SUO: Re: Proposed SUO Content Outline
Frederick N. Chase wrote:
>
> John Sowa wrote, in part:
>
> "... which axioms have consequences
> that can peacefully coexist with the
> consequences of all the other axioms
> in the very big SUO pot. ..."
>
> I think I saw in John's message several
> possible notions of "peacefully coexist":
> 1) There are no contradictions provable.
>
> 2) If there are contradictions, they are benign
> (See, e.g., below).
>
> 3) There exists a set of assumptions such that there
> are no contradictions, at least fatal ones.
>
> Since Jon and John seem to agree that, as a practical matter,
> we shouldn't aspire to 1) then I wonder if item 2) can be
> developed along the following lines.
>
> Having a means for articulation of domains of
> interest for subtending ontologies or theories:
>
> Theory X
> Uses Y, Z
> ...
> End Theory
>
> (Thank you, Matthew)
>
> provides a context for identifying
> an inconsistency as benign versus fatal.
>
> A theory X is not fatally affected by contradictions
> implied by (axioms of) a theory Q it does not use.
>
> -- Fred Chase
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Fred,
I sense that there is some sort of a radical misunderstanding
between us here, so ineluctable that I cannot even figure out
any way to formulate it -- but I will go ahead and try anyway.
Any contradiction is fatal to a classical deductive system.
But demonstrative reasoning, as it is embodied in a deductive system,
is not all there is to doing inquiry, yet another name for "science".
The other brands of reasoning that we nevertheless have to do,
like it or not, require non-demonstrative forms of inference
that people of a certain up-braiding have come to classify
as "abductive" and "inductive" inference. What it means
to be non-demonstrative is that there is a certain risk
of going wrong, and it is true that such a risk has the
potential of being fatal. Wouldn't it be luverly if
we had a choice? We don't. That is called reality.
The fact that any deductive system big enough to count
perforce encounters a whole new world of difficulties,
remarkably reminiscent of those that we countenance
when we face up to any order of natural phenomena,
well, that was a surprisingly difficult thing
to prove -- to some people! -- but the hard
part of the proof is really just the fact
that one is trying to prove it internal
to a system that is built on the premiss
of trying to deny it. The sorts of folks
for whom this particular circumstance was
always so obvious that it needed no proof,
like Pythagoras, Plato, or Peirce, already
had another way of describing the situation,
which had a lot to do with how the "Theorem"
got its name -- they taught that mathematics,
despite the illusions of the many, was really
an empirical, "hands-on", observational science,
and ipso facto, cannot be done by pure deduction.
In our enlightened modern times, this suddenly got
to be news -- to some people! -- and so it took all
the dogged persistence and all the recursive skill
of a person like Gödel, who, being an unabashedly
Platonic realist never doubted the matter in the
first place, to prove it to the satisfaction of
all those benighted denizens of the PriMat Land.
Alright, I think that I probably ought to pause to see
if we have reached any kind of a new understanding now.
Let me know,
Jon Awbrey
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