Rich Cooper wrote: (01)
>Example: Let the two theories be:
>TrueS(F,x) := <expression1 of terminals x>;
>FalseS(F,x) := <expression2 of terminals x>;
[ such that . . . ] (02)
> 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).
>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),
The overspecified sentences are true in the AND (and therefore false in
This means that TrueS and FalseS are inconsistant for F(x). Thus either
inconsistant, or TrueS and FalseS are themselves inconsistant. (03)
> or well specified with
>current knowledge (false in the xor).
The well specified sentences are true in the XOR; they are either known
to be true in F(x)
or known to be false in F(x), but not both.
The underspecified sentences are false in both the AND and the XOR (true
in the NOR) (04)
>Rich AT EnglishLogicKernel DOT com
>[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,
> 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?
doug foxvog doug@xxxxxxxxxx http://ProgressiveAustin.org (06)
"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
- Dr. Martin Luther King Jr.
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