Changing the topic to reflect the narrow focus here: (01)
Query for Chris Menzel re: his reply to Cory C: (02)
>
> 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).
>
> Yours in excruciating correctness,
>
> Chris Menzel
> (03)
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. 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 - 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. (04)
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? (05)
Pat (06)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (07)
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