Re: [ontolog-forum] Inconsistent Theories

[Jawit Kien <jawit.kien@xxxxxxxxx>]

So to be a bit pedantic but to show the confusion that can happen when trying to examine something
that is only loosely defined...

On Mon, Feb 8, 2010 at 1:54 PM, Rich Cooper <rich@xxxxxxxxxxxxxxxxxxxxxx> wrote:

Example: Let the two theories be:

TrueS(F,x) := <expression1 of terminals x>;
FalseS(F,x)       := <expression2 of terminals x>;

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).

Presumably "terminal vector" means something to Rich. Since it doesn't mean
anything to me, I'll make the assumption that if we have airplane terminals,
it also makes sense to talk about boat or spaceship terminals,
If an airplane terminal is a location that people arrive and depart from
when they are traveling by airplane, then boat or spaceship terminals could
be places that either float in water or float in space and allow transport by their
corresponding mode of transportation.  If they float, then these terminals
could move, and hence don't have to be anchored in place.
It would then make sense to say these terminals have a vector,
since a vector is made up of a direction and an angle. 
When an object moves along a vector such as the unspecified terminals
would move along, it would make sense to talk about the various positions
or locations in 3-spaces which the terminal and group them as a set F.
So the predicate TrueS() would map these sets and various vectors
and would hold when a particular terminal x and a particular set were observed.
Presumably each observation would qualify as a "Thing", as you mentioned.

I'm not sure what F(x) means. I guess it means a function that when given a
direction and angle would produce a set of locations.  and your FalseS ()
predicate would give you the infinite number of locations/positions which do not
correspond to any vectors that the terminal moves along.

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).

I agree this whole example feels incredibly underspecified.
I have no idea how the FO ideas could clarify it, or even if they would even have
an impact on these questions.

I expect that if you have the terminal at any known point, there may be an infinite
number of vectors it can move along, but there still will be plenty that it don't lie
on those vectors.  When you are talking about xor'ing infinite sets, I expect you
will find that your intuitions don't guide you well.

JK
 

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

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).
>
> Yours in excruciating correctness,
>
> Chris Menzel
>

  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.

  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?

Pat


Patrick Cassidy
MICRA, Inc.
908-561-3416
cell: 908-565-4053
cassidy@xxxxxxxxx




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