[Scl] Re: Logic, Topic Maps, and RDF

[John F. Sowa]

Enrico,

CL does allow quantified variables to range over relations.

EF> .... OWL-lite and OWL-DL are subsets of FOL,
 > but OWL-full is not: there are higher order features
 > which are not directly expressible in FOL. By the way,
 > strictly speaking, the same holds for RDFS: the model theory
 > of RDFS is not first order, even though I suspect that an
 > easy encoding of RDFS graphs in FOL theories is possible.

As I wrote in a response to Jon Awbrey, the CL model theory
is purely first order, but it supports quantification over
relations by the simple absence of any syntactic distinction
between individuals and relations.

An individual is simply an entity that cannot be used to
relate other entities.  If you write r(x,y) where r is
not a relation, there is no syntax error.  You have simply
made a false statement.

The lack of a syntactic distinction between individuals
and relations may seem confusing to logicians who have
made that distinction all their lives.  But actually,
the lack of such a distinction makes the CL model theory
simpler than Tarski's version, which has separate sets
for individuals and relations.

And by the way, the CL model theory also allows you to
make metalevel statements about relations and how they
relate relations to individuals.  Russell's paradox (which
should be called Zermelo's paradox) does not arise because
any statement of the following form is simply false:

    There is a relation R that is true of every relation
    that is not true of itself.

The reason why this statement is false is that the CL
model theory assumes a single domain D over which all
quantifiers range.  For any fixed domain D, it is not
possible to find any R for which the above statement
is true.  Ergo, the statement is false -- no paradox.

And if you want to have a "stratified" theory that
distinguishes levels, such as a set of objects, a set
of relations over objects, a set of metarelations over
relations, metametarelations over metarelations, etc.,
you can do so.  What you get is a subset of full CL.
That means your language is not as expressive as CL,
but it is still within the CL family.

By the way, that is what I am doing to bring CGs into
the CL family:  the version of CGs that everybody has
been using does distinguish relations from individuals.
So that language will exprss a subset of full CL.  But
I am also defining a CL version of CGs, which does not
make that distinction.  It is identical in expressive
power to full CL, and it is a superset of the current
version of CGs.  For example, the following CG says
that the cat Yojo is on a mat:

    [Cat: Yojo]->(On)->[Mat].

This statement is perfectly well formed in the usual CG
language.  But the following statement is not well formed
because it uses the individual Yojo as a type, the dyadic
relation On as a type, and the type Cat as a dyadic
relation:

    [On: Mat]->(Cat)->[Yojo].

However, in the unconstrained version of CGs, which is
identical in expressive power to full CL, this statement
is well formed, but false.  For most purposes, I would
recommend that people use the more constrained version.
But it might be necessary to use unconstrained CGs
to express unconstrained statements in other languages,
such as full OWL, for example.

Re complexity:  Murray said that we need a "CL for Dummies".
I would qualify that -- we need many documents:

  1. A precise formal description of CL for logicians.

  2. An ISO standard for implementers, which must include
     guidelines on how to relate the CL abstract syntax
     to the concrete syntax of any CL language.

  3. Books at all levels from dummy to expert on the
     concrete languages of the CL family.

No "dummies" are expected to use CL.  In fact, no one
is expected to use CL directly because it does not have
any concrete syntax.  The dummies books need to be
written for the languages that people are actually
expected to use.

John