Example: Let the two theories be: (01)
TrueS(F,x) := <expression1 of terminals x>;
FalseS(F,x) := <expression2 of terminals x>; (02)
So that there are two theories: TrueS(F,x) is the set of Things which are
believed (with current knowledge) to be in the set F for terminal vector x,
while FalseS(F,x) is the set of Things which are believed NOT to be in F(x). (03)
Only a physicist can believe both true and false theories in a single
situation and not think they need refining. Then if there are any Things in
both sets, something in expression1(x) is inconsistent with something in
expression2(x). The xor of the two sets (TrueS xor FalseS) identifies those
Things which are over specified (true in the xor), or well specified with
current knowledge (false in the xor). (04)
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
-----Original Message-----
From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Patrick Cassidy
Sent: Monday, February 08, 2010 10:24 AM
To: '[ontolog-forum] '
Subject: [ontolog-forum] Inconsistent Theories (05)
Changing the topic to reflect the narrow focus here: (06)
Query for Chris Menzel re: his reply to Cory C: (07)
>
> 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
> (08)
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. (09)
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? (010)
Pat (011)
Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx (012)
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