Chris: (01)
This entire thread highlights my basic assertion that the context of
"perceiver" is really a core principle to qualify any assertion. (02)
When you state x=y, if I am only perceiving a two dimensional characters
of an alphabet written on a paper, I disagree since x is clearly x and y
is clearly y. From my point of perception, I can clearly see that an x
looks different that a y and if I do not know they are variables for
something else, I would have to perceive them as different. (03)
If x represents something else, as does y, then it may be true that x =
y pending on the point of view. If x and y are different monikers for
the same object, then it is true if I perceive only the object itself.
However, if I perceive the object plus the distinction that one group of
people refer to it as x and another refer to it as y, then I could
(probably contentiously) argue that from their perception, x and y are
not the same since x and y have different names. (04)
Nevertheless, your logic is equally infallable. Anyone perceiving
ID_FOL: x = y.. will agree (05)
It would have been cool to have this knowledge to argue with algebra
teachers in high school 25 years ago. I could have been (more)
disruptive by challenging their assertions ;-) (06)
Duane (07)
Chris Menzel wrote: (08)
>On Fri, Jun 17, 2005 at 12:26:04PM -0700, Duane Nickull wrote:
>
>
>>Chris Menzel wrote:
>>
>>
>>
>>>ID: if x = y, then anything true of x is true of y.
>>>
>>>
>>>
>>Disagree.
>>
>>
>
>Well, we haven't really fixed the context rigorously enough for there to
>be anything to disagree about. In the context of standard first-order
>logic with its usual extensional semantics, the above principle is
>rendered schematically as follows:
>
> For any well-formed formula A not containing the variable y, the
> following is an axiom:
>
> ID_FOL: x = y -> (A -> A'),
>
> where A' is the result of replacing every free occurrence of x in A
> with an occurrence of y.
>
>And the fact is that, on the standard, extensional semantics for
>first-order logic, every instance of ID_FOL is valid. That's not
>something about which one can rationally disagree; it's just a
>mathematical fact about first-order languages and their models.
>
>What one might disagree with is that the principle ID (properly
>formalized) is valid in every logical context. And that is exactly what
>the counterexamples I mentioned were designed to show -- ID fails, or at
>least appears to fail, in contexts involving belief and necessity (among
>others).
>
>
>
>>Most things may be the same but it is still instance y as opposed to
>>instance x, therefore his axiom has a logic error.
>>
>>
>
>You lost me there, I'm afraid.
>
>
>
>>They are still two different things. It may be better to state:
>>
>>if x = y, then x is y and both should be called x.
>>
>>
>
>That rather seriously confuses the *value* of the variable "x" with "x"
>itself. It also seems implausible -- Mark Twain is Sam Clemens, but I
>don't see any particular reason why he *should* be called by either
>handle. (And aside from these points, the development of theories that
>include semantical notions like "called" are fraught with difficulty.)
>
>
>
>>Also - an important consideration of context is perceiver.
>>
>>
>
>Yes, that can be important. That is one reason there is a lot of active
>research on formal theories of context at the moment.
>
>
>
>>To you and I, a coffee table is a solid item, to a neutrino, it is a
>>lot of open space interspersed with a few bits of solid matter...
>>
>>
>
>True enough, so a neutrino's ontology of your front room will probably
>look a lot different than your front room ontology! Note, though, that
>the notion of context is often just assumed in the background of an
>ontology -- we often create ontologies *from the perspective of a given
>perceiver or set of perceivers*, e.g., domain experts in a mfg shop
>floor. For these cases, the notion of context needn't play an explicit
>role. More and more, though, we have a need to represent multiple
>ontologies arising out of different contexts within a *single*
>framework, and this requires explicit mechanisms that enable us to group
>pieces of information according to context -- and that's exactly what
>formal theories of context are attempting to provide.
>
>Cheers!
>
>Chris Menzel
>
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