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Re: [ontolog-forum] Ontology, Information Models and the 'Real World': C

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Mon, 28 May 2007 11:28:46 -0400
Message-id: <465AF52E.1000102@xxxxxxxxxxx>
Folks,    (01)

I have a web page that gives a brief review and summary
of some basic topics and definitions in math and logic:    (02)

    Mathematical Background    (03)

Following is the table of contents (each section takes about
3 or 4 pages, if printed out).  After that is an excerpt from
Section 5 on relations.    (04)

John Sowa
_____________________________________________________________    (05)

    1. Sets, Bags, and Sequences
    2. Functions
    3. Lambda Calculus
    4. Graphs
    5. Relations
    6. Representing Relations by Graphs
    7. Lattices
    8. Propositional Logic
    9. Predicate Logic
   10. Axioms and Proofs
   11. Formal Grammars
   12. Game Graphs
   13. Model Theory
   14. References    (06)

An excerpt from Section 5:    (07)

The following table lists some common types of relations, an axiom that 
states the defining constraint for each type, and an example of the 
type. The symbol  represents an arbitrary dyadic relation.    (08)

   Type                 Axiom                      Example
   ----          -----                      -------
   Reflexive     (Ax)xx                    x is as old as y
   Irreflexive  (Ax)not(xx)               x is the mother of y
   Symmetric    (Ax,y)(xy -> yx)         x is the spouse of y
   Asymmetric   (Ax,y)(xy -> not yx)     x is the husband of y
   Antisymmetric (Ax,y)(xy & yx -> x=y)   x was present at y's birth
   Transitive    (Ax,y)(xy & yz -> xz)   x is an ancestor of y    (09)

The symbol A, called the universal quantifier, may be read "for every" 
or "for all".  It is discussed further in Section 9 on predicate logic. 
Some important types of relations -- such as partial order, linear 
order, and equivalence -- satisfy two or more of the above axioms:    (010)

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