Blog:Definition: idea, logic, practise

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Purpose to collect community knowledge.

Here in articles and comments, we may collect our ideas, thoughts, and experience on this crucial topic.

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:-)

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