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SOL as isomorphic interpretation of theories in a FOL

Last post 06-02-2008 10:27 by Brian Nalewajek. 2 replies.
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  • 05-31-2008 6:19

    SOL as isomorphic interpretation of theories in a FOL

    Hi there,

    I believe OF is as good a place as any if I want to find logicians.

    Does anyone know how well known it is that theorems of 'Second Order Logic' may be expressed/interpreted as isomorphic interpretations of theorems expressed in First Order Logic.

    i.e. Is there good documentary evidence that although Godel's incompleteness theorems apply to SOLs/Higher order Logics, that isomorphism doesn't just stop there. i.e In that Godel Numbers are few within the countless stream of natural numbers, there are (sound) isomophic interpretations within the countless streams of (symbols expressed as) theorems of First Order Logic that (in their interpretation) are theorems of a SOL, and so the best (or worst, take your pick) that may be said of FOL is that it may 'consistently model an inconsistent model'.

    How well known is that? If you know, and know any good papers that point that out, can you let me know.

    Or is it as simple as understanding that 'any' UOD may be modelled in a FOL, and within an inconsistent theory (as a UoD), you can prove anything.
    i.e. it's so trivial, that nobody writes about it.

    I don't mind if this is a dumb question. I would just like to hear what others have to say.

    Just as a heads up. I'm not looking to debate this. So for those who don't accept Godel...i'm not in that camp to start with, and i'm not jumping ship.
    I'm just interested in what other people 'think' about it. I'll accept whatever you have to say (as your opinion)...

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  • 05-31-2008 22:26 In reply to

    Re: SOL as isomorphic interpretation of theories in a FOL

    Hi there,

    I received confirmation of this off-line, so for those interested:

    "FOL is undecidable, and since ORM covers FOL, ORM is undecidable too. In practice this turns out to be not a major problem."

    I have no confirmation for how well this is known as 'common knowledge', but experience tells us that nothing is less common.

    So, we should draw from this that it is up to to the 'modeller' (or some sought of proof validation regime. i.e. it need not be a person) to verify that the UoD represented within an Object-Role Model is consistent, because ORM may surely model an inconsistent theory.

    I hope this is helpful and welcome information within the community.

    Best regds,
    Victor

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  • 06-02-2008 10:27 In reply to

    Re: SOL as isomorphic interpretation of theories in a FOL

     

    Hi Victor,

    Thanks for starting this thread on the nature of the logical connection between ORM models and the target domain UofD.  Your statement/question should be addressed not only by ORM modelers, but by the more general fact-based modeling community.

    You've couched your concern in terms of First Order Logic, and other mathematical theorems and principles.  While I appreciate the exactness of arguments in these terms (and envy those with the discipline to utilize those terms with nearly the same exactness); It might help to broaden the discussion by using terminology more accessible to the general readership here - where appropriate.

    Let me try to paraphrase the central concern you put forward: A limitation of the logical basis for Object Role Modeling (and other similar fact-based approaches), raises concerns about the degree of correctness of a resultant model of an actual target domain (Universe of Discourse - UoD, or UofD).  The concern is that while a model may be technically correct, and even if it correctly models the UofD as stipulated; the result may perpetuate logical inconsistencies existent in the target domains's UofD.

     A couple of weeks ago, Dr. Nijssen suggested a multi-approach modeling exercise, against a common target domain, to compare the techniques and results.  I thought that a good idea, and offered to help create the framework for the exercise.  One question that came to mind was on how to evaluate the correctness of submitted models?  How do I assess the correctness of the models I create?  In ORM, we can appeal to authority to see if we correctly followed the methodology, as we can to the creators of whichever tool we might use.  Yet, that still seems to leave other important aspects of correctness unaddressed.

    In order to organize my own thoughts and concerns on the matter, I started an article on the correctness of ORM models.  I haven't finished the article - but if/when I'll put it up on my site, and offer a copy or link to OF.  In general, though, the article explores the degrees of correctness of a model (beginning with syntactical and grammatical correctness), and proceeds through degrees of fidelity of the model to the target domain.  I don't presume to be an authority on any or all of the levels (we have access to the best here anyway), but became more satisfied that I could delineate the limitations of each form or degree of correctness, as the article progressed.  As I looked at higher degrees of correctness, I realized that I had to have a better grasp of the nature of a Universe of Discourse.

    A definition of a Universe of Discourse (a search will find several) is one thing; but I feel a more comprehensive understanding is essential to contemplating higher degrees of model correctness.  That was the reason I posted the thread on nature of UofDs, in these forums.  I wasn't looking for a definition, but wanted to hear from other modelers how they interpret and use the definition they have, in making judgments and assessments during the modeling process, as well as in their criteria of resultant model correctness.  For myself, I think a target domain for fact-based modeling can't be seen as an entity, completely separate from, and unaffected by a modeling process.

    Getting a feeling for the nature of a UofD, and for the degrees of correctness for a modeling of a domain; I think I can come to terms with the concern you raised about the limitations to the logical basis of ORM models.  Yes, properly formed ORM models can contain logical inconsistencies - but can those errors be located and managed?.  The general limit of all logical disciplines is in their application.  A properly formed syllogism is a powerfully demonstrated statement of self consistent truth, a triumph or reasoning.  Yet, they all rely on the acceptance of some premise: is Socrates indeed a man? Are all men mortal?  The fact that ORM is based on FOL (and that FOL has the qualities it has), is tremendously useful.  That it has the limitations of all of its kind shouldn't be surprising.  One great advantage of the sound logical basis for ORM is that it can show where any inconsistencies won't be found - as in the Windows game Minesweeper, knowing where the mines can't be is a big part of knowing where they are.

    I hope we can continue the thread you started; get others to add their perspectives, comments and questions.

    BRN..

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