SUO: Re: Logic, Topic Maps, and RDF
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John F. Sowa wrote:
>
> 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.
John,
Honestly, this worries me a lot.
When you say "r(x, y)" is a false statement,
this is to say that r is false of <x, y>.
Do you mean to say that "app(r, <x, y>)" is false?
That seems like another way of saying "syntax error".
And it requires a rather heavy type-checking regime.
Aside from all this, one cannot pull the same
applicative and currying tricks with relations
that one can pull with functions. One can only
do that with the indicator functions of relations,
and this has consequences that you do not seem to
be taking into account.
Then again, the absolute atomist's distinction between individuals and relations,
that you ceremoniously oust by the front door, is all the more self-deceptively
brought back in by the back door when you say something like this:
JS: An individual is simply an entity that cannot be used to relate other entities.
Is this a property that the putative entity has independently of discursive context?
Then again, as became evident the last time that you said this stuff about S/CL,
you still maintain the distinction between connectives and predicates, which is
a distinction with no pragmatic difference, and generates a host of further
confusions among the communicants of GOL.
Jon Awbrey
> 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
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