another important domain for possible applications
of semantics is the methodology of science... The semantics of
scientific language should be simply included as a part in the
methodology of science...

One of the main problems of the methodology of empirical science
consists in establishing conditions under which an empirical theory or
hypothesis should be regarded as acceptable. This notion of
acceptability must be relativized to a given stage of the development of
a science (or to a given amount of presupposed knowledge)...

Section 22

As regards the applicability of semantics to mathematical sciences and
their methodology, i.e., to metamathematics, we are in a much more
favorable position than in the case of empirical sciences...

Pat Hayes

unread,
Feb 5, 2018, 11:11:40 AM2/5/18
to ontolog-forum, Nicola Guarino
Hi Nicola

Yes, of course the interpretation mapping itself is not part of the world being described: it is the semantic mapping from language expressions into that world. I hope nobody interpreted any thing I have said as disagreeing with this.

I believe I can summarize your point as follows: small theories (small ontologies, with only a few axioms) cannot fully capture a complicated world, because they allow non-standard models, ie interpretations which satisfy the theory but are not correct, i.e. they are not parts of any intended world, or the names of the theory are not correctly interpreted. This is of course true, and IMO one of the most useful methodological aspects of Tarskian model theory when developing ontologies. As I suggested in the ‘naive physics manifesto’, longer ago than I care to remember, an excellent way to critique a proposed ontology is to deliberately look for non-standard models, as they vividly reveal gaps in what one might call ‘coverage’ of the axioms. (I also used the blocks world as an example, as we all did :-) And then, as you say, the ontology can be improved by filling the gaps in its descriptive powers which are thus revealed, by enriching its expressibility. (As a later example, the ‘time catalog’ noted that all temporal ontologies (that did not explicitly mention durations of intervals) had all their truths preserved when the entire infinite time-line is projected into the unit interval, so they could not possibly account for the future being unbounded. I learned that trick from van Bentham, by the way.)

But, in order to do this kind of stuff, we are assuming that our formalism, the language of our ontology sentences, has a Tarskian semantics in the first place. Looking for nonstandard models is a quintissential application of model theory. Without that theory, the whole idea of a model, and hence of a nonstandard model, does not make sense. Which is why I defend it whenever it is attacked. Mo