I would like to clarify on some remarks recently made about Common Logic. (01)
On 5/14/13 12:00 PM, John F Sowa wrote:
> 5. Common Logic has only one domain of quantification,
* A CL interpretation does indeed have one domain of quantification
(called the universe or domain of discourse). The interpretation of a
particular CL text may require the consideration of more than one domain
of quantification , due to portions ("modules" in the current version)
being required to satisfy interpretations created by restricting
(shrinking) the domain of quantification. This has been shown to be
syntactic sugar - it is always possible to simply add a condition to the
quantification in order to restrict it to a certain subset of the domain
of quantification. (02)
In addition a CL interpretation allows for reference to entities that
are outside of the domain of quantification. This larger set of entities
is called the universe of reference. (03)
> and the
> CL syntax allows any entity to be used as a function or
> relation with any number of arguments.
* A Cl interpretation associates a name with one entity in the domain of
reference. Each CL interpretation also associates one function and one
relation with every entity in the domain of reference. The same name is
used to indicate the application of this function or relation,
distinguished by the syntactic context in which the name appears. This
is not, in my view, the same as the entity being "used" as a function or
relation - that suggests that the entity has some intrinsic functional
or relational characteristics, when in fact the association of an entity
with a function or relation is created by the interpretation. A
different interpretation could associate a different function or
relation with the same entity. (04)
> In that sense,
> one could say that CL has an implicit one-category ontology.
I would say that CL has an implicit five-category ontology. There are
the three categories of entity, function and relation. The category of
entity is further divided into the domain of discourse and the
complement of the domain of discourse within the universe of reference. (05)
The functions and relations can also be partitioned into discourse and
non-discourse. While it is possible to have the same function/relation
associated with a discourse entity and also with a non-discourse entity,
any function/relation associated with some entity in the domain of
discourse could be considered a discourse function/relation, due to it
being "reachable" by a quantification such as (06)
(exists (x) (= x(a) b) ) (07)
This gives six categories ( discourse and nondiscourse relations,
functions and entities). However the last of these (nondiscourse
entities) is irrelevant from a practical standpoint, as a text that
refers to a nondiscourse entity (in an equation, or as a function or
predicate argument) should[1] never be satisfiable. (08)
CL does not impose any explicit ontology on entities. It is up to the
author whether to include axioms that require a particular entity or
group of entities to be the same as the function or relation associated
with them. (09)
[1] The published CL standard has some holes in it so that this
obligation is not adequately met. The upcoming revision of the CL
standard (http://ontolog.cim3.net/cgi-bin/wiki.pl?CommonLogic_V2) will
address this issue. (010)
Tara (011)
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