SUO: Re: More terminology
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Pat & All,
For the definitions of terms like "class", "set", "function", "relation",
and so on, you might consider referring to a standard text like this one:
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| John L. Kelley, 'General Topology'.
| Appendix on Axiomatic Set Theory.
SET. Set Theory
01. http://suo.ieee.org/ontology/msg04082.html
Appendix. Elementary Set Theory
02. http://suo.ieee.org/ontology/msg04083.html
A.1. The Classification Axiom Scheme
03. http://suo.ieee.org/ontology/msg04084.html
04. http://suo.ieee.org/ontology/msg04086.html
05. http://suo.ieee.org/ontology/msg04088.html
A.2. Elementary Algebra of Classes
06. http://suo.ieee.org/ontology/msg04089.html
07. http://suo.ieee.org/ontology/msg04091.html
08. http://suo.ieee.org/ontology/msg04092.html
09. http://suo.ieee.org/ontology/msg04093.html
10. http://suo.ieee.org/ontology/msg04094.html
A.3. Existence of Sets
11. http://suo.ieee.org/ontology/msg04095.html
12. http://suo.ieee.org/ontology/msg04096.html
13. http://suo.ieee.org/ontology/msg04097.html
A.4. Ordered Pairs: Relations
14. http://suo.ieee.org/ontology/msg04098.html
15. http://suo.ieee.org/ontology/msg04099.html
A.5. Functions
16. http://suo.ieee.org/ontology/msg04100.html
...
Links 2 through 16 of the above material are
selected and transcribed into plaintext from:
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
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Patrick Cassidy wrote:
>
> Terminology management:
>
> John F. Sowa wrote:
> > Pierre,
> >
> > PG>Are you pissed? I apologize.
> >
> > No. I'm not pissed. I'm just frustrated with the endless
> > rehashing of debates about terminology.
> >
> I agree that we need a standardized terminology list for
> this discussion group. I seem to recall that one of us proposed
> to set up such a list over a year ago, but don't recall
> whether anything concrete was decided.
> Can we use this thread to decide how to create a standard
> terminology for this group? Does anyone have experience setting
> up a WIKI? If not, I will volunteer to maintain a terminology
> page on my site (simple list-nothing fancy) until someone
> else sets up a better hypertext version.
>
> As one starting point, I would suggest a careful definition of
> the terms "class" and "relation" and "predicate" (which
> may require definition of "term" and "sentence").
>
> For "class" and "relation" I would prefer the usage that is
> given for KIF classes. It may not be the same as VNBG classes,
> but I think it is the most common use of the term - no?
> The definitions from the KIF site are attached below. Any
> dissenters? (silly question?) If the majority prefer this
> definition, and there is a minority that prefer a different
> usage, I would suggest creating a different term for the
> alternative usage.
>
> Pat
>
> --
> =============================================
> Patrick Cassidy
>
> MICRA, Inc. || (908) 561-3416
> 735 Belvidere Ave. || (908) 668-5252 (if no answer)
> Plainfield, NJ 07062-2054 || (908) 668-5904 (fax)
>
> internet: cassidy@micra.com
> =============================================
> from:
> http://www-ksl.stanford.edu/knowledge-sharing/ontologies/html/frame-ontology/CLASS.html
>
> CLASS
> Documentation:
> A class can be thought of as a collection of individuals. Formally, a
> class is a unary relation, a set of tuples (lists) of length one. Each
> tuple contains an object which is said to be an instance of the class.
> An individual, or object, is any identifiable entity in the universe
> of discourse (anything that can be denoted by a object constant in
> KIF), including classes themselves.
>
> The notion of CLASS is introduced in addition to the relation
> vocabulary because of the importance of classes and types in knowledge
> representation practice. The notion of class and relation are merged
> to unify relational and object-centered representational conventions.
> Classes serve the role of `sorts' and `types'.
>
> There is no first-order distinction between classes and unary
> relations. One is free to define a second-order predicate that makes
> the distinction. For example, (predicate C) could mean that the unary
> relation C should be thought of more as a property than as a
> collection of individuals over which one might quantify some
> statement. Logically, all such predicates would still be instances of
> the metaclass CLASS.
>
> The fact that an object i is an instance of class C is denoted by the
> sentence (C i). One may also use the equivalent form (INSTANCE-OF i
> C). This is not equivalent to (MEMBER i C).
> An instance of a class is not a set-theoretic member of the class;
> rather, the tuple containing the instance is a element of the set of
> tuples which is a relation.
>
> The definition of a class is a predicate over a single free variable,
> such that the predicate holds for instances of the class. In other
> words, classes are defined intentionally. Two separately-defined
> classes may have the same extension (in this case they are = to each
> other). It is possible to define a class by enumerating its instances,
> using KIF's set operations. For example, (define-class primary-color
> (?color)
> (member ?color (set red green blue)))
> Subclass-Of: Relation
>
> ========================
> RELATION
>
> Documentation:
> A relation is a set of tuples that represents a relationship among
> objects in the universe of discourse. Each tuple is a finite, ordered
> sequence (i.e., list) of objects. A relation is also an object itself,
> namely, the set of tuples. Tuples are also entities in the universe of
> discourse, and can be represented as individual objects, but they are
> not equal to their symbol-level representation as lists.
>
> By convention, relations are defined intensionally by specifying
> constraints that must hold among objects in each tuple. That is, a
> relation is defined by a predicate which holds for sequences of
> arguments that are in the relation.
>
> Relations are denoted by relation constants in KIF. A fact that a
> particular tuple is a member of a relation is denoted by
> (<relation-name> arg_1 arg_2 .. arg_n), where the arg_i are the
> objects in the tuple. In the case of binary relations, the fact can be
> read as `arg_1 is <relation-name> arg_2' or `a <relation-name> of
> arg_1 is arg_2.' The relation constant is a term as well, which
> denotes the set of tuples.
> Subclass-Of: Set
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