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Re: Identity implies logic

From: Seth Russell <seth@robustai.net>
Date: Fri, 28 Sep 2001 10:12:18 -0700
Message-ID: <000201c14841$9c988580$657ba8c0@c1457248a.sttls1.wa.home.com>
To: "Pat Hayes" <phayes@ai.uwf.edu>
Cc: "RDF-LOGIC" <www-rdf-logic@w3.org>
Re: Identity implies logicSeth: It seems to me that if the law of identity
does not hold in a context, then the Law of the Excluded Middle (LEM) and
the Law of Non Contradiction (LNC) are irrelevant and cannot be used in that
context.

Pat Hayes: What law of identity are you referring to, exactly?

Seth: the formula referred to as 'Identity' on the page:
http://www.math.psu.edu/simpson/papers/philmath/node9.html
and the one alluded to by Quine when he said "...whoever denies the law of
excluded middle changes the subject."   I'm taking 'changing the subject' to
mean that Identity does not hold inside a particular context.

Pat Hayes: Why? I don't see the connection. (Presume you mean
LEM = P or (not P)
LNC = not (P and (not P))
right?

Seth: right, didn't I make that clear in my mentograph?
http://robustai.net/mentography/lawsOfLogic.gif

Pat Hayes:  These amount to the same thing in classical logic.)

Seth: Well, ok, (though i'd like to see a proof that LEM can be derived from
LNC just to close the gap in my education).   But can we rely on such a
classical  logic on the web ... i'm not at all sure myself whether this wild
west is so tamable ... are you?   Perhaps more flexible choices will be
needed ... like the ones scoped by Peter Suber at:
http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm

Seth: For example:  If we have two different computers hooked up to the
semantic web and in one's database an apple is only known as 'apple' and the
other it is only known as 'orange',

Pat Hayes:  Wait. How could you know that? That is, A is talking about
things called A#apple, and B is talking about things called B#orange. How
could you (or anyone) know they are supposed to be the same things?

Seth:  Well we humans do this all the time.  I'm relatively sure that there
are stochastic processes that can ascertain after a number of  interactions
that two tokens refer to the same thing within a reasonable margin of error.

Seth:  then logical consequences that are inferred from the combination of
those two databases in that regard might  be erroneous.

Pat Hayes:  If you only make valid inferences, they will never be erroneous
(unless you make them from erroneous premises, of course, but the logic
can't check that for you.)  However, in general, one might be able to infer
something from (A and B) that cannot be inferred from either A or B alone,
so combining information from disparate sources is a risky business. It also
suggests opportunities, though. ( Rather like life, right?)

Seth:  Right :)  But I am in no way arguing against combining information,
rather I am arguing for systems which will allow us to combine information
and be aware of the risks entailed by drawing inferences from the
combination.    Since information on the web *is* context sensitive, not
having a way to transmit information about the context of a collection of
statements, seems to me to be a big missing piece in the puzzle.  For
example, I would like to be able to transmit the fact that an agent
considers a set of statements to be truth functional, with Identity and
therefore LEM can be used in the collection.   Doesn't the MT just take that
as an assumption?   How do we say: collection A is just a conglomeration of
whatever collected by whomever, and collection B is tight and logical and we
can believe its entailments ?

Pat Hayes:  However, I can't quite see how one could get errors from your
example. The problem seems more to be the other way round: If A says
A#apples are red, you ought to be able to infer that B#oranges are red too,
but you can't. Your inferential abilities are too weak, not too strong.

Seth: Point taken, but with a slightly different example we could arrive at
an error.  How about if a class were confused with its subclass, or a map
with its  territory?

Seth:   In KIF I think this would be expressed something like:
  (=>
      (holdsIn contextX IDENTITY)
      (and
          (holdsIn contextX LNC)
          (holdsIn contextX LEM)
  )

Pat Hayes: Not sure what that is supposed to mean.

Seth:  Well could you help me with saying that correctly in KIF ?  But
informally it means to me what Quine stated above about LEM not applying
when people change the subject.  Thing is, people on the web are *always*
changing the subject.

Seth: Don't you think that a model theory for RDF should take this basic
assumption about when to apply strong logic into consideration?

Pat Hayes:  The model theory just says how RDF graphs can be interpreted,
and thereby how to infer valid conclusions. Not sure what you mean by strong
logic, but validity of entailment *always* applies.

Seth:  Well I'm still digesting the MT [1] and I don't yet know what you
mean by "entailment *always* applies"
 .... but it scares me.

[1] http://www.w3.org/TR/2001/WD-rdf-mt-20010925/

Seth Russell
Received on Friday, 28 September 2001 13:19:14 UTC

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