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What is a definition

This is also from our g-group: <Alex:I hope Jon Awbrey will change this text as he needs.> "RM: What is your view of definitions?

[Jon_Awbrey]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 "

Reducing definitions

From our google group: "[Alex:]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.

[JFS:]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"."

Many-sorted logics

This topic is not about definitions, but something not far from, as we may think that sorts are definable somehow in logic without sorts. The topic itself arose in our google group as follows:

"[Alex:]> 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".

[JFS:]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 "

Context:-)

3 Statement

2018-04-18 The Possibility of a Shared Ontology

As has been discussed several times there seems to be agreement that we can't expect a single, shared ontology for complex domains or across many domains.

But yes we need shared conceptualizations first as part of this "mutual understanding".

We might start by agreeing on some of the lower-hanging problems involved such as the sub-set of spatial–temporal dynamics of systems...or what we understand by "sustainable systems." It seems to involve some interactions between natura, technical and social systems, and with how those interactions affect the challenge of system operation over time. This group may be focused on application systems and supporting socio-tech issues.

Ref: PNAS - Proceedings of the National Academy of Science, 2010. Sustainability science section of website and [1]

2018-04-17 Comments on the slides and a strange proposal on context

What is Context?

Context is the property of an object and a non object that enables an agent to identify an object as a unique entity or something separate from the non object. Notice that words come in pairs, it is contrast that perception is based on, and a dual nature of concepts rules the mind. That enables comparison, the building block of learning, acquiring practical knowledge and abstract thinking. I interpret the world in terms of a lean upper ontology, the first three concepts of which must be familiar from upper ontology.

Context is an abstract noun. As an abstract noun it is either an object, or a property or a relation.

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Second Statement

Henson Graves STATEMENT:

Here are some subtopics of the overall problem of how to realize the axiom set-theory-interpretation paradigm in everyday practice.

1. What is the tradeoff between KR languages in which classification is by sets or types which are terms vs those which classification is represented by predicates?

2. What are the appropriate modular units for reuse?

3. How can formal approaches best exploit graphics based modeling languages?

The paradigm that I am suggesting is exactly the same one that is in John's diagram. For this audience I will stick to logician's terminology. Engineers develop artifacts which are naturally axiom sets even if they don't always recognize it. The axiom sets generate a theory depending on the logic the axiom set is embedded in. The axiom sets are finitely presented. The theory generated by the axiom set are the conclusions which can be derived from the axioms in the logic. As any axiom set or theory there are natural definitions of a valid model of the theory generated by the axiom set. Depending on the axiom set there may possibly be both finite or infinite models. Following John's diagram there are two general kinds of models, the ones in the physical world and simulation models. To me what is important in both cases is not finiteness of the model but its constructivity. This requires a lot more discussion which I will not do here. Following the usual logic paradigm you expect the reasoning to be "true" in all valid models, at least when the theory is sound.

For now only one more idea.

The axioms sets that engineers produce generally have a lot more valid models than the engineer intended. So a lot of practical engineering work is identifying the additional assumptions needed to constrain the valid models. One could view these additional assumptions as "context" for the axiom sets. That is in some domain all of these assumptions could be added automatically to a proof assistant (aka engineer's model development tool) without the axiom set developer even being aware of it. For example in the paper reference above we were modeling both the aircraft and its operational environment. For the axiom set to generate the 3D simulations that we produced we had to add an immense amount of assumptions about physics regarding aerodynamics, sensor performance etc.

Initial Statement

Alex Shkotin STATEMENT:

ONTOLOGY=FORMAL_THEORY+FINITE_MODEL

Maxima: "Ontology" is a mirage made by great DL logics in the minds of experts of Expert Systems and WWW.

But ontology, unlike Expert System knowledge, must be presented to public/customer as a text for a universal processor like Protege+Hermit or Jena if you prefer batch mode

{:-)

Till now the main idea is that

Ontology is a formal text keeping together elements of Formal Theory and Finite Model in a form suitable for Reasoner - usually DL one. A Reasoner can check automatically consistency, perform classification and do other great reasoning services.

To clarify term "engineer's models (aka axiom sets)" let's look at a situation like this: we have

a) a formal theory (some axioms and definitions) for particular application area like "air vehicle model-based design" mentioned by Henson Graves, see [1].

b) a finite model of theory (a) keeping a particular engineer's model in a DB.

If we need to reason about this (b) model under (a) axioms we convert axioms of formal theory to axioms of OWL-DL and we convert model itself to OWL-DL axioms and this latter set is known as "engineer's models (aka axiom sets)".

The point is that when we work with the finite model itself we need another kind of language (not OWL) to perform "engineering" task: for ex. how many wings do this aircraft have?

And we need a language to handle these models.

These models are finite as they have a finite number of elements. They use numbers but a finite number of them each.

The famous example of a finite model is a graph as still a math object but very useful for modeling. DB is not a math object, unfortunately. If we add a label possibility to graph we should describe it rigorously, and then we may get RDF.

One of the first examples of using the graph as a model is Seven Bridges of Königsberg[2].


[1] Air Vehicle Model-Based Design and Simulation Pilot, by Henson Graves, Stephen Guest, Jeff Vermette, Yvonne Bijan, Harold Banks, Greg Whitehead, Bill Ison. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.638.8184

[2] Seven Bridges of Königsberg. https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg


Article 4

This is the fourth test article.

Article 3

This is yet another example of a blog article.

Ontology-Context-Semantics

This theme considers the relationships between Ontology, Context and Semantics.

triangle.jpg

Semantics

Semantics

This is the study of meaning, and it is a very broad area of study. Moreover, different fields have different concerns:

  • Linguistics
  • Computer Science
  • Logic
  • Psychology

Data Semantics

This is a narrower than general semantics, but even this divides into at least two possibilities:

  • Traditional database integration
  • Ontology

Semantics generally depends on context, so it might be useful to consider both context and semantics.

Context

Context is a very broad area. Specialized areas within this theme include:

  • Situation Awareness
  • Linguistic Contexts
  • Philosophical Aspects
  • Logical Aspects

Here are some conferences dealing with context:

Article2

Test of blog article posting. This can be edited.

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Article1

This is a test of article creation.

Hybrid System

Reasoning With Neural Tensor Networks for Knowledge Base Completion by Socher, Chen, Manning and Ng

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