ONT Re: General Design
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John, Rich, et al.
The axioms get in the way when you try to put them first.
The axioms are the cart and experience is the horse.
I guess recalcitrant experience is the donkey or mule.
The "logical" order of axioms and experience, as if the
concepts generated the data, is a put-up job. It is the
order that you rearrange the refined axioms and the raw
experience into, but only "after the fact", so to speak,
after you finish extracting the axioms that usefully
summarize a wide range of experience. But this is
not the natural order for discovering the axioms.
Or maybe axioms are the cart and experiences are the cartload.
It can a take a couple hundred or a couple thousand years of
experience with a motley crue of crude examples before some
people will start to notice that the same darn patterns of
thought tend to keep turning up in what appears to be the
darndest places, and the useful sorts of axioms only but
gradually percolate themselves to the top of the heap.
But even this is a retrospective illusion. At the
beginning of the story you don't even know what
the "examples" are supposed to be examples of.
This is the way that it has been with all the best axiom sets that I know,
like those that define graphs, groups, and other mathematical materials.
Now, once you abstract a useful set of axioms, it is possible to engage
in a certain amount of free variation with them, and it is possible to
hit on some lucky finds that way, but this sort of "luck" all depends
on the mega-years of perspiration that went into finding the initial
set of principles.
It is necessary to think about abstract structures as independent spaces,
formal or mathematical models that are "models" in one sense because they
are defined by axioms, but "models" in another sense because they analogue
an aspect of real experience.
But these models can at best be approximations to reality, and so there is
an unavoidable risk in using them, not that there is an alternative for us!
The act of creating or discovering a model is what is known as "abductive"
reasoning. It becomes necessary to distinguish "thinking within the model" --
thinking inside the box, if you will -- from "thinking of/about the model",
all the sorts of reasoning of that it takes to come up with the axioms in
the first place, and all the sorts of reasoning that it takes to evaluate
the utility of the corresponding model for any exterior/ulterior purpose.
But the catch is this: Thinking within the model can be done by deductive reasoning,
and maybe supported someday by the ilks of theorem-provers that we all know and love,
but thinking up the model to start with, and testing the model in the final analysis,
except for mere logical consistency, just cannot be covered by deductive means alone.
This is something that will just have to be taken into account
in designing the logical architecture of ontology using systems.
Jon Awbrey
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John F. Sowa wrote:
>
> Rich,
>
> JS> The axioms get in the way.
>
> RC> Does that mean 'The (top level) axioms get in the way'?
> >
> > If so, then the top level is the empty level, as you suggested
> > earlier. And any conceivable context of objects, properties
> > and axioms could be the next level below the empty top level?
>
> The top node is true of everything. Therefore, the only axioms
> that are associated with it are the tautologies. They can't
> get in the way of anything.
>
> The organization of any hierarchy is determined by the dependencies
> between nodes: category X is beneath category Y iff the existence
> of something of type X implies that it is also of type Y. Any
> category that has no such dependencies is could be placed at a
> level immediately beneath the top. The analysis of the
> dependencies determines the hierarchy.
>
> John
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