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Re: Identity implies logic

From: Pat Hayes <phayes@ai.uwf.edu>
Date: Fri, 28 Sep 2001 15:40:44 -0500
Message-Id: <p05101031b7da883f2544@[205.160.76.184]>
To: "Seth Russell" <seth@robustai.net>
Cc: 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

which is, in KIF,

(forall (?x) (iff (A ?x)(A ?x)))

OK.

>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.

I would advise extreme caution in re-interpreting what Quine said to 
fit your (or anyone's) pet theory.

>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

No, that was not clear from the mentograph.

>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'

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

>   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.)

>   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.)

>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.

>
>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.

>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?

To get errors (rather than inabilities to make inferences) you need 
to somehow convey 'too much' information rather than 'too little'. I 
really don't see how that could happen unless the overall logic were 
non-monotonic.

>
>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 ?

The problem isn't with the KIF, but what (holdIn contextX ...) is 
supposed to mean. (What are contexts?) 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.

>  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. (You really have to read 
Quine carefully, he's like a very sharp knife.  Try reading 'Set 
Theory and its Logic' to see a master at work.)

>  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.

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.

Pat Hayes


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Received on Friday, 28 September 2001 16:40:53 UTC

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