On Mon, 2010-02-08 at 13:24 -0500, Patrick Cassidy wrote:
> Changing the topic to reflect the narrow focus here:
> Query for Chris Menzel re: his reply to Cory C:
> > On Feb 8, 2010, at 10:43 AM, Cory Casanave wrote:
> > > ...
> > > Considering the "Pat Axiom": That if 2 theories can be shown to be
> > incompatible they must share some concepts - intuitively obvious but
> > I have never seen it made explicit, thanks!
> > Actually, the "Pat axiom" needs a couple small qualifications.
> > First, each of the two theories in question has to be consistent.
> > An inconsistent theory is incompatible with every theory, regardless
> > of any concept overlap. Second, the theories must not put
> > incompatible conditions on the number of things that exist. In
> > first-order logic (with identity), it is possible to express that
> > there are only N things, for any natural number N. So if T1 says
> > "There are exactly three things" and T2 says "There are exactly four
> > things", they will be incompatible even if they share no concepts
> > (though I suppose one could say in this case that they share the
> > concept of identity).
> I'd like to get this right, so I could use some additional
> clarification: If two theories assert that there are different numbers
> of "things" then it seems to me that these must refer to instances of
> the same category to be inconsistent. (01)
Pretty clearly. (02)
> Even though the category is not mentioned in the axioms, the
> implication of the (English language) interpretation is that the
> "thing" category is everything that could possible exist - (03)
Well, no, it's just everything that does, in fact, exist (according to
the theory). Note that, typically, a theory simply will not state how
many things exist. So this gap in your "axiom" is more just a fussy
theoretical one than one that has any serious practical upshot. (04)
> and that would be the category of which the "things" are instances.
> It seems that these assertions have to be made with respect to the
> same context, and if the context is the whole universe of all possible
> things that might exist, then that would specify the category
> intended. (05)
Well, that assumes that the theories themselves some way of expressing
contexts. But if they do, well, then yeah. (06)
> As you can see, I am quite unfamiliar with this level of abstract
> thinking. So, tell me, is there some way to avoid specifying at least
> implicitly the category of "things" referenced and still conclude that
> those theories are inconsistent? (07)
I don't know how you would do that. Even theories with context
mechanisms usually have some way of expressing the broadest possible
context (e.g., the context where x=x for all x) and hence can express
the idea of "thing" in the most general sense. (08)
I have to say again, I don't think there's much practical upshot here.
Seems pretty unlikely to me that any real world ontology is going to be
interested in declaring that there exists some specific finite number of
Chris Menzel (010)
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