Re: Fwd: SUO Quo Vadis
On Wed, 21 Dec 2005, Avril Styrman wrote:
> Date: Wed, 21 Dec 2005 16:09:43 +0200
> From: Avril Styrman <Avril.Styrman@helsinki.fi>
> To: John F. Sowa <sowa@BESTWEB.NET>
> Cc: Rob Freeman <lists@CHAOTICLANGUAGE.COM>,
> "pevnev@juno.com" <pevnev@juno.com>, SCOTT@DE.IBM.COM,
> standard-upper-ontology@IEEE.ORG
> Subject: Re: Fwd: SUO Quo Vadis
>
> John, some more questions.
>
> > > D.M.Armstrong's Theory of Universals 1978 19.VI:
> > >
> > > "One hand washes another; both wash the rest of the body.
> > > Perhaps the trickiest sort of case is that where a person
> > > loves or hates himself. But even here genuine self-relation
> > > seems avoidable. If a man loves himself, then it is not
> > > that self-loving state which he loves, but other aspect
> > > of himself. It is possible that he should love the self-
> > > loving state, but this seems to demand a new, second-
> > > order, loving state which is distinct from the original one."
> >
> > That's an interesting solution to the apparent paradox of
> > loving and hating being antonyms. You could also solve that
> > problem by saying that loving and hating are not true antonyms.
> > In any case, that's irrelevant to the structure of the lattice.
>
> I can't see the connection with antonyms. You have a category
> called Theory inside a hierarchy of theories. That category
> cannot describe all theories, because then it would describe
> itself, which causes a vicious infinite regress. What does
> the category called Theory then describe in your generalization
> hierarchy of theories?
>
>
> > > BOT describes a contradiction, a round square.
> >
> > Yes. It includes all possible contradictions. So what?
>
> The idea of TOP containing all possible truths and BOT
> containing all possible contradictions is clear, but the
> actual definition of an absurd BOT is impossible.
I believe a simple approach to this is to understand a theory as the
deductive closure of a set of sentences and the organization of the
lattice of theories to be based on the number of models that satisfy a
theory.
If TOP is the set of all logical truths, then that theory is satisfied by
every model.
If BOT is an inconsistent theory, then it is satisfied by no models.
Every theory in between TOP and BOTTOM in the lattice will have contingent
statements that are satisfied in some but not all models.
The ordering can be based on whether the set of models of one theory is a
superset of the set of models for another theory. Alternatively, we could
cast the ordering in terms of entailment since if T1 entails T2, then the
models of T2 is a superset of the models of T1.
Since, BOT entails every theory, it's in the right position.
Since, every theory entails TOP, it's in the right position.
If one theory (T1) included all the axioms of another theory (T2) and
more, T1 would entail T2 as well. So, T2 would be 'between' T1 and TOP.
Given the assumption of deductive closure, we can take this one step
further and state that there is only 1 inconsistent theory and that it
contains all sentences of the language and that all other theories
(elements of the lattice) are consistent.
John C.