Re: Identity implies logic

From: "Pat Hayes" <phayes@ai.uwf.edu>

> >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
>
> which is, in KIF,
>
> (forall (?x) (iff (A ?x)(A ?x)))
>
> OK.

OK.  But now I see the wisdom of your first question, because there seem to
be two identities here that we need be concerned with .. the identity of the
predicates (all triples are unique) and the identity of the nodes (all nodes
are unique) .  Could the identity of the nodes be expressed as follows?

(forall (?x) (iff (?x) (?x))  .. but that looks weird ... oh well I'm lost
here.

> >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?
>
> No, that was not clear from the mentograph.

I've relabeled my graph using Suber's terms (which I would now like to stick
with for the duration of any dialogue on this subject).   Please reload and
take another look at:

http://robustai.net/mentography/lawsOfLogic.gif

Obviously I haven't defined context in the graph, but outside of that could
you tell me what is unclear?  Especially since it just represents an
instance of the kind of graphs that your MT has supposedly given meaning to.

> >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).
>
> By deMorgan's law:
> not (P and (not P))  =  (not P) or (not (not P))
> By double negation:
> (not P) or (not (not P))  =  (not P) or P
> and then permute the 'or'

Woops, maybe we better back up here ... let me quote from Suber at:
http://www.earlham.edu/~peters/courses/logsys/pnc-pem.htm

     "Under De Morgan's theorem, the PNC can be transformed
      into the PEM and vice versa, but this only shows that
      De Morgan's Theorem presupposes the PEDC.
      (Logics that deny the PEM must deny some forms
      of De Morgan's theorem.)

      "The PNC and PEM need not be equivalent in n-valued
      logics when n > 2"  ...

     "If we use a standard two-valued logic, the three principles
     are already present even if they do not appear as axioms.
     The three principles can be proved in such a logic, but
     any such proof would be viciously circular."

So I take it that we need to keep PNC and PEM separate until we commit to
classical binary logic and committing to binary logic is not something we
can just do willy nilly for everybody with a single stroke of the pen .. can
we ... should we ?   Let me ask you a straight question:  Does the MT commit
us to (classical?) binary logic ?

> Now, what was the connection between LEM/LNC and Identity, again?

Identity => PEM and PNC

But I'm not sure that this shouldn't be:

Identity <=> PEM and PNC

Could you help me with that little detail ?

> >   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?
>
> Yes, we can rely on it precisely because it is the most reliable
> logic around (which is why it is 'classical'). Well, one might make a
> case that intuitionistic logic is safer, I guess, but I don't think
> that the SW is going to be much concerned with issues that arise in
> the foundations of mathematics. (In any case, RDF is intuitionistic,
> since it has no negation.)

Well then, sorry,  I use the wrong word when I asked my question with the
term 'rely'.

> >   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
>
> Maybe, but I see no reason to think so.  I once started to write a
> thesis on multivalued logics, but gave up when I proved that they
> could all be trivially expressed in 2-valued logics, and provided no
> useful expressive extension. (There are some useful applications of a
> particular 3-valued logic in computability theory, but not for
> general reasoning; and some versions of type theory use a 4-valued
> logic for mathematical reasons, but this is really 2 truth values
> plus a 'top' and 'bottom' to make a simple lattice.)

It seems to me that the usefulness of multi valued logics is just where we
cannot use 2 valued logic; and we cannot use two valued logic in a context
where we are not sure if we are talking about the same things; which, if you
are talking to guys like me, is most of the time.

> >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.
>
> Nonsense. That would require magical powers. Here are two tokens:
> AAAA
> BBBB
> which I am using to refer to some things. Go ahead, figure out
> whether they refer to the same thing or not. Use whatever means you
> like, stochastic or otherwise.  You can make as many copies of this
> message as you want. Give me a call when you know the answer.

You missed the word 'interactions' in my response above.

> >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 ?
>
> OK, I'm sympathetic to your general concern. But we have to stop
> hedging somewhere. How are you going to transmit those facts you
> mention, unless you have a formalism to do it in? And what
> descriptive powers will that formalism have? At some point you just
> have to say: here is a standard descriptive language to get y'all
> started transmitting facts to one another, and here is its semantics.
> It just IS that way, OK? If you don't like it, invent a different
> standard and maybe more people will use yours. RDF/S and DAML+OIL are
> first steps in this direction.

OK, RDF is just fine with me ... with or without a MT.

> >Seth:  Well could you help me with saying that correctly in KIF ?
>
> The problem isn't with the KIF, but what (holdIn contextX ...) is
> supposed to mean. (What are contexts?)

I've changed the graph to read ContextCave so as not to confuse what I am
saying with other definitions of context.  A ContextCave is just a set of
statements collected by some agent in some matter.


>Also, all three of these are
> elementary theorems in KIF, so its kind of dumb to say in KIF that
> they might not be true.

Hmmm ... where can I see those theorems expressed ?

> >  But
> >informally it means to me what Quine stated above about LEM not applying
> >when people change the subject.
>
> That isn't what Quine says in that quote.

Well, yes,  I turned it around ... I take it that Quine was railing against
Deviant Logics but in doing so he expressed in a clear manner just what the
nature of such logics are ... perhaps I should reproduce the entire
paragraph here:

  "By reasoning of a couple of pages back, whoever denies
   the law of excluded middle changes the subject.  This is
   not to say he is wrong in so doing.  In repudiating
   'p or ~p' he is indeed giving up classical negation, or
   perhaps alternation, or both; and he may have
   his reasons."

> Bear in mind the caveats about assuming that information is
> unchanging, temporal snapshots, and so on.  RDF (and its semantics)
> are really only a very simple first step, not the final answer to all
> the semantic web's problems.

Yes I see you have nicely excised the bug-a-boo of time from the MT.  But
time is not the only problem, Identity is a problem too .... isn't it ?

Seth Russell

Received on Sunday, 30 September 2001 13:42:12 UTC