Chris,
Thanks, that is getting closer to specifics, but I am still unclear
exactly where the logical inconsistencies lie.
The issue that I would consider a problem is not just that people can
create logically incompatible theories - that is absolutely certain and
trivial to demonstrate. The question is whether, if those theories are
actually logically incompatible, they can or cannot always be expressed with
some common set of primitives. I will illustrate the trivial case. (01)
Theorem 1: P(x)
Theorem 2: not P(x) (02)
Both of these inconsistent theories use the same primitives (P, x, not,
and the logical meaning of the predicate relation), and therefore we can
describe the relation between them (and prove the inconsistency). Neither
theorem would be part of the ontological commitment of the FO, but a
description of the predicate P and a definition of x, along with the logical
primitives could be - and the theorems could be *described* by that FO, as
theorems whose truth value is only asserted in some context that is not the
FO's ontological commitment. Those primitives (P, x, not) would not be
inconsistent with each other. The basic principle is that we should be
able, in an ontology, to describe a theory without asserting that it is
necessarily true. The things that are necessarily true would be the
ontological commitment of the set of primitives chosen.
Thus, we can have primitives in the FO such as object, time interval,
spatial interval, mass, energy, real numbers, and force, and generate from
these (and probably some others as well) the Newtonian mechanics and
Einsteinian space-time theory. Neither of those theories would be part of
the ontological commitment (not asserted to be necessarily true or false),
but they can be described by the FO and one can make assertions regarding
the circumstances under which the predictions of those theories conform
(within experimental error) to the results of measurements. People (some at
least) can grasp the difference between describing a model and asserting
that it is useful versus asserting that a particular model is necessarily
true in all possible worlds, and I feel confident that we can structure our
ontologies so that they can represent that difference too.
I also have no doubt that mathematicians can describe fascinating
theories that axiomatize inconsistent *models* of something or other. The
question I do not have the background to answer is whether, given any two
theories that are demonstrably logically inconsistent, can both always be
described using the same set of elements, including the logical primitives?
In the case of the example you mentioned, where the axiom of choice is
replaced by a contradictory axiom, would it not be possible to *describe*
both the axiom of choice and its opposite with a common set of elements,
without asserting that either axiom is true? The axioms themselves need not
be in the FO, just those elements required to state the axioms. Then the
contradictory axioms can be asserted, one in each of the incompatible
theories, using those primitives. Then the different set theories, having
the contradictory axioms which are not themselves in the FO, would (if I
understand correctly) be logically inconsistent, but still *describable*
with the same set of primitives (including the descriptions of the axiom of
choice and its contradictory axiom). (03)
The reason I suspect that the composability holds generally, aside from
not having seen examples showing otherwise, is that it seems to me, if one
can prove a logical inconsistency of two theories, they must be described by
the same set of elements (plus the logical elements), or else the logical
contradiction could not be proved. The set of elements that can describe
those models would be the set that *could* be in the FO (unless they prove
inconsistent with something else in the FO)
Now I freely admit that with my impoverished mathematical background, I
may well have missed some very important point(s), obvious to
mathematicians. I would sincerely be happy to be disabused of this error,
since I would thereby learn something that I think is important for an
understanding of the question of conceptual primitives - to learn the
limitations of that tactic. If there are limitations, this will indicate
the choices that must be made to in order to apply the tactic of conceptual
primitives to the problem of semantic interoperability. (04)
Could you show me provably logically inconsistent theories whose
component elements must themselves be necessarily inconsistent? If
inconsistent theories have contradictory axioms, the question is whether
those contradictory axioms can or cannot be composed of elements with a
consistent meaning. (05)
Pat (06)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (07)
> -----Original Message-----
> From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-
> bounces@xxxxxxxxxxxxxxxx] On Behalf Of Christopher Menzel
> Sent: Wednesday, February 03, 2010 5:55 PM
> To: [ontolog-forum]
> Subject: Re: [ontolog-forum] Foundation ontology, CYC, and Mapping
>
> On Wed, 2010-02-03 at 13:36 -0500, Patrick Cassidy wrote:
> > Rob,
> > [RF] > > Examples of incompatible theories: the axiomatic set
> theories of
> > > > mathematics.
> > >
> > The math theories you mention may or may not be all describable in
> terms
> > of the same primitives - but I can't visualize the incompatibilities
> you are
> > referring to (and decide whether they are not expressible by common
> > primitives) unless I can see the logical axioms that demonstrate a
> logical
> > contradiction. My first attempt to find them came up null.
>
> It is very easy to find examples: ZF set theory is incompatible with
> AFA, the theory that results from removing the axiom of foundation from
> ZF and replacing it with an anti-foundation axiom that asserts the
> existence of, e.g., self-membered sets. Again, ZF + the axiom of
> choice
> is inconsistent with ZF + the axiom of determinacy.
>
> There is really only one axiom common to all set theories, viz.,
> extensionality: sets with the same members are identical and perhaps
> that is the only primitive principle you need. That said, virtually
> all
> set theories in broad use seem to be based in a common set of
> intuitions
> and share a pretty solid axiomatic core -- Kripke-Platek set theory
> would probably be a good shot at that core: KP is basically ZF without
> the axioms of infinity and power set and with weaker versions of the
> axiom schemas of separation and replacement. (See Barwise, _Admissible
> Sets and Structures_ to see KP in action (more exactly KPU, KP + "there
> are things other than sets")).
>
> There are much sharper incompatibilities between ZF-style theories and
> theories based on Quine's NF (notably, in these theories, there is a
> set
> of all sets) but NF-style theories are not in wide use and are mostly
> of
> theoretical interest (see, e.g., Thomas Forster's excellent book _Set
> Theory with a Univerals Set).
>
> Chris Menzel
>
>
>
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