On Dec 18, 2010, at 11:07 AM, John F. Sowa wrote: Chris and Leo, It's extremely confusing to use technical terms from some system S when talking about another system in which they are not defined. CM: The central trick here is that, in the semantics of Common Logic,
classes *have* extensions but are not *identified* with their
extensions in; this makes it easy to allow a class to be in its
own extension without any need for non-well-founded set theory.
Pat used the same idea when he developed the semantics of RDF.
Section 6.2 of IS 24707 defines the CL semantics. That section does not use the word 'class'. Using the word 'class' in discussing CL semantics adds more confusion than it could ever clarify.
Well, I was talking to Leo, who's really smart and not prone to confusion. :-) Granted, I was speaking informally about those objects in a model whose relation extensions contain objects in the domain only, no n-tuples. That said, I'm not sure it is possible, when discussing any technical subject matter informally, to avoid saying things that might be confusing for people who don't bother to study and learn the subject matter in question.
The CL standard does use the word 'class' in a nontechnical sense
on page vi of the introduction:
"Common Logic has some novel features, chief among them being
a syntax which is signature-free and permits 'higher-order'
constructions such as quantification over classes or relations
while preserving a firstorder model theory, and a semantics
which allows theories to describe intensional entities such
as classes or properties."
The word 'property' is another technical-sounding word that
is not used in CL.
I'd suggest it be added, defined simply to be an object whose relation extension contains only objects, not n-tuples.
I recommend that the introduction avoid using any technical or
technical-sounding words that are not used in CL itself. Such
terms may be used in an informative passage about using CL to
support other languages, but only when the CL terms and the
technical terms of the other languages are kept distinct.
I suggest that the above sentence be shortened to
"Common Logic has some novel features, chief among them being
a syntax which is signature-free and permits 'higher-order'
constructions such as quantification over relations while
preserving a first-order model theory."
I'd suggest:
"Common Logic has some novel features. Chief among these are (a) a syntax that is signature-free and permits 'higher-
order' constructions such as predicate quantification; (b)
a first-order model theory that allows interpretations in which properties and relations can be understood intensionally, in the sense that, in those interpretations, relations with the same extensions needn't be identical; and (c) CL's semantics permits interpretations in which relations exist in the domain of quantification."
Or something like that. Leo: Are classes intensional objects (I guess, like types)? Could
you point me to where in CL this is discussed?
CM: The semantics allows for objects in the class role (more generally, the
relation role) to be intensional, in the sense that there are CL models
in which coextensional classes are not identical.
Since CL semantics does not have classes,
Well, I disagree with your saying that without qualification. It is most certainly the case that relation extensions of some objects in an interpretation can be restricted to classes. I suggest that this sentence be restated:
"CL semantics allows relations to be intensional, in the sense
that there are CL models in which coextensional relations are not
identical. In languages that use the terminology of classes,
a class could be represented by a CL relation. Such a class
could then have intensional identity criteria."
I like that passage. :-) It's a bit redundant, given the passage above, however.
-chris
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